Theory of hierarchically-organized neuronal oscillator dynamics that mediate rodent rhythmic whisking

Abstract

Rodents sense their environment through coordinated, rhythmic orofacial motor actions, including whisking. The whisking oscillator lies in the medulla, consists of inhibitory neurons, and can free-run or be paced by breathing. Despite knowledge of the underlying anatomy and physiology, the mechanisms to generate rhythmic whisking remain to be understood. We surmount this challenge and formulate and analytically solve a hierarchical model of the whisking circuit. The model posits, consistent with experiment, that there are two subpopulations of oscillator neurons, with stronger connections between, as opposed to within, each subpopulation. We show that connections between the subpopulations support rhythmicity, while connections within a subpopulation induce variable spike timing that enhances the dynamic range of rhythmicity. Free-running oscillations are rapidly reset by inhalation, which paces whisking by disinhibiting vibrissa motoneurons. Our model provides a computational framework to support longstanding observations of concurrent autonomous and driven rhythmic motor actions that comprise behaviors.

1. INTRODUCTION

Motor output is the manifestation of an animal’s behavior. How is this controlled through internal brain dynamics? The movement of an animal’s body may be considered in terms of overlapping sequences of individual motor actions [Tinbergen (1951)]. Each action is a movement primitive that involves the stereotypic activation of different muscles by premotor nuclei, along with potential adjustments of their activation by proprioceptive feedback [von Holst (1939/1973). The coordination of motor actions into behaviors nominally occurs through pre-premotor (pre²motor) nuclei, the next level in a hierarchy of neuronal control [Weiss (1941), Mussa-Ivaldi and Bizzi (2000), Moore et al. (2014b)]. Pre²motor nuclei are distributed across a plethora of motor centers, from the neocortical pre²motor outputs of infragranular projection neurons that appear to govern complex sequencing down to pre²motor nuclei in thee brainstem and spinal cord that generate repetitive movements. Far from a rigid hierarchy, pre²motor nuclei can initiate behaviors that involve strong as well as partial coordination among motor actions [Kaplan et al. (2020)].

The behaviors that comprise active orofacial sensation involve vibrissa touch, lingual touch, and smell. The underlying motor actions are driven by premotor nuclei in the brainstem for the control of whisking, licking, nose twitching, heading, and sniffing, the rapid breathing that occurs when a rodent forages or is otherwise attentive [ Lund et al. (1998), Deschênes et al. (2012), Kurnikova et al. (2017), McElvain et al. (2018), Ruder and Arber (2019)]. Of particular interest, orofacial motor actions have a rhythmic component and, further, are arranged in a tractable hierarchy [McElvain et al. (2018), Wallach et al. (2020)]. Inhalation, controlled by the preBötzinger complex (pBötC) in the medulla, drives sniffing and further acts as a pre²motor master oscillator to many premotor nuclei that are involved in rhythmic orofacial motor actions [Moore et al. (2013), Kleinfeld et al. (2014)].

Rodents can rhythmically whisk over a broad range of frequencies [Berg and Kleinfeld (2003), Carvell and Simons (1990)], yet whisking is typically paced by breathing [Moore et al. (2013), Ranade et al. (2013), Welker (1964)] (Figure 1A). During sniffing, the breathing and whisking are phase-locked at a one-to-one ratio (Figure 1B). Whisking may also free-run between breaths, but the whisking cycle is rapidly reset by the onset of inhalation (Figure 1C,D). Beyond whisking per se, past work has identified the population of neurons that are both necessary and sufficient for whisking. These cells lie in a region denoted the ventral intermediate reticular zone (vIRt) that is adjacent to the pBötC [Moore et al. (2013), Deschênes et al. (2016b)]. Past work further identified the pattern of connectivity and the sign of their synaptic interactions [Moore et al. (2013), Deschênes et al. (2016b), Takatoh et al. (2013), Takatoh et al. (2021), Takatoh et al. (2022)].

Figure 1. Analysis of existing and new data related to breathing and whisking.

(A) Typical experimental set-up for recording from head-restrained rodents.

(B,C) Time series of breathing (red) and whisking (blue) for a head-restrained rat [Moore et al. (2013)].

(D) Annotated whisking and breathing. The values ${\theta }_1^{\text{amp}},{\theta }_2^{\text{amp}}$, and ${\theta }_3^{\text{amp}}$ denote the amplitudes of the first, second and third whisks within the same breathing cycle.

(E) Time series of breathing (red) and whisking (blue) for a free moving rat.

(F) Semi-logarithmic plot of the number of breaths for which there are $N_{\mathrm{W}}$ whisking cycles. The dotted lines are exponential fits with slopes of 0.95 (restrained) and 0.92 (free).

(G) Amplitude of the first whisk ${\theta }_1^{\text{amp}}$ within a breathing cycle versus $N_{\mathrm{W}}$. Bars denote the standard error.

(H) Probability density function of $\mathrm{l}\mathrm{o}\mathrm{g}\left({\theta }_{I+1}^{\mathrm{a}\mathrm{m}\mathrm{p}}/{\theta }_i^{\text{amp}}\right)$ versus ${\theta }_i^{\text{amp}}$ for $i=1,2,3$. The normalization of each distribution is ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} d\left[\mathrm{l}\mathrm{o}\mathrm{g}\left({\theta }_{I+1}^{\mathrm{a}\mathrm{m}\mathrm{p}}/{\theta }_i^{\mathrm{a}\mathrm{m}\mathrm{p}}\right)\right]\mathrm{l}\mathrm{o}\mathrm{g}\left({\theta }_{I+1}^{\mathrm{a}\mathrm{m}\mathrm{p}}/{\theta }_i^{\mathrm{a}\mathrm{m}\mathrm{p}}\right)=1$.

(I) Modulation depth of the spike rates of vIRt neurons. Black circles are from intracellular recordings with lightly anesthetized rats [Deschênes et al. (2016b)]. Green circles represent new data from awake mice with extracellular electrodes and optical tagging. The depth is calculated from the spike rates as $<($ minimum – maximum $)/$ average $>$ where the averaging is on a per cycle basis. Data plotted as a function of the phase in the whisk cycle at which the spike rate is maximal.

(J) New analysis to determine the coefficient of variation ${\mathrm{C}\mathrm{V}}_2$ as a function of the phase in the whisk cycle. Same data and notations as in panel I.

The goal of the present work is to decipher the underlying mechanisms for rhythmic whisking and for the partial coordination [von Holst (1939/1973)] of whisking and breathing. We seek an analytically tractable model that encompasses past experimental results as well as provides insights to the hierarchical control of whisking from the pre²motor level down to the motion of the vibrissae.

1.1. Detailed background

Results from a variety of experimental procedures in both adult rats and mice indicate that the vast majority of neurons that comprise the vIRt release the inhibitory transmitter GABA or glycine. Juxtacelleular recording from vIRt neurons, followed by in situ hybridization, showed that nine of ten cells were GAGBergic while one was glutaminergic [Deschênes et al. (2016b)]; i.e., 0.9 of vIRt neurons are inhibitory. Retrograde labeling experiments from the vFMN to the vIRT using the organic dye fluoro-gold, followed by in situ hybridization (≈ 300 neurons), showed that 0.86 of double labeled cells are glycinergic, 0.54 are GABAergic, and 0.13 are glutaminergic [Moore et al. (2013)]; i.e., 0.9 of vIRt neurons are inhibitory. Transsynaptic retrograde labeling experiments based on glycoprotein-deleted rabies [Takatoh et al. (2021)], followed by in situ hybridization (≈ 600 neurons), found that 0.67 are GABAergic and 0.30 are glutaminergic [Takatoh et al. (2022)]; i.e., 0.7 of vIRt neurons are inhibitory. Yet silencing of these glutaminergic neurons did not effect whisking, implying that more than 0.7 of the rhythm generating neurons are inhibitory. In our model, we take the vIRt to contain solely inhibitory cells.

Neurons in the vIRt are observed to spike either in phase with vibrissa protraction or with in phase with vibrissa retraction [Moore et al. (2013), Deschênes et al. (2016b), Takatoh et al. (2022)]; we label these two populations vIRt^pro and vIRt^ret, respectively. The neurons that spike in phase with retraction form the dominant premotor input to the vibrissa facial motor neurons (vFMNs) that drive the intrinsic muscles in the mystacial pad and protract the vibrissae [Deschênes et al. (2016b)]. Thus the premotor vIRt^ret input acts to shunt or negate motoneuron output. This implies that the onset of whisking requires a concomitant excitatory input to the vFMN. Lastly, there is no evidence that individual vIRt neurons produce intrinsic periodic oscillations at the whisking frequency.

Selective physical or chemical lesion of neurons [Deschênes et al. (2016b)], or near complete inactivation of vIRt synapses by the expression of tetanus light-chain in parvalbumin neurons [Takatoh et al. (2022)], abolishes whisking on the side of the lesion. These procedures maintain modulation of the set-point [Kleinfeld et al. (2015), Takatoh et al. (2022)]. in other experiments, pharmacological activation of the vIRt induces long epochs of continuous whisking on the ipsilateral side in the lightly anesthetized rodent [Moore et al. (2013), Moore et al. (2014a)]. These results indicate that the vIRt is both necessary and sufficient to generate the whisking rhythm independent of breathing.

A breathing cycle in rodents may initiate one whisking cycle that is locked to breathing (Text Box 1) or multiple whisking cycles. For the case of several whisking cycles within a single breathing cycle, measurements with head-fixed rodents show that the onset of the first whisk is locked to the onset of inhalation, but that of the following whisks slowly dephase and their amplitude is reduced. The whisks are resynchronized with breathing at the next inhalation [Moore et al. (2013), Deschênes et al. (2016b)] (Figure 1E). This is an example of partial coordination and the occurrence of intervening whisks between bouts of inhalation implies that both inhibitory input from the pBötC, among inhibitory interactions within the vIRt, contribute to whisking rhythmogenesis. Of interest, a molecular manipulation to diminish the conductance of inhibitory inputs solely to vIRt^ret cells reduces the amplitude of whisking, quenches the occurrence of intervening whisks, yet preserves the locking of whisking to breathing [Takatoh et al. (2022)].

Text Box 1. Description of whisking when 1:1 with sniffing.

When whisking and sniffing approximately synchronize at a 1:1 ratio during exploration, a special case but one that is behaviorally relevant. it is useful to describe the vibrissa angle, $\theta \left(t\right)$, by an idealized representation [Hill et al. (2011)]. Here

$$\theta (t)={\theta }^{\text{set}}(t)-{\theta }^{\text{amp}}(t)[1+\text{cos}\varphi (t)]$$ (1)

where ${\theta }^{\text{set}}$ is the slowly varying set-point, which corresponds to full protraction of the vibrissae, and ${\theta }^{\text{amp}}$ is the slowly varying amplitude of the rhythmic sweep. The rapidly varying phase in the whisk cycle, denoted $\varphi \left(t\right)$, is found from

$${⟨\frac{d\varphi (t)}{dt}⟩}_t=2\pi f_{\text{pBötC}},$$ (2)

where $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ is the frequency of the pBötC input to the vIRt^ret, and also the whisking frequency in this case, and $⟨\dots {⟩}_t$ means time-averaging. These model parameters can be compared to values of ${\theta }^{\text{set}},{\theta }^{\mathrm{a}\mathrm{m}\mathrm{p}}$, and $\varphi \left(t\right)$ that are extracted from experimental measurements of whisking through a Hilbert transform [Hill et al. (2011)].

With respect to descending control, vIRt neurons receive excitatory inputs from superior colliculus [Kaneshige et al. (2018), McElvain et al. (2018)], deep cerebellar nucleus [Takatoh et al. (2021), Takatoh et al. (2022)], and a midbrain reticular region near the red nucleus [Takatoh et al. (2022)]. Recordings from photo-tagged vIRt cells reveals a transition from either quiescence or tonic spiking to rhythmic bursts of spikes [Takatoh et al. (2022)]. It is an open issue as to whether the same input to the vIRt depolarizes the vFMNs.

1.2. Computational challenges

The experimental observations raise computational challenges. We ask: (i) How does the rodent generate robust, rhythmic whisking that can lock to breathing as well as free-run between breaths? This addresses the issue of partial correlation of the whisking and breathing. (ii) How can an oscillator that generates periodic bursts of action potentials be constructed from solely inhibitory neurons? Networks with all inhibitory synapses typically produce two types of bursting patterns. One type is network-wide asynchronous bursts of spikes [Golomb (2007), Golomb and Rinzel (1993)], which is not observed for the vIRt. Another type is bursts of spikes that alternate among different clusters of neurons whose membership may evolve over time [Golomb and Rinzel (1994), Golomb et al. (2001)]. In contrast, the vIRt oscillator requires two fixed populations of neurons that alternately produce bursts of spikes. (iii) How can we achieve high variability in the firing patterns of vIRt neurons, recalling that variability is associated asynchronous dynamics van Vreeswijk and Sompolinsky (1996), van Vreeswijk and Sompolinsky (1998)] as opposed to oscillations. Further, what improvements in functional output are achieved, if any, by variability in the output of the circuit [Pehlevan and Sompolinsky (2014)]?

To address these questions, we construct a circuit model of the vIRt that incorporates the known anatomical connections, spiking and synaptic properties of the associated neurons, and the interaction of the vIRt with the pBötC and the vFMN. We formulate our model with conductance-based equations for “realistic” neurons. Yet, as our goal is to identify key parameters that control rhythmic whisking and the interaction of the pBötC and vIRt oscillators, we apply averaging of the spiking output and reduce this model to analytical expressions for the time-dependent spike rate of populations of the underlying neurons.

2. RESULTS

2.1. Experiments

Two hypotheses may explain the generation and synchrony of bursts of spikes in vIRt neuronal populations. One is that the input from pBötC is necessary to drive and synchronize bursts of spikes in vIRt neurons, which implicitly entrains whisking to the breathing rhythm. Consequently, whisking oscillations should vanish, or at least decay, after each input from the pBötC. A second hypothesis is that rhythmic bursts of spikes are generated and synchronized by the inhibitory interactions among vIRt neurons. Under this scenario, synchrony of spiking between individual neurons within the vIRt can prevail without input from the pBötC; external input is necessary only to lock the whisk cycle to the breathing rhythm. These hypotheses are not exclusive. Whisking is synchronized to breathing, yet whisking can be pharmacologically induced independent of breathing [Moore et al. (2013), Deschênes et al. (2016b), Moore et al. (2014a)]. Furthermore, whisking can occur for many cycles after onset of inhalation [Moore et al. (2013), Deschênes et al. (2016b)]; in this case the amplitude of the so-called intervening whisks appear to decay following inhalation [Figure 7 in Deschênes et al. (2016b)]. Thus it is possible that the input from the pBötC synchronizes neurons the vIRt, yet in addition these neurons may also synchronize on their own.

2.1.1. Intervening whisks

To resolve the interplay of mechanisms that drive rhythmic whisking, we re-analyzed the data on whisking by head-restrained rats [Moore et al. (2013)] (Figure 1B,C), and obtained and analyzed new data on whisking by unrestrained rats (Figure 1D). Angular amplitudes of consecutive whisks during one breathing cycle are denoted by ${\theta }_1^{\text{amp}},{\theta }_2^{\text{amp}},\dots ,{\theta }_{N_{\mathrm{W}}}^{\mathrm{a}\mathrm{m}\mathrm{p}}$ (Figure 1E), where $N_{\mathrm{W}}$ is the number of whisks in a breathing cycle. We observe that the number of breathing cycles that includes $N_{\mathrm{W}}$ whisking cycles decreases exponentially, with almost identical decay constants for head-restrained and freely-moving rats (Figure 1F). Further, ${\theta }_1^{\text{amp}}$ is essentially the same for the two behavioral conditions, with ${\theta }_1^{\text{amp}}\approx {15}^{°}$ (Figure 1G). These findings add to past work that showed a slight reduction in whisking frequency with head-fixation [Berg and Kleinfeld (2003)]. Thus head-fixation quantitatively but not qualitatively affects the amplitude of whisking.

We computed the probability distribution function (PDF) of the ratio ${\theta }_{i+1}^{\text{amp}}/{\theta }_i^{\text{amp}}$ as a means to quantify the average dependence of the amplitude on the index, i (Figure 1H). We observe that the amplitude of the second whisk after a breath, i.e., ${\theta }_2^{\text{amp}}$, is smaller than the amplitude directly affected by input from the pBötC, i.e., ${\theta }_1^{\text{amp}}$. The amplitude of successive whisks asymptotes to a constant value, approximated by ${\theta }_4^{\text{amp}}/{\theta }_1^{\text{amp}}=0.43$ for head-restrained rats and ${\theta }_4^{\text{amp}}/{\theta }_1^{\text{amp}}=0.64$ for freely-moving rats. The diminished steady-state amplitude applies to whisks generated solely by the vIRt, as opposed to driven by the pBötC.

2.1.2. Analysis of spiking variability

The variability in spike timing of neurons in a circuit provides clues to the nature and extent of the underlying synaptic connections [Rosenbaum et al. (2017)]. We previously reported on neurons that preferred to spike upon retraction versus protraction of the vibrissae in rats [Deschênes et al. (2016b), Deschênes et al. (2016a)] (Figure 1I). We now calculated the spike-to-spike variability of individual vIRt^pro and vIRt^ret neurons during the inter-spike intervals within bursts (Figure 1J). Our metric is the coefficient of variation ${\mathrm{CV}}_2$ [Holt et al. (1996)]. We find that ${\mathrm{C}\mathrm{V}}_2=0.75\pm 0.05$ (mean ± SEM). We also calculate ${\mathrm{C}\mathrm{V}}_2$ from extracellular recording from inhibitory neurons in the vIRT of mice [Takatoh et al. (2022)], and find that ${\mathrm{C}\mathrm{V}}_2=0.55\pm 0.02$. These relatively large values imply that the timing of spikes from individual vIRt neurons is irregular during a burst.

2.2. Circuit model

2.2.1. Architecture

Our model of the circuit for rhythmic whisking consists of the vIRt per se and the motoneurons in the vFMN that drive the intrinsic vibrissa muscles, along with rhythmic input from the pBötC and constant input from a high-order excitatory drive (Figure 2; Star Methods). In the absence of a connectome for the vIRt oscillator, we base the architecture on the observed segregation of vIRt neurons into two functional clusters, i.e., the vIRt^ret and vIRt^pro subpopulations (Figure 2A). There are N neurons in each population. The probability that a neuron from one population is inhibited by a neuron from the same or the second population is given by the fraction $K/N$, where K is smaller than but of order N. We take the architecture to be symmetric, so the strength of individual synapses within a subpopulation of vIRt neurons, i.e., between pairs of vIRt^ret neurons and between pairs of vIRt^pro neurons, is $g_{\text{intra}}/K$ [Amit (1989)]. Similarly, we assume that the strength of individual synapses between neurons in the two subpopulations, i.e., between a pair of vIRt^ret and vIRt^pro neurons, is symmetric with $g_{\text{inter}}/K$.

Figure 2. Architecture of the brainstem circuit model for whisking.

(A) The neuronal-level circuit for conductance-based modeling. The triangles are neurons. The blue and red colors denote inhibitory and excitatory connections, respectively. The currents $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ and $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}$ represent constant external depolarizing input to the vIRt and vFNM neuronal populations, respectively. Conductances between neuronal pairs are denoted by $g_{\text{intra}}/K$ for pairs of neurons that belong to the same subpopulation, by $g_{\text{inter}}/K$ for neuronal pairs from two different subpopulations, and by $g_{\mathrm{F}\mathrm{r}}/K$ for vIRt^ret-to-vFNM connections. The amplitude of the square-wave pBötC-to-vIRt^ret input is denoted by $g^{\mathrm{r}\mathrm{B}}$.

(B) Schematic of the different currents in the cellular modelfor vIRt cells; the same model applies to vFM neurons with possibly different conductances. The currents are a leak current, ${\mathrm{I}}_{\mathrm{L}}$, the transient sodium current, $I_{\mathrm{N}\mathrm{a}}$, the delayed rectifier potassium current, $I_{\mathrm{K}\mathrm{d}\mathrm{r}}$, the persistent sodium current $I_{\mathrm{N}\mathrm{a}\mathrm{P}}$, the mixed cation h-current, $I_{\mathrm{h}}$, an M-type ${\mathrm{K}}^{+}$ current, $I_{\text{adapt}}$, external excitatory currents from other brain areas, $I_{\text{ext}}$ and synaptic currents $I_{\mathrm{s}\mathrm{y}\mathrm{n}}$ that comprise $I_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}},I_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}$ and $I^{\mathrm{r}\mathrm{B}}$ or $I^{\mathrm{F}\mathrm{r}}$. Details in Star Methods.

(C) Spiking rate versus $I_{\text{ext}}^{\mathrm{r}}$ curves of the single vIRt neuron model for values of $g_{\text{adapt}}$ that range from 0 to 7 mS/cm² (solid lines). The insert is for $g_{\text{adapt}}=6\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ with the spike rate versus $I_{\mathrm{e}\mathrm{x}\mathrm{t}}$ curve as solid and a linear approximation as dotted, as used in the rate model.

(D) The force developed by an intrinsic muscle that moves vibrissae as a function of the average firing rate of vFMN neurons, $M^{\mathrm{F}}$. The solid black line denotes simulation results, and the dotted red line denotes a fit. Details in Star Methods.

(E) Schematic of the rate model in which each population of neurons is described by an activation variable that controls its synaptic outputs, and another activation variable that controls the adaptation current. The interactions $J_{\text{intra}}$ and $J_{\text{inter}}$ replace $g_{\text{intra}}$ and $g_{\text{inter}}$, respectively.

The observation that the vIRt^ret and vIRt^pro subpopulations are active in anti-phase supports the choice that vIRt^ret neurons, but not the vIRt^pro neurons, receive rhythmic inhibitory input from the pBötC. Since the vIRt neurons are inhibitory, this is consistent with potential synchrony of breathing and whisking. The vIRt^ret neurons rhythmically inhibit neurons in the vFMN. In addition, all neurons receive a constant depolarizing input, denoted $I_{\text{ext}}^{\mathrm{r}}$ for the vIRt^ret and vIRt^pro subpopulations and $I_{\text{ext}}^{\mathrm{F}}$ for the vFMN population. This completes the neuronal circuit (Figure 2A).

2.2.2. Single cell and vibrissa dynamics

Single neurons in the vIRt and the vFMN are represented by conductance-based schemes (Figure 2B). In addition to spike-generating ionic channels, we require that vIRt posses slow internal dynamics as a means to produce bursts of spikes. The parsimonious solution is to include a slow ${\mathrm{K}}^{+}$-channel that causes spike-frequency adaptation [Soloduchin and Shamir (2018)]. The conductances and adaptation time constants of this channels are $g_{\text{adapt}}^{\mathrm{r}}$ and ${\tau }_{\mathrm{a}}^{\mathrm{r}}$ for all vIRt neurons and $g_{\text{adapt}}^{\mathrm{F}}$ and ${\tau }_{\mathrm{a}}^{\mathrm{F}}$ for vFMN. These currents naturally occur in vFMN neurons [Magarinos-Ascoñe et al. (1999), Nguyen et al. (2004)].

