Inspiratory and sigh breathing rhythms depend on distinct cellular signaling mechanisms in the preBötzinger complex

Abstract

Breathing behavior involves the generation of normal breaths (eupnea) on a timescale of seconds and sigh breaths on the order of minutes. Both rhythms emerge in tandem from a single brainstem site but whether and how a single cell population can generate two disparate rhythms remains unclear. We posit that recurrent synaptic excitation in concert with synaptic depression and cellular refractoriness gives rise to the eupnea rhythm, whereas an intracellular calcium oscillation that is slower by orders of magnitude gives rise to the sigh rhythm. A mathematical model capturing these dynamics simultaneously generates eupnea and sigh rhythms with disparate frequencies, which can be separately regulated by physiological parameters. We experimentally validated key model predictions regarding intracellular calcium signaling. All vertebrate brains feature a network oscillator that drives the breathing pump for regular respiration. However, in air-breathing mammals with compliant lungs susceptible to collapse, the breathing rhythmogenic network may have refashioned ubiquitous intracellular signaling systems to produce a second slower rhythm (for sighs) that prevents atelectasis without impeding eupnea.

Graphical Abstract

INTRODUCTION

Central pattern generator (CPG) circuits produce the rudimentary oscillation and motor pattern for rhythmic behaviors. In mammals, depending on context, the locomotor CPG produces either walking, running, or bounding where each form of locomotion is mutually exclusive (Bellardita & Kiehn, 2015; Kiehn, 2016; Grillner & El Manira, 2020); the oromotor CPG produces either chewing, lapping, or swallowing where, again, one form of ingestive behavior precludes the others (Westberg & Kolta, 2011; Costa, 2018). The breathing CPG, however, generates two rhythms in tandem: one for eupnea to ventilate the lungs on a second-to-second basis (~3 Hz in rodents to ~0.2 Hz in humans) and another for sighs to optimize pulmonary function with periodicity on the order of minutes (0.5–5 min⁻¹ in rodents to ~0.2 min⁻¹ in humans) (Li & Yackle, 2017). We investigated the mechanisms underlying these two rhythms to elucidate the neural origins of breathing and advance understanding of CPGs in general.

Eupnea and sigh are both forms of inhalation, and both emanate from the preBötzinger Complex (preBötC) of the lower brainstem (Lieske et al., 2000; Ruangkittisakul et al., 2008; Li et al., 2016; Del Negro et al., 2018; Severs et al., 2022). The active phase of eupnea, inspiration, initiates due to recurrent synaptic excitation among glutamatergic preBötC interneurons (Wallén-Mackenzie et al., 2006; Carroll & Ramirez, 2013; Kam et al., 2013b; Ashhad & Feldman, 2020; Ashhad et al., 2023) with a facilitatory role for burst-generating intrinsic conductances (Koizumi & Smith, 2008; Ramirez & Baertsch, 2018; Yamanishi et al., 2018; Phillips et al., 2022). Inspiration terminates due to refractory mechanisms that include synaptic depression and recruitment of neural activity-dependent outward currents (Del Negro et al., 2009; Krey et al., 2010; Guerrier et al., 2015; Kottick & Del Negro, 2015) and inactivation of a persistent inward current (Butera et al., 1999; Del Negro et al., 2002a; Ptak et al., 2005; Pace et al., 2007a; Yamanishi et al., 2018).

The sigh mechanism involves intracellular calcium (Ca²⁺) signaling. First, drugs that target voltage-gated Ca²⁺ channels and intracellular Ca²⁺ release, as well as Ca²⁺ chelators perturb sighs (Lieske et al., 2000; Lieske & Ramirez, 2006a; Toporikova et al., 2015; Morgado-Valle et al., 2022; Severs et al., 2023). Further, the sigh rhythm can be accelerated by group I metabotropic glutamate receptors, M3 acetylcholine receptors, and bombesin-like peptide receptors all of which act via G_q/11 and phospholipase C to produce inositol 1,4,5-trisphosphate-mediated intracellular Ca²⁺ release (Lieske & Ramirez, 2006b; Tryba et al., 2008; Ramos-Álvarez et al., 2015; Li et al., 2016). Finally, sigh rhythm slows down or stops following G_βγ-mediated downregulation of P/Q-type Ca²⁺ channels (Lieske & Ramirez, 2006b).

A previous cellular model of the preBötC that incorporated intracellular Ca²⁺ signaling identified a region of parameter space in which a sigh-like rhythm could emerge (Jasinski et al., 2013). Although that model generally supports an intracellular signaling-related origin for sigh rhythm, there are two incongruities. First, that model generates sigh-like rhythm only in a limited region (~15%) of its parameter space. Second, the cycle period of the sigh-like rhythm was too fast (5–20 s) to credibly explain the periodicity of sigh rhythms in vivo and in vitro, which typically measure in the range 0.5–5 min (i.e., 30–300 s), although a two-timescale-reduction version of the model generates more realistic sigh periodicity (Wang & Rubin, 2017). An explanation for sigh rhythmogenesis must account for its robustness throughout parameter space and its frequency, which is slower than eupnea (inspiratory rhythm) by an order of magnitude or more.

Another model of inspiratory and sigh rhythmogenesis postulated a discrete neuron population dedicated to sigh rhythmogenesis via Ca²⁺ oscillations that depend on hyperpolarization-activated mixed cationic current (I_h), and another neuron population that generates inspiratory rhythm based on cellular voltage-dependent persistent Na⁺ current (I_NaP). The two distinct populations synchronize through chloride-mediated synaptic inhibition (Toporikova et al., 2015). The veracity of that model depends on four principles: i) the existence of dedicated sigh and inspiratory neuron populations, ii) I_h being sigh rhythmogenic, iii) I_NaP being inspiratory rhythmogenic, and iv) synaptic inhibition synchronizing the two populations. Yet, all four stipulations are inconsistent with physiological evidence.

First, the literature does not support dichotomous eupnea- and sigh-specialized neuron populations in the preBötC; rather, the consensus is that rhythmogenic neurons belong to the same, or extensively overlapping, preBötC network(s) (Lieske et al., 2000; Lieske & Ramirez, 2006a, 2006b; Tryba et al., 2008; Viemari et al., 2013). However, if there are discrete eupnea- and sigh-dedicated cell populations, then one population may be non-neural, consisting instead of astrocytes (Severs et al., 2023). Second, an inspiratory rhythmogenic role for I_NaP is still a matter of considerable debate, with a branch of genetic knockout, and knockdown perturbations (da Silva et al., 2023) and pharmacological experiments (Del Negro et al., 2002b; Peña et al., 2004; Del Negro et al., 2005; Pace et al., 2007a) unsupportive, and a parallel branch of modeling, pharmacology, and optogenetic experiments supportive (Koizumi & Smith, 2008; Ausborn et al., 2018; Yamanishi et al., 2018; Phillips et al., 2019, 2022; Phillips & Rubin, 2019). Third, blocking I_h stops sigh rhythm in the embryonic preBötC (Toporikova et al., 2015), but this observation has not been corroborated in the preBötC postnatally (Thoby-Brisson et al., 2000). Lastly, chloride-mediated synapses are not necessary to synchronize inspiratory and sigh oscillations in a slice model of breathing rhythmogenesis, except in a limited time window during embryonic and early postnatal development when the chloride electrochemical gradient is reversed and GABA_A and glycinergic synapses are functionally excitatory (Ritter & Zhang, 2000; Borrus et al., 2020).

We exploited rhythmically active slice preparations postnatally to address the unresolved issues recapped above, constructed a mathematical model of inspiratory and sigh rhythmogenesis, and then evaluated its testable predictions. We posit that a recurrent excitation-based network oscillator generates inspiration while interactions between plasma membrane Ca²⁺ fluxes and Ca²⁺ excitability of the endoplasmic reticulum (ER) give rise to sighs. These disparate mechanisms enable inspiratory and sigh rhythms to operate quasi-independently within a single neuronal population tasked with maintaining functionality of compliant mammalian lungs and adjusting breathing for different physiological contexts.

METHODS

Ethical approval and animal use

We the authors read, and understand, the ethical principles under which the journal operates. Our research complied with the animal ethics checklist in the editorial (Grundy, 2015). The Institutional Animal Care and Use Committee at our institution, William & Mary, approved these protocols (#2022-0054, title: Cellular and synaptic mechanisms of breathing and locomotion in mammals, IACUC-2022-07-29-15755-cadeln), which conform to the policies of the Office of Laboratory Animal Welfare (National Institutes of Health, Bethesda, Maryland, USA) as well as the guidelines of the US National Research Council (National Research Council, 2011). Mice (described below) were maintained on a 12-hour light / 12-hour dark cycle at 23° C and were fed ad libitum with free access to water. The mice were provided with several forms of enrichment including opaque igloo shelters, wood blocks, nest materials, and weekly peanuts in the shell.