The spike-rate versus input current relation, i.e., $f\text{-}{I}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ curve, for vIRt neurons with various values of $g_{\text{adapt}}^{\mathrm{r}}$ show that the $f\text{-}{I}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ curve rises linearly just above the onset of spiking for a sufficiently large value of $g_{\text{adapt}}^{\mathrm{r}}$ [Shriki et al. (2003)] (Figure 2C). The transformation from the spiking patterns of vFNM neurons to vibrissa movement is calculated with the use of a simplified version of a mechanical model [Simony et al. (2010)] (Figure 2D). We exploit the slow time-scale of the adaptation current, the self-consistent assumptions of asynchronous spike activity, and the near-constant external inputs to the neurons to develop a rate model for spiking (Figure 2E).

2.3. Whisking paced by rhythmic inhibition from the pBötC

A first mode of whisking makes use of the vIRt^ret as a relay and filter. The synchronous drive of whisking by input from pBötC to the vIRt^ret, that, in turn, inhibits neurons in the vFMN, i.e., “inhibition-of-inhibition”, provides near synchrony of inhalation and whisking (Figure 1B). Beyond this qualitative description, we analyzed a minimal model with only feedforward connections from the pBötC to the vIRt^ret and from the vIRt^ret to the vFMN (Figure 3A). Numerical simulation of the conductance-based equations for sets of physiologically plausible parameters show that vIRt^ret neurons are silenced during spiking of the pBötC (Figure 3B,C). This silencing leads to heightened spiking in vFNM neurons and concurrent large amplitudes of rhythmic whisking (Figure 3B–D).

Figure 3. Dynamics of a conductance-based and rate-based feed-forward circuit for whisking.

(A) Schematic of the conductance-based circuit with input from the pBötC to the vIRt^ret and from the vIRt^ret to the vFMN to drive whisking. The vIRt^pro subpopulation plays no role. We further set $g_{\text{intra}}=0$. We chose $g^{\mathrm{r}\mathrm{B}}=0.5\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\mathrm{F}\mathrm{r}}=3\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,I_{\text{ext}}^{\mathrm{r}}=20\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2,I_{\text{ext}}^{\mathrm{F}}=3.1\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2,K=25,g_{\text{adapt}}^{\mathrm{r}}=7\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\text{adapt}}^{\mathrm{F}}=0.3\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,N=100,T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=200\mathrm{m}\mathrm{s},T_{\text{rand}}=10\mathrm{m}\mathrm{s},\mathrm{\Delta }{T}_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}=70\mathrm{m}\mathrm{s}$. Details of all calculations are in Star Methods.

(B) Time courses of spiking for an example neuron from the vIRt^ret and vFMN subpopulations.

(C) Rastergrams across twenty neurons in the simulated subpopulations.

(D) The calculated vibrissa angle $\theta \left(t\right)$, calculated with the model of the motor plant and common parameters.

(E) The reduced rate-based circuit with input from the pBötC to the vIRt^ret and from the vIRt^ret to the vFMN to drive whisking.

(F-G) Circuit properties as the synaptic conductance from the pBötC is increased. Parameters are as in panel A, except that $g^{\mathrm{r}\mathrm{B}}$ is varied. Properties are computed using three modeling strategies. First, by numerical simulations of the conductance-based model (solid black line and circles). Simulations were carried out over five realizations for each parameter set, and the error bars denote standard deviation. Second, by numerical simulations of the rate model equations (thick cyan line). Third, from analytical solution of the rate model (thin orange line). The average number of spikes produced by the vIRt^ret neurons per whisk is shown in panel F. The average number of spikes produced by vFMN neurons per whisk is shown in panel G. The average amplitude of each whisk is shown in panel H.

2.3.1. Conversion to a rate model

We reduced the conductance equations to a rate model to gain insight into the behavior of the feedforward circuit (Text Box 2). This is a crucial aspect of our analysis, as it provides the framework to determine how different aspects of whisking depend on specific cellular and extrinsic parameters, both here and later in the description of autonomous oscillations. The dynamics of the vIRt^ret and vFNM subpopulations are each controlled by two equations: one equation describes the activation rate of the synaptic inputs and the other the magnitude of the adaption current. We find that the average number of spikes produced during the time interval of pBötC activity by a vIRt^ret neuron is an approximately linearly decreasing function of the pBötC to vIRt total synaptic strength, $g^{\mathrm{r}\mathrm{B}}$, until the number reaches zero (Equations 6 and 7) (Figure 3E,F; analysis is confirmed by numerical simulations of the conductance-based model). Conversely, the activity of the vFMN neurons are a linearly increasing function of $g^{\mathrm{r}\mathrm{B}}$ between a threshold value and a saturation value of $g^{\mathrm{r}\mathrm{B}}$ (Figure 3G). Lastly, the whisking amplitude increases with increasing values of $g^{\mathrm{r}\mathrm{B}}$, above threshold, until a saturated value is reached (Figure 3H).

Text Box 2. Rate description of the feed-forward circuit.

Rate equations are derived by summing the fast postsynaptic synaptic currents to each neuron and averaging over all presynaptic neurons [Shriki et al. (2003), Golomb et al. (2006), Hayut et al. (2011)]. They contain effective synaptic weights, denoted by “J”, that are proportional to synaptic conductances as well as the difference between the average membrane potential and the synaptic reversal potential.

Effective interactions.

The rate equations for the vIRt and vFMN neurons are in terms of an effective interaction for the fast synapses, denoted $J^{\mathrm{F}\mathrm{r}}$ and given by (Equation 49 in Star Methods)

$$J^{\text{Fr}}=-g^{\text{Fr}}{\langle {\langle V^{\text{F}}(t)-V_{\text{GABA}}\rangle }_t\rangle }_i$$ (3)

where $g^{\mathrm{F}\mathrm{r}}$ is the conductance of the vIRt^ret inputs to vFMN neurons, $V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}$ is the reversal potential for ${\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_A$ receptors, and ${⟨⟨\dots {⟩}_t⟩}_i$ means population- and time-averaged. Typical values of ${⟨{⟨V_{\mathrm{F}}\left(t\right)-V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}⟩}_t⟩}_i$, found numerically, are −27 to −20 mV. $J^{\mathrm{F}\mathrm{r}}$ has units of current. Effective coefficients that represent the increase of adaptation with firing rate are obtained from the conductances of adaptation currents, i.e.,

$$J_{\text{a}}^{\text{F}}={\gamma }^{\text{F}}{g}_{\text{adapt}}^{\text{F}}$$ (4)

for neurons in the vFMN and

$$J_{\text{a}}^{\text{r}}={\gamma }^{\text{r}}{g}_{\text{adapt}}^{\text{r}}$$ (5)

for neurons in the vIRt, where the constants ${\gamma }^{\mathrm{F}}$ and ${\gamma }^{\mathrm{r}}$ are determined by a linear fit of the spike rate versus $I_{\mathrm{e}\mathrm{x}\mathrm{t}}$ curves (Figure 2C) and $J_{\mathrm{a}}^{\mathrm{F}}$ and $J_{\mathrm{a}}^{\mathrm{r}}$ have units of charge. Lastly, the inhibitory input from the pBötC is given by

$$I^{\text{rB}}=g^{\text{rB}}{\langle {\langle V(t)-V_{{\text{GABA}}_{\text{A}}}\rangle }_t\rangle }_i$$ (6)

and has units of current. The input is in the form of a square-wave with frequency $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ and active duration $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$.

Spike counts.

Which parameters control the amplitude of a whisk? The amplitude depends on the number of spikes produced by vIRt^ret neurons during period $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. We denote this number $N_{\text{spikes}}^{\mathrm{r}}$ and consider the formula for $N_{\text{spikes}}^{\mathrm{r}}$ in the simplifying limit that the frequency of whisking is low, i.e., $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}\ll 1/{\tau }_{\mathrm{a}}^{\mathrm{r}}$ and $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}\ll 1/{\tau }_{\mathrm{a}}^{\mathrm{F}}$, where ${\tau }_{\mathrm{a}}^{\mathrm{r}}$ and ${\tau }_{\mathrm{a}}^{\mathrm{F}}$ are the time constant of the adaptation currents in ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\text{ret}}$ neurons and vFMN respectively. Here

$$\begin{gathered} N_{\text{spikes}}^{\text{r}}=\frac{{\tau }_{\text{a}}^{\text{r}}{\beta }^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}} \\ \times \left\{\frac{\text{Δ}{T}_{\text{pBötC}}}{{\tau }_{\text{a}}^{\text{r}}}\left(I_{\text{ext}}^{\text{r}}-I_0^{\text{r}}\right)-I^{\text{rB}}\left[\frac{\text{Δ}{T}_{\text{pBötC}}}{{\tau }_{\text{a}}^{\text{r}}}+\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\left(1-e^{-\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)\text{Δ}{T}_{\text{pBötC}}/{\tau }_{\text{a}}^{\text{r}}}\right)\right]\right\}, \end{gathered}$$ (7)

where $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ is the external excitatory current, $I_0^{\mathrm{r}}$ is a threshold current, and ${\beta }^{\mathrm{r}}$ is the neuronal “gain” in the rate equations that relates the intracellular input to the spike-rate, with units of rate/current or 1/charge.

Equation 7 for $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{r}}$ highlights the competition between the excitatory drive, i.e., $\left(I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I_o^{\mathrm{r}}\right)$, and the rhythmic inhibition from the pBötC. When $I^{\mathrm{r}\mathrm{B}}$ is moderate in magnitude, the vIRt^ret neurons cease to spike just after the beginning of the pBötC activity, and when it is sufficiently large the vIRt^ret neurons are silent during the whole pBötC activity period. Lastly, in the high gain limit, the spike count is independent ${\beta }^{\mathrm{r}}$.

Turning to the motor output, the number of spikes produced by vFMN neurons is zero below a minimum value of $g^{\mathrm{r}\mathrm{B}}$. The number reaches a saturation level if and when vIRt^ret neurons become completely silent during pBötC activity. Between these two values the number, denoted $N_{\text{spikes}}^{\mathrm{F}}$, increases monotonically, and near linearly as a function of $g^{\mathrm{r}\mathrm{B}}$. The whisking amplitude depends on the muscular force that moves the vibrissae and this too increases monotonically with $N_{\text{spikes}}^{\mathrm{F}}$.

A further insight provided by the rate analysis (Equation 7) is that the response time of the feed-forward circuit is reduced from the time constant of the slow adaptation, i.e., ${\tau }_{\mathrm{a}}^{\mathrm{r}}$, to ${\tau }_{\mathrm{a}}^{\mathrm{r}}/\left(1+{\beta }^{\mathrm{r}}{J}_{\mathrm{a}}^{\mathrm{r}}\right)$. While the slow adaption current will be shown to be critical for the generation of oscillations by the full vIRt circuit, the slow-time is accelerated for feed-forward transmission. This leads to the rapid inhibition of vIRt^ret cells and the near synchrony of inhalation and whisking.

2.4. Rhythmic whisking without input from the pBötC

We begin by checking the viability of the vIRt circuit alone to oscillate and drive rhythmic whisking via the projection from vIRt^ret neurons to those in the vFMN (Figure 4A); this subsystem is solely responsible for intervening whisks. A minimal circuit contains only connections between the vIRt^ret and vIRt^pro subpopulations, i.e., $g_{\text{inter}}\ne 0$ but $g_{\text{intra}}=0$. Individual neurons in this two subpopulation system can generate alternating burst of spikes (Figure 4B). The network robustly oscillates at physiological frequencies, as seen in the spike raster plots and in the rhythmic cycling of the calculated vibrissa motion (Figure 4C,D). Of note, the two subpopulations alternate between regular spiking and quiescence; variability within the spike trains is small (Supplementary Figure 1). This minimal circuit captures the desired rhythmic output of vIRt neurons, but not the observed variability (Figure 1J).

Figure 4. Dynamics of conductance-based circuits without pBötC input to the vIRt.

(A) Schematic of the circuit with connections only between vIIRt^ret and vIRt^pro subpopulations, for which we set $g_{\text{intra}}=0$. We chose $g_{\text{inter}}=6\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\text{adapt}}=7\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,I_{\mathrm{e}\mathrm{x}\mathrm{t}}=20\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2,K=25$, and $N=100$.

(B) Time courses of spiking for an example neuron from the vIIRt^ret, vIIRt^pro, and vFMN subpopulations.

(C) Rastergrams across twenty neurons in the simulated subpopulations.

(D) The calculated vibrissa angle $\theta \left(t\right)$ with the model of the motor plant.

(E) Schematic of the circuit with connections both between vIRt^ret and vIRt^pro subpopulations and within each subpopulation, for which we set $g_{\text{intra}}=12\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ and $g_{\text{inter}}=20\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$; other parameters as in panel A

(F) Time courses of spiking for an example neurons for the vIRt^ret, vIRt^pro, and vFMN subpopulations. Properties are computed using three modeling strategies as in Figure 3F–G.

(G) Rastergrams across twenty neurons in the simulated subpopulations.

(H) The calculated vibrissa angle $\theta \left(t\right)$.

The spiking patterns of vIRt neurons becomes irregular when intraneuronal connections are added within each subpopulation, i.e., $g_{\text{inter}}\ne 0$ and $g_{\text{intra}}\ne 0$, with ${\mathrm{C}\mathrm{V}}_2=0.5$ for the example of Figure 4E,F. Neurons in the two subpopulations spike throughout both halves of the whisking cycle (Figure 4G), but with a bias so that the summed activity across many neurons leads to a rhythmic albeit noisy trajectory of whisking (Figure 4H). This simulated activity is similar to most experimentally observed trajectories [Moore et al. (2013), Deschênes et al. (2016b)] (Figure 1B–D). Based on these numerical results, we focus on a circuit for the vIRt that includes both inter- and intra-neuronal connections.

2.4.1. Conversion to a rate model

The analysis of the rate equations (Text Box 3) show that there are three forms of output dynamics as a function of the difference in conductances

$$\text{Δ}{g}_{\text{syn}}=g_{\text{inter}}-g_{\text{intra}}.$$ (8)

Text Box 3. Rate description of the vIRt oscillator.

There are three operating domains of the network as a function of the difference in effective interactions (Star Methods), i.e.,

$$\text{Δ}{J}_{\text{syn}}\equiv J_{\text{inter}}-J_{\text{intra}}$$ (9)

where, as in Text Box 2,

$$J_{\text{inter}}=-g_{\text{inter}}{\langle {\langle V^{\text{vIRt}}(t)-V_{{\text{GABA}}_{\text{A}}}\rangle }_t\rangle }_i$$ (10)

and

$$J_{\text{intra}}=-g_{\text{intra}}{\langle {\langle V^{\text{vIRt}}(t)-V_{{\text{GABA}}_{\text{A}}}\rangle }_t\rangle }_i,$$ (11)

which both $J_{\text{inter}}$ and $J_{\text{intra}}$ have units of current.

Uniform state.

For values of $\mathrm{\Delta }{J}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ below a threshold denoted $J_{\mathrm{t}\mathrm{r}}$, with

$$J_{\text{tr}}=\frac{1}{{\tau }_s{\beta }^{\text{r}}}\left[1+\frac{{\tau }_s}{{\tau }_{\text{a}}^{\text{r}}}\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)\right],$$ (12)

both of the vIRt subpopulations are active [Soloduchin and Shamir (2018)]. The condition $\mathrm{\Delta }J<J_{\mathrm{t}\mathrm{r}}$ leads to uniform activity throughout the entire network. The average activity of each subpopulation, denoted by $M^{\mathrm{r}}$ or $M^{\mathrm{p}}$, is

$$M^{\text{r}}=M^{\text{p}}=\frac{{\beta }^{\text{r}}\left(I_{\text{ext}}^{\text{r}}-I_0^{\text{r}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}\left(J_{\text{intra}}+J_{\text{inter}}\right)},$$ (13)

where $M^{\mathrm{r}}$ and $M^{\mathrm{p}}$ have units of rate, the external input $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ is taken to be the same for vIRt^ret and vIRt^pro neurons, and $I_0^{\mathrm{r}}$ is a threshold current. The output is maintained even if adaptation is removed, i.e., $J_{\mathrm{a}}^{\mathrm{r}}$ goes to zero, and the net activity is nominally the ratio between external input that drives the neurons and inhibitory interactions that quench activity.

Oscillatory state.

For values of $\mathrm{\Delta }{J}_{\text{syn}}$ above $J_{\text{tr}}$ but below a second, larger threshold, denoted $J_{\text{det}}$ with

$$J_{\text{det}}=\frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_s{\beta }^{\text{r}}}$$ (14)

the network activity oscillates between the vIRt^ret and vIRt^pro subpopulations. The width of this region is

$$\text{Δ}{J}_{\text{syn}}=J_{\text{det}}-J_{\text{tr}}\underset{{\tau }_s/{\tau }_{\text{a}}^{\text{r}}\to 0}{\to }\frac{J_{\text{a}}^{\text{r}}}{{\tau }_s}.$$ (15)

Oscillations will occur even if $J_{\text{intra}}=0$, i.e., for a network with connections only between the two subpopulations of viRt neurons.

The period of the rhythm, $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$, is found from the solution of a transcendental equation with two dimensionless parameters and the dimensionless ratio $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}/{\tau }_{\mathrm{a}}^{\mathrm{r}}$ (Equation 132), which is exact in the limit of ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$, i.e.,

$$\frac{{\tau }_{\text{s}}{J}_{\text{inter}}}{J_{\text{a}}^{\text{r}}}=\frac{\left(\frac{1+\tilde{J}}{\tilde{J}}\right)\left[1-e^{-\left(\frac{2+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]-\left[1-e^{-\left(\frac{1+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]e^{-\left(\frac{1}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}}{\left(\frac{1+\tilde{J}}{\tilde{J}}\right)\left[1-e^{-\left(\frac{2+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]-\left[1-e^{-\left(\frac{1+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]}$$ (16)

with

$$\tilde{J}\equiv \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}.$$ (17)

Our formulas (Equations 16 and 17) highlights five features of the oscillatory output of the vIRt. First, the period is independent of the strength of the external input, $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ (Figure 5E). Second, the period scales linearly with the adaptation time-constant, i.e., $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\propto {\tau }_{\mathrm{a}}^{\mathrm{r}}$. Third, $1/T_{\text{vIRt}}$ diverges as $\mathrm{\Delta }{J}_{\text{syn}}\to J_{\text{tr}}$ from above, and goes to zero as $\mathrm{\Delta }{J}_{\text{syn}}\to J_{\text{det}}$ from below (Figure 5D). Fourth, $1/T_{\text{vIRt}}$ decreases monotonously with $J_{\text{inter}}$ (Equations 140 to 147). Lastly, in the high gain limit, the period is independent of the gain, ${\beta }^{\mathrm{r}}$.

The average spike rate is per cycle, which controls the amplitude of a whisk, is given by

$$\begin{gathered} {⟨M^r\left(t\right)⟩}_t=\frac{\tilde{J}}{1+\tilde{J}}\frac{\left(I_{\text{ext}}^{\text{r}}-I_0^{\text{r}}\right)}{J_{\text{a}}^{\text{r}}} \\ \times \left\{\frac{1}{2}+\frac{\tilde{J}}{1+\tilde{J}}\frac{{\tau }_{\text{a}}^{\text{r}}}{T_{\text{vIRt}}}\frac{\left[1-e^{-\left(\frac{1}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]\left[1-e^{-\left(\frac{1+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}\right]}{1-e^{-\left(\frac{2+\tilde{J}}{2}\right)T_{\text{vIRt}}/{\tau }_{\text{a}}^{\text{r}}}}\right\} \end{gathered}$$ (18)

and, like $N_{\text{spikes}}^{\mathrm{r}}$ (Equation 7), is linearly proportional to $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I_0^{\mathrm{r}}$ and, in the high gain limit, is independent of ${\beta }^{\mathrm{r}}$.

Bistable state.

For large values of $\mathrm{\Delta }{J}_{\text{syn}}$ with $\mathrm{\Delta }{J}_{\mathrm{s}\mathrm{y}\mathrm{n}}>J_{\text{det}}$, the network dynamics are bistable with only one of two subpopulations active with firing rate

$$M^{\text{r}}=\frac{{\beta }^{\text{r}}\left(I_{\text{ext}}^{\text{r}}-I_0^{\text{r}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}$$ (19)

and

$$M^{\text{p}}=0$$ (20)

or vice versa. Thus, the average activity is linearly proportional to the external input minus the current threshold, $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I_0^{\mathrm{r}}$, in all regions.

Numerical solution of the rate equations with the realistic values ${\tau }_s=10\mathrm{m}\mathrm{s}$ and ${\tau }_{\mathrm{a}}^{\mathrm{r}}=83$ $\mathrm{m}\mathrm{s}$ yields results that are close to the values found in the limit ${\tau }_{\mathrm{a}}^{\mathrm{r}}\gg {\tau }_s$ (Figure 5). Noting that

$$\text{Δ}{g}_{\text{syn}}\propto \text{Δ}{J}_{\text{syn}},$$ (21)

we present our results in the main text in terms of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$, where $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}=g_{\text{inter}}-g_{\text{intra}}$, to enable direct comparisons between calculated results from the rate equations and the results from simulations of the conductance-based equations. Similarly, $g_{\mathrm{d}\mathrm{e}\mathrm{t}}\propto J_{\mathrm{d}\mathrm{e}\mathrm{t}}$ and $g_{\mathrm{t}\mathrm{r}}\propto J_{\mathrm{t}\mathrm{r}}$.

When $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ is smaller than a critical value, denoted $g_{\mathrm{t}\mathrm{r}}$, neurons within both the vIRt^ret and vIRt^pro subpopulations are tonically active [Soloduchin and Shamir (2018)]. Thus the vIRt does not drive rhythmic whisking. In contrast, when $\mathrm{\Delta }{g}_{\text{syn}}$ is larger than $g_{\text{tr}}$ but smaller than a second critical value, denoted $g_{\text{det}}$, neurons within the vIRt^ret and vIRt^pro subpopulations alternate in their activity and the output of the vIRt^ret oscillates with a period of $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$. The period is independent of the excitatory external drive current, $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ (Equations 16 and 17). Further, in the physiologically relevant limit of ${\tau }_{\mathrm{a}}^{\mathrm{r}}\gg {\tau }_s$, where ${\tau }_s$ is the time-constant of synapses throughout the vIRt network, the period is proportional to ${\tau }_{\mathrm{a}}^{\mathrm{r}}$ (Equations 8 and 9). Finally, as $\mathrm{\Delta }{g}_{\text{syn}}$ increases so that it surpasses $g_{\text{det}}$, the output ceases to oscillate and becomes bistable. Here, vIRt^ret neurons are predominantly active while vIRt^pro neurons are predominantly quiescent, or vice versa, so that output from the vIRt^ret can only shift the set-point of the vibrissae. In all cases, the rate of spiking is proportional to the value of the external drive current that exceeds a threshold, denoted $I_0^{\mathrm{r}}$ (Equations 13 , 17 – 19).

The analysis of the rate equations yields insights to the dependence of whisking on network parameters (Figure 5A). The average activity of the network, M, is a weakly decreasing function of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ within the uniform and oscillatory regions, i.e., $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}\le g_{\text{det}}$ (Figure 5B). There are, however, differences in the calculated output between the conductance-based formalism and the rate model. For $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}>g_{\text{det}}$, the bistable output obtained in simulations of the conductance equations, with sparse connectivity and heterogeneity in neuronal intrinsic properties occurs for larger values of $\mathrm{\Delta }{g}_{\text{syn}}$ than predicted by the rate equations. The discrepancy is related to the irregular spiking (Figures 4E–H). While the average activity across the vIRt^ret and vIRt^pro subpopulations is well predicted from the rate formulation as a function of both $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ (Figure 5B) and $I_{\mathrm{e}\mathrm{x}\mathrm{t}}$ (Figure 5C, with $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ just shy of the value of $g_{\mathrm{d}\mathrm{e}\mathrm{t}}$), the vIRt^ret and vIRt^pro neurons do not spike completely in alternation. Thus, for $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}>g_{\text{det}}$, there is a wide range of values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ for which one neuronal subpopulation is more active than the other and oscillations remain. The difference in activity between the more and the less active subpopulation increases with $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ until only one subpopulation is active and the network is solely bistable.

Figure 5. Dynamical properties of circuits without pBötC input to the vIRt.

(A) Schematic of the circuit. The dynamics are calculated for $g_{\text{intra}}=12\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\text{adapt}}^{\mathrm{r}}=7\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\text{adapt}}^{\mathrm{F}}=0.3\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ and $I_{\text{ext}}^{\mathrm{F}}=3.1\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2$. The excitatory input fixed at $I_{\text{ext}}^{\mathrm{r}}=20\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2$ and $g_{\text{inter}}$ is varied in terms of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}=g_{\text{inter}}-g_{\text{intra}}$ for panels B, D, F, H, and J). $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ is fixed at $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}=8\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ (black arrow in in panel L) with $I_{\text{ext}}$ varying for panels C, E, G, I, and K. Properties are computed using three modeling strategies as in Figure 3F–G, and we use the same notation.

(B,C) The average spike rate $⟨M{⟩}_i$. The values for the vIRt^ret and vIRt^pro are equal in the uniform state, as the neuronal subpopulations are constantly spiking, and in the symmetric oscillatory state, since the neuronal subpopulations are alternately active. As of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ increases, one subpopulation becomes more active than the other and the less active subpopulation becomes silent at large values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$. The rate model exhibits a transition from a symmetric oscillatory state to a bistable state at a value of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}=g_{\mathrm{d}\mathrm{e}\mathrm{t}}$ while the actual transition in the conductance-based model occurs when of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ has further increased. The values of $g_{\mathrm{t}\mathrm{r}}$ and $g_{\mathrm{d}\mathrm{e}\mathrm{t}}$ are defined in Star Methods.

(D,E) The whisking frequency $1/T_{\text{vIRt}}$.

(F,G) The average whisking amplitude $<{\theta }^{\text{amp}}{>}_t$. Analytical results are computed in panel G for the uniform and the bistable state.

(H,I) The whisking set-point ${<{\theta }^{\text{set}}>}_t$. Analytical results are computed in panel I for the uniform and the bistable state.

(J,K) The coefficient of variation ${⟨{\mathrm{C}\mathrm{V}}_2⟩}_i$ calculated solely from the conductance-based equations.

(L) Phase diagram showing the three dynamical regimes, uniform (left), oscillatory (middle) and bistable (right) computed using the conductance-based model. Values of the coefficient of variation ${⟨{\mathrm{C}\mathrm{V}}_2⟩}_i$ calculated for several values of $\mathrm{\Delta }{g}_{\text{syn}}$ and $g_{\text{intra}}$ are written.

(M) The coefficient of variation ${⟨{\mathrm{C}\mathrm{V}}_2⟩}_i$ calculated as a function of K. The grey ribbon denotes typical experimentally-measured values for ${\mathrm{C}\mathrm{V}}_2$ (Figure 1J)

The frequency of whisking starts at high rates (Equations 16 and 17) for low values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$. It then decreases with increasing values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ (Figure 5D) over the full range of the oscillatory output. The frequency is calculated to be independent of the external input (Equation 16), thus separating the period of whisking from the amplitude of whisking (Equation 18). However, simulation of the current-based model shows that the period is a weakly increasing function of the external input, i.e., a factor of 1.4 over the full range of input current (Figure 5E), for realistic parameters.

The output from the vIRt^ret subpopulation is used to drive model motoneurons in the vFMN, whose spiking is subject to adaptation. The output from the motoneurons, in turn, serves as input to a model of the musculature in the mystacial pad and thus drives whisking. The whisking amplitude, ${\theta }^{\text{amp}}$ (Equation 1), is at or near zero in the uniform region and increases sharply when $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ exceeds $g_{\mathrm{t}\mathrm{r}}$. for which the output of the vIRt oscillates, until the amplitude reaches a saturation value (Figure 5F). In addition, the whisking amplitude is a monotonically increasing function of $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$, up to saturation, consistent with the increase in activity (Equation 18) with increasing external input (Figure 5G).

We now turn to the set-point of the vibrissae. For the uniform region, the inhibitory output from the vIRt initially counters the excitatory input to the vFMN and the vibrissae are almost fully retracted. The vibrissae gradually protract and, upon the onset of oscillations, the set-point reaches the full protracted position concurrent with saturation of the amplitude of whisking (Figure 5H). This set-point is maintained until the onset of bistability; depending on the state of the vIRt^ret subpopulation, the set-point either continues to be maintained or returns to fully a retracted position. The set-point is largely unaffected by changes in the external input to the vIRt (Figure 5I). The maximal value of the set-point during the oscillatory state is larger than the value when the vIRt^ret are the active subpopulation in a bistable region since the adaptation current does not reach its maximal value at the beginning of the active phase within an oscillatory period. All told, the set-point a function of the external input to the vFMN and both non-rhythmic and rhythmic component of output of the vIRt.

2.4.2. Variability in spike timing

Variability, as measured by ${\mathrm{C}\mathrm{V}}_2$ (Figure 1I), is outside of the rate formalism we apply. In contrast, numerical simulations of the conductance-based equations yield variable spike rates as a “finite size” effect that originates from the modest number of synaptic inputs to a cell, i.e., the values for the circuit parameters includes $K=$ 25 (Figure 4E–H) as a realistic estimate. This significantly contributes to nonzerovalues of ${\mathrm{C}\mathrm{V}}_2$ that extends across all values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ and external input $I_{\mathrm{e}\mathrm{x}\mathrm{t}}$ (Figure 5J,K). The circuit produces a state in which one sub-population fires more than the other one in a broader range of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$, as $g_{\text{intra}}$ increases and the variability increases (Supplementary Figure 1B,C, Figure 5B,C,L). There is therefore a quantitative relationship between range of oscillations and the value of $g_{\text{intra}}$: large values of $g_{\text{intra}}$ give rise to a larger values of variability. The implication is that variability increases the resilience of the oscillatory output.

Spiking variability during bursts, measured by ${\mathrm{C}\mathrm{V}}_2$, decreases as K increases while the ratio $K/N$ remains unchanged because fluctuations in the input synaptic conductances to a neuron are averaged out (Figure 5M). ${\mathrm{C}\mathrm{V}}_2$ does not go to zero for larger K because the inter-spike intervals during bursts decrease with time due to the adaptation current even without any fluctuations in the synaptic inputs (Supplementary Figure 1B,C). The measured ${\mathrm{C}\mathrm{V}}_2$ (Figure 1J) is consistent with the value we consider, i.e., $K=25$.

2.5. Composite system of pBötC input and vIRt oscillations

Up to now we separately analyzed the two sources of rhythmic whisking, i.e., feedforward drive by the pBötC and whisking driven by internal vIRt dynamics. Although the coupling from the pBötC to the vIRt is likely to be strong, the composite system is expected to operate in a manner largely predicted from the two mechanism for rhythmic whisking, i.e., drive from the pBötC and autonomous oscillations, since there is no known feedback from the vIRt to the pBötC [Deschênes et al. (2016b)]. To further understand the behavior of the composite system, we first simulated the conductance-based equations for the full circuit (Figures 2A). Choosing parameters equal to those of solely the vIRt circuit (Figure 4E–H), we see that the vIRt^ret neurons are silenced by activity in the pBötC, the vIRt^pro neurons are excited, and the vFMN neurons are excited (Figure 6A,B). This leads to protraction of the vibrissae with amplitude ${\theta }_1^{\text{amp}}$ in a manner that is nearly synchronous with inhalation (Figure 6B,C). The amplitude of this whisk is invariably larger than those of the subsequent intervening whisks, i.e., ${\theta }_2^{\text{amp}},{\theta }_3^{\text{amp}},\dots$, that are generated by the internal vIRt dynamics (Figure 6C).

Figure 6. Dynamics of a conductance-based circuits with pBötC input to the vIRt.

See Figure 2A for the schematic. We used the same parameters as the simulation without pBötC input (Figure 4E) plus $g^{\mathrm{r}\mathrm{B}}=0.5\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=700\mathrm{m}\mathrm{s},T_{\text{rand}}=150\mathrm{m}\mathrm{s}$, and $\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=70\mathrm{m}\mathrm{s}$.

(A) Time courses of spiking for an example neurons for the vIRt^ret, vIRt^pro, and vFMN subpopulations.

(B) Rastergrams across twenty neurons in the simulated subpopulations.

(C) The calculated vibrissa angle $\theta \left(t\right)$.

(D) The calculated whisking amplitude as a function of $g_{\text{inter}}$. The first whisk is driven by the pBötC input and subsequent intervening whisks, with $i=2\text{-}4$, are driven by internal vIRt dynamics. Error bars in D-F denote SD.

(E). The calculated whisking amplitudes as a function of $g^{\mathrm{r}\mathrm{B}}$.

(F). The slowing down of whisking for the intermediate whisks is shown by plotting ${<t_{w,i+1}-t_{w,i}>}_t$ as a function of $g_{\text{inter}}$.

2.5.1. Dependence of amplitudes and time differences on synaptic strengths

We performed numerical studies of the composite system to test aspects of the coordination of the pBötC-driven and vIRt-generated dynamics. First, we observed that the average amplitude of the first whisk, ${\theta }_1^{\text{amp}}$, is affected mainly by the pBötC-induced inhibition-of-inhibition and thus essentially independent of $g_{\text{inter}}$ (Figure 6D). It increases approximately linearly with increasing values of $g^{\mathrm{r}\mathrm{B}}$ until saturation is observed (Figure 6E). This dependency resembles that for the circuit with solely feedforward inhibition (Figure 3H). The non-zero value of ${\theta }_1^{\mathrm{a}\mathrm{m}\mathrm{p}}$ for $g^{\mathrm{r}\mathrm{B}}=0$ reflects the non-zero average amplitude of the rhythmic whisks that occur from the vIRt circuit dynamics and is similar to the amplitudes of intervening whisks ${\theta }_i^{\text{amp}}$, that are almost independent of $g^{\mathrm{r}\mathrm{B}}$ for $i\ge 3$ (Figure 6E). In contrast to ${\theta }_1^{\text{amp}}$, the average amplitude of intervening whisks increases with increasing values of $g_{\text{inter}}$ until saturation is reached (Figure 6D); this is consistent with the increase in whisking amplitude with increasing values of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$ in isolated vIRt circuits (Figure 5J). The amplitude ${\theta }_2^{\text{amp}}$ is somewhat larger than ${\theta }_3^{\text{amp}}$, consistent with the experimental observations in head-restrained rats (Figure 1H).

Lastly, we studied the instantaneous timing between whisks. The time $t_{\mathrm{w},i}$ is defined as the time of the maximal whisking angle during the i-th whisk in the breathing cycle. The average value of the difference in time between consecutive whisks, denoted $<t_{\mathrm{w},i+1}-t_{\mathrm{w},i}{>}_t$, is observed to increase with $g_{\text{inter}}$ (Figure 6F). This matches expectations for oscillations that are generated by the vIRt (Figure 5F).

2.5.2. Phase resetting

In general, output from the pBötC not only drives protraction of the vibrissae through the pBötC→vIRt^ret→vFMN relay (Figure 3A,E), but impacts the timing of the vIRt oscillator and results is partial coordination among the two oscillators. A central question is how individual whisking responses depend on the timing between the time of the maximum protraction angle of the previous whisk, designated $t_{\mathrm{w},1}$, and the onset time of pBötC activity, designated $t_{\mathrm{B}}$ (Figure 7A). We quantify the timing of the whisk immediately following the onset time of pBötC activity by the time of its maximum protraction angle, designated $t_{\mathrm{w},2}$. When a protraction is quickly followed by inhalation, the subsequent protraction occurs rapidly (Figure 7B, top line). This result matches experimental observations [Moore et al. (2013)] (Figure 1A,B), where this rapid succession is referred to as a “double pump” [Towal and Hartmann (2008), Deutsch et al. (2012)]. In contrast, when inhalation starts during protraction, the peak of the subsequent protraction is delayed (Figure 7B, bottom line). The relation between protraction events and inhalation across all data shows that the time between successive whisks, designated $\mathrm{\Delta }{t}_{\mathrm{w},21}=t_{\mathrm{w},2}-t_{\mathrm{w},1}$, varies linearly with the time between the onset of inhalation and the previous protraction, designated $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}=t_{\mathrm{B}}-t_{\mathrm{w},1}$. Formally,

$$\text{Δ}{t}_{\text{w},21}=s_{\text{w},21}\text{Δ}{t}_{\text{Bw},1}+\text{Δ}{t}_{\text{w},21}(0)$$ (22)

where the data yield a slope of $s_{\mathrm{w},21}=0.82$ and an intercept of $\mathrm{\Delta }{t}_{\mathrm{w},21}\left(0\right)=88\mathrm{ms}$ (Figure 7C). Thus a whisk is either advanced or delayed relative to the normal whisking period depending on the value of $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$.

Figure 7. Effects of pBötC activity on the timing and amplitude of the subsequent whisk.

(A). Cartoon to define the symbols.

(B). Stimulations of the conductance model to show the vibrissa angle $\theta \left(t\right)$ for two aspects of the timing of breathing relative to whisking (Figure 4). The common vertical dotted line, labeled $t_{\mathrm{w},1}$, denotes the peak of a whisk just prior to input from the pBötC. The top trace is an example of input from the pBötC just after the time of the peak and the bottom trace is an example of relatively late input. Model parameters: $g^{\mathrm{r}\mathrm{B}}=0.5\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=700\mathrm{m}\mathrm{s},T_{\text{rand}}=150\mathrm{m}\mathrm{s}$ (defined in Star Methods), $\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=70\mathrm{m}\mathrm{s}$.

(C). Experimental results from head-restrained rats [Moore et al. (2013)] of the total time between protraction, $\mathrm{\Delta }{t}_{\mathrm{w},21}$, as a function of the time of inhalation after the last protraction, $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$. The gray line is the linear fit over the full range. The mean is over all trials.

(D). The calculated total time $\mathrm{\Delta }{t}_{\mathrm{w},21}$ as a function of $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$ for two values of input conductance from the pBötC, i.e., $g^{\mathrm{r}\mathrm{B}}=0.5\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ (top) and $0.05\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ (bottom). The gray line is the linear fit over the range $40\mathrm{m}\mathrm{s}<\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}<140\mathrm{m}\mathrm{s}$. For $g^{\mathrm{r}\mathrm{B}}=0.05\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ and small values of $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$, the deduced values of $\mathrm{\Delta }{t}_{\mathrm{w},21}$ may be large since the first increase in the vibrissa angle caused by input from the pBötC may be too small to be detected as a separate new whisk.

(E). The slope $s_{\mathrm{w},21}$ (top) and the intercept $\mathrm{\Delta }{t}_{\mathrm{w},21}\left(0\right)$ (bottom) as a function of $g^{\mathrm{r}\mathrm{B}}$. Error bars denote SD. The dotted lines denote the experimental values from panel D.

(F). The whisking amplitude ${<{\theta }_1^{\text{amp}}>}_t$ versus breathing frequency $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. The top panel shows the frequency computed from experimental data from head-restrained rats [Moore et al. (2013)]. The grey area highlights the range of exploratory sniffing frequencies. The bottom panel shows the frequency calculated as a function of breathing frequency for two values of $\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=20\mathrm{m}\mathrm{s}$ (bottom lines) and 70 ms (top lines), and $T_{\text{rand}}=10$ (dotted lines) and 70 ms (solid lines).

For the limiting case of an arbitrarily large value of input from the pBötC, the vibrissae asymptote to their maximum position while the pBötC is active and reach maximum protraction just as the pBötC stops spiking, i.e., for $\mathrm{\Delta }{t}_{\mathrm{w},2\mathrm{B}}=\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ where $\mathrm{\Delta }{t}_{\mathrm{w},2\mathrm{B}}\equiv t_{\mathrm{w},2\mathrm{B}}-t_{\mathrm{B}}$ is the time between the onset of inhalation and the next protraction, and $\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ is the width of the inhibitory input from the pBötC to vIRt^ret neurons. Thus $t_{\mathrm{w},2}$ occurs at $t_{\mathrm{B}}+\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. In this limiting case, $\mathrm{\Delta }{t}_{\mathrm{w},21}=\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}+\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ so that the slope of the $\mathrm{\Delta }{t}_{\mathrm{w},21}$ versus $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$ line is $s_{\mathrm{w},21}=1$ and the intercept is $\mathrm{\Delta }{t}_{\mathrm{w}2,1}(0)=\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ (Equation 22). More realistically, the delay $\mathrm{\Delta }{t}_{\mathrm{w},2\mathrm{B}}$ is expected to decrease as $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$ increases in value, since the vibrissae will be part-way to maximal protraction; this reduces the value of the slope.

Simulation of the conductance-based equations show indeed that $\mathrm{\Delta }{t}_{\mathrm{w},21}$ is proportional to $\mathrm{\Delta }{t}_{\mathrm{B}1}$ (Figure 7D). The slope of the line for $\mathrm{\Delta }{t}_{\mathrm{w},21}$ versus $\mathrm{\Delta }{t}_{\mathrm{B}\mathrm{w},1}$ is greater for larger values of $g^{\mathrm{r}\mathrm{B}}$; it starts from 0 for $g^{\mathrm{r}\mathrm{B}}=0$ and increases to a saturation value of $s_{\mathrm{w},21}=0.87$, which is close to the experimental value (Figure 7C,E). The intercept $\mathrm{\Delta }{t}_{\mathrm{w},21}\left(0\right)$ is equal to $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ for $g^{\mathrm{r}\mathrm{B}}=0$ and decreases to a saturation value of $\mathrm{\Delta }{t}_{\mathrm{w},21}\left(0\right)=$ $85\mathrm{m}\mathrm{s}$; this is approximately the time needed to generate a whisk when the onset of pBötC activity arrives just after the previous maximum protraction. The conclusion from this analysis is that the experimental data (Figure 7C) imply that input from the pBötC to the vIRt^ret is strong, such that the pBötC largely resets the state of the vIRt oscillator, with $g^{\mathrm{r}\mathrm{B}}>0.2\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$.

The input from the pBötC also affects the average amplitude of the following whisk, ${<{\theta }_1^{\text{amp}}>}_t$. Experimental measurements of ${<{\theta }_1^{\text{amp}}>}_t$ versus the breathing frequency, $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$, reveal that $<{\theta }^{\text{amp}}{>}_t$ reaches a maximal value within the 4-10 Hz range of exploratory whisking (Figure 7F). This dependence is captured by the conductance-based model (Figure 1F) when the input to vIRt^ret neurons from the pBötC has a duration of $\mathrm{\Delta }{t}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}\approx 70\mathrm{m}\mathrm{s}$; see also Supplementary Figure 2. This corresponds to a boost in amplitude of protraction by feedforward inhibition from the pBötC that is timed to the inactivation of vIRt^ret neurons.

2.5.3. Response to suppressed inhibitory input to vIRt^ret neurons

Regular whisking in mice is perturbed through the expression of a gephyrin-specific ubiquitin ligase, designated GFE3 Gross et al. (2016), in vIRt^ret neurons. This manipulation suppresses the postsynaptic inhibitory current. The consequences are that no intervening whisks occur, the dominant whisking- breathing ratio is almost always one-to-one, and the whisking amplitude decreased by about a factor of two [Takatoh et al. (2022)].

Can we account for the change caused by GFE3 expression in vIRt^ret neurons with the current model for whisking (Figure 2A)? Toward this goal, we assume that GFE3 expression decreases the postsynaptic inhibitory conductance for all inputs to each vIRt^ret neuron by a common factor of x, where $x\in [0,1]$ and

$$g^{\text{rb}}\leftarrow x{g}^{\text{rb}},g^{\text{rr}}\leftarrow x{g}^{\text{intra}},\text{and}{g}^{\text{rp}}\leftarrow x{g}^{\text{inter}},$$ (23)

while all other conductances remain unchanged (Figure 8A). Numerical simulations of the conductance-based equations show that as x varies from one to zero, the amplitude of intervening whisks diminishes until intervening whisks disappear below a critical value, that is, $x\approx 0.6$ in Figure 8B. Mechanistically, the generation of oscillations by mutual inhibition between the vIRt^ret and vIRt^pro subpopulations of neurons is disrupted since GFE3 diminishes the strength of the vIRt^pro-to-vIRt^ret but not vIRt^ret-to-vIRt^pro pathway.

Figure 8. Analysis of perturbations to vIRt^ret synapses and summary of brainstem control of whisking.

(A) Diagram for dynamics of the vibrissa oscillator with the vIIRt^ret under partial gephyrin degradation to weaken all ${\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}$-ergic inhibitory inputs to vIRt^ret neurons (yellow circle). We assume that all synapses in the vIRt^ret are equally weakened by a factor of x. A. We chose $g_{\text{intra}}=12\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g_{\text{inter}}=20\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,g^{\mathrm{r}\mathrm{B}}=0.5\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$, $g_{\text{adapt}}=7\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,I_{\text{ext}}^{\mathrm{r}}=20\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2,I_{\text{ext}}^{\mathrm{F}}=3.1\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2,K=25$ and $N=100$ for our simulations.

(B) The amplitude of the four consecutive whisks after the onset of pBötC activity, denoted by $i=1,2,3,4$, as a function of x (Equation 23). Error bars denote SD. The yellow band in B-D indicates the fraction estimated from the data in [Takatoh et al. (2022)].