Multi-photon experiments employed Cre-driver mice generated by inserting an IRES-CRE-pGK-Hygro cassette in the 3’ untranslated region of the Developing brain homeobox 1 (i.e., Dbx1) gene, which we refer to as Dbx1^Cre mice (Bielle et al., 2005) (IMSR Cat# EM:01924, RRID:IMSR_EM:01924). We obtained Dbx1^Cre mice from Prof. J. Corbin at Children’s National Hospital (Washington, DC, USA). We crossed female Dbx1^Cre mice with males from a reporter strain featuring Cre-dependent expression of the fluorescent Ca²⁺ indicator GCaMP6f dubbed Ai148 by the Allen Institute (Daigle et al., 2018) (IMSR Cat# JAX:030328, RRID:IMSR_JAX:030328). Ai148 mice were obtained from Jackson Labs (Bar Harbor, Maine, USA). Their offspring, Dbx1;Ai148 mice, expressed GCaMP6f in Dbx1-derived cells (Kottick et al., 2017).

Breathing-related measurements in vitro

Mouse pups aged postnatal day 0 to 4 of both sexes were anesthetized by hypothermia until there was no evidence of tail-pinch or paw-pinch withdrawal reflexes, which assured a sufficient plane of anesthesia, and then killed by thoracic transection. Their neuraxes were removed in artificial cerebrospinal fluid (aCSF) containing (in mM): 124 NaCl, 3 KCl, 1.5 CaCl₂, 1 MgSO₄, 25 NaHCO₃, 0.5 NaH₂PO₄, and 30 dextrose equilibrated with 95% O₂-5% CO₂, pH 7.4. Once isolated, the neuraxes were glued to an agar block and then cut in the transverse plane to obtain a single 500-μm-thick slice that exposed the preBötC at its rostral face. Atlases for wild-type and Dbx1 reporter mice show that the loop of the inferior olive and the semi-compact division of the nucleus ambiguus collocate with the preBötC during early postnatal development (Ruangkittisakul et al., 2014). Slices were then perfused with aCSF at 28° C in a recording chamber below a fixed-stage microscope.

We elevated extracellular K⁺ concentration ([K⁺]_o) to 9 mM to increase preBötC excitability (Funk & Greer, 2013). Inspiratory-related motor output was recorded from the hypoglossal (XII) nerve rootlets, which were captured in transverse slices along with the XII motoneurons and their axon projections to the nerve rootlets, using suction electrodes and a differential amplifier. We obtained field potential recordings by forming a seal over the preBötC with a suction electrode at the rostral slice surface. Amplifier gain was set at 2000. Signals were acquired digitally at 1 kHz while low-pass filtering at 300 Hz. XII and preBötC bursts were full-wave rectified and smoothed for display and quantitative analyses of burst events.

To apply drugs locally in the preBötC, we fabricated micropipettes from borosilicate glass (OD: 1.5 mm, ID: 0.86 mm) and filled them with either thapsigargin or xestospongin (see below). Two pipettes were inserted into the preBötC on both sides of slice preparations. We microinjected the drugs using 7–9 psi pressure pulses lasting 10–20 ms in duration, delivered at a frequency of 5 Hz. Pipettes for local drug application in the preBötC precluded simultaneous local field recordings; we monitored preBötC activity in those experiments only via XII nerve recordings.

Multi-photon imaging

We imaged cytosolic Ca²⁺ concentration in neurons contained in slices from Dbx1;Ai148 mice using a multiphoton laser-scanning confocal microscope (Thorlabs, Newton, New Jersey, USA) equipped with a water immersion objective (20x, 1.0 numerical aperture). Illumination was provided by an ultrafast tunable laser with a power output of 950 mW at 940 nm, 80-MHz pulse frequency, and 100-fs pulse duration (Coherent Chameleon Discovery, Santa Clara, California, USA). We scanned Dbx1;Ai148 mouse slices over the preBötC and collected time series images using a photomultiplier tube detector at 15 Hz. Each frame reflects one-way raster scans with a resolution of 256 × 256 pixels (116 × 116 μm). Fluorescence data were collected using Thorlabs LS 4.1 software and then analyzed using MATLAB 2021a (MathWorks, Natick, Massachusetts, USA, RRID:SCR_001622).

First, we calculated the average fluorescence intensity for all pixels in each frame of the time series. The mean fluorescence intensity was used as an index of overall network activity during the time series. The bursts of fluorescence intensity were periodic and their cycle periods were normally distributed. We used the 95% confidence interval (CI) of cycle periods to define the high frequency (short cycle period) and low frequency (long cycle period) limits of a window in frequency space. Next, we performed fast Fourier transforms of the time series for each pixel. The maximum power from the previously defined window in frequency space was mapped to the corresponding pixel in a new, processed two-dimensional image.

We calculated the mean and standard deviation of the power from each pixel in the new processed image (e.g., Fig. 7A). Rhythmically active pixels showed greater than the average power compared to non-rhythmic pixels. All pixels with intensity less than mean + 2*SD were set to zero. The remaining contiguous pixel sets, whose area exceeded 8 μm², were retained as regions of interest (ROIs). The Ca²⁺ fluorescence changes within those ROIs, obtained from the original time series, were reported using the equation $\frac{(F_i-F_o)}{F_o}$, i.e., $\frac{\Delta F}{F_o}$, where $F_i$ was the instantaneous average fluorescence intensity for all pixels within a given ROI and $F_o$ was the average fluorescence intensity of all pixels within that same ROI averaged over the entire time series.

Figure 7.

Effects of P2 receptor antagonists on inspiratory and sigh rhythms. A, slice preparation visualized under fluorescent illumination with a GFP filter set (left). Sites colocalized with the preBötC include the semi-compact nucleus ambiguus (scNA) and the inferior olivary nucleus (IO_loop). A bounding box centered on the preBötC is magnified and shown below. Pseudo-color pixel intensity indicates inspiratory rhythmicity obtained by frequency domain analyses (see Methods). Inspiratory Dbx1 preBötC neurons are enumerated according to the traces in B. B, Ca²⁺ fluorescence changes in Dbx1 preBötC neurons with XII motor output. Unmarked bursts are inspiratory; σ indicates sigh bursts. The cocktail of P2 receptor antagonists includes PPADS (50 μM), suramin (50 μM), TNP-ATP (10 μM), MRS2179 (10 μM), and MRS2578 (10 μM). The 40% calibration bar refers to fluorescence changes (ΔF) with respect to baseline fluorescence (F₀), i.e., ΔF/F₀, and it applies to all cellular imaging traces. C and D, mean inspiratory (C) and sigh (D) frequency for each slice tested in control, P2 antagonists (N = 8 total), and washout. This group analysis includes N = 4 slices exposed to the P2 receptor blocker cocktail (above) and N = 4 slices exposed to 20 μM MRS2279. Filled circles (red) show the mean frequency. Vertical bars (red) show 95% confidence intervals (CIs) for the means. Black dotted horizontal lines demarcate the means of control and P2 antagonist conditions. The rightmost plot shows the effect size (labeled ‘difference’ on the x-axis), which includes the mean change (Δ mean on the y-axis) for each individual slice (gray) and for the group (red) with 95% CIs. Note: when the 95% CIs extend beyond the boundaries for the mean values in control and experimental conditions (P2 antagonists), then the statistical test is not significant (NS) at $\alpha <0.05$.

We created cycle-triggered averages of the fluorescence activity within the ROIs for inspiratory bursts, but we omitted the sigh events. We defined a 3-s window triggered by, and centered on, the XII motor output and averaged the inspiratory bursts for each ROI.

Inspiratory burst and sigh burst detection

We distinguished a sigh burst from an inspiratory burst in the preBötC field recordings by measuring burst area and the duration of the interval between the putative sigh burst and the subsequent inspiratory burst. Sigh bursts were typically larger in area than inspiratory bursts or, in the case of doublet bursts, there were two bursts co-occurring with less than 1 s pause between them (Li et al., 2016; Borrus et al., 2020). Additionally, a prolonged interval between the putative sigh and the following inspiratory burst, typically 1.3x the average cycle period for inspiratory bursts, confirmed that an event in question was indeed a sigh burst.

Pharmacology

We employed the following drugs to block neuron-glia signaling: PPADS (50 μM), suramin (50 μM), TNP-ATP (10 μM), MRS2179 (10 μM), and MRS2578 (10 μM). We applied MRS2279 (20 μM) in separate experiments. We used thapsigargin (10 or 100 μM) and xestospongin C (1 μM) to interrogate intracellular Ca²⁺ sequestration and release. We employed ZD7288 (50 μM) to block I_h. Thapsigargin, xestospongin, and ZD7288 were dissolved in dimethyl sulfoxide (DMSO) to generate stock solutions. Final concentration of DMSO in aCSF never exceeded 1% by volume. All drugs were obtained from Millipore Sigma (Burlington, Massachusetts, USA).

Activity model of inspiratory rhythm

The preBötC core was modeled as a three-variable dynamical system,

$${\tau }_a\frac{da}{dt}=a_{\infty }\left(w\cdot s\cdot a-\theta \right)-a$$

$${\tau }_s\frac{ds}{dt}=s_{\infty }\left(a\right)-s$$

$${\tau }_{\theta }\left(a\right)\frac{d\theta }{dt}={\theta }_{\infty }\left(a\right)-\theta ,$$

where $a$ represents network activity, $s$ represents synaptic depression and activity-dependent outward currents ($s=1$ indicates the absence of depression, while $s=0$ corresponds to full depression), and $\theta$ represents cellular adaptation broadly defined. The sigmoidal steady-state functions $a_{\infty }$ and $s_{\infty }$ are given by $a_{\infty }\left(x\right)={\left\{1+e^{4\left({\gamma }_a-x\right)/k_a}\right\}}^{-1}$ and $s_{\infty }\left(a\right)={\left\{1+e^{4\left({\gamma }_s-a\right)/k_s}\right\}}^{-1}$. Because $k_a$ is positive, $a_{\infty }$ is a monotone increasing function of $x=w\cdot s\cdot a-\theta$ (Fig. 1) where $w$ is the synaptic gain, the triple product $w\cdot s\cdot a$ is the aggregate synaptic drive as well as Ca²⁺-activated nonspecific cation current (I_CAN), which is offset by cellular adaptation, $\theta$. Conversely, $k_s$ is negative and $s_{\infty }$ is a monotone decreasing function of $a$ (Fig. 2A). The steady-state level of cellular adaptation,

$${\theta }_{\infty }\left(a\right)={\left\{1+e^{4\left({\gamma }_{\theta }-a\right)/k_{\theta }}\right\}}^{-1},$$

is an increasing function of $a$ because $k_{\theta }$ is positive (Fig. 2A). The time constant for cellular adaptation given by

$${\tau }_{\theta }\left(a\right)={\left({\tau }_{\theta }^{max}-{\tau }_{\theta }^{min}\right)\left\{1+e^{4\left({\gamma }_{{\tau }_{\theta }}-a\right)/k_{{\tau }_{\theta }}}\right\}}^{-1}+{\tau }_{\theta }^{min}$$

is also a function of $a$ (Fig. 2B). The numeral 4 that appears in these sigmoidal functions makes $k_a,k_s,k_{\theta }$ and $k_{{\tau }_{\theta }}$ the inverse of the slopes of $a_{\infty },s_{\infty },{\theta }_{\infty }$ and ${\tau }_{\theta }$ at their respective half-maxima.

Figure 1.

Steady-state population firing rates $a_{\infty }$ as a function of synaptic depression ($s$). Stable and unstable steady states are shown by filled and open circles, respectively. In the absence of synaptic depression ($s=1$) there is a steady state near the maximal population firing rate ($a\approx 1$). For intermediate synaptic depression ($s=0.4$) the network is bistable. For highly depressed synapses ($s=0.2$) there is a stable steady state at low population firing rate ($a\approx 0$). Parameters: $w=1,{\gamma }_a=0.18,k_a=0.2$ from Table 1.

Figure 2.

A and B, synaptic depression ($s$) and cellular adaptation ($\theta$) as functions of network activity ($a$) via the steady-state functions, $s_{\infty }$ and ${\theta }_{\infty }$ (A) and exponential relaxation times, ${\tau }_s$ and ${\tau }_{\theta }$ (B). C, nullclines for state variables $s$ and $a$; the $a$ nullcline is shown for three different values of $\theta$. Green curve pertains to $s$; all red curves pertain to $a$.

For a fixed amount of cellular adaptation ($\theta ={\theta }_0$), steady-state network activity ($\overline{a}$) solves $\overline{a}=a_{\infty }\left(w\cdot \overline{s}\cdot \overline{a}-{\theta }_0\right).$ Network activity can have 1–3 steady states depending on $s$. For intermediate values of $s$, the network is bistable (Fig. 1). Increasing $\theta$ shifts the steady-state network activity (i.e., the intersection of $s$ and $a$ nullclines) to the right (Fig. 2C). That is, an increase in cellular adaptation (larger $\theta$) can be offset by decreased synaptic depression (larger $s$). Parameters for the activity model of inspiratory rhythm (Table 1) were chosen to ensure that $\theta$ accumulates rapidly in the active phase (large $a$), but $\theta$ recovers slowly during the silent phase when network activity is low, i.e., ${\tau }_{\theta }$ is a decreasing function of $a$ (Fig. 2B).

Table 1.

Standard parameters for inspiratory model (dimensionless).

Symbol	Definition	Value
$w$	network connectivity	1
${\gamma }_a$	synaptic drive for half-maximal network activity $a_{\infty }$ (i.e., average firing threshold)	−0.3
$k_a$	reciprocal of slope of $a_{\infty }$ at half maximum	0.2
${\tau }_a$	network recruitment time constant	0.15
${\gamma }_s$	network activity for half-maximal $s_{\infty }$	0.14
$k_s$	reciprocal of slope of $s_{\infty }$ at half maximum	−0.08
${\tau }_s$	time constant of synaptic depression	0.75
${\gamma }_{\theta }$	network activity for half-maximal ${\theta }_{\infty }$	0.15
$k_{\theta }$	reciprocal of slope of ${\theta }_{\infty }$ at half maximum	0.2
${\tau }_{\theta }^{max}$	maximum of time constant of cellular adaptation	6
${\tau }_{\theta }^{min}$	minimum of time constant of cellular adaptation	0.15
${\gamma }_{{\tau }_{\theta }}$	network activity for half-maximal ${\tau }_{\theta }\left(a\right)$	0.3
$k_{{\tau }_{\theta }}$	reciprocal of slope of ${\tau }_{\theta }\left(a\right)$ at half maximum	−0.5

Calcium handling and the sigh rhythm

The Ca²⁺ subsystem was modeled via two state variables for cytosolic Ca²⁺ concentration ($c$) and total Ca²⁺ concentration ($c_{tot}$). Their ordinary differential equations are $dc/dt=j_{er}+j_{pm}$ and $d{c}_{tot}/dt=j_{pm}$, where $j_{pm}= {ȷ}_0+{ȷ}_{a}a-v_{out}{c}^4/\left({\kappa }_{out}^4+ c^4\right)$ and $j_{er}= \left[v_{ip3r}{f}_{\mathit{\text{open}}}\left(c\right)+v_{\mathit{\text{leak}}}\right]\left[c_{er}-c\right]-v_{\mathit{\text{serca}}}{c}^2/\left({\kappa }_{\mathit{\text{serca}}}^2+c^2\right)$ with $c_{er}=\left(c_{tot}-c\right)/\rho$ and $f_{\mathit{\text{open}}}\left(c\right)={\left[1+e^{4\left({\gamma }_m-c\right)/k_m}\right]}^{-1}{\left[1+e^{4\left({\gamma }_h-c\right)/k_h}\right]}^{-1}$. The parameters $v_{ip3r}$ and $v_{\mathit{\text{leak}}}$ are rate constants for Ca²⁺-induced Ca²⁺ release and a passive leak, both with a driving force given by the concentration gradient across the ER membrane ${(c}_{er}-c)$. The parameter $v_{\mathit{\text{serca}}}$ is the maximal rate of the reuptake flux mediated by sarco-endoplasmic reticulum Ca²⁺ ATPases (SERCA). The parameter $v_{out}$ is the maximum rate of extrusion of Ca²⁺ by plasma membrane Ca²⁺ ATPases (PMCA). The terms ${ȷ}_0+{ȷ}_{a}a$ represent Ca²⁺ currents, the first term is ${ȷ}_0$ is the background Ca²⁺ influx rate akin to a leakage current I_Ca(leak) and the second term is a linear function of the network activity $a({ȷ}_a$ is a proportionality constant) and their product reflects voltage-gated current I_CaV. Parameters for the Ca²⁺ system are listed in Table 2.

Table 2.

Standard parameters for Ca²⁺ subsystem.