(C) The times between average successive whisks ${<t_{\mathrm{w},\mathrm{i}+1}-t_{\mathrm{w},\mathrm{i}}>}_t$ as a function of as a function of x.

(D) Dependence of ${⟨{\mathrm{C}\mathrm{V}}_2⟩}_i$ of vIRt^ret neurons on the value of x.

(E) Summary of the circuitry that underlies the vibrissa motor plant. The vIRt drives rhythmic motion, and the shape of the waveform is set by the motor plant and feedback pathway that change the input to intrinsic and extrinsic motoneurons, and thus contact forces, upon touch. Figures updated from published summary [Bellavance et al. (2017), Kleinfeld and Deschênes (2011)].

A seeming paradox is that the average time difference between consecutive whisks, i.e., ${⟨t_{\mathrm{w}(i+1)}-t_{\mathrm{w}\mathrm{i}}⟩}_t$, increases for $i=1$ and only weakly decreases $i=2$ and 3 as x decreases (Figure 8C). Further, the amplitude ${\theta }_1^{\text{amp}}$ first increases then decreases as x varies from one to zero. The primary reason for these increases is that intervening whisks leave the vibrissae, on average, at a protracted angle when the pBötC inhibition begins (Figure 7B, top), opposed to the more retracted initial angle when intervening whisks are blocked by GFE3. A secondary reason is that intervening whisks lead to greater adaptation of the facial motoneurons in comparison to a state of silent vFMN neurons. As x continues to decrease, ${\theta }_1^{\text{amp}}$ eventually is diminished to zero as a consequence of weak disinhibition of vIRt^ret neurons. The observed elimination of intervening whisks and a two-fold decrement in ${\theta }_1^{\text{amp}}$ correspond to $x\approx 0.3$. We predict that blocking inhibition among the vIRt^ret neurons will decreases spiking irregularity (Figure 8D).

3. DISCUSSION

We delimited and evaluated two mechanisms and their interplay that drive rhythmic spiking in neurons of the vibrissa oscillator, the vIRt, in the medulla. The vIRt is comprised predominantly of inhibitory neurons. Our model (Figure 2A,E) explains published [Moore et al. (2013), Towal and Hartmann (2006), Ranade et al. (2013), Berg and Kleinfeld (2003)] and new (Figure 1D,F–H) behavioral data on whisking in terms of the underlying anatomy and physiology [Moore et al. (2013), Deschênes et al. (2016b), Takatoh et al. (2013)], including newly reanalyzed variability (Figure 1I) . The first mechanism is feedforward control by strong inhibitory input from the pBötC inhalation oscillator to one subpopulation of neurons in the vIRt (Figure 3). This input results in disinhibition of the neurons in the facial nucleus that control the intrinsic muscles and leads to protraction of the vibrissae. The second mechanism is autonomous generation of rhythmic spiking activity across two subpopulations of neurons in the vIRt (Figures 4 and 5). More than the classic half-center oscillator scheme, in which reciprocal inhibition between two neurons [Brown (1911)] plus a mechanism for cellular or synaptic adaptation [Marder and Calabrese (1996), Satterlie (1985), Isett et al. (2018)] leads to oscillations, here the autonomous generation of oscillations requires two subpopulations of neurons (Figure 2A,E). The connections within each subpopulation lead to variability in neuronal spike times (Figures 4E–H). This in turn increased the dynamic range of the rhythmic output and makes the rhythm robust against changes in network parameters, such as the specific value of synaptic conductances (Figure 5F,L,M).

The interaction of the two mechanisms for spiking by neurons in the vIRt is observed when the breathing rate is slow (Figure 1C–H). The first whisk in a breathing cycle is driven by output from the pBötC and has the greatest amplitude (Figure 1E). The subsequent whisks are driven exclusively by the vIRt. The ratio of the interval between breaths to the interval between whisks is non-integer [Moore et al. (2013), Deschênes et al. (2016b)]. This is consistent with the relatively fixed period of the vIRt oscillator (Figure 5G) and the apparent lack of feedback from whisking, either corollary discharge from the vIRt or reafferent whisking input from the trigeminus to the breathing complex [Moore et al. (2015)]. Further, resetting the state of vIRt neurons by strong input from pBötC leads to non-monotonic protractions, such events may appear as “double pumps” in vibrissa touch [Towal and Hartmann (2006), Deutsch et al. (2012)] (Figures 6C and 7B). As a general issue, the drive of vibrissa movement by incommensurate rhythmic signals from breathing and vIRt oscillators results in a pattern of partial correlation between breathing and whisking, in line with the ideas of von Holst for the interactions of rhythmic motor actions across both active sensing as well as locomotion [von Holst (1939/1973)].

Beyond the classic half-center oscillator, past work with all-inhibitory networks focused on three architectures. The first architecture is small loops of neurons with an odd number of cells in the loop [Kling and Szekely (1968), Ádám and Kling (1971)], analogous to a ring oscillator in electronics. A pulse of spikes will propagate around the network. This circuit provides the basic motif for the generation of the swim rhythm in the leech [Stent et al. (1978)]. The second architecture is populations of inhibitory neurons without spatial structure to their connectivity. Such homogeneous networks, for the case of all-to-all coupling, may show synchronous spiking. This circuit provides the basic motif for the generation of fast gamma rhythms in mammalian cortex [Whittington et al. (1995)], although the circuit in neocortex may incorporate other cell types [Börgers et al. (2005)]. All inhibitory networks can produce seemingly complex spiking behavior as well. One instantiation is that groups of neurons that tend to spike together coalesce in clusters that spike alternately over time [Golomb and Rinzel (1994)]. A second instantiation is networks in which all of the neurons spike asynchronously [Golomb and Hansel (2000), Neltner et al. (2000)]. The third architecture, of relevance to the present work, makes use of two subpopulations of neurons, each with an adaptation current, to generate alternating burst of spikes. This architecture has been used to model rhythmic spiking by spinal cord circuits involved in locomotion [Ausborn et al. (2018)]. Guided by the observation [Moore et al. (2013), Deschênes et al. (2016b)] of two subpopulations of neurons in the vIRt, we incorporated and advanced such an architecture by incorporating two strengths of inhibition (Figure 2A).

We model the vIRt using conductance-based equations and map the all-inhibitory network onto rate equations [Shriki et al. (2003), Golomb et al. (2006), Hayut et al. (2011)]. Each neuronal population is represented by two variables: a fast synaptic variable and a slow adaptation variable. To map the conductance-based equations onto rate equations, we estimate the difference between the average neuronal membrane potential and the reversal potential of the synapses. Therefore, such rate equations are a good approximation for excitatory systems, where the average membrane potential lies far from the synaptic reversal potential for glutamine-ergic synapses. In inhibitory networks, the average membrane potential lies close to the reversal potential for ${\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}$-ergic synapses. Nonetheless, we find that the average difference between the neuronal membrane potential and the synaptic reversal potential can be accurately calculated and the reduction holds (Figure 5).

Five general lessons about neuronal dynamics from our two-population circuit (Figure 2) follow from the analysis of the rate model. The dynamic properties of the network is shown to have three regions of output as a function of the difference between the strengths of inter- and intra-population synaptic conductances, $\mathrm{\Delta }{g}_{\text{syn}}$ (Equations 9 and 11) (Figure 5B,D,L). Only the central region leads to rhythmic bursts of spikes across the two subpopulations (Figure 5). Interestingly, the oscillatory region of this model is similar to that used to understand the perceptual issue of binocular rivalry within in cortical circuits [Wilson (2003), Shpiro et al. (2007), Soloduchin and Shamir (2018)]. A decision in the context of rivalry can be presented by a transition from the oscillatory to the bistable state. A second general second lesson is that the frequency of whisking and the amplitude of a whisk are decoupled. Within the region for vIRt-driven whisking, the frequency of oscillations is directly tied to the time-scale of the adaptation current (Equations 16 and 17). In contrast, the whisking frequency is independent of the external input, which gates the oscillations, as well as controls the amplitude of whisking for both pBötC-drive (Equation 7) and intrinsic-driven whisking (Equation 18). The third lesson is that the range of the region for vIRt-driven whisking, controlled by the value of $\mathrm{\Delta }{g}_{\mathrm{s}\mathrm{y}\mathrm{n}}$, is proportional to the strength of the adaptation current.

A fourth general lesson is that the extent of synaptic connectivity within each subpopulation of inhibitory neurons balances the trade-off between variability in spike rate and the range of conductances that support whisking. The variability is a so-called “finite size” effect (Figure 5M). On the one hand, we scale the synaptic conductance between two neurons by the number of connections, i.e., as $1/K$, according to the “weak synapses scenario” [Pehlevan and Sompolinsky (2014)]. This assures that the intrinsic properties, and specifically adaptation, are not diminished by averaging. On the other hand, we choose the value of K to be nominally consistent with anatomical data [Moore et al. (2013)] as opposed to letting $K\to \mathrm{\infty }$. This yields a value of ${\mathrm{C}\mathrm{V}}_2$ that nominally matches experimental values (Figure 5D,L,M). Unfortunately, we cannot determine an analytical expression for the variability with the current methods. Such an analysis may, in principle, be carried out for the “strong synapses scenario” [Pehlevan and Sompolinsky (2014)], for which scaling of the synaptic input by $1/\sqrt{K}$, and variability emerges in the limit $K\to \mathrm{\infty }$ and $N\to \mathrm{\infty }$ but with a constant ratio of $K/N$ [van Vreeswijk and Sompolinsky (1996), Renart et al. (2010)]. Nonetheless, numerical simulations show that the variability leads to an increase in dynamic range of the oscillatory region (Supplementary Figure 1D, Figure 5D,L).

A final general lesson from our model follows from the absence of feedback from the vIRT^ret or vIRT^pro neurons to the pBötC. The direct modulation of vIRT^ret output from the pBötC is strong and leads to a resetting of the rhythm. Thus the absence of feedback implies that the rhythmic dynamics of the vibrissae can be largely predicted as the combination of drive to the vIRT^ret neurons by the pBötC and autonomous oscillations generated within the vIRt. The need for orofacial acts to protect the patency of the airway suggests that unidirectional drive from the pBötC may be a general medullary design rule.

The whisking oscillator circuit is a major component of a larger circuit for active sensing with vibrissae. The rhythmicity of inspiration, via the pBötC, and the vIRt, define the period and shape of the unperturbed waveform for a whisk through drive to the intrinsic muscles for each vibrissa. Further, the pBötC and the Bötzinger complex (BötC) activate protraction and retraction extrinsic muscles in the mystacial pad, which shift the pivot point for the vibrissae [Hill et al. (2008), Moore et al. (2013)]. Beyond these internal signals, vibrissa touch leads to rapid feedback that modifies the shape of the whisking waveform [Nguyen and Kleinfeld (2005), Bellavance et al. (2017)]. Feedback is mediated by both excitatory and inhibitory projections from spinal trigeminal nuclei to the extrinsic muscles (Figure 8E); in this manner the spinal trigeminal nuclei act as premotor nuclei. Rapid feedback can lead to a transient decrease in contact force, such that the contact force per whisk has a double peak. This modulation is posited to play a role in texture discrimination by slip-stick friction [Ritt et al. (2008), Lottem and Azouz (2009), Schwarz (2016), Isett et al. (2018)]. Rapid feedback also minimizes the contact time [Bellavance et al. (2017)]. Finally, feedback also occurs on the time-scale of multiple whisks through a shift in the set-point of whisking [Towal and Hartmann (2006), Mitchinson et al. (2007)] and, in principle, could involve high-order input to the vIRT^ret.

6. STAR METHODS

6.1. RESOURCE AVAILABILITY

6.1.1. Lead Contact

Further information and requests for resources should be directed to and will be fulfilled by Dr. David Golomb (golomb@bgu.ac.il).

6.1.2. Materials Availability

This study did not generate new unique reagents.

6.1.3. Data and Code Availability

The data sets supporting the current study, and an associated “read me” file, are available at https://datadryad.org/xxx. The code for the models is available at https://github.com/XXX.

6.2. NETWORK MODEL

Subpopulation indices: $\mu$ and $\nu$, coded as $\mathrm{r}\leftarrow {\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{r}\mathrm{e}\mathrm{t}},\mathrm{p}\leftarrow {\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{p}\mathrm{r}\mathrm{o}},\mathrm{F}\leftarrow \mathrm{v}\mathrm{F}\mathrm{M}$, and $\mathrm{B}\leftarrow$ pBötC.

6.2.1. Glossary

$N^{\mu }$

number of neurons in the μ-th subpopulation, $\mu =\mathrm{r},\mathrm{p},\mathrm{F}$, or B.

$K^{\mu \nu }$

The average number of synaptic connections from ν-th subpopulation to μ-th subpopulation.

${\tilde{C}}_{ij}^{\mu \nu }$

connectivity matrix element between j-th neuron in ν-th subpopulation to i-th neuron in μ-th subpopulation.

C

membrane capacitance per unit area.

$V_i^{\mu }$

membrane potential of the i-th neuron from the μ-th subpopulation.

$I_{\mathrm{i}\mathrm{o}\mathrm{n},i}^{\mu }$

intrinsic ionic current from a specific type of the i-th neuron from the μ-th subpopulation. “ion” can be $\mathrm{L},\mathrm{N}\mathrm{a},\mathrm{N}\mathrm{a}\mathrm{P},\mathrm{K}\mathrm{d}\mathrm{r}$, adapt, or h.

$g_{\text{ion}}^{\mu }$

intrinsic ionic conductance from a specific ionic type for the μ-th subpopulation. “ion” can be L, $\mathrm{N}\mathrm{a},\mathrm{N}\mathrm{a}\mathrm{P},\mathrm{K}\mathrm{d}\mathrm{r}$, adapt, or h.

$\mathrm{\Delta }{g}_{\mathrm{L}}$

width of the distribution of the leak conductance.

$\mathrm{\Delta }{g}_{\text{adapt}}$

width of the distribution of the adaptation conductance.

$V_{\text{rev}}$

reversal potential of an ionic current. The current can be ${\mathrm{N}\mathrm{a}}^{+},{\mathrm{K}}^{+},\text{h},{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}$ (for GABA-mediated synapses) or Glu (for glutamine-ergic synapses).

$I_{\text{ext}}^{\mu }$

external input to a neuron from the μ-th subpopulation.

$I_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \nu }$

synaptic input from all the neurons from the ν-th subpopulation to the i-th neuron from the μ-th subpopulation.

m

activation variable of $I_{\mathrm{N}\mathrm{a}}$.

h

inactivation variable of $I_{\mathrm{N}\mathrm{a}}$.

p

activation variable of $I_{\mathrm{N}\mathrm{a}\mathrm{P}}$.

n

activation variable of $I_{\mathrm{K}\mathrm{d}\mathrm{r}}$.

z

activation variable of $I_{\text{adapt}}$.

r

activation variable of $I_{\mathrm{h}}$.

${\mathrm{v}\mathrm{a}\mathrm{r}}_{\mathrm{\infty }}^{\mu }\left(V\right)$

steady-state value of an activation or an inactivation variable as a function of V. “var” may be h, p, $\mathrm{n},\mathrm{z}$, or r.

${\tau }_{\text{var}}^{\mu }$

time constant of an activation or an inactivation variable. “var” may be $\mathrm{h},\mathrm{p},\mathrm{n}$, z, or r.

$G_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \nu }$

total synaptic conductance from all contributing neurons in ν-th subpopulation to neuron i in μ-th subpopulation.

$g^{\mu \nu }$

average synaptic conductance from all neurons in ν-th subpopulation ν onto a neuron in μ-th subpopulation.

$g_{\text{intra}}\equiv g^{\mathrm{r}\mathrm{r}}=g^{\mathrm{p}\mathrm{p}}$

average synaptic conductance from all the neurons within the vIRt^ret subpopulation onto to one neuron in the vIRt^ret subpopulation. Ditto for the vIRt^pro subpopulation.

$g_{\text{inter}}\equiv g^{\mathrm{r}\mathrm{p}}=g^{\mathrm{p}\mathrm{r}}$

average synaptic conductance from all of the neurons within the vIRt^ret subpopulation onto to one neuron in the vIRt^pro subpopulation. And visa versa.

${\tilde{s}}_j^{\nu }$

a synaptic variable from presynaptic neuron j in the ν-th subpopulation.

$t_{j,k}^{\nu }$

the time of the k-th spike of the j-th neuron in the ν-th subpopulation.

${\tau }_s^{\nu }$

synaptic decay time constant.

l

index of pBötC period of activity.

$t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$

starting time of activity in the l-th pBötC cycle.

$T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t},l}$

duration of the l-th pBötC cycle, composed of an active and a silent phase. If these durations are constant, then $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}=T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$.

$f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=1/T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$

frequency of pBötC input to the vIRt system where $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}={⟨T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}⟩}_l$

$t_l^{\mathrm{B}}$

starting time of the l-th period of the pBötC.

$\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$

duration of the active state of the pBötC.

$T_{\text{rand}}$

variability of the time period of the pBötC.

$h^{\mathrm{B}}$

function denoting the dependency of the pBötC activity on time.

${\left[{\mathrm{C}\mathrm{a}}^{2+}\right]}_i$

normalized intracellular $\left[{\mathrm{C}\mathrm{a}}^{2+}\right]$ in the i-th neuron.

$C_k$

coefficients used for calculating ${\left[{\mathrm{C}\mathrm{a}}^{2+}\right]}_i$.

${\tau }_{\mathrm{w}\mathrm{r}}$

rise time constant of $\left[{\mathrm{C}\mathrm{a}}^{2+}\right]$ inside a cell.

${\tau }_{\mathrm{w}\mathrm{c}}$

decay time constants of $\left[{\mathrm{C}\mathrm{a}}^{2+}\right]$ inside a cell.

$r_0$

amplitude of ${\mathrm{C}\mathrm{a}}^{2+}$ entry to a cell.

F

normalized muscle force.

$A_0$

coefficient controlling muscle force.

$\theta$

vibrissa sweep angle.

${\tau }_{\mathrm{w}\mathrm{m}}$

time scale of the dependency of vibrissa angle on the muscle force.

$A_1$

coefficient controlling the dependency of vibrissa angle on the muscle force.

$M^{\mu }$

population-average firing rate of the μ-th subpopulation.

${\mathrm{I}\mathrm{S}\mathrm{I}}_{\text{max},\text{median}}^{\mu }$

the maximum value of the ISI for each neuron within the μ-th subpopulation is calculated and ${\mathrm{I}\mathrm{S}\mathrm{I}}_{\text{max},\text{median}}^{\mu }$ is the median of these maximal values.

$f_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$

frequency of vIRt-induced bursting oscillations, with $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}=1/f_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$

${\mathrm{C}\mathrm{V}}_2^{\mu }$

coefficient of variation of the μ-th subpopulation Holt et al. (1996).

${\theta }_{\text{sd}}$

the standard deviation of $\theta \left(t\right)$.

${\theta }^{\text{set}}$

the local maximum of $\theta \left(t\right)$.

${\theta }^{\text{amp}}$

the whisking amplitude.

$t_{\mathrm{w},\mathrm{m}\mathrm{i}\mathrm{n}}$

time of the global minimum of $\theta \left(t\right)$.

$s^{\mu }$

population-averaged synaptic variable of the μ-th subpopulation (rate model).

$a^{\mu }$

population-averaged adaptation variable of the μ-th subpopulation (rate model).

$a_0^{\mu }$

$a^{\mu }$ for $t=0^{+}$, just after the onset of pBötC activity.

${\beta }^{\mu }$

slope of the f-I curve for the μ-th subpopulation (rate model).

$I_0^{\mu }$

current threshold of a neuron from the μ-th subpopulation (rate model).

$I_{\text{ext}}^{\mu }$

external current that drives a neuron from the μ-th subpopulation (rate model).

${\tilde{I}}_{\text{ext}}^{\mu }$

external current into a neuron from the μ-th subpopulation minus the current threshold (rate model).

$J_{\mathrm{a}}^{\mu }$

adaptation strength (rate model).

${\tau }_a^{\mu }$

adaptation time constant (rate model).

$J^{\mu \nu }$

effective synaptic coupling between the ν-th and μ-th subpopulations (rate model).

${\epsilon }^{\mu }$

+1 if the μ-th subpopulation has excitatory synaptic output and −1 if it has inhibitory output.

$J_{\text{intra}}\equiv J^{\mathrm{r}\mathrm{r}}=J^{\mathrm{p}\mathrm{p}}$

effective synaptic self-coupling within the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{r}\mathrm{e}\mathrm{t}}$ or the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\text{pro}}$ subpopulations.

$J_{\text{inter}}\equiv J^{\mathrm{p}\mathrm{r}}=J^{\mathrm{r}\mathrm{p}}$

effective synaptic self-coupling between the vIRt^ret and the vIRt^pro subpopulations.

${\gamma }^{\mu }$

ratio between effective interactions $J_{\mathrm{a}}^{\mu }$ (rate model) and synaptic conductances $g_{\text{adapt}}^{\mu }$ (CB model).

$I_{\text{intra}}^{\mathrm{C}\mathrm{B}}$

average synaptic current from a vIRt subpopulation onto a neuron from the same subpopulation (CB model).

$I_{\text{intra}}^{\text{rate}}$

average synaptic current from a vIRt subpopulation onto a neuron from the same subpopulation (rate model).

$I^{\mu \mathrm{B}}$

current from the pBötC during its active phase to the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{r}\mathrm{e}\mathrm{t}}$ subpopulation.

$F_{\text{fit}}$

fit of the muscle force.

$P_{\text{fit}}$

polynomial used to define $F_{\text{fit}}$ in terms of the parameters $A_{\mathrm{L}},M_{\mathrm{L}}^{\mathrm{F}},M_1^{\mathrm{F}},M_2^{\mathrm{F}},B_2$, and $B_3$.

${\hat{M}}^{\mu }$

^βμ × the total current into a neuron from μ-th subpopulation minus the current threshold (rate model).

$a_{\text{func}}^{\mathrm{r}}$

function defined by Equation 70.

$N_{\text{spikes}}^{\mu }$

number of spikes fired by a neuron from the μ-th subpopulation during pBötC activity.

$N_{\text{spikes;func}}^{\mathrm{r}}$

spike function defined by Equation 71.

$\tilde{A},\tilde{B},\tilde{C},\tilde{D}$

terms defined by Equation 74.

$t_{\mathrm{M}>0}^{\mathrm{F}}$

the time below which ${\hat{M}}_{\mathrm{F}}(t)\ge 0$.

$N_{\mathrm{M}>0}^{\mathrm{F}\mathrm{r}}$

function defined by Equation 81.

$T_{\mathrm{M}=0}^{\mathrm{r}}$

the time below which $M^{\mathrm{r}}\left(t\right)=0$.

$\tilde{\mathrm{\Gamma }}$

term defined by Equation 100.

$F_m$

proportionality constant in Equation 101.