Symbol	Definition	Value	Units
$v_{ip3r}$	rate constant of Ca2+ release	20	s−1
$v_{\mathit{\text{leak}}}$	rate constant of Ca2+ leak	0.25	s−1
$v_{\mathit{\text{serca}}}$	maximum rate of SERCA pumps	60	μM s−1
${\kappa }_{\mathit{\text{serca}}}$	half maximum for SERCA pumps	0.3	μM
$\rho$	ER/cytosol effective volume ratio	0.15	-
${\gamma }_m$	activation of intracellular Ca2+ channels	0.25	μM
$k_m$	reciprocal of slope of $m_{\infty }$ at half maximum	0.16	μM−1
${\gamma }_h$	activation of intracellular Ca2+ channels	0.3	μM
$k_h$	reciprocal of slope of $h_{\infty }$ at half maximum	−0.24	μM−1
${ȷ}_0$	background Ca2+ influx rate	0	μM s−1
${ȷ}_a$	Ca2+ influx rate proportionality constant	0.1	μM s−1
$v_{out}$	maximum rate of PMCA pumps	0.4	μM s−1
${\kappa }_{out}$	half maximum for PMCA pumps	0.3	μM
${\lambda }_c$	maximum Ca2+-dependent increase of activity	1.5	-
${\gamma }_c$	threshold for Ca2+-dependent increase of activity	0.33	μM
$k_c$	reciprocal slope of Ca2+-dependent increase of activity	0.05	μM

Bistability of cytosolic [Ca²⁺] and relaxation oscillator dynamics

Relaxation oscillations of the Ca²⁺ subsystem underlying the sigh rhythm can be understood by considering how $c$ dynamics depend on $c_{tot}$. If $c_{tot}$ were constant, then the following scalar ordinary differential equation for a closed-cell model would apply,

$$\frac{dc}{dt}=h\left(c\right)=\left[v_{ip3r}{f}_{\mathit{\text{open}}}\left(c\right)+v_{\mathit{\text{leak}}}\right]\left(c_{er}\left(c\right)-c\right)-v_{\mathit{\text{serca}}}{c}^2/\left({\kappa }_{\mathit{\text{serca}}}^2+c^2\right).$$

Using three different values for $c_{tot}$, Fig. 3A plots the ER Ca²⁺ release flux ($j_{rel}$) and the ER Ca²⁺ influx ($j_{\mathit{\text{serca}}})$ as a function of $c$. For intermediate values of $c_{tot}$ the bell-shaped release flux $j_{rel}$ is balanced by the sigmoidal reuptake flux $j_{\mathit{\text{serca}}}$ at three different values of $c$. Fig. 3B shows the net ER flux $h\left(c\right)=j_{rel}-j_{\mathit{\text{serca}}}$ as a function of $c$. For the intermediate value of $c_{tot}$, the net ER flux, $h\left(c\right)=j_{rel}-j_{\mathit{\text{serca}}}$, intersects the horizontal axis three times ($dc/dt=0$, Fig. 3B). The slope of the derivative $h\prime \left(c\right)$ at the three steady states shows that the low and high steady states are stable ($h\prime \left(c\right)<0$) while the middle steady state is unstable ($h\prime \left(c\right)>0$). When the fluxes representing Ca²⁺ release and reuptake from the ER are augmented by plasma membrane fluxes, $j_{pm}=j_{in}-j_{out}$, the changes in $c_{tot}$ cause periodic gain and loss of steady states via saddle-node bifurcations, which results in a relaxation oscillation that drives the sigh rhythm.

Figure 3.

Ca²⁺ fluxes in the closed-cell model (see Methods) in which $c_{tot}$ is treated as a parameter. A, fluxes associated with IP₃ receptor-mediated Ca²⁺ release ($j_{rel}$ blue) and SERCA pumps ($j_{\mathit{\text{serca}}}$ red) as functions of cytosolic Ca²⁺ concentration ($c$). B, phase diagram of the closed-cell model for three values of $c_{tot}$: 0.5, 1.0 μM, and 1.7 μM dotted, solid, and dashed lines respectively). Parameters are listed in Table 2.

Coupling of the inspiratory and sigh rhythms

The fast ($a,s,\theta$) and slow ($c,c_{tot}$) subsystems of the activity model are bi-directionally coupled such that inspiratory and sigh rhythms emerge from a single population. Inspiratory to sigh (fast to slow) coupling: episodic network activity ($a$) influences the dynamics of intracellular Ca²⁺ via the plasma membrane Ca²⁺ influx rate $j_{in}={ȷ}_0+{ȷ}_{a}a$. Sigh to inspiratory (slow to fast) coupling: cytosolic Ca²⁺ ($c$) activates I_CAN, modeled abstractly using a two-term network activity function, $a_{\infty }\left(x,c\right)=a_{\infty }^x\left(x\right)+a_{\infty }^c\left(c\right)$, where $a_{\infty }^x\left(x\right)={\left[1+e^{4\left({\gamma }_a-x\right)/k_a}\right]}^{-1}$ and $a_{\infty }^c\left(c\right)={\lambda }_c{\left[1+e^{4\left({\gamma }_c-c\right)/k_c}\right]}^{-1}$ depend on synaptic drive ($x=w\cdot s\cdot a-\theta$) and cytosolic Ca²⁺ ($c$), respectively. Figure 4 plots the network activity nullcline ($da/dt=0$) given by $s=\left[4\left({\gamma }_a+\theta \right)-k_{a}ln\left(1/\left[a-a_{\infty }^c\left(c\right)\right]-1\right)\right]/4wa$ for a range of cytosolic [Ca²⁺] ($0.05\le c\le 0.35\mu M$). Upon ER Ca²⁺ release and during the active phase of the Ca²⁺ oscillation, an upward shift of the $a$ nullcline (magenta) accounts for the increase in network activity mediated by the Ca²⁺-activated cationic current.

Figure 4.

The influence of cytosolic [Ca²⁺] ($c$) on the network activity nullcline ($\dot{a}=0$) given by $a=a_{\infty }\left(w\bullet s\bullet a-\theta ,c\right)$ for fixed $\theta$ and $c$. The synaptic depression nullcline ($\dot{s}=0$) given by $s=s_{\infty }\left(a\right)$ is shown in green. Parameters: $w=1,\theta =0.18-{\gamma }_a=0.48,{\lambda }_c=0.8,{\gamma }_c=0.35,k_c=0.05$ from Tables 1 and 2.

Spiking model of constituent preBötC neurons

We extended the activity model to include spiking and Ca²⁺ dynamics for $N$ distinct neurons with excitatory interactions mediated by a physiologically plausible network topology. The state of each neuron was modeled using $d{a}_i/dt=-a_i/{\tau }_a,d{s}_i/dt=\left[s_{\infty }\left(a_i\right)-s_i\right]/{\tau }_s,d{\theta }_i/dt=\left[{\theta }_{\infty }\left(a_i\right)-{\theta }_i\right]/{\tau }_{\theta }\left(a_i\right),d{c}_i/dt=j_i^{er}+j_i^{pm}$, and $d{c}_i^{tot}/dt=j_i^{pm}$, where $c_i^{er}=\left(c_i^{tot}-c_i\right)/\rho$ and $j_i^{pm}= {ȷ}_0+{ȷ}_a{a}_i+{ȷ}_x{x}_i-{v_{out}c}_i^4/\left({\kappa }_{out}^4+ c^4\right)$. The excitatory drive $x_i={\sum }_{j=1}^N{w}_{ij}{s}_{j }{a}_j$ in $j_i^{pm}$ was calculated as a sum over presynaptic neurons where ${ȷ}_a$ and ${ȷ}_x$ determine the relative contributions of pre- and postsynaptic Ca²⁺ influx, respectively. Spiking was implemented by incrementing each activity variable $a_i$ according to a time-inhomogeneous Poisson process with an instantaneous rate proportional to $a_{\infty }\left(x_i-{\theta }_i,c_i\right)=a_{\infty }^x\left(x_i-{\theta }_i\right)+a_{\infty }^c\left(c_i\right)$, with a maximum firing frequency of 100 Hz prior to release of ER Ca²⁺.

Network connectivity was determined by an $N\times N$ weighted adjacency matrix $W =\left(w_{ij}\right)$. The element $w_{ij}>0$ if neuron $i$ was postsynaptic to neuron $j$ and zero otherwise. Neurons were not self-excitatory ($w_{ii}=0$). The connectivity for each simulation was modeled as a random directed Erdös-Réyni graph with parameters $N=500$ and connection probability of $p=0.1$. The adjacency matrix of this graph was normalized to obtain a matrix of synaptic weights $W=\left(w_{ij}\right)$ with unit row and column sums (${\sum }_{i=1}^N{w}_{ij}={\sum }_{j=1}^N{w}_{ij}=1$). This ensured that the stochastic network dynamics did not depend on network size provided $N$ was sufficiently large. The population model is both stochastic and heterogeneous. The ${\alpha }_{\infty }\left(x\right)$ curve is sigmoidal, which represents heterogeneity of thresholds for action potential firing, i.e., rheobase is smoothed due to heterogeneity Spikes occur via a random Poisson process with rate proportional to $a_i\left(t\right)$. Connectivity is random, with each neuron projecting to 33 ± 5.5 others. Cellular heterogeneity is produced by assigning a range of neuronal excitabilities, governed by ${\gamma }_a$, the half-maximum of the dependence of steady-state discharge rate on synaptic drive, with mean −0.3 and standard deviation 0.05. Parameters for the spiking model are listed in Table 3.

Table 3.

Parameters for the spiking network model of inspiratory rhythmogenesis.

Symbol	Definition	Value	Units
$N$	network size (number of neurons)	500	-
$p$	probability that neuron $i$ excites neuron $j(i\ne j)$	0.1	-
${ȷ}_0$	background Ca2+ influx rate	0.05	μM/s
${ȷ}_a$	proportionality constant for presynaptic Ca2+ influx	0	μM/s
${ȷ}_x$	proportionality constant for postsynaptic Ca2+ influx	0.3	μM/s

Numerical simulations and data analysis

We used MATLAB 2021a and XPPAUT software to simulate and analyze ordinary differential equation models. Numerical integration was performed using Euler’s method with a time step of 0.01–0.25 ms in MATLAB. XPPAUT was used with default solver settings. The code and equations are in the public repository on Model DB (Accession No. 267252).

Vertical error bars show 95% confidence intervals in all figures, not standard deviation or standard error. We performed null hypothesis statistical tests including paired t-tests using two tails and linear regression. We report effect size as the difference of the means with 95% confidence intervals, that is, estimation statistics (Cumming & Calin-Jageman, 2017; Calin-Jageman & Cumming, 2019).

RESULTS

Inspiratory and sigh rhythms can be separately modulated

Sighs in vivo exhibit a biphasic pattern: there is a normal breath at tidal volume followed by an augmented second part of the breath (Cherniack et al., 1981; Li & Yackle, 2017). That biphasic pattern often manifests in slice preparations that retain the preBötC, remain rhythmically active in vitro, and generate breathing-related motor output via the hypoglossal (XII) nerve. preBötC field recordings in slices from Wistar and Sprague-Daley rats as well as CD-1 mice often display that biphasic pattern. However, sigh bursts measured in slices obtained from mouse strains with a C57BL/6 background typically register as long-duration augmented bursts or doublet bursts (Fig. 5A, B). Regardless of burst pattern, every sigh is followed by a pronounced post-sigh apnea, which aids in detection. Here, working with a C57BL/6 mouse strain, we detected augmented singlet or doublet bursts from preBötC field recordings and XII output – which were invariably followed by post-sigh apneas – and registered those events as fictive sighs (Fig. 5B, C).

Figure 5.

Slice preparations from neonatal mice generate inspiratory and sigh rhythms. A, rostral side view of a slice with characteristic landmarks including the 4th ventricle, XII nucleus (dorsal and medial), inferior olive (loop division), i.e., IO_loop. XII nerve root (ventral), shown with bounding boxes that approximate the borders of the preBötzinger complex (preBötC). B, preBötC field and XII nerve root recordings showing inspiratory rhythm and sighs ($\sigma$). The expanded box shows 1 s centered on the sigh in the recording bout shown. C, slow sweep speed recording of just XII nerve root emphasizing post-sigh apneas after each sigh burst ($\sigma$). The expanded box shows 16 sec roughly centered on the second sigh in the recording bout shown. The scale bar in panel A refers to 500 μm (A), 2 s (B), and 15 s (C).

We monitored inspiratory and sigh frequency via preBötC field recordings and XII motor output while manipulating the baseline membrane potential of preBötC neurons by changing artificial cerebrospinal fluid (aCSF) extracellular K⁺ concentration ([K⁺]_o) (Fig. 6A). Mean inspiratory frequency measured 0.139 ± 0.0441 Hz (N = 19) at 9 mM [K⁺]_o aCSF. Decreasing [K⁺]_o slowed inspiratory frequency incrementally such that at 3 mM [K⁺]_o it measured 0.00889 ± 0.0112 Hz (N = 9) or zero (N = 11) (Fig. 6B). These data align with previous studies showing that baseline membrane excitability governs inspiratory frequency (Johnson et al., 2001; Del Negro et al., 2009; Doi & Ramirez, 2010), and are consistent with either a recurrent excitation-based network oscillator as its core underlying mechanism (Kam et al., 2013b; Guerrier et al., 2015; Kallurkar et al., 2020; Ashhad & Feldman, 2020) or an excitatory network also incorporating I_NaP pacemaker properties (Koizumi & Smith, 2008; Ausborn et al., 2018; Phillips et al., 2019, 2022). We did not analyze the fraction of subthreshold ‘burstlets’, in which there is detectable activity at the field recording that does not result in a motor output burst (Kam et al., 2013a; Kallurkar et al., 2020), because the activity model incorporates burstlets within the preinspiratory phase of the cycle and does not produce standalone burstlets. Mean sigh frequency measured 0.663 ± 0.167 min⁻¹ (0.0111 ± 0.00278 Hz, N = 13) at 9 mM [K⁺]_o. Decreasing [K⁺]_o slowed the sigh frequency incrementally such that at 3 mM it measured 0.303 ± 0.130 min⁻¹ (or 0.00505 ± 0.00216 Hz, N = 7) (Fig. 6C). Sigh rhythm was 19-fold less sensitive to changes in [K⁺]_o than inspiratory rhythm (the relative frequency increase was a line with slope m = 21.683, which was significantly different from a slope of zero, p = 0.000358, r² = 0.94, N = 10) (Fig. 6D).

Figure 6.

modulation of inspiratory and sigh rhythms. A, preBötC field and XII nerve recordings of inspiratory and sigh rhythms in slices at 9 and 3 mM extracellular K⁺ concentration in the aCSF ([K⁺]_o). Sigh events are indicated by σ (also E). B and C, inspiratory (B) and sigh (C) frequencies plotted as a function of aCSF [K⁺]_o. Gray circles show individual slices; red squares show mean frequency with red vertical bars showing 95% confidence intervals (CIs) (N = 19). D, mean inspiratory frequency plotted versus corresponding mean sigh frequency for different aCSF [K⁺]_o. m = 18.46 indicates the slope of the regression line, which was non-zero by linear regression (p = 0.000358). E, preBötC field and XII nerve recordings of inspiratory and sigh rhythms in slices in 9 mM [K⁺]_o ACSF before and after neuromedin-B (NMB) application. F and G, sigh and inspiratory frequencies plotted at different NMB concentrations. Gray circles show individual slices; red squares show mean frequency with red vertical bars showing 95% CIs (N = 7). H, mean inspiratory frequency plotted versus corresponding mean sigh frequency for different NMB concentrations. $m\approx 0$ indicates the negligible slope of the regression line, which could not be distinguished from a slope of zero (p = 0.841).

Neuromedin B (NMB) is a bombesin-like peptide associated with sigh regulation (Li et al., 2016; Morgado-Valle et al., 2022). Sigh frequency increased from 0.578 ± 0.307 min⁻¹ (0.00963 ± 0.00511 Hz) in control to 1.00 ± 0.443 min⁻¹ (0.0167 ± 0.00739 Hz) following 10 nM NMB and to 1.37 ± 0.659 min⁻¹ (0.0228 ± 0.0110 Hz) following 30 nM NMB (Fig. 6G). The inspiratory frequency measured 0.165 ± 0.0554 Hz in control, and it remained unaffected by 10 nM NMB (0.182 ± 0.0640 Hz) or 30 nM NMB (0.170 ± 0.0528 Hz) (Fig. 1F). Comparing the changes in inspiratory vs. sigh frequency yielded a line with a negligible slope (the relative frequency change was a line with slope m = −0.0405, which was not significantly different from a slope of zero, p = 0.841, N = 7) (Fig. 6H), indicating that sigh frequency can be modulated with minimal impact on inspiratory frequency. We note that extremely high (50 nM) concentrations of bombesin accelerate inspiratory frequency (Morgado-Valle et al., 2022). These data suggest that inspiratory and sigh rhythms can be quasi-independently modulated and thus probably have different underlying rhythmogenic mechanisms.

Purinergic signaling is not necessary for sigh rhythm generation

It is possible that two discrete rhythmogenic mechanisms are embodied in different cell populations: one neural and the other non-neural. Astrocytes can generate intracellular Ca²⁺ oscillations and gliotransmission via purinergic P2 receptors modulates inspiratory preBötC rhythms (Huxtable et al., 2010; Okada et al., 2012; Rajani et al., 2016). Therefore, it is conceivable that astrocytes generate sigh rhythm and communicate it to preBötC neurons via purinergic P2 receptors. Specifically P2Y₁ receptors may be important for sigh rhythmogenesis in vitro (Severs et al., 2023).