$I_{\text{onset}}^{\mathrm{r}\mathrm{B}}$

the $I^{\mathrm{r}\mathrm{B}}$ value above which the vibrissae begin to oscillate.

$I_{\mathrm{s}\mathrm{a}\mathrm{t}}^{\mathrm{r}\mathrm{B}}$

saturation value of $I^{\mathrm{r}\mathrm{B}}$.

$T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}$

time for maximal protraction.

$A_{\text{stab}}$

4 × 4 stability matrix.

$A_{\text{stab},\pm }$

reduced, $2\times 2$ stability matrices.

$\overrightarrow{a}$

two-dimensional vector, used for stability calculation.

$J_{\text{tr}}$

trace of the matrix $A_{\text{stab},-}$.

$J_{\text{det}}$

determinant of the matrix $A_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b},-}$.

$\tilde{J}$

expression defined in Equation 124.

$J_{\text{nor}}$

expression defined in Equation 140.

$\tilde{E}\left(T_{\text{vIRt}}\right)$

expression defined in Equation 142.

$\tilde{F}\left(T_{\text{vIRt}}\right)$

expression defined in Equation 144.

${⟨\cdots ⟩}_t$

average over time.

${⟨\cdots ⟩}_i$

average over the neurons in a population.

x

factor of blockade of inhibition in the vIRt^ret

$\varphi$

phase of whisking oscillations.

$t_{\mathrm{w},i}$

the time of the maximal whisking angle during the i-th whisk in the breathing cycle.

6.2.2. Network architecture

The circuit model includes three subpopulations of neurons: inhibitory glycineergic/GABAergic vIRt^ret and vIRt^pro subpopulation and cholinergic vFMN motoneurons; see “Detailed background” in the Introduction (Figure 2A,E). vIRt^ret neurons correspond to vIRt^PV neurons in [Takatoh et al. (2022)]. The letters r, p, F, and B denote the vIRt^ret, vIRt^pro, vFMN, and preBöt neuronal subpopulations respectively; mathematical symbols have these letters as superscripts. The vIRt^ret and vIRt^pro subpopulations have the same properties unless otherwise stated. Quantities defining both the vIRt^ret, and vIRt^prp subpopulations are denoted by “r”.

6.2.3. Conductance-based (CB) network model

The numbers and strengths of synaptic connections from vIRt neurons to other vIRt neurons and to vFMN neurons are known only in an approximate manner. The intrinsic properties of vIRt neurons and the synaptic connectivity within the vIRt can only be estimated. Our strategy is to define a set of parameter values, named “reference parameter set”, close to measured values when known Golomb et al. (1994), Gutnisky et al. (2017)]. We vary parameters to explore the role of those parameters on the system dynamics. The number of vIRt neurons is estimated to be on the order of 100 to 1000 [Moore et al. (2013)] and about 60 vFMN neurons project to an intrinsic muscle connecting two vibrissae [Guest et al. (2018)].

6.2.3.1. Sizes and connectivity

The number $N^{\mu }$ denotes the number of neurons in the μ-th subpopulation.

• We choose $N^{\mathrm{r}}=N^{\mathrm{p}}=N^{\mathrm{F}}=100$.

Simulations show that taking a larger value of N has a small effect on the results of this manuscript. The probability that a neuron from the ν-th presynaptic subpopulation forms a synapse on a neuron from the μ-th postsynaptic subpopulation is governed by a uniform distribution with value $K^{\mu \nu }/N^{\nu }$ [Golomb and Hansel (2000)] (Figure 2A). The number $K^{\mu \nu }$ denotes the average number of inputs from the ν-th subpopulation to the μ-th subpopulation.

• We set $K^{\mu \nu }=25$ for all $\mu$ and $\nu$ that $K^{\mu \nu }\ne 0$, i.e., $\mu =\mathrm{r}$, p, or F and $\nu =\mathrm{r}$ or p.

We define the matrix ${\tilde{C}}_{ij}^{\mu \nu }$ to be 1 if the j-th neuron from the ν-th subpopulation projects to the i-th neuron from the μ-th subpopulation, and 0 otherwise, such that ${⟨{\tilde{C}}_{ij}^{\mu \nu }⟩}_{i,j}=K^{\mu \nu }/N^{\nu }$.

6.2.3.2. Dynamics of single vFMN neurons

The single-compartment conductance-based model of a motoneuron in the vFMN is based on [Harish and Golomb (2010), Golomb (2014)]. The model includes the transient ${\mathrm{Na}}^{+}$ current, $I_{\mathrm{N}\mathrm{a}}$, the persistent sodium current, $I_{\mathrm{N}\mathrm{a}\mathrm{P}}$, and the delayed rectifier ${\mathrm{K}}^{+}$ current, $I_{\mathrm{K}\mathrm{d}\mathrm{r}}$, that generate spikes (Figure 2B). The slowly-activating current $I_{\text{apapt}}$, for example, an M -type ${\mathrm{K}}^{+}$ current, generates the adaptation and afterhyper-polarization observed in motoneurons [Meunier and Borejsza (2005)]. The model also includes the leak current $I_{\mathrm{L}}$ and the $\mathrm{H}\mathrm{C}\mathrm{N}$ or h-current $I_{\mathrm{h}}$ [Nguyen et al. (2004)]. The current balance equation for the membrane potential of the i-th neuron, $V_i^{\mathrm{F}}$, where $i=1,\dots ,N^{\mathrm{F}}$, is

$$C\frac{d{V}_i^{\text{F}}}{dt}=-I_{\text{L},i}^{\text{F}}-I_{\text{Na},i}^{\text{F}}-I_{\text{NaP},i}^{\text{F}}-I_{\text{Kdr},i}^{\text{F}}-I_{\text{adapt},i}^{\text{F}}-I_{\text{h},i}^{\text{F}}-I_{\text{ext}}^{\text{F}}-I_{\text{syn},i}^{\text{Fr}}$$ (24)

where V is the membrane potential of the neuron and $\mathrm{C}=1\mu \mathrm{F}/{\mathrm{c}\mathrm{m}}^2$ is the membrane capacitance of the neuron.

The following equations and parameters for the ionic currents are implemented.

Leak current $I_{\mathrm{L},i}^{\mathrm{F}}$

$$I_{\text{L},i}^{\text{F}}=g_{\text{L},i}^{\text{F}}\left(V-V_{\text{L}}\right)$$ (25)

• $g_{\mathrm{L},i}^{\mathrm{F}}$ is taken at random for each neuron, i, from a uniform distribution between $g_{\mathrm{L}}-\mathrm{\Delta }{g}_{\mathrm{L}}$ and $g_{\mathrm{L}}+\mathrm{\Delta }{g}_{\mathrm{L}}$. The random component prevents spurious synchronization of neuronal spikes.

• We use $g_{\mathrm{L}}=0.12\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2,\mathrm{\Delta }{g}_{\mathrm{L}}=0.06\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$, and $V_{\mathrm{L}}=-70\mathrm{m}\mathrm{V}$.

The indices i and F are dropped for simplicity.

Transient ${\mathrm{N}\mathrm{a}}^{+}$ current, $I_{\mathrm{N}\mathrm{a}}$:

$$I_{\text{Na}}(V,h)=g_{\text{Na}}{m}_{\infty }^3(V)h\left(V-V_{\text{Na}}\right)$$ (26)

• $m_{\mathrm{\infty }}\left(V\right)=1/\left[1+\exp \left(-\left(V-{\theta }_m\right)/{\sigma }_m\right)\right]$.

• $dh/dt=\left[h_{\mathrm{\infty }}\left(V\right)-h\right]/{\tau }_h\left(V\right)$.

• $h_{\mathrm{\infty }}\left(V\right)=1/\left[1+\exp \left(-\left(V-{\theta }_h\right)/{\sigma }_h\right)\right]$.

• ${\tau }_h\left(V\right)={\tau }_{h0}/\left[\exp \left(V-{\theta }_h\right)/{\sigma }_{h1}+\exp \left(-\left(V-{\theta }_h\right)/{\sigma }_{h2}\right)\right]$.

• We use $g_{\mathrm{N}\mathrm{a}}=100$ mS/cm², $V_{\mathrm{N}\mathrm{a}}=55$ mV, ${\theta }_m=-28$ mV, ${\sigma }_m=7.8$ mV, ${\theta }_h=-50$ mV, ${\sigma }_h=-7$ mV, ${\tau }_{h0}=30$ ms, ${\sigma }_{h1}=15$ mV, and ${\sigma }_{h2}=16$ mV.

Persistent ${\mathrm{N}\mathrm{a}}^{+}$ current, $I_{\mathrm{N}\mathrm{a}\mathrm{P}}$:

$$I_{\text{NaP}}(V)=g_{\text{NaP}}{p}_{\infty }(V)\left(V-V_{\text{Na}}\right)$$ (27)

• $p_{\mathrm{\infty }}\left(V\right)=1/\left[1+\exp \left(-\left(V-{\theta }_p\right)/{\sigma }_p\right)\right]$.

• We use $g_{\mathrm{N}\mathrm{a}\mathrm{P}}=0.04$ mS/cm², ${\theta }_p=-53$ mV, and ${\sigma }_p=5$ mV.

Delayed rectifier ${\mathrm{K}}^{+}$ current, $I_{\mathrm{K}\mathrm{d}\mathrm{r}}$:

$$I_{\text{Kdr}}(V,n)=g_{\text{Kdr}}{n}^4\left(V-V_{\text{K}}\right)$$ (28)

• $dn/dt=\left[n_{\mathrm{\infty }}\left(V\right)-n\right]/{\tau }_n\left(V\right)$.

• $n_{\mathrm{\infty }}\left(V\right)=1/\left[1+\exp \left[-\left(V-{\theta }_n\right)/{\sigma }_n\right]\right]$,

• ${\tau }_n\left(V\right)={\tau }_{n0}/\left[\exp \left(\left(V-{\theta }_{n0}\right)/{\sigma }_{n1}\right)+\exp \left(-\left(V-{\theta }_{n0}\right)/{\sigma }_{n2}\right)\right]$.

• We use $g_{\mathrm{K}\mathrm{d}\mathrm{r}}=20$ mS/cm², $V_{\mathrm{k}}=-90$ mV, ${\theta }_n=-23$ mV, ${\sigma }_n=15$ mV, ${\tau }_{n0}=7$, ${\theta }_{n0}=-40$ mV, ${\sigma }_{n1}=40$ mV and ${\sigma }_{n2}=50$ mV.

Adaptation current, $I_{\text{adapt}}$:

$$I_{\text{adapt}}(V,z)=g_{\text{adapt}}z\left(V-V_{\text{K}}\right)$$ (29)

• $dz/dt=\left[z_{\mathrm{\infty }}\left(V\right)-z\right]/{\tau }_z$

• $z_{\mathrm{\infty }}\left(V\right)=1/\left[1+\exp \left(-\left(V-{\theta }_z\right)/{\sigma }_z\right)\right]$

• We use $g_{\text{adapt}}=0.3$ mS/cm², ${\theta }_z=-45$ mV, ${\sigma }_z=4.25$ mV, ${\tau }_z=75$ ms

Hyperpolarization-activated h-current $I_{\mathrm{h}}$:

$$I_{\text{h}}(V,r)=g_{\text{h}}r\left(V-V_{\text{h}}\right)$$ (30)

• $dr/dt=\left[r_{\mathrm{\infty }}\left(V\right)-r\right]/{\tau }_r\left(V\right)$.

• $r_{\mathrm{\infty }}\left(V\right)=\left[1+\exp \left(-\left(V-{\theta }_r\right)/{\sigma }_r\right)\right]$.

• ${\tau }_r\left(V\right)={\tau }_{r0}/\left[\exp \left(\left(V-{\theta }_{n1}\right)/{\sigma }_{n1}\right)+\exp \left(-\left(V-{\theta }_{n2}\right)/{\sigma }_{n2}\right)\right]$.

• We use $g_{\mathrm{h}}=0.05$ mS/cm², $V_{\mathrm{h}}=-27.4$ mV, ${\theta }_r=-83.9$ mV, ${\sigma }_r=-7.4$ mV, ${\tau }_{r0}=6000$ ms, ${\theta }_{n1}=-140$ mV, ${\theta }_{n2}=-40$ mV, ${\sigma }_{n1}=21.6$ mV, and ${\sigma }_{n2}=22.7$ mV.

External depolarizing current $I_{\text{ext}}$:

• vFMN neurons are excited and depolarized by excitatory [Hattox et al. (2002)] and neuromodulatory [Cramer and Keller (2006)] inputs from several brain areas. We assume that those inputs evolve slowly with time and treat them as a constant external depolarizing current that enables vFMN neurons to spike in the presence of rhythmic inhibitory input.

• We use $I_{\mathrm{e}\mathrm{x}\mathrm{t}}=3.1\mu \mathrm{A}/{\mathrm{c}\mathrm{m}}^2$.

External Inhibitory currents $I_{\text{syn}}$:

• The inhibitory synaptic input current from the activity of the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{r}\mathrm{e}\mathrm{t}}$ neurons is $I_{\mathrm{s}\mathrm{y}\mathrm{n}}^{\mathrm{F}\mathrm{r}}$. It is discussed in detail below.

6.2.3.3. Dynamics of single vIRt neurons

The model of a vIRt neuron is derived from the vFMN neuron model. The current balance equation for the membrane potential of the i-th neuron, $V_i^{\mu }$, where $\mu =\mathrm{r}$ or p and $i=1,\dots ,N_{\mu }$, is

$$C\frac{d{V}_i^{\mu }}{dt}=-g_{\text{L},i}^{\mu }-I_{\text{Na},i}^{\mu }-I_{\text{NaP},i}^{\mu }-I_{\text{Kdr},i}^{\mu }-I_{\text{adapt},i}^{\mu }-I_{\text{ext}}^{\text{r}}-I_{\text{syn},i}^{\mu \text{r}}-I_{\text{syn},i}^{\mu \text{p}}-I_{\text{syn},i}^{\mu \text{B}}.$$ (31)

The parameter modifications with respect to the vFMN model are:

• The adaptation conductance $g_{\text{adapt}}$ is taken at random from a uniform distribution between $g_{\text{adapt}}-\mathrm{\Delta }{g}_{\text{adapt}}$ and $g_{\text{adapt}}+\mathrm{\Delta }{g}_{\text{adapt}}$, where $g_{\text{adapt}}=7$ mS/cm² and $\mathrm{\Delta }{g}_{\text{adapt}}=3$ mS/cm². The random component prevents spurious synchronization of neuronal spikes.

• We now use ${\theta }_z=-28$ mV, ${\sigma }_z=-3$ mV, and ${\tau }_z=83\mathrm{ms}$ in Equation (29).

• vIRt neurons receive inputs from several brain areas [McElvain et al. (2018)], that are represented here by an external current $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}=20$ μA/cm² for both vIRt sub-populations.

• The synaptic currents $I_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \mathrm{r}},I_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \mathrm{p}}$ and $I_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \mathrm{B}}$ are inhibitory inputs currents that originate from the activity of the vIRt^ret, vIRt^pro and pBötC neuronal subpopulations respectively.

6.2.3.4. Inhibitory synaptic currents

The rhythmic synaptic inputs that we consider are all inhibitory. The net such input that neuron i from subpopulation $\mu$ receives from all contributing neurons in presynaptic subpopulation $\nu$ is given by the sum:

$$I_{\text{syn},i}^{\mu \nu }=G_{\text{syn},i}^{\mu \nu }(t)\left(V_i^{\mu }-V_{{\text{GABA}}_{\text{A}}}\right).$$ (32)

• The total conductance $G_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \nu }$ for $\mu =\mathrm{F}$, p, or r and $\nu =\mathrm{p}$ or r is:

  $$G_{\text{syn},i}^{\mu \nu }(t)=\frac{g_{\mu \nu }}{K_{\mu \nu }}\sum _{j=1}^{N_{\nu }}{\tilde{C}}_{ij}^{\mu v}{\tilde{s}}_j^v(t).$$ (33)

  Unless noted, we assume symmetry in connectivity within the vIRt. Thus $g_{\mathrm{r}\mathrm{r}}=g_{\mathrm{p}\mathrm{p}}$ and $g_{\mathrm{r}\mathrm{p}}=g_{\mathrm{r}\mathrm{p}}$. We scale the conductance strength as $1/K$ in order to keep the mean synaptic input approximately fixed as the number of neurons is varied.

• We define $g_{\text{intra}}\equiv g^{\mathrm{r}\mathrm{r}}=g^{\mathrm{p}\mathrm{p}}$ and $g_{\text{inter}}\equiv g^{\mathrm{r}\mathrm{p}}=g^{\mathrm{p}\mathrm{r}}$.

• The pre-synaptic activation variable, ${\tilde{s}}_j^{\nu }$ from neuron j in the ν-th subpopulation is:

  $${\tilde{s}}_j^{\nu }=\sum _k\text{exp}\left[-\left(t-t_{j,k}^{\nu }\right)/{\tau }_s^{\nu }\right]\text{Θ}\left(t-t_{j,k}^{\nu }\right)$$ (34)

  where $\mathrm{\Theta }$ is the Heaviside function.

• The j-th pre-synaptic neuron fires at times $t_{j,k}^{\nu }$, where k is the spike index.

• The variable ${\tau }_s^{\nu }$ is the time constant of synaptic decay.

• The pBötC inhibitory input to the vIRt sub-populations, i.e., the total conductance $G_{\mathrm{s}\mathrm{y}\mathrm{n},i}^{\mu \nu }$ for $\mu =\mathrm{F},\mathrm{p}$, or r and $\nu =\mathrm{B}$, is modeled as lumped oscillating square-wave pulses, i.e.,

  $$G_{\text{syn},i}^{\mu \text{B}}(t)=g^{\mu \text{B}}{h}^{\text{B}}(t-t_{\text{pBötC},l})$$ (35)

  where $h^{\mathrm{B}}\left(t\right)$ is a periodic sequence of rectangular pulses. The l-th cycle starts at time $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l},l=1,2,\dots$, and lasts for a duration of $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$, such that $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l+1}=t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}+T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$, where $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$ is the duration between the starting times of the l-th and the $(l+1)$-th cycles of the pBötC, and $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t},1}=0$.

• The pBötC is active during a time interval $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ following the cycle onset and, clearly, $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}>\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. The activity function of the pBötC is therefore where, for the l-th cycle, $h_{\mathrm{B}}(t-t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l})=1$ for $0\le t-t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}\le \mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ and is 0 for $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}\le t-t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}\le T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$.

• The values of $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$ are picked up at random from a uniform distribution between $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}-T_{\text{rand}}/2$ and $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}+T_{\text{rand}}/2$.

• We choose ${\tau }_s^{\nu }={\tau }_s=10\mathrm{m}\mathrm{s},V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}=-80\mathrm{m}\mathrm{V},T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=700\mathrm{m}\mathrm{s},T_{\text{rand}}=150\mathrm{m}\mathrm{s}$, and $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=70\mathrm{m}\mathrm{s}$ unless otherwise stated.

• The pBötC average frequency is $f_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}=1/T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$.

6.2.3.5. Vibrissa movement

The transformation from firing of vFMN neurons to vibrissa movement is computed based on a simplified version of a model of vibrissa movement Simony et al. (2010). The normalized ${\mathrm{C}\mathrm{a}}^{2+}$ concentration in the muscle cells that belong to a motor unit, denoted ${\left[{\mathrm{C}\mathrm{a}}^{2+}\right]}_i$, is determined by the presynaptic motoneuron. The value of ${\left[{\mathrm{C}\mathrm{a}}^{2+}\right]}_i$ for the interval $t_{i,k}^{\mathrm{F}}\le t<t_{i,k+1}^{\mathrm{F}}$, where $t_{i,k}^{\mathrm{F}}$ is the time of the k-th spike of the i-th vFMN neuron, is

$${\left[{\text{Ca}}^{2+}\right]}_i(t)=\frac{r_0{\tau }_{\text{wc}}}{{\tau }_{\text{wc}}-{\tau }_{\text{wr}}}\left[e^{-\left(t-t_{i,k}^{\text{F}}\right)/{\tau }_{\text{wc}}}-e^{-\left(t-t_{i,k}^{\text{F}}\right)/{\tau }_{\text{wr}}}\right]+C_k{e}^{-\left(t-t_{i,k}^{\text{F}}\right)/{\tau }_{\text{wc}}},$$ (36)

where

$$C_k=\{\begin{array}{ll}1& ,k=1\\ C_{k-1}{e}^{-\text{Δ}{t}_{i,k-1}/{\tau }_{\text{wc}}}+\frac{r_0{\tau }_{wc}}{{\tau }_{\text{wc}}-{\tau }_{\text{wr}}}\left(e^{-\text{Δ}{t}_{i,k-1}^{\text{F}}/{\tau }_{\text{wc}}}-e^{-\text{Δ}{t}_{i,k-1}^{\text{F}}/{\tau }_{\text{wr}}}\right)& ,k>1\end{array}$$ (37)

with $\mathrm{\Delta }{t}_{i,k}^{\mathrm{F}}=t_{i,k+1}^{\mathrm{F}}-t_{i,k}^{\mathrm{F}}$.

• The dependence of the normalized muscle force F on $\left[{\mathrm{C}\mathrm{a}}^{2+}\right]$ is

  $$F\left({\left[{\text{Ca}}^{2+}\right]}_i\right)=A_0\frac{{\left[{\text{Ca}}^{2+}\right]}_i^4}{1+{\left[{\text{Ca}}^{2+}\right]}_i^4}.$$ (38)
• The vibrissa angle dynamics are over-damped and thus follow the muscle force according to a first-order linear differential equation

  $$\frac{d\theta }{dt}=-\frac{\theta }{{\tau }_{\text{wm}}}+A_1\sum _{i=1}^{N_{\text{F}}}F$$ (39)

  where the sum is over all the motor units that are innervated by vFMN neurons and belong to the same muscle. This approximation to the motion becomes exact for small whisking angles.

• Following Simony et al. (2010), we take $r_0=1.9,{\tau }_{\mathrm{w}\mathrm{r}}=5\mathrm{m}\mathrm{s}$ and ${\tau }_{\mathrm{w}\mathrm{c}}=6\mathrm{m}\mathrm{s}$. We also take $A_0=1\mathrm{m}\mathrm{g}\times \mathrm{m}\mathrm{m}/{\mathrm{m}\mathrm{s}}^2,{\tau }_{\mathrm{w}\mathrm{m}}=20\mathrm{m}\mathrm{s}$, and $A_1={12}^{\circ }\mathrm{m}\mathrm{s}/(\mathrm{m}\mathrm{g}\times \mathrm{m}\mathrm{m})$; the latter choice makes the variation of $\theta$ within the experimentally observed range.

6.2.4. Schema for a rate model

We reduce the CB model of the vIRt network to a rate model. The reduction is based on [Shriki et al. (2003), Golomb et al. (2006), Hayut et al. (2011)]. The connections between the $CB$ and rate description is in terms of the time-averaged firing rates of all the neurons in the μ-th subpopulation, where averaging occurs over long times, i.e., over many whisking cycles during oscillatory states, and over the neurons in a sub-population. This average is given by the parameter $M^{\mu }$. For the rate model, $M^{\mu }$, is assumed to depend on the effective input current to the neuron as a threshold-linear function

$$M^{\mu }={\beta }^{\mu }{\left[\sum _{\nu }{J}^{\mu \nu }{\epsilon }^{\nu }{s}^{\nu }-a^{\mu }+I_{\text{ext}}^{\mu }-I_0^{\mu }\right]}_{+}.$$ (40)

The components of this formalism are:

• The parameter ${\beta }^{\mu }$ is the slope of the single-neuron f-I curve for a fixed value of $a^{\mu }$. This curve is assumed to be linear above current threshold, defined by

  $${[x]}_{+}=\{\begin{array}{ll}0\hfill & ,x\le 0\hfill \\ x\hfill & ,x>0\hfill \end{array}.$$ (41)

• The coefficient $J^{\mu \nu }$ is the effective synaptic coupling between the ν-th and the μ-th subpopulations. We define $J_{\text{intra}}\equiv J^{\mathrm{r}\mathrm{r}}=J^{\mathrm{p}\mathrm{p}}$ and $J_{\text{inter}}\equiv J^{\mathrm{r}\mathrm{p}}=J^{\mathrm{p}\mathrm{r}}$.

• The sign of the coupling is

  $${\epsilon }^{\nu }=\{\begin{array}{ll}+1\hfill & ,\text{excitatory synapses}\hfill \\ -1\hfill & ,\text{inhibitory synapses}\hfill \end{array}.$$ (42)
• The population-averaged synaptic variable $s^{\nu }$ are dynamical variables of the system, evolving according to

  $$\frac{d{s}^{\nu }}{dt}=-\frac{s_{\nu }}{{\tau }_s}+M^{\nu }$$ (43)

  where $\nu =\mathrm{p}$ or r is the index of the presynaptic subpopulation and we took ${\tau }_s$ $\equiv {\tau }_s^{\mathrm{r}}={\tau }_s^{\mathrm{p}}$.
• The population-average of the activation variable of the adaptation current, $I_{\text{adapt}}$ (Equation 29), evolves according to Ben-Yishai et al. (1995)

  $$\frac{d{a}^{\mu }}{dt}=\frac{-a^{\mu }+J_{\text{a}}^{\mu }{M}^{\mu }}{{\tau }_{\text{a}}^{\mu }},$$ (44)

  where $J_{\mathrm{a}}^{\mu }$ is the adaptation strength and ${\tau }_{\mathrm{a}}^{\mu }$ is the adaptation time constant that is set to be equal to ${\tau }_z^{\mu }$ of the μ-th subpopulation. The original rate formalism [Shriki et al. (2003), Golomb et al. (2006), Hayut et al. (2011)] was developed for only slow variations in population firing activity. However, $a^{\mu }$ evolves slower than the other variables. Thus we consider $a^{\mu }$ as a separate dynamical variable for each subpopulation.
• The parameter $I_0^{\mu }$ is the current threshold of a neuron from the μ-th subpopulation (insert in Figure 2C). We define

  $${\tilde{I}}_{\text{ext}}^{\mu }\equiv I_{\text{ext}}^{\mu }-I_0^{\mu }$$ (45)

  and take ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}={\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{p}}$.

6.2.4.1. Reduction of a CB model to a rate model

To fit the rate model to the CB model, we compute the f-I curve of the single neuron model for several values of $g_{\text{adapt}}^{\mu }$ (Figure 2C). Such f-I curves are close to linear functions for large values of $g_{\text{adapt}}^{\mu }$ [Shriki et al. (2003)].

• For a single isolated neuronal subpopulation with external input $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mu }$ and without any synaptic coupling to itself or to other subpopulations, we compute the average spike rate at steady state from the rate model, (Equations 43, 40, and 44) [Hayut et al. (2011)]

  $$M^{\mu }=\frac{{\beta }^{\mu }{\left[I_{\text{ext}}^{\mu }-I_0^{\mu }\right]}_{+}}{1+{\beta }^{\mu }{J}_{\text{a}}^{\mu }}.$$ (46)
• Under the additional assumptions that the adaptation constant $J_{\mathrm{a}}^{\mu }$ in the rate model (Equation 44) is proportional to $g_{\text{adapt}}^{\mu }$ in the CB model (Equation 29), and that $I_0^{\mu }$ in the rate model is the value of $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mu }$ in the CB model for the onset of spiking of the single neuron model, we have

  $$M^{\mu }=\frac{{\beta }^{\mu }{\left[I_{\text{ext}}^{\mu }-I_0^{\mu }\right]}_{+}}{1+{\beta }^{\mu }{\gamma }^{\mu }{g}_{\text{adapt}}^{\mu }}$$ (47)

  where ${\gamma }^{\mu }$ is the proportionality constant between $J_{\mathrm{a}}^{\mu }$ and $g_{\text{adapt}}^{\mu }$.

• For the vIRt neuron model, we fit the right-hand-side of Equation 47 to the f-I curves or, here, $M\text{-}{I}_{\mathrm{e}\mathrm{x}\mathrm{t}}$, curves for $g_{\text{adapt}}^{\mathrm{r}}=3,\dots ,7\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ and $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$ between $I_0^{\mathrm{r}}$ $=0.29$ and 20 μA/cm² (Figure 2C). We obtain: ${\beta }^{\mathrm{r}}=0.0175{\mathrm{c}\mathrm{m}}^2/\mathrm{ms}\cdot \mu \mathrm{A}$ and ${\gamma }^{\mathrm{r}}=0.0247\mathrm{m}\mathrm{s}\cdot \mathrm{m}\mathrm{V}$.

• For the vFMN neuron model, we fit the right-hand-side of Equation 47 for $g_{\text{adapt}}^{\mathrm{F}}=0.3$ and $0.6\mathrm{m}\mathrm{S}/{\mathrm{c}\mathrm{m}}^2$ and $I_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}$ between $I_0^{\mathrm{F}}=0.46$ and $6\mu \mathrm{A}/{\mathrm{cm}}^2$. We obtain: ${\beta }^{\mathrm{F}}=0.0305{\mathrm{cm}}^2/\mathrm{m}\mathrm{s}\cdot \mu \mathrm{A}$ and ${\gamma }^{\mathrm{F}}=0.061\mathrm{m}\mathrm{s}\cdot \mathrm{m}\mathrm{V}$.

• To estimate $J^{\mu \nu }$, we note that the current originating from the ν-to-μ coupling during steady-state is $I_{\text{rate}}^{\mu \nu }=-J^{\mu \nu }{\tau }_s{M}^{\nu }$ (Equations 43 and 40). In the CB model, the population-average of the same current is, ignoring correlations (Equations 31 to 34) [Argaman and Golomb (2018)],

  $$-I_{\mathrm{CB}}^{\mu \nu }=-g^{\mu \nu }{\tau }_s{\langle {\langle {\tilde{s}}^{\nu }\rangle }_t\rangle }_i{\langle {\langle V^{\mu }\left(t\right)-V_{{\text{GABA}}_{\text{A}}}\rangle }_t\rangle }_i$$ (48)

  where ${⟨⟨\cdots {⟩}_t⟩}_i$ denotes time- and population-averaging.
• The condition to obtain $I_{\text{rate}}^{\mu \nu }=-I_{\mathrm{C}\mathrm{B}}^{\mu \nu }$ is

  $$J^{\mu \nu }=g^{\mu \nu }{\langle {\langle V^{\mu }(t)-V_{{\text{GABA}}_{\text{A}}}\rangle }_t\rangle }_i.$$ (49)

  To estimate ${⟨{⟨V^{\mu }\left(t\right)-V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}⟩}_t⟩}_i$, we simulate the single neuron model without external input and average $V^{\mu }\left(t\right)$ over a long time. We find that the typical values of ${⟨{⟨V^{\mu }\left(t\right)-V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}⟩}_t⟩}_i$ are 27 mV for the vIRt model $(\mu =\mathrm{r}$ or p; Equation 31) and 20 mV for the vFMN model ($\mu =\mathrm{F}$; Equation 24). These values correspond to the average membrane potential of a single neuron driven by the current $I_{\text{ext}}^{\mu }$, ignoring the membrane potential during spikes above $-25\mathrm{m}\mathrm{V}$.

• The synaptic currents are given by

  $$I_{\text{syn},i}^{r\text{B}}(t)=I^{\text{rB}}{h}^B\left(t-t_{\text{pBötC},l}\right)$$ (50)

  where $I^{\mathrm{r}\mathrm{B}}=g^{\mathrm{r}\mathrm{B}}{⟨{⟨V^{\mathrm{r}}\left(t\right)-V_{{\mathrm{G}\mathrm{A}\mathrm{B}\mathrm{A}}_{\mathrm{A}}}⟩}_t⟩}_i$ (Equations 31, 32, and 35).

6.2.4.2. Rate model for vibrissa movement

We reduce the mechanical model (Equations 36 to 39) to a rate representation by computing the time-average of the muscle force, denoted F, as a function of the firing rate $M^{\mathrm{F}}$ for a tonically-firing vMNF neuron (Equations 36 to 38).

• The force is found by multiplying the spike rate of a single neuron by the number of vFMN neurons, i.e., $N^{\mathrm{F}}$. It increases supra-linearly with $M^{\mathrm{F}}$ for moderate $M^{\mathrm{F}}$ values and saturates at high $M^{\mathrm{F}}$ values due to the nonlinear dependence of F on ${\left[{\mathrm{C}\mathrm{a}}^{2+}\right]}_i$ (Equation 38) (Figure 2D).

• To quantify the dependency of F on $M_{\mathrm{F}}$, we use the function:

  $$F_{\text{fit}}\left(M^{\text{F}}\right)=A_L\text{log}\left(1+\frac{M^{\text{F}}}{M_{\text{L}}^{\text{F}}}\right)+A\frac{P_{\text{fit}}\left(M^{\text{F}}\right)}{1+P_{\text{fit}}\left(M^{\text{F}}\right)}$$ (51)

  where

  $$P_{\text{fit}}\left(M^{\text{F}}\right)=\frac{M^{\text{F}}}{M_1^{\text{F}}}+B_2{\left(\frac{M^{\text{F}}}{M_2^{\text{F}}}\right)}^2+B_3{\left(\frac{M^{\text{F}}}{M_3^{\text{F}}}\right)}^3.$$ (52)

  Fitting Equation 52 to the curve obtained by simulations for $M^{\mathrm{F}}$ between 0 and 500 Hz yields the following parameters: $A_L=1.02\mathrm{mg}\cdot \mathrm{mm}/{\mathrm{ms}}^2$, $M_{\mathrm{L}}^{\mathrm{F}}=77{\text{s}}^{-1}$, $M_1^{\mathrm{F}}=526{\text{s}}^{-1}$, $M_2^{\mathrm{F}}=612{\text{s}}^{-1}$, $M_3^{\mathrm{F}}=460{\text{s}}^{-1}$, $A=9.23\mathrm{mg}\cdot \mathrm{mm}/{\mathrm{ms}}^2$, $B_2=-23$, and $B_3=152$. The fitted curve is almost indistinguishable from the simulation results (Figure 2D).

• In the rate representation, the vibrissa angle (Equation 39) becomes

  $$\frac{d\theta }{dt}=-\frac{\theta }{{\tau }_{\text{wm}}}+F_{\text{fit}}\left(M^{\text{F}}\right).$$ (53)
• For the special case of $M_{\mathrm{F}}$ constant in time, the vibrissa set-point angle is (Equations 51 and 53)

  $${\theta }_{\text{max}}={\tau }_{\text{wm}}{F}_{\text{fit}}\left(M^{\text{F}}\right).$$ (54)

6.2.4.3. Analytical solution of the rate model for the pBötC-vIRtt^ret-vFMN circuit

We consider a feed-forward pBötC-vIRt^ret-vFMN circuit without any inhibitory connections between vIRt neurons.

• We assume that the pBötC input to the vIRt^ret (Equations 35 and 50) is periodic $\left(T_{\text{rand}}=0\right)$ and is equal to

  $$I_{\text{syn}}^{\text{rB}}(t)=I^{\text{rB}}{h}^{\text{B}}(t-n{T}_{\text{pBötC}})$$ (55)

  for $n{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}\le t<(n+1)T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$.
• The average activity of the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\mathrm{r}\mathrm{e}\mathrm{t}}$ subpopulation (Equations 43–44) becomes:

  $$M^{\text{r}}={\left[{\widehat{M}}^{\text{r}}\right]}_{+}$$ (56)

  $${\widehat{M}}^{\text{r}}={\beta }^{\text{r}}{\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}{h}^{\text{B}}(t-n{T}_{\text{pBötC}})-a^{\text{r}}\right]}_{+}$$ (57)

  $${\dot{s}}^{\text{r}}=-\frac{s^{\text{r}}}{{\tau }_s}+M^{\text{r}}$$ (58)

  $${\dot{a}}^{\text{r}}=\frac{-a^{\text{r}}+J_{\text{a}}^{\text{r}}{M}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}.$$ (59)
• Similarly, the average activity of the vFMN subpopulation (Equations 43–44) becomes:

  $$M^{\text{F}}={\left[{\widehat{M}}^{\text{F}}\right]}_{+}$$ (60)

  $${\widehat{M}}^{\text{F}}={\beta }^{\text{F}}{\left[{\tilde{I}}_{\text{ext}}^{\text{F}}-J^{\text{Fr}}{s}^{\text{r}}-a^{\text{F}}\right]}_{+}$$ (61)

  $${\dot{s}}^{\text{F}}=-\frac{s^{\text{F}}}{{\tau }_s}+M^{\text{F}}$$ (62)

  $${\dot{a}}^{\text{F}}=\frac{-a^{\text{F}}+J_{\text{a}}^{\text{F}}{M}^{\text{F}}}{{\tau }_{\text{a}}^{\text{F}}}.$$ (63)

  Note that the values of ${\hat{M}}^{\mathrm{r}}$ and ${\hat{M}}^{\mathrm{F}}$ can be either positive or negative.

• We solve Equations 56 to 63 using the approximation ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$ and ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{F}}$. This implies

  $$s^{\text{r}}={\tau }_s{M}^{\text{r}}$$ (64)

  and

  $$s^{\text{F}}={\tau }_s{M}^{\text{F}}.$$ (65)

• The network is feed-forward and we therefore first compute the values of $M^{\mathrm{r}}$ and $a^{\mathrm{r}}$ and then the values of $M^{\mathrm{F}}$ and $a^{\mathrm{F}}$. We assume that the pBötC switches from silent to active at $t=0$, after it has been oscillating for at least one cycle.

• Case of a slow breathing rate. This occurs when ${\tau }_{\mathrm{a}}^{\mathrm{r}}\ll T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}-\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ and ${\tau }_{\mathrm{a}}^{\mathrm{F}}\ll T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}-\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. The values of the activities $a^{\mathrm{r}}$ and $a^{\mathrm{F}}$ take on the steady state values, denoted by $a_0^{\mathrm{r}}$ and $a_0^{\mathrm{F}}$ respectively. $a_0^{\mathrm{r}}$ is found from Equations 56–59 with $h_{\mathrm{B}}\left(0^{-}\right)=0$, so that

  $$a_0^{\text{r}}=\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}.$$ (66)
  Similarly, $a_0^{\mathrm{F}}$ is found from Equations 60 – 63, so that

  $$a_0^{\text{F}}=0.$$ (67)

• For faster breathing rates, we start with a particular value of $a_0^{\mathrm{r}}$, compute $a^{\mathrm{r}}(T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})$ as described below, and find the steady-state value of $a_0^{\mathrm{r}}$, i.e., the value obtained after many breathing periods by solving $a^{\mathrm{r}}(T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})=a_0^{\mathrm{r}}$. Using this value of $a_0^{\mathrm{r}}$, we compute $a_0^{\mathrm{F}}$ by solving the equation $a^{\mathrm{F}}(T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})=a_0^{\mathrm{F}}$. In principle, the stability of this solution for $a_0^{\mathrm{r}}$ and $a_0^{\mathrm{F}}$ should be calculated. Comparison with simulation, however, shows that the solution is always stable, and we do not carry out the stability calculation here. We limit our calculation to the case where the vFMN subpopulation is silent before each duration of pBötC activity, i.e., $M_0^{\mathrm{F}}\left(t\right)=0$ for $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}<t<T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. The calculation depends on the values of ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I^{\mathrm{r}\mathrm{B}}$ and $a_0^{\mathrm{r}}$. We consider three cases and several subcases (Supplemental Figure 3).

• A. ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I^{\mathrm{r}\mathrm{B}}>a_0^{\mathrm{r}}$:

  Here ${\hat{M}}_0^{\mathrm{r}}>0$ and $M^{\mathrm{r}}\left(t\right)>0$ during the time of pBötC activity, i.e., $0\le t\le$ $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$, because $a^{\mathrm{r}}\left(t\right)$ decreases with increasing t. From Equations 57 and 59,

  $$a^{\text{r}}(t)=a_{\text{func}}^{\text{r}}\left(t,0,a_0^{\text{r}}\right)$$ (68)

  for $0\le t\le \mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$, where

  $$\begin{gathered} a_{\text{func}}^{\text{r}}(t,t_{\text{begin}},a_{\text{begin}}^{\text{r}})=\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}} \\ +\left[a_{\text{begin}}^{\text{r}}-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\right]e^{-\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)\left(t-t_{\text{begin}}\right)/{\tau }_{\text{a}}^{\text{r}}}. \end{gathered}$$ (69)
  Integrating $M_{\mathrm{r}}\left(t\right)$ from $t=0$ to the integration time $T_{\text{integ}}=\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ yields that the total number of spikes during the time interval $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ is

  $$N_{\text{spikes}}^{\text{r}}=N_{\text{spikes};\text{func}}^{\text{r}}(\text{Δ}{T}_{\text{pBötC}},a_0^{\text{r}})$$ (70)

  where

  $$\begin{gathered} N_{\text{spikes;func}}^{\text{r}}(T_{\text{integ}},a_{\text{begin}}^{\text{r}})={\int }_0^{T_{\text{integ}}}dt{M}^{\text{r}}(t) \\ =\frac{T_{\text{integ}}{\beta }^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}} \\ -{\beta }^{\text{r}}\left(a_{\text{begin}}^{\text{r}}-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\right)\frac{{\tau }_{\text{a}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\left(1-e^{-\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)T_{\text{integ}}/{\tau }_{\text{a}}^{\text{r}}}\right) \end{gathered}$$ (71)
  The value of ${\hat{M}}^{\mathrm{F}}$ at time $t=0^{+}$ is

  $${\widehat{M}}^{\text{F}}\left(0^{+}\right)={\beta }^{\text{F}}{\left[{\tilde{I}}_{\text{ext}}^{\text{F}}-J^{\text{Fr}}{\tau }_s{\beta }^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}-a_0^{\text{r}}\right)-a_0^{\text{F}}\right]}_{+}.$$ (72)

  We consider two subcases:

• A1. Subcase of $\mathbf{A}$ with ${\hat{M}}^{\mathrm{F}}\left(0^{+}\right)<=0$, i.e., ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}-{\tau }_s{\beta }^{\mathrm{r}}{J}^{\mathrm{F}\mathrm{r}}\left({\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I^{\mathrm{r}\mathrm{B}}-a_0^{\mathrm{r}}\right)-a_0^{\mathrm{F}}<0$:

  Here we assume that $a_0^{\mathrm{F}}=0$, even if breathing is fast, and show that this assumption is self-consistent. Under this assumption, ${\hat{M}}^{\mathrm{F}}(t)$ will not be larger because $a^{\mathrm{r}}$ decreases with time and, as a result, $M^{\mathrm{r}}\left(t\right)>M^{\mathrm{r}}\left(0\right)$. Therefore, $a^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})=0,N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{F}}=0$, the vibrissa does not move and ${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0$. During the time interval between $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ and $T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}},a^{\mathrm{F}}$ only decreases (see below). Therefore, $a^{\mathrm{F}}(T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})=0$ and the assumption $a_0^{\mathrm{F}}=0$ is self-consistent.

• A2. Subcase of A with ${\hat{M}}^{\mathrm{F}}\left(0^{+}\right)>0$:

  To compute ${\hat{M}}^{\mathrm{F}}(t)$ for $t>0$, we substitute $a^{\mathrm{r}}\left(t\right)$ (Equation 59 and $M^{\mathrm{r}}\left(t\right)$ (Equation 57) in Equations 63 and 61 and obtain

  $$\begin{gathered} \frac{d{a}^{\text{F}}}{dt}=\frac{-a^{\text{F}}+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}\left({\tilde{I}}_{\text{ext}}^{\text{F}}-{\tau }_s{J}^{\text{Fr}}{M}^{\text{r}}-a^{\text{F}}\right)}{{\tau }_{\text{a}}^{\text{F}}} \\ =-\tilde{A}+\tilde{B}{a}^{\text{F}}+\tilde{C}{e}^{-\tilde{D}t} \end{gathered}$$ (73)

  where

  $$\begin{gathered} \tilde{A}\equiv \frac{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}{{\tau }_{\text{a}}^{\text{F}}}, \\ \tilde{B}\equiv \frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}{{\tau }_{\text{a}}^{\text{F}}}\left[{\tilde{I}}_{\text{ext}}^{\text{F}}-\frac{{\tau }_s{\beta }^{\text{r}}{J}^{\text{Fr}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\right], \\ \tilde{C}\equiv a_0^{\text{r}}-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}, \\ \tilde{D}\equiv \frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}. \end{gathered}$$ (74)
  The solution of Equation 73 is

  $$a^{\text{F}}(t)=\frac{\tilde{B}}{\tilde{A}}+\left(a_{\text{begin}}^{\text{F}}-\frac{\tilde{B}}{\tilde{A}}-\frac{\tilde{C}}{\tilde{A}-\tilde{D}}\right)e^{-\tilde{A}\left(t-t_{\text{begin}}\right)}+\frac{\tilde{C}}{\tilde{A}-\tilde{D}}{e}^{-\tilde{D}\left(t-t_{\text{begin}}\right)}$$ (75)

  where $t_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}=0$ and $a_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}^{\mathrm{F}}=a_0^{\mathrm{F}}$. The activity ${\hat{M}}^{\mathrm{F}}\left(t\right)$ is

  $${\widehat{M}}^{\text{F}}(t)={\beta }^{\text{F}}\left\{{\tilde{I}}_{\text{ext}}^{\text{F}}-J^{\text{Fr}}{\tau }_s{\beta }^{\text{r}}\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}-a^{\text{r}}(t)\right]-a^{\text{F}}(t)\right\}$$ (76)

  where

  $$a^{\text{r}}(t)=a_{\text{func}}^{\text{r}}\left(t,0,a_0^{\text{r}}\right).$$ (77)

  Equations 74 to 77 are obtained from Equations 57–63 and 70 in a self-consistent manner so long as ${\hat{M}}^{\mathrm{F}}(t)>0$.