We monitored inspiratory and sigh rhythms in preBötC neurons derived from progenitors that express the transcription factor Dbx1 (hereafter: Dbx1 neurons), which comprise the inspiratory rhythmogenic preBötC core (Bouvier et al., 2010; Gray et al., 2010; Wang et al., 2014; Vann et al., 2016, 2018; Koizumi et al., 2016; Cui et al., 2016). We crossed Dbx1^Cre mice (Bielle et al., 2005) with Ai148 reporter mice that express membrane-bound GCaMP6f in a Cre-dependent manner (Daigle et al., 2018). Their offspring express GCaMP6f in Dbx1-derived cells (Fig. 7A). Multi-photon imaging of Dbx1;Ai148 mouse slices produced measurable Ca²⁺ transients simultaneously recorded with XII motor output (Figs. 7B and 8). Although a dominant fraction of Dbx1-derived cells are astrocytes (Kottick et al., 2017), we are confident that we analyzed Dbx1 neuronal activity because: recorded-cell diameters equaled or exceeded 25 μm, characteristic of neurons, whereas astrocyte cell bodies typically measure ~10 μm in diameter; Cre-dependent fluorophore expression in astrocytes mostly resides in the processes, producing diffuse fluorescence that is normalized during background correction; and astrocyte Ca²⁺ transients typically perdure for several inspiratory cycles and thus do not synchronize with the inspiration (Härtel et al., 2009) but see (Okada et al., 2012). We computed cycle-triggered averages (CTAs) of inspiratory XII bursts and Ca²⁺ transients to substantiate the XII-synchronized rhythmicity of Dbx1 preBötC neurons even when fluorescent signal-to-noise ratio was minimal (Fig. 7B, cta). Neither a cocktail of P2 receptor antagonists (50 μM PPADS, 50 μM suramin, 10 μM TNP-ATP, 10 μM MRS2179, and 10 μM MRS2578, N =4) (Fig. 7B) nor the highly selective P2Y₁ receptor antagonist 20 μM MRS2279 (N = 4) (Fig. 8) modified the frequency of inspiratory or sigh rhythms, so we pooled the data. Inspiratory rhythm measured 0.239 ± 0.0610 Hz in control vs. 0.263 ± 0.0639 Hz after P2 receptor blockade (p = 0.288, N = 8) (Fig. 7C). Sigh rhythm measured 0.798 ± 0.338 min⁻¹ (0.00399 ± 0.00102 Hz) in control vs. 0.684 ± 0.229 min⁻¹ (0.00438 ± 0.00107 Hz) after P2 receptor blockade (p = 0.398, N = 8) (Fig. 7D). Blocking purinergic P2 receptor-mediated signaling did not perturb the sigh rhythm, diminishing the likelihood that purinergic transmission is obligatory and that astrocytes are sigh rhythmogenic, which does not preclude a regulatory role for astrocytes.

Figure 8.

Effect of specific P2Y1 receptor antagonist MRS2279 on inspiratory and sigh rhythms. Ca²⁺ fluorescence changes in Dbx1 preBötC neurons with XII motor output (red). Three conditions are shown: control, 20 μM MRS2279, and washout. The 40% calibration bar refers to fluorescence changes (ΔF) with respect to baseline fluorescence (F₀), i.e., ΔF/F₀, and it applies to all cellular imaging traces. This experiment was performed in N = 4 slices, group data in Fig. 7C and D.

Sigh and inspiratory rhythms arise from the same neuron population

We next assessed whether there are separate neuron populations with independent inspiratory and sigh rhythmicity in vitro. The probability of finding discrete rhythmic populations was low (Lieske et al., 2000; Lieske & Ramirez, 2006b, 2006a; Ruangkittisakul et al., 2008; Tryba et al., 2008; Viemari et al., 2013). Nevertheless, one modeling study attributes inspiratory and sigh mechanisms to a single population (Jasinski et al., 2013) whereas a contemporary study assigns inspiratory and sigh mechanisms to separate populations (Toporikova et al., 2015).

We monitored inspiratory and sigh rhythms in glutamatergic Dbx1-derived preBötC neurons using multiphoton imaging of GCaMP6f-expressing neurons while monitoring XII output in Dbx1;Ai148 slices. 208 of 209 Dbx1 preBötC neurons from nine slices were active during both inspiratory and sigh bursts; one Dbx1 preBötC neuron was active only during sigh bursts. Our observations, and the existing literature, provide no new evidence that would support the existence of dichotomous inspiratory and sigh-dedicated neuronal populations. Rather, the most parsimonious interpretation of these data aligns with the consensus that both rhythms emerge from the same neuronal population, which is glutamatergic and derived from Dbx1-expressing precursors (Bouvier et al., 2010; Gray et al., 2010; Picardo et al., 2013; Wang et al., 2014; Koizumi et al., 2016; Cui et al., 2016; Ramirez & Baertsch, 2018).

An excitatory network oscillator generates inspiratory rhythm

We model the essential dynamics of inspiration using a three-tuple of state variables $\left(a, s, \theta \right)$ (Tabak & Rinzel, 2005). $a$ reflects aggregate activity of the preBötC core, whose interconnected constituent rhythmogenic neurons produce regenerative synaptic excitation during the interburst interval that leads to inspiratory burst generation (Wallén-Mackenzie et al., 2006; Carroll & Ramirez, 2013; Kam et al., 2013b, 2013a; Ashhad & Feldman, 2020).

Inspiratory bursts terminate due to synaptic depression, outward currents recruited by neural activity (Del Negro et al., 2009; Krey et al., 2010; Guerrier et al., 2015; Kottick & Del Negro, 2015), and slow (~500 ms) inactivation of a persistent Na⁺ current (Butera et al., 1999; Del Negro et al., 2002a; Ptak et al., 2005; Pace et al., 2007a; Yamanishi et al., 2018). State variable $s$ captures the collective dynamics of these conjoint pre- and postsynaptic refractory mechanisms. The preBötC core remains refractory following burst termination (during the phase postinspiration); exogenous stimulation cannot generate a subsequent inspiratory burst (Kottick & Del Negro, 2015; Baertsch et al., 2018). The existence of an absolute refractory period is recognized in both reduced preparations in vitro as well as intact rodents in vivo (Alsahafi et al., 2015; Baertsch et al., 2018; Vann et al., 2018).

Following the refractory period, which overlaps with postinspiration, a burst or breath can be evoked exogenously, or it may occur spontaneously (as shown in Fig. 9A). The spontaneous rhythm in vitro shows no correlation between the interburst interval duration (T_n) and the magnitude of the subsequent inspiratory burst (A_n+1) (Fig. 9B upper). These data indicate that recovery from refractoriness does not govern forthcoming bursts. Then what does influence the magnitude of the impending burst if not recovery from refractoriness? Another key observation provides a clue: the distribution of interburst intervals is Gaussian (i.e., normal, Fig. 9B lower). These data suggest that the forthcoming burst is governed by a deterministic (i.e., time-dependent) process subject to noise fluctuations. However, the dynamics must be different than those of $s$ because we know $s$ recovers prior to the next burst. Therefore, we introduced a new state variable ($\theta$) that reflects broadly defined cellular adaptation, which amalgamates pre- and postsynaptic processes distinct from those incorporated in $s$ (see Discussion for physiological interpretations of cellular adaptation $\theta$).

Figure 9.

Inspiratory rhythm generation in the activity model. A, inspiratory rhythm in vitro showing preBötC field and XII recordings, illustrating cycle period (T_n) and subsequent burst area (A_n+1). B, plot of A_n+1 vs. T_n (upper) and a histogram of T_n (lower) for the experiment. C, inspiratory rhythm in the model showing the three-tuple of state variables ($a, s, \theta$) for the inspiratory subsystem in a time series and a $(s,\theta )$ phase plane. D, plot of A_n+1 vs. T_n (upper) and a histogram of T_n (lower) for the model.

Figure 9C shows a time series of spontaneous inspiratory rhythm in the model with baseline noise. Inspiratory bursts are reflected by peaks in $a$. Studying the trajectory in $\left(s,\theta \right)$ phase space illustrates that $s$ decreases during the burst (synapses depress, outward currents activate, persistent inward currents inactivate) faster than $\theta$ increases (cellular adaptation sets in) (Fig. 9C, clockwise trajectory). The refractory period begins when $s$ is at a minimum and $\theta$ is at a maximum. During the interburst interval, $s$ recovers faster than $\theta$ declines. The system becomes burst-capable as $s$ recovers, but the next burst occurs spontaneously only after $\theta$ declines sufficiently. The inspiratory rhythm model recapitulates the lack of a relationship between cycle period (T_n) and magnitude of the subsequent inspiratory burst (A_n+1) as well as normally distributed cycle times (Fig. 9D). Absent $\theta$, the model remains rhythmic, but T_n governs A_n+1, which does not match the experimental data. The $\left(a, s, \theta \right)$ system encapsulates preBötC inspiratory burst dynamics such that it is unnecessary to explicitly model each constituent neuron to examine the mechanisms underlying two rhythms in one population.

Intracellular Ca²⁺ oscillations produce sigh rhythm

Sigh rhythm is weakly voltage-dependent and Dbx1 preBötC neurons show sigh-related Ca²⁺ transients that do not depend on gliotransmission (Figs. 6–8), which suggests that neuronal Ca²⁺ oscillations produce sigh bursts. An existing model of sigh-related embryonic Ca²⁺ oscillations depends on I_h (Toporikova et al., 2015). We tested the veracity of that mechanism in the postnatal context by applying the selective I_h blocker ZD7288 (50 μM) to postnatal slices, which had no consistent or statistically significant effect on sigh frequency (0.572 ± 0.336 min⁻¹ [0.00954 ± 0.00560 Hz] in control vs 0.732 ± 0.322 min⁻¹ [0.0121 ± 0.00537 Hz] in ZD7288, paired t-test p = 0.145, N = 11). Those data suggest that postnatal sigh rhythmogenesis does not depend on I_h (Fig. 10A, B).

Figure 10.

Effects of ZD7288 on inspiratory and sigh rhythms. A, preBötC field and XII nerve recordings in slices. Sigh events are indicated by σ. B, mean sigh and inspiratory frequency for each slice tested in control, ZD7288, and washout (N = 11). Open circles (gray) reflect individual slices. Filled circles (red) show the mean frequencies. Vertical bars (red) show 95% confidence intervals (CIs) for the means. Black dotted horizontal lines demarcate the means of control and ZD7288 conditions. The rightmost plot shows the effect size, which includes the mean change (Δ mean on the y-axis) for each individual slice (gray) and for the group (red) with 95% CIs. Note: when the 95% CIs extend beyond the boundaries for the mean values in control and experimental conditions (ZD7288), then the statistical test is not significant (NS) at $\alpha <0.05$.

Note, blocking I_h either does not affect (Mironov et al., 2000; Burgraff et al., 2022) or moderately speeds up (Thoby-Brisson et al., 2000) inspiratory burst frequency in slices from postnatal mice. Here, ZD7288 did not affect inspiratory rhythm (0.194 ± 0.0764 Hz in control vs 0.185 ± 0.0452 Hz in ZD7288, paired t-test p = 0.624, N = 11) (Fig. 10B).

We formulated a mathematical model wherein the sigh rhythm emanates from a single glutamatergic preBötC neuron population without obligatory roles for I_h or synaptic inhibition (Lieske et al., 2000; Borrus et al., 2020). The sigh subsystem is a two-tuple of state variables, $(c,c_{tot})$, which tracks cytosolic and total Ca²⁺ in constituent neurons where $c$ is cytosolic Ca²⁺ and $c_{tot}$ is the sum of $c$ and Ca²⁺ stored in the ER ($c_{er}$). See (De Young & Keizer, 1992; Keizer et al., 1995; Friel & Chiel, 2008) for origins and review of intracellular Ca²⁺ dynamics modeled in this fashion.

Figure 11A shows a schematic of the sigh subsystem. (Because the inspiratory subsystem is modeled with abstract variables – activity ($a$), refractoriness ($s$), and adaptation ($\theta$) – that subsystem is difficult to render in a biophysically meaningful schematic.) Inspiration periodically activates voltage-gated Ca²⁺ currents yet cytosolic Ca²⁺ ($c$) increases only minimally during each cycle because the ER sequesters most of the entering Ca²⁺ via sarco-endoplasmic reticulum Ca²⁺ ATPase (SERCA) pumps (Keizer et al., 1995; Friel & Chiel, 2008). $c_{tot}$ increases in step with inspiratory rhythm as the ER fills up (Fig. 11B). In $\left(c,c_{tot}\right)$ phase space, the trajectory follows the left (low $c$) branch of the $c$ nullcline. When the system reaches its left knee (Fig. 11C, ✱) the replete ER releases Ca²⁺ and the trajectory moves rightward rapidly to the high $c$ branch of the $c$ nullcline (upper purple trajectory). The resulting increase in $c$ evokes Ca²⁺-activated nonspecific cation current (I_CAN), which is ubiquitously expressed in preBötC neurons (Pace et al., 2007b; Koizumi et al., 2018; Picardo et al., 2019). I_CAN activation underlies the sigh burst, which manifests as a prominent peak in the $a$ time series. During the sigh burst, plasma membrane Ca²⁺ ATPase (PMCA) pumps extrude cytosolic Ca²⁺; $c$ and $c_{tot}$ both decrease (Fig. 11C, lower purple trace), and the system returns to the left (low $c$) branch of the $c$ nullcline. The extraordinary activity $a$ during a sigh burst decreases $s$ below the nadir, and increases $\theta$ beyond the peak, that these variables typically reach during inspiratory cycles (Fig. 11B). $s$ and $\theta$ take longer to recover from their extrema, which explains the post-sigh apnea, i.e., the prolonged interval that follows a sigh burst.

Figure 11.

Inspiratory and sigh rhythms in the full model. A, schematic diagram of the Ca²⁺ subsystem showing the critical channels, pumps, and intracellular stores. I_Ca(leak) corresponds to ${ȷ}_0$ and I_CaV corresponds to ${ȷ}_{a}a$ in the model. I_CAN is built in the function $a_{\infty }\left(x\right)$. Dashed arrows show Ca²⁺ fluxes. PMCA and SERCA are defined in the main text. From the model, $\frac{v_{out}{c}^4}{\left({\kappa }_{out}^4+ c^4\right)}$ and $\frac{v_{\mathit{\text{serca}}}{c}^2}{\left({\kappa }_{\mathit{\text{serca}}}^2+c^2\right)}$ reflect PMCA and SERC fluxes, respectively. IP₃R flux is a given by $j_{er}$ as described in Methods (see Calcium handling and the sigh rhythm). B, time series of state variables $(a,s,\theta ,c,c_{tot})$. C, sigh trajectory in $(c,c_{tot})$ phase space showing $c$ and $c_{tot}$ nullclines; ✱ marks onset of the sigh burst. Color coding in C matches time series trajectories in B. D and E, inspiratory (D) and sigh (E) frequencies for different levels of baseline excitability (${\gamma }_a$, D) or different maximum IP₃R conductance ($v_{ip3r}$, E). Gray squares show single simulations; red lines show a linear regression with slopes given by m.

The $(c,c_{tot})$ system represents Ca²⁺ dynamics within a representative neuron. Since our model tracks aggregate activity $a$, it assumes that intracellular Ca²⁺ oscillations are synchronized across the network. To validate the assumption that intracellular Ca²⁺ dynamics will synchronize, we simulated 500 discrete neurons, which are heterogeneous and possess action potential-generating capabilities as well as internal Ca²⁺ dynamics as described in Methods (see Spiking model of constituent preBötC neurons) (Fig. 12). We coupled them using generic non-NMDA-like ionotropic glutamate receptor-mediated synapses and randomized initial $c_{tot}$ for each neuron using a normal distribution ($\mu = 1, \sigma = 0.05)$. Excitatory synaptic interactions suffice to synchronize the Ca²⁺ oscillations of the constituent neurons (Fig. 12). The population model with 500 discrete units was only used to test the synchronization of intracellular Ca²⁺ oscillations; the activity model was used for all other tests, including those in the next two subsections.

Figure 12.

Network activity and dynamics of intracellular [Ca²⁺] for a spiking model of inspiratory and sigh rhythms. The network consists of 500 neurons and is randomly interconnected with 6.5% probability. Time series traces (blue) show average network values of activity rate ($a$), cytosolic Ca²⁺ ($c$), and total Ca²⁺ ($c_{tot}$). The x-axis is the same for all plots, see 50 s calibration bar (bottom). Boxed-in plots show the individual activity rate ($a$), cytosolic Ca²⁺ ($c$), and total Ca²⁺ ($c_{tot}$) across the 500 neurons, pseudo-color scaling is shown (right). Note: this network of 500 discrete neurons was only employed in this figure, all other simulations and analyses employed the activity model.