  We define the time $t_{{\mathrm{M}}^{\mathrm{F}}>0}$ to be the end of the time interval during which ${\hat{M}}^{\mathrm{F}}(t)>$ 0, starting from $t=0$. If

  $${\widehat{M}}^{\text{F}}(\text{Δ}{T}_{\text{pBötC}})<0,$$ (78)

  we solve the equation ${\hat{M}}^{\mathrm{F}}\left(t_{\mathrm{M}>0}^{\mathrm{F}}\right)=0$ numerically and find its solution $t_{\mathrm{M}>0}^{\mathrm{F}}$. If the inequality 78 holds, ${\hat{M}}^{\mathrm{F}}(t)\ge 0$ for $0\le t\le t_{\mathrm{M}>0}^{\mathrm{F}}$ and ${\hat{M}}^{\mathrm{F}}(t)\ge 0$ for $t>t_{\mathrm{M}>0}^{\mathrm{F}}$. Otherwise, $t_{\mathrm{M}>0}^{\mathrm{F}}=\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. The value $a^{\mathrm{F}}\left(t_{\mathrm{M}>0}^{\mathrm{F}}\right)$ is given by Equation 75 for $t=t_{\mathrm{M}>0}^{\mathrm{F}}$. If $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}>t_{\mathrm{M}>0}^{\mathrm{F}}$,

  $$a^{\text{F}}(\text{Δ}{T}_{\text{pBötC}})=a^{\text{F}}\left(t_{\text{M}>0}^{\text{F}}\right)e^{-\left(T_{\text{pBötC}}-t_{\text{M}>0}^{\text{F}}\right)/{\tau }_{\text{a}}^{\text{F}}}$$ (79)
  The number of spikes $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{F}}$ is computed by integrating $M^{\mathrm{F}}$ from 0 to $t_{\mathrm{M}>0}^{\mathrm{F}}$, i.e.,

  $$N_{\text{spikes}}^{\text{F}}=N_{\text{spikes};{\text{M}}^{\text{r}}>0}^{\text{F}}\left(t_{\text{M}>0}^{\text{F}},a_0^{\text{r}},a_0^{\text{F}}\right)$$ (80)

  where

  $$\begin{gathered} N_{\text{spikes};{\text{M}}^{\text{r}}>0}^{\text{F}}\left(T_{\text{integ}},a_{\text{begin}}^{\text{r}},a_{\text{begin}}^{\text{F}}\right)={\int }_0^{T_{\text{integ}}}dt{M}^{\text{F}}\left(t,a_{\text{begin}}^{\text{r}},a_{\text{begin}}^{\text{F}}\right) \\ ={\beta }^{\text{F}}\{{\tilde{I}}_{\text{ext}}^{\text{F}}{T}_{\text{integ}}-\frac{T_{\text{integ}}{\tau }_s{\beta }^{\text{r}}{J}^{\text{Fr}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right) \\ +\frac{{\tau }_s{\beta }^{\text{r}}{J}^{\text{Fr}}{\tau }_{\text{a}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\left[a_{\text{begin}}^{\text{r}}-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\left({\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}\right)\right]\left(1-e^{-\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)T_{\text{integ}}/{\tau }_{\text{a}}^{\text{r}}}\right) \\ +\frac{\tilde{B}}{\tilde{A}}{T}_{\text{integ}}+\left(a_{\text{begin}}^{\text{F}}-\frac{\tilde{B}}{\tilde{A}}-\frac{\tilde{C}}{\tilde{A}-\tilde{D}}\right)\frac{1}{\tilde{A}}\left(1-e^{-\tilde{A}{T}_{\text{integ}}}\right) \\ +\frac{\tilde{C}}{\tilde{D}(\tilde{A}-\tilde{D})}\left(1-e^{-\tilde{D}{T}_{\text{integ}}}\right)\}. \end{gathered}$$ (81)

• B.$0<{\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I^{\mathrm{r}\mathrm{B}}<a_0^{\mathrm{r}}$:

  Here $M^{\mathrm{r}}\left(t\right)=0$ for $0\le t\le T_{{\mathrm{M}}^{\mathrm{r}}=0}$, where

  $$T_{{\text{M}}^{\text{r}}=0}={\tau }_{\text{a}}^{\text{r}}\text{log}\left(\frac{a_0^{\text{r}}}{{\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}}\right).$$ (82)

• B1. Subcase of $\mathbf{B}$ for $T_{{\mathrm{M}}^{\mathrm{r}}=0}<\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$:

  For $0\le t\le T_{{\mathrm{M}}^{\mathrm{r}}=0}$,

  $$a^{\text{r}}(t)=a_0^{\text{r}}{e}^{-t/{\tau }_{\text{a}}^{\text{r}}}.$$ (83)
  For $T_{{\mathrm{M}}^{\mathrm{r}}=0}\le t\le \mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$,

  $$a^{\text{r}}(t)=a_{\text{func}}^{\text{r}}\left(t,T_{{\text{M}}^{\text{r}}=0},a^{\text{r}}\left(T_{{\text{M}}^{\text{r}}=0}\right)\right).$$ (84)
  The number of spikes fired by the vIRt^ret during the duration of pBötC activity is

  $$N_{\text{spikes;func}}^{\text{r}}=N_{\text{spikes;func}}^{\text{r}}\left(\text{Δ}{T}_{\text{pBötC}}-T_{{\text{M}}^{\text{r}}=0},0\right).$$ (85)
  From Equations 57 and 59, for $0\le t\le T_{{\mathrm{M}}^{\mathrm{r}}=0}$, we find

  $$a^{\text{F}}(t)=a_{\text{func}}^{\text{F}}(t)$$ (86)

  where

  $$a_{\text{func}}^{\text{F}}\left(T_{\text{integ}}\right)=\frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}+\left[a_0^{\text{F}}-\frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}\right]e^{-\tilde{A}{T}_{\text{integ}}}$$ (87)

  and à is defined by Equation 74. As described for Case A2 above, we compute $a^{\mathrm{F}}\left(t\right)$ and ${\hat{M}}^{\mathrm{F}}(t)$ for $t>T_{{\mathrm{M}}^{\mathrm{r}}=0}$ using Equations 75 and 76 for $t_{\text{begin}}=T_{{\mathrm{M}}^{\mathrm{r}}=0}$ and $a_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}^{\mathrm{F}}=a^{\mathrm{F}}\left(T_{{\mathrm{M}}^{\mathrm{r}}=0}\right)$.

• B1.1. Sub-subcase of $\mathbf{B}1$ for ${\hat{M}}^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})<0$:

  $a^{\mathrm{F}}\left(T_{{\mathrm{M}}^{\mathrm{r}}=0}\right)$ is given by Equations 86 and 87. We compute $t_{{\mathrm{M}}^{\mathrm{F}}>0}$ as described for Case A2. The value ${a^{\mathrm{F}}}_{\left(t_{{\mathrm{M}}^{\mathrm{F}}>0}\right)}$ given by Equation 75 with $t_{\text{begin}}=T_{{\mathrm{M}}^{\mathrm{r}}=0}$ and $a_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}^{\mathrm{F}}=a^{\mathrm{F}}\left(T_{\mathrm{M}=0}\right)$. Lastly, $a^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}})$ is computed using Equation 79.

  To compute $N_{\text{spikes}}^{\mathrm{F}}$, we first compute the number of spikes fired by vFMN neurons during the time interval $0\le t\le T_{{\mathrm{M}}^{\mathrm{r}}=0}$, during which there is no inhibition from the vIRt^ret and $M^{\mathrm{r}}=0$. For this purpose, we integrate Equation. 61 and use Equations 86 and 87. We add this value to the number of spikes by the vFMN when there is some inhibition from the vIRt^ret. The sum of the spikes in these two time intervals is

  $$\begin{gathered} N_{\text{spikes}}^{\text{F}}=N_{\text{spikes};{\text{M}}^{\text{r}}=0}^{\text{F}}\left(T_{{\text{M}}^{\text{r}}=0},a_0^{\text{F}}\right) \\ +N_{\text{spikes};{\text{M}}^{\text{r}}>0}^{\text{F}}\left(t_{{\text{M}}^{\text{F}}>0}-T_{{\text{M}}^{\text{r}}=0},a_0^{\text{r}}{e}^{-t_{{\text{M}}^{\text{r}}=0}/{\tau }_{\text{a}}^{\text{r}}},a^{\text{F}}\left(T_{{\text{M}}^{\text{r}}=0}\right)\right) \end{gathered}$$ (88)

  where

  $$\begin{gathered} N_{{\text{spikes;M}}^{\text{r}}=0}^{\text{F}}\left(T_{\text{integ}},a_{\text{begin}}^{\text{F}}\right)={\beta }^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}{T}_{\text{integ}} \\ -\frac{{\tau }_{\text{a}}^{\text{F}}{\beta }^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}\left(a_{\text{F},\text{beg}}-\frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}\right)\left(1-e^{-\left(1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}\right)T_{\text{intrg}}/{\tau }_{\text{a}}^{\text{F}}}\right). \end{gathered}$$ (89)

• B1.2. Sub-subcase of B1 for ${\hat{M}}_{\mathrm{F}}\left(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\text{ö}\mathrm{t}\mathrm{C}}\right)\ge 0$:

  Here $a^{\mathrm{F}}\left(T_{{\mathrm{M}}^{\mathrm{r}}=0}\right)$ is given by Equations. 86, 87, and $a^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})$ is given by Equation 75 with $t_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}=T_{{\mathrm{M}}^{\mathrm{r}}=0}$ and $a_{\mathrm{b}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{n}}^{\mathrm{F}}=a^{\mathrm{F}}\left(T_{{\mathrm{M}}^{\mathrm{r}}=0}\right).N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{F}}$ is given by the equation

  $$\begin{gathered} N_{\text{spikes}}^{\text{F}}=N_{{\text{spikes;M}}^{\text{r}}=0}^{\text{F}}\left(T_{\text{Mr}=0},a_{\text{F}0}\right) \\ +N_{\text{spikes};{\text{M}}^{\text{r}}>0}^{\text{F}}\left(\text{Δ}{T}_{\text{pBöt}}-T_{{\text{M}}^{\text{r}}=0},a_0^{\text{r}}{e}^{-t_{{\text{M}}^{\text{r}}=0}/{\tau }_{\text{a}}^{\text{F}}},a^{\text{F}}\left(T_{{\text{M}}^{\text{r}}=0}\right)\right). \end{gathered}$$ (90)

• B2. Subcase of $T_{{\mathrm{M}}^{\mathrm{r}}=0}\ge \mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$:

  Here

  $$a^{\text{r}}(\text{Δ}{T}_{\text{pBötC}})=a_0^{\text{r}}{e}^{-\text{Δ}{T}_{\text{pBöt}}/{\tau }_{\text{a}}^{\text{r}}},$$ (91)

  $$a^{\text{F}}(\text{Δ}{T}_{\text{pBötC}})=a_{\text{func}}^{\text{F}}(\text{Δ}{T}_{\text{pBötC}}),$$ (92)

  where $a_{\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}}^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})$ is given by Equation 87 and

  $$N_{\text{spikes}}^{\text{r}}=0,$$ (93)

  $$N_{\text{spikes}}^{\text{F}}=N_{\text{spikes};{\text{M}}^{\text{r}}=0}^{\text{F}}(\text{Δ}{T}_{\text{pBötC}},a_0^{\text{F}}),$$ (94)

  where $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s};{\mathrm{M}}^{\mathrm{r}}=0}^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})$ is given by Equation 89.

• C. Case of ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-I^{\mathrm{r}\mathrm{B}}<0$:

  $a^{\mathrm{r}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}),a^{\mathrm{F}}(\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}),N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{r}}$, and $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{s}}^{\mathrm{F}}$ are given by Equations 91, 92, 93, and 94 respectively.

• In all cases (A-C), the value of $a^{\mathrm{r}}(T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}})$ is

  $$a^{\text{r}}(T_{\text{pBötC}})=\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}+\left[a_0^{\text{r}}\left(\text{Δ}{T}_{\text{pBötC}}\right)-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}\right]e^{-\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\right)\left(T_{\text{pBötC}}-\text{Δ}{T}_{\text{pBötC}}\right)/{\tau }_{\text{a}}^{\text{r}}}.$$ (95)
  Since we consider the case where $M_{\mathrm{F}}=0$ when the pBötC is silent, i.e., for $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}<0<T_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$,

  $$a^{\text{F}}(T_{\text{pBötC}})=a^{\text{F}}(\text{Δ}{T}_{\text{pBötC}})e^{-\left(T_{\text{pBötC}}-\text{Δ}{T}_{\text{pBöC}}\right)/{\tau }_{\text{a}}^{\text{r}}}.$$ (96)

6.2.4.4. Analytical calculation of the vibrissa set-point angle at rest and at saturation for the pBötC-vIRt^ret-vFMN circuit

We compute the onset value of $I^{\mathrm{r}\mathrm{B}},I_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}^{\mathrm{r}\mathrm{B}}$, above which the vibrissae start to oscillate, along with the saturation value $I_{\mathrm{s}\mathrm{a}\mathrm{t}}^{\mathrm{r}\mathrm{B}}$ above which the whisking amplitude saturates at a maximum value (Figure 3H). We perform this calculation for low-frequency whisking, for which the values of the adaptation $a_0^{\mathrm{r}}$ and $a_0^{\mathrm{F}}$ are given by Equations. 66 and 67.

• The vibrissa is at rest $(\theta =0)$ if ${\hat{M}}^{\mathrm{F}}\le 0$ during the whole period of pBötC activity. Since $a^{\mathrm{r}}$ decreases and $M^{\mathrm{r}}$ increases with time after the onset of pBötC activity (Equation 70), we look for the value of $I^{\mathrm{r}\mathrm{B}}$ value for which ${\hat{M}}^{\mathrm{F}}\left(0^{+}\right)=0$. From Equation 61, this means that

  $${\tilde{I}}_{\text{ext}}^{\text{F}}-J^{\text{Fr}}{\tau }_s{\widehat{M}}^{\text{F}}\left(0^{+}\right)=0$$ (97)
  Substituting Equation 57 and the steady-state value of $s^{\mathrm{r}}$ (Equation 58) in Equation 97, we find that the value $I^{\mathrm{r}\mathrm{B}}$ above which $M^{\mathrm{F}}$ is not indentically zero and the vibrissa starts to oscillate is:

  $$I_{\text{onset}}^{\text{rB}}={\tilde{I}}_{\text{ext}}^{\text{r}}-a_0^{\text{r}}-\frac{{\tilde{I}}_{\text{ext}}^{\text{F}}}{{\tau }_s{\beta }^{\text{r}}{J}^{\text{Fr}}}.$$ (98)
• The saturation value of the amplitude of $\theta$ during rhythmic whisking is obtained for the $I^{\mathrm{r}\mathrm{B}}$ value above which $T_{{\mathrm{M}}^{\mathrm{r}}=0}=\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$ (Equation 82). Using Equations 87, 67, and 61 for $M^{\mathrm{r}}=0$, we obtain

  $$M^{\text{F}}(t)={\beta }^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}\left[1+\frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}{e}^{-\widetilde{\text{Γ}}t}\right].$$ (99)

  where

  $$\widetilde{\text{Γ}}=\frac{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}{{\tau }_{\text{a}}^{\text{F}}}.$$ (100)
• The dependency of the force term on $M^{\mathrm{F}}$ is nonlinear (Equations 51 and 53). Yet we can approximate it by linear dependency

  $$\frac{d\theta }{dt}=-\frac{\theta }{{\tau }_{\text{wm}}}+F_m{M}^{\text{F}}$$ (101)

  where $F_m$ is a proportionality constant.
• The time $T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}$ for maximal protraction is determined by calculating the maximal value of $\theta \left(t\right)$ computed by solving Equations 101, 99, and 100 for the initial condition $\theta \left(0\right)=0$. This time is found to be independent of $F_m$, i.e.,

  $$T_{\theta ,\text{max}}=\frac{{\tau }_{\text{wm}}}{\widetilde{\text{Γ}}{\tau }_{\text{wm}}-1}\text{log}\left[\frac{{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}\widetilde{\text{Γ}}{\tau }_{\text{wm}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}-\widetilde{\text{Γ}}{\tau }_{\text{wm}}}\right].$$ (102)

  This calculation is valid if $T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}\le \mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$. Otherwise, we replace $T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}$ with $\mathrm{\Delta }{T}_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C}}$.

• To compute the value $I_{\mathrm{s}\mathrm{a}\mathrm{t}}^{\mathrm{r}\mathrm{B}}$ above which saturation is obtained, we look for the value of $I^{\mathrm{r}\mathrm{B}}$ for which $M^{\mathrm{r}}\left(t\right)=0$ for $0\le t\le T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}$ based on self-consistency. Since $M^{\mathrm{r}}=0$ for this time interval, $a^{\mathrm{r}}=a_0^{\mathrm{r}}{e}^{-t/{\tau }^{\mathrm{r}}}$ (Equation 59). Note that so long as (Equations 57 and 66)

  $${\tilde{I}}_{\text{ext}}^{\text{r}}-I^{\text{rB}}-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{F}}}{e}^{-t/{\tau }_{\text{a}}^{\text{r}}}\le 0,$$ (103)

  we have $M^{\mathrm{r}}=0$ and

  $$I_{\text{sat}}^{\text{rB}}={\tilde{I}}_{\text{ext}}^{\text{r}}\left(1-\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{F}}}{e}^{-T_{\theta ,\text{max}}/{\tau }_{\text{a}}^{\text{r}}}\right).$$ (104)

• The value of $M^{\mathrm{F}}$ at saturation is computed by substituting $t=T_{\theta ,\mathrm{m}\mathrm{a}\mathrm{x}}$ in Equation 99. The whisking amplitude at saturation is given by the steady-state value of Equation 54, ${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}={\tau }_{\mathrm{w}\mathrm{m}}{F}_{\mathrm{f}\mathrm{i}\mathrm{t}}\left(M_{\mathrm{s}\mathrm{a}\mathrm{t}}^{\mathrm{F}}\right)$.

6.2.4.5. Analytical solution of the rate model for the vIRt^ret-vIRt^pro circuit

The dynamics of the two vIRt sub-population are described in terms of the population-averaged synaptic variables (Equation 43) and adaptation variables (Equations 44). In our analysis, we take ${\beta }^{\mathrm{r}}={\beta }^{\mathrm{p}}$ for simplicity.

• For the case of a symmetric circuit composed of two vIRt sub-populations without pBötC input, this description involves four differential equations:

  $$\begin{gathered} {\dot{s}}^{\text{r}}=-\frac{s^{\text{r}}}{{\tau }_s}+M^{\text{r}} \\ =-\frac{s^{\text{r}}}{{\tau }_s}+{\beta }^{\text{r}}{\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{intra}}{s}^{\text{r}}-J_{\text{inter}}{s}^{\text{p}}-a^{\text{r}}\right]}_{+}, \end{gathered}$$ (105)

  $${\dot{a}}^{\text{r}}=\frac{-a^{\text{r}}+J_{\text{a}}^{\text{r}}{\beta }^{\text{r}}{\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{intra}}{s}^{\text{r}}-J_{\text{inter}}{s}^{\text{p}}-a^{\text{r}}\right]}_{+}}{{\tau }_{\text{a}}^{\text{r}}},$$ (106)

  $$\begin{gathered} {\dot{s}}^{\text{p}}=-\frac{s^{\text{p}}}{{\tau }_s}+M^{\text{p}} \\ =-\frac{s^{\text{p}}}{{\tau }_s}+{\beta }^{\text{r}}{\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{inter}}{s}^{\text{r}}-J_{\text{intra}}{s}^{\text{p}}-a^{\text{p}}\right]}_{+}, \end{gathered}$$ (107)

  $${\dot{a}}^{\text{p}}=\frac{-a^{\text{p}}+J_{\text{a}}^{\text{r}}{\beta }^{\text{r}}{\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{inter}}{s}^{\text{r}}-J_{\text{intra}}{s}^{\text{p}}-a^{\text{p}}\right]}_{+}}{{\tau }_{\text{a}}^{\text{r}}}.$$ (108)

  Several types of fixed-point (FP) solutions are possible for Equations 105 through 108.

• Uniform state: Neurons in the two subpopulations fire at the uniform state, for which $M^{\mathrm{p}}>0$ and $M^{\mathrm{r}}>0$ with Soloduchin and Shamir (2018):

  $$\begin{gathered} M^{\text{r}}=M^{\text{p}} \\ =\frac{{\beta }^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}\left(J_{\text{intra}}+J_{\text{inter}}\right)}. \end{gathered}$$ (109)
  If ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}<J^{\mathrm{F}\mathrm{r}}{\tau }_s{M}^{\mathrm{r}}$ (Equations 61 and 63), $M^{\mathrm{F}}=0$ and the set-point vibrissa angle, ${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, is zero. Otherwise,

  $$M^{\text{F}}=\frac{{\beta }^{\text{F}}\left({\tilde{I}}_{\text{ext}}^{\text{F}}-{\tau }_s{J}^{\text{Fr}}{M}^{\text{r}}\right)}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}.$$ (110)
  The stability matrix of this FP is determined by considering the vIRt circuit only (Equations 105 to 108):

  $$A_{\text{stab}}=-\left(\begin{array}{cccc}\frac{1}{{\tau }_s}+{\beta }^{\text{r}}{J}_{\text{intra}}& {\beta }^{\text{r}}& {\beta }^{\text{r}}{J}_{\text{intra}}& 0\\ \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{J}_{\text{intra}}}{{\tau }_{\text{a}}^{\text{r}}}& \frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}& \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{J}_{\text{intra}}}{{\tau }_{\text{a}}^{\text{r}}}& 0\\ {\beta }^{\text{r}}{J}_{\text{intra}}& 0& \frac{1}{{\tau }_s}+{\beta }^{\text{r}}{J}_{\text{intra}}& {\beta }^{\text{r}}\\ \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{J}_{\text{intra}}}{{\tau }_{\text{a}}^{\text{r}}}& 0& \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}{J}_{\text{intra}}}{{\tau }_{\text{a}}^{\text{r}}}& \frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}\end{array}\right).$$ (111)
  Substituting eigenvectors of the form $(\overrightarrow{a},\pm \overrightarrow{a}{)}^{\mathrm{T}}$, where $\overrightarrow{a}$ is a two-dimensional vector, yields two, $2\times 2$ matrices:

  $$A_{\text{stab},\pm }=-\left(\begin{array}{cc}\frac{1}{{\tau }_s}+{\beta }^{\text{r}}\left(J_{\text{intra}}\pm J_{\text{inter}}\right)& {\beta }^{\text{r}}\\ \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_a^{\text{r}}}\left(J_{\text{intra}}\pm J_{\text{inter}}\right)& \frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}\end{array}\right).$$ (112)
  Stability depends on the two eigenvalues having negative real parts, which is implies that the anti-symmetric mode $(\overrightarrow{a},-\overrightarrow{a}{)}^{\mathrm{T}}$ is the least stable. The conditions for receiving two eigenvalues with negative real parts are $\mathrm{Tr}\left(\left(A_{\text{stab},-}\right)<\right.$ 0 and $\mathrm{Det}\left(A_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b},-}\right)>0$, for which:

  $$J_{\text{inter}}-J_{\text{intra}}<J_{\text{tr}}$$ (113)

  with

  $$J_{\text{tr}}\equiv \frac{1}{{\beta }^{\text{r}}}\left(\frac{1}{{\tau }_s}+\frac{1}{{\tau }_{\text{a}}^{\text{r}}}+\frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\tau }_{\text{a}}^{\text{r}}}\right)$$ (114)

  and

  $$J_{\text{inter}}-J_{\text{intra}}<J_{\text{det}}$$ (115)

  with

  $$J_{\text{det}}\equiv \frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\beta }^{\text{r}}{\tau }_s}.$$ (116)

  If ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$ then $J_{\mathrm{t}\mathrm{r}}<J_{\text{det}}$, and the uniform FP will become unstable when $J_{\text{inter}}-J_{\text{intra}}$ increases above $J_{\text{tr}}$. Since $J_{\text{tr}}>0,J_{\text{inter}}$ should be larger than $J_{\text{intra}}$ to destabilize the FP.

• Bistable state: Two stable symmetric FP coexists in the bistable state, one with $M^{\mathrm{p}}>0$ and $M^{\mathrm{r}}=0$, and the second with $M^{\mathrm{r}}>0$ and $M^{\mathrm{p}}=0$. In the first case,

  $$M^{\text{p}}=\frac{{\beta }^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}$$ (117)

  is nonzero if ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}-J_{\text{inter}}{s}^{\mathrm{p}}<0$. From Equation 117, this condition becomes

  $$J_{\text{inter}}-J_{\text{intra}}>\frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\beta }^{\text{r}}{\tau }_s}$$ (118)

  or, from Equation 116,

  $$J_{\text{inter}}-J_{\text{intra}}>J_{\text{det}}.$$ (119)
  With $M^{\mathrm{r}}=0$, the value of $M^{\mathrm{F}}$ for is ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}>0$ (Equations 61 and 63),

  $$M^{\text{F}}=\frac{{\beta }^{\text{F}}{\tilde{I}}_{\text{ext}}^{\text{F}}}{1+{\beta }^{\text{F}}{J}_{\text{a}}^{\text{F}}}.$$ (120)
  For the other bistable state ,

  $$M^{\text{r}}=\frac{{\beta }^{\text{r}}{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}$$ (121)

  and $M^{\mathrm{p}}=0$. Similar to the uniform case, ${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0$ if ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{F}}<J^{\mathrm{F}\mathrm{r}}{\tau }_s{M}^{\mathrm{r}}$. Otherwise, $M^{\mathrm{F}}$ is determined by Equation 110, and there is bistability of two protracted values of ${\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ as a result of the vIRt dynamics, determined by Equation 54.

• Oscillatory state: This is the state of interest for whisking. No stable FP is found for $J_{\text{inter}}-J_{\text{intra}}$ between $J_{\text{tr}}$ (Equation 114) and $J_{\text{det}}$ (Equations 116 and 118). A limit cycle exists in this regime, in which the two vIRt subpopulations spike in alternation. Each population is active during a half time period $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}/2$, and is silent during the following half time period.

  • Following Soloduchin and Shamir (2018), we compute the amplitude and $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ of the system in the limit ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$. We define the time $t=0$ as the time when the ${\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}^{\text{ret}}$ subpopulation begins to be active and the vIRt^pro subpopulation becomes silent. During the time interval $0\le t\le T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}/2$, the active state of vIRt^ret subpopulation, $s^{\mathrm{r}}$, reaches a quasi-steady-state value that evolves on the slow time-scale of order ${\tau }_{\mathrm{a}}^{\mathrm{r}}$ according to

    $$s^{\text{r}}(t)=\frac{{\tau }_s{\beta }^{\text{r}}\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-a^{\text{r}}(t)\right]}{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}.$$ (122)
    The adaptation variable $a^{\mathrm{r}}$ evolves according to Equations 44 and 122, i.e.,

    $${\dot{a}}^{\text{r}}=\frac{\tilde{J}}{{\tau }_{\text{a}}^{\text{r}}}{\tilde{I}}_{\text{ext}}^{\text{r}}-\frac{1+\tilde{J}}{{\tau }_{\text{a}}^{\text{r}}}{a}^{\text{r}}$$ (123)

    where

    $$\tilde{J}\equiv \frac{{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}.$$ (124)
    Defining $a_0^{\mathrm{r}}=a^{\mathrm{r}}(t=0)$, the solution of Equation 123 is

    $$a^{\text{r}}(t)=\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}+\left(a_0^{\text{r}}-\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}\right)e^{-(1+\tilde{J})t/{\tau }_{\text{a}}^{\text{r}}}.$$ (125)
    The firing rate is $M^{\mathrm{r}}=0$ during the time interval $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}/2\le t\le T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ and leads to

    $$a^{\text{r}}(t)=a^{\text{r}}\left(\frac{T_{\text{vIRt}}}{2}\right)e^{-\left(t-T_{\text{vIRt}}/2\right)/{\tau }_{\text{a}}^{\text{r}}}.$$ (126)
    Using periodicity, i.e., $a^{\mathrm{r}}\left(T_{\text{vIRt}}\right)=a_0^{\mathrm{r}}$, we obtain

    $$a_0^{\text{r}}=\frac{\tilde{J}}{1+\tilde{J}}\frac{e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{\tilde{I}}_{\text{ext}}^{\text{r}}$$ (127)

    and, together with Equation 125, we obtain

    $$a_{\text{r}}\left(T_{\text{vIRt}}/2\right)=\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}+\left(a_0^{\text{r}}-\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}\right)e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}.$$ (128)
    To compute $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$, we utilize the switch in $M^{\mathrm{p}}$ from zero to a positive value at $t=T_{\text{vIRt}}/2$, i.e.,

    $${\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{inter}}{s}^{\text{r}}\left(T_{\text{vIRt}}/2\right)-a_0^{\text{r}}=0.$$ (129)
    Inserting Equation 122 into Equation 129 leads to an equation for $a_0^{\mathrm{r}}$, i.e.,

    $${\tilde{I}}_{\text{ext}}^{\text{r}}-J_{\text{inter}}\frac{{\tau }^s{\beta }^{\text{r}}}{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}\left[{\tilde{I}}_{\text{ext}}^{\text{r}}-a^{\text{r}}\left(T_{\text{vIRt}}/2\right)\right]-a_0^{\text{r}}=0.$$ (130)
    The combination of Equations 130, 127, and 128 define an implicit transendental relation for the bursting $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ of the vIRt oscillator, i.e.,

    $$\begin{gathered} J_{\text{inter}}=\frac{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}{{\tau }_s{\beta }^{\text{r}}}\times \\ \frac{\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}\right)\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]-{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{\left(1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}\right)\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]-{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]} \end{gathered}$$ (131)
    Equation 131 can be written in terms of two dimensionless parameters, $\tilde{J}$ and ${\tau }_{\mathrm{s}}{J}_{\text{inter}}/J_{\mathrm{a}}^{\mathrm{r}}$, and the ratio $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}/2{\tau }_{\mathrm{a}}^{\mathrm{r}}$:

    $$\frac{{\tau }_{\text{s}}{J}_{\text{inter}}}{J_{\text{a}}^{\text{r}}}=\frac{\left(\frac{1+\tilde{J}}{\tilde{J}}\right)\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]-\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{\left(\frac{1+\tilde{J}}{\tilde{J}}\right)\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]-\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]}$$ (132)

    The period scales linearly with ${\tau }_{\mathrm{a}}^{\mathrm{r}}$. It is independent of the effective input, ${\tilde{I}}_{\text{ext}}^{\mathrm{r}}$.

  • For $T_{\text{vIRt}}\to \mathrm{\infty }$, Equations 127 and 125 yield

    $$a_0^{\text{r}}=0$$ (133)

    and

    $$a^{\text{r}}\left(T_{\text{vIRt}}/2\right)=\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}.$$ (134)
    Substituting Equations 133, 122, and 124 in Equation 129, we obtain

    $$J_{\text{inter}}-J_{\text{intra}}=\frac{1+{\beta }^{\text{r}}{J}_{\text{a}}^{\text{r}}}{{\beta }^{\text{r}}{\tau }_s}.$$ (135)

    The period $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ diverges at the maximal value that $J_{\text{inter}}-J_{\text{intra}}$ may take for a limit cycle. Above that value, the system settles in a bistable state (Equation 118).

  • For $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$, Equation 127 yields

    $$a_0^{\text{r}}=\frac{\tilde{J}}{2+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}\text{.}$$ (136)
    Substituting this value in Equation 130 and using Equation 128, the condition for receiving very small $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ becomes

    $$J_{\text{inter}}-J_{\text{intra}}=\frac{1}{{\beta }^{\text{r}}{\tau }_s}.$$ (137)

    This condition coalesces with Equation 114 for ${\tau }_s\ll {\tau }_{\mathrm{a}}^{\mathrm{r}}$. Thus for $J_{\text{inter}}-$ $J_{\text{intra}}$ just above the value where the uniform state becomes unstable, a limit cycle with a very short time period, on the time scale of ${\tau }_{\mathrm{a}}^{\mathrm{r}}$, will emerge.

  • To compute the total number of spikes fired by a neuron during the active phase, we substitute Equations 122 and 125 in Equation 105. For the half-cycle for which the vIRt^ret subpopulation is active, we obtain

    $$M^{\text{r}}(t)=\frac{{\beta }^{\text{r}}}{1+{\tau }_s{\beta }^{\text{r}}{J}_{\text{intra}}}\left[\frac{{\tilde{I}}_{\text{ext}}^{\text{r}}}{1+\tilde{J}}-\left(a_0^{\text{r}}-\frac{\tilde{J}}{1+\tilde{J}}{\tilde{I}}_{\text{ext}}^{\text{r}}\right)e^{-(1+\tilde{J})t/{\tau }_{\text{a}}^{\text{r}}}\right].$$ (138)
    Using Equation 127, we find that the time-average firing $⟨M_{\mathrm{r}}\left(t\right)⟩$ is given by

    $$\begin{gathered} {⟨M^{\mathrm{r}}\left(t\right)⟩}_t=\frac{1}{T_{\text{vIRt}}}{\int }_0^{T_{\text{vIRt}}}dt{M}^r\left(t\right) \\ =\frac{1}{2}\frac{\tilde{J}}{1+\tilde{J}}\frac{I_{\text{ext}}^{\text{r}}}{J_{\text{a}}^{\text{r}}} \\ \times \left\{1+\frac{\tilde{J}}{1+\tilde{J}}\left(\frac{2{\tau }_{\text{a}}^{\text{r}}}{T_{\text{vIRt}}}\right)\frac{\left[1-e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]}{\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]}\right\}. \end{gathered}$$ (139)

    Note that ${⟨M^{\mathrm{r}}\left(t\right)⟩}_t$ is proportional to ${\tilde{I}}_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\mathrm{r}}$.

6.2.4.6. The oscillation period T_vIRt increases monotonically with J_inter

We show that $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ increases monotonically with increasing values of $J_{\text{inter}}$ for $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>$ 0.

• We write Equation 132 in terms of normalized variables, i.e.,

  $$J_{\text{nor}}=\frac{1-\tilde{E}\left(T_{\text{vIRt}}\right)e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{1-\tilde{E}\left(T_{\text{vIRt}}\right)}$$ (140)

  where

  $$J_{\text{nor}}\equiv \frac{{\tau }_{\text{s}}{J}_{\text{inter}}}{J_{\text{a}}^{\text{r}}}$$ (141)

  and

  $$\tilde{E}\left(T_{\text{vIRt}}\right)=\frac{\tilde{J}}{1+\tilde{J}}\frac{1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}.$$ (142)

  Clearly, $0<\tilde{E}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)<1$.

• We show that $\tilde{E}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)$ increases with $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$. Differentiating $\tilde{E}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)$ with respect to $T_{\text{vIRt}}$, we obtain

  $$\frac{d\tilde{E}\left(T_{\text{vIRt}}\right)}{d{T}_{\text{vIRt}}}=\frac{\tilde{J}}{2{\tau }_{\text{a}}^{\text{r}}(1+\tilde{J})}\frac{e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}}{{\left[1-e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]}^2}\tilde{F}\left(T_{\text{vIRt}}\right)$$ (143)

  where

  $$\tilde{F}\left(T_{\text{vIRt}}\right)=\left[(1+\tilde{J})-(2+\tilde{J})e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}+e^{-(2+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]$$ (144)
  Therefore, we need to show that $\tilde{F}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)>0$ for positive $T_{\text{vIRt}}$. We note that

  $$\tilde{F}(0)=0$$ (145)
  Differentiating Equation 144 with respect to $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$, we obtain

  $$\frac{d\tilde{F}\left(T_{\text{vIRt}}\right)}{d{T}_{\text{vIRt}}}=\frac{2+\tilde{J}}{2{\tau }_{\text{a}}^{\text{r}}}{e}^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\left[1-e^{-(1+\tilde{J})T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]$$ (146)

  This means that $d\tilde{F}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)/d{T}_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$ for $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0,\tilde{F}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)$ is an increasing function of $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$, and, because of Equation 145, is positive. We have therefore shown that $d\tilde{E}\left(T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}\right)/d{T}_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$ for $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$

• To show that $d{J}_{\text{nor}}/d{T}_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$, we differentiate Equation 140 with respect to $T_{\text{vIRt}}$ and obtain

  $$\frac{d{J}_{\text{nor}}}{d{T}_{\text{vIRt}}}=\frac{\frac{d\tilde{E}}{d{T}_{\text{vIRt}}}\left[1-e^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}\right]+\frac{\tilde{E}}{2{\tau }_{\text{a}}^{\text{r}}}{e}^{-T_{\text{vIRt}}/\left(2{\tau }_{\text{a}}^{\text{r}}\right)}(1-\tilde{E})}{{\left[1-\tilde{E}\left(T_{\text{vIRt}}\right)\right]}^2}$$ (147)

  Since, $d\tilde{E}/d{T}_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$ and $0<\tilde{E}<1$ for $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}>0$, the numerator is positive, $d{J}_{\text{nor}}/d{T}_{\text{vIRt}}>0$. From Equations 140 and 141, we obtain that $d{T}_{\text{vIRt}}/d{J}_{\text{inter}}>0$.

6.3. NUMERICAL METHOD DETAILS

Simulations of the conductance-based model were performed using the fourth-order Runge-Kutta method with time step $\mathrm{\Delta }\mathrm{t}=0.01\mathrm{m}\mathrm{s}$. Simulations with smaller $\mathrm{\Delta }t$ reveal similar statistics of neuronal firing patterns, such as spike rates M, averaged over many whisking cycles, or ${\mathrm{C}\mathrm{V}}_2$. Differences between individual voltage time courses, however, diverged over large integration time interval. Statistics were computed after removing an initial transient of $1\mathrm{s}$, over an interval of $6\mathrm{s}$ (Figures 2C,D,3B–D,F–H, 4B–D,F–H, 5B–M, 6A–F and 8B–D; Supplementary Figure 1B–K) or 60 s (Figure 7D,E,F; Supplementary Figure 2A,B). In numerical experiments to establish the dependence on system parameters (Figures 3F–H, 5B–M, 6D–F, 7E,F, 8B–D; Supplementary Figure 1B–K), each data point is computed by averaging over five network realizations.

Simulations of the rate model (Figures 3F–H, 5B–I; Supplementary Figure 1B–I) were performed using the fourth-order Runge-Kutta method with time step $\mathrm{\Delta }t=0.02\mathrm{m}\mathrm{s}$.

6.4. EXPERIMENTAL SUBJECT DETAILS

The Institutional Animal Care and Use Committee at the University of California San Diego approved all protocols. Two, male Long Evans male rats, aged 9 months, were maintained in standard cages on a natural light-dark cycle. For surgery, rats were anesthetized with isoflurane (Butler Schein). Body temperature was monitored and maintained at ${37}^{\circ }\mathrm{C}$. Subcutaneous injections of $5\mathrm{\%}(\mathrm{w}/\mathrm{v})$ glucose in saline were given every 2 h for rehydration. Buprenorphine (0.02 mg/kg; Butler Schein) was administered i.p. for post-operative analgesia.

6.5. EXPERIMENTAL METHOD DETAILS

Breathing signals were acquired by measuring respiration-related temperature changes. We implanted a thermistor (GAG22K7MCD419, TE Connectivity) in the nasal cavity McAfee et al. (2016). The change in temperature leads to a change in resistance of the thermistor, and was converted to a the period of a digital pulse generator a CMOS timers (ICM7555 in astable mode, Renesas electronics). The clock’s pulse period is thus proportional to the change in nasal cavity temperature. The timer signal pulse train was sampled at $20\mathrm{k}\mathrm{H}\mathrm{z}$ and logged on a computer using the LabChart acquisition system (AD Instruments). The pulse frequency is measured by detecting individual pulses’ onset and converting inter-pulse intervals to the frequency.The underlying signal was filtered between 1 and $15\mathrm{H}\mathrm{z}$ with a 3-pole Butterworth filter that was run backwards and forwards in time.

Head and vibrissa movements were recorded using a high-speed video camera (Basler axA204 −90umNIR). A 1445 by 485 pixel $(24.5\mathrm{c}\mathrm{m}\times 8.2\mathrm{c}\mathrm{m})$ area was captured at 350 frames/s. During the recording sessions all vibrissae except C1, C2, and C3 were trimmed. The animal’s head and the nose were marked by adhering a two white ellipsoidal beads, $5.4\mathrm{m}\mathrm{m}$ long by $3.8\mathrm{m}\mathrm{m}$ diameter, to the skin surface parallel to the rostro-caudal axis of the rat’s head using super glue (Loctite 495). The markers’ placement improved the robustness of the tracking algorithm [Clack et al. (2012), Mathis et al. (2018)] and facilitated intrasubject comparisons. After successfully tracking the head and nose position, in each frame, the orientation angle and head markers position was used to place a square $(500\times 500$ pixel) region of interest around the animal’s head. After further cropping , the underlying image is split in half, i.e., right and left part of the face, to make it compatible with a vibrissa tracking algorithm [Clack et al. (2012)] .

6.6. QUANTIFICATION AND STATISTICAL ANALYSIS

6.6.1. Vibrissa tracking

We used DeepLabCut [Mathis et al. (2018)] to detect the position of each individual vibrissa base in single frame. Custom written scripts in MATLAB (MathWorks) and Python 3.6.9 were used to carry out and combine various tracking steps.

6.6.2. Population- and time-averaged quantities

The model parameter $M_{\mu }$ is the time-averaged firing rates of all the neurons in the μ-th subpopulation, where averaging occurs over many whisking cycles at the oscillatory state. We characterize several dynamical states of the vIRt subpopulations without pBötC input (Figure 5). In the silent state, neurons in the two subpopulations do not fire. In the uniform state, neurons in the two subpopulations fire tonically. In the bistable state, one subpopulation is active and neurons fire tonically, and the second subpopulation is silent. This state coexists with another state in which the two subpopulations switch roles. In the oscillatory state, the two subpopulations are active and silent alternately and neurons exhibit bursting behavior.

To find out if an active network is in a oscillatory state, we compute the maximal inter-spike interval (ISI) for each neuron in a subpopulation, and ${\mathrm{ISI}}_{\text{max,median}}^{\mu }$ denotes the median, over the subpopulation, of the maximum ISI for all the firing neurons in the population. If, for the two vIRt subpopulations, ${\mathrm{ISI}}_{\text{max,median}}^{\mu }$ is larger than twice the average ISI of these subpopulations, the vIRt network is considered to exhibit bursting. Otherwise, it is considered to be tonically spiking. The subpopulation-bursting time period $T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$ is determined based on auto- and cross-correlation analysis of the subpopulation firing activity.

For tonic firing, the coefficients of variation of and ${\mathrm{C}\mathrm{V}}_{2,i}^{\mu }$ of the i-th neuron in the μ-th subpopulation are computed from the definition of ${\mathrm{C}\mathrm{V}}_2$ of a neuron [Holt et al. (1996)]. In bursting states, ISI are included in the calculation of ${\mathrm{C}\mathrm{V}}_{2,i}^{\mu }$ only if they are smaller than $0.4T_{\mathrm{v}\mathrm{I}\mathrm{R}\mathrm{t}}$. By doing this, we consider the variation of ISIs only within each burst, and exclude ISIs that belong to different bursts. Therefore, neurons that do not have at least one burst with 3 spikes do not have a defined ${\mathrm{C}\mathrm{V}}_{2,i}^{\mu }$ value. The value ${\mathrm{C}\mathrm{V}}_2^{\mu }$ is calculating by averaging ${\mathrm{C}\mathrm{V}}_{2,i}^{\mu }$ over all the neurons in the μ-th subpopulation for which this value is defined.

6.6.2.1. Extracting whisking amplitudes

We extract the amplitude of the successive whisking amplitudes within a breathing cycle from simulations by comparing the vibrissa trace $\theta \left(t\right)$ with the initiation times of the pBötC activity $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$. We begin by computing the standard deviation of $\theta \left(t\right)$, ${\theta }_{\text{sd}}$, along the entire simulation time interval, and the global minimum of $\theta \left(t\right)$, obtained in $t_{\mathrm{w},\text{min}}$. We then begin from the time of the global minimum, and search for the local maxima and minima of $\theta \left(t\right)$ alternately for t either smaller or larger than $t_{w,\mathrm{m}\mathrm{i}\mathrm{n}}$. We consider only the local extrema that their absolute difference from the previous extrema is larger than $0.8\times {\theta }_{\text{sd}}\left(0.1\times {\theta }_{\text{sd}}\right.$ (Figures 5M, 8B,C). For each pBötC cycle starting at $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$, we find all the whisks such that their maximal $\theta$ occurs between $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l}$ and $t_{\mathrm{p}\mathrm{B}\mathrm{ö}\mathrm{t}\mathrm{C},l+1}$, and these whisks are defined to belong to this breathing cycle. The whisk amplitude is half of the difference between the values of $\theta$ at its local maximal value and at the previous local minimal value.

Similar calculations are carried out for whisking signals obtained experimentally. Following [Moore et al. (2013)], the initiation time of the pBötC activity measured experimentally, $t_{\mathrm{p}\mathrm{B}\mathrm{o}\mathrm{t}\mathrm{C},l}$, are defined to precede the maximal value of the breathing signal from the thermocouple by $30\mathrm{m}\mathrm{s}$.