Inspiratory and sigh rhythms can be separately modulated in the model

We tested whether the model of inspiratory and sigh rhythms could recapitulate the effects of changing [K⁺]_o or adding bombesin-like peptides that together suggest a dissociation of inspiratory and sigh rhythmogenic mechanisms. The change in cellular excitability that results from manipulating [K⁺]_o in the aCSF in vitro could be modeled by changing ${\gamma }_a$, the input-output function attributable to a leak current that determines how close baseline membrane potentials are to the spike threshold. Inspiratory model frequency ranged from quiescence to ~0.25 Hz as ${\gamma }_a$ varied from 0.1 to 0.4 (akin to varying [K⁺]_o 3 to 9 mM) whereas sigh frequency in the model ranged from 0.38 to 1.08 min^–1 (i.e., 0.006–0.018 Hz). Inspiratory rhythm is 19-fold more sensitive to changes in excitability than sigh rhythm, consistent with experiment (compare Figs. 11D and 6D).

We simulated the effects of bombesin-like peptides at the final stage of their Gq-linked signaling cascade (Ramos-Álvarez et al., 2015) by increasing the inositol 1,4,5-trisphosphate (IP₃) receptor release rate via parameter $v_{ip3r}$. Doing so accelerated sigh frequency from 0.6 min^–1 (0.01 Hz) to 10.8 min^–1 (0.18 Hz) without affecting the inspiratory frequency, akin to the NMB experiments (compare Figs. 11E to 6H). The model reproduces separate mechanisms that independently modulate inspiratory and sigh rhythms.

Having benchmarked the model to reproduce inspiratory and sigh frequency modulation by changes in cellular excitability, we next turned our attention to novel non-intuitive predictions regarding sigh frequency modulation via intracellular signaling pathways.

Disrupting SERCA pumps increases the frequency of sigh rhythm and then stops it

Partially blocking SERCA pumps (i.e., decreasing $v_{\mathit{\text{SERCA}}}$ 50% from 60 to 30 s⁻¹) counterintuitively increased sigh frequency but decreased sigh magnitude (Fig. 13A). Both effects are a consequence of how SERCA activity influences the $c_{tot}$ threshold that compels ER Ca²⁺ release, marked by the left knee of the $c$ nullcline (see Fig. 11C ✱). Sigh frequency is determined by the time required to refill the ER, which depends on the vertical separation between the knees of the $c$ nullcline. Reducing $v_{\mathit{\text{SERCA}}}$ by 50% decreases their vertical separation and thus speeds-up sigh rhythm (Fig. 13B). Further reducing $v_{\mathit{\text{SERCA}}}$ by 80% to 12 s⁻¹ stops sigh rhythmogenesis because the left knee of the $c$ nullcline crosses the vertical $c_{tot}$ nullcline, which now intersects the $c$ nullcline on its positively sloped left branch, i.e., oscillations cease via a Hopf bifurcation (Fig. 13B and C, hb point).

Figure 13.

Role of SERCA pumps in sigh rhythm. A, attenuation of SERCA pumps in the model speeds-up or stops sigh rhythms. Filled circles on the $\int a$ axis show magnitude (area) of $a$ for each burst. B, model behavior in the $(c,c_{tot})$ phase plane for corresponding values of $v_{\mathit{\text{serca}}}$. Magenta shows the $c_{tot}$ nullcline; cyan shows the $c$ nullcline. Shaded areas indicate limit cycles (direction clockwise); filled circle indicates a stable steady state (no oscillations). C, bifurcation diagrams for $c_{tot},c$, and cycle period (T) versus the parameter $v_{\mathit{\text{serca}}}$ plotted logarithmically in descending order from 100 to 9 μM-s⁻¹. HB indicates a Hopf bifurcation point. $v_{\mathit{\text{serca}}}$ values from the time series (A) and phase planes (B) are indicated on the abscissa. D, preBötC field and XII nerve recordings before and after application of 10 μM thapsigargin. Sighs are denoted by σ. E and F, mean sigh frequency (E) and amplitude (F) for each slice tested in control, 10 μM thapsigargin, and washout (N = 11 total). Open circles (gray) reflect individual slices. Filled circles (red) show the mean frequency (E) and amplitude (F). Vertical bars (red) show 95% confidence intervals (CIs) for the means. Black dotted horizontal lines demarcate the means of control and 10 μM thapsigargin conditions. The rightmost plots show the effect size, which includes the mean change (Δ mean on the y-axis) for each individual slice (gray) and for the group (red) with 95% CIs. Note: the 95% CIs do not extend beyond the boundaries for the mean values in control and experimental conditions (10 μM thapsigargin), which indicates that the statistical tests are significant at $\alpha <0.05$, in this case p = 0.00902 (sigh frequency, N = 11) and p = 0.0294 (sigh amplitude, N = 5). G, XII nerve recording during local preBötC application of 100 μM thapsigargin. H, mean sigh frequency for each slice tested in control, 100 μM thapsigargin, and washout. The rightmost plots show the effect size, which includes the mean change (Δ means on the y-axis) for each individual slice (gray) and for the group (red) with 95% CIs (N = 4). Note: the 95% CIs do not extend beyond the boundaries for the mean values in control and experimental conditions (100 μM thapsigargin), which indicate that the statistical test is significant at $\alpha <0.05$, in this case p = 0.00335.

To examine these effects experimentally, we applied 10 μM thapsigargin via the bath to partially block SERCA pumps while performing preBötC field and XII nerve root recordings. There was no effect on inspiratory rhythm (0.145 ± 0.060 Hz in control vs. 0.142 ± 0.052 Hz in 10 μM thapsigargin, p = 0.871, N = 11). Sigh frequency increased from 0.672 ± 0.454 min⁻¹ (0.0112 ± 0.00757 Hz) in control to 1.11 ± 0.879 min⁻¹ (0.0185 ± 0.0147 Hz) in 10 μM thapsigargin (paired t-test, p = 0.00902, N = 11). Sigh burst magnitude in preBötC field recordings decreased by 32% (24 ± 13 mV-s in control vs. 16 ± 11 mV-s in thapsigargin, paired t-test, p = 0.0294, N = 5) (Fig. 13D–F). The effects on sigh frequency (0.75 ± 0.56 min⁻¹, i.e., 0.0125 ± 0.00933 Hz) and magnitude (19 ± 11 mV-s) were reversible. These experimental results matched the model predictions for partial SERCA blockade.

Next, we fully blocked SERCA pumps by injecting a high dose of thapsigargin (100 μM) bilaterally into the preBötC. Local microinjection was important in this context because neurons outside of preBötC that are retained in slices, like the raphé obscurus, are tonically active and help maintain preBötC excitability (Ptak et al., 2009). By microinjecting locally in the preBötC, we avoided what would have been widespread SERCA blockade that could have impacted the preBötC rhythmogenic network via indirect effects. The drawback was that local injection precluded simultaneous preBötC field potential recording. Therefore, we measured XII motor output only and focused on the effects of 100 μM thapsigargin on sigh frequency since premotor neurons postsynaptic to the preBötC core may have masked or filtered preBötC activity and prevented accurate measurement of sigh magnitude. To establish that our injection micropipettes correctly targeted the preBötC we first injected a bolus of high potassium aCSF into the preBötC bilaterally. Potassium rapidly and reversibly increased inspiratory frequency whereas 1% DMSO aCSF, the vehicle, had no effect (Fig. 14). Thapsigargin (100 μM) had no effect on inspiratory rhythm (0.182 ± 0.0681 Hz in control vs. 0.224 ± 0.0644 Hz in 100 μM thapsigargin, p = 0.0905, N = 4) but it stopped the sigh rhythm (N = 3) or decreased it by 81% from 1.515 ± 0.257 min⁻¹ (0.0253 ± 0.00428 Hz) in control to 0.254 min⁻¹ (0.00423 Hz) (N = 1) (Fig. 13G, H), which recovered in washout (1.180 ± 0.0375 min⁻¹, i.e., 0.0196 ± 0.000625 Hz).

Figure 14.

Validation of bilateral microinjection methods. Bilateral microinjection of [K⁺] into the preBötC transiently increases respiratory frequency, which recovers fully in several minutes. Bilateral microinjection of the vehicle dimethyl sulfoxide (DMSO) is without effect.

Blocking IP₃ receptors diminishes the frequency of sigh rhythm and then stops it

Attenuating IP₃ receptor-mediated Ca²⁺-induced Ca²⁺ release rate ($v_{ip3r}$) decelerates or stops the sigh rhythm (Fig. 15A). Decreasing $v_{ip3r}$ ≤40% elevates the critical $c_{tot}$ that compels ER Ca²⁺ release (see Fig. 11C ✱), which enhances the vertical separation between the knees of the $c$ nullcline and slows-down sigh rhythm (Fig. 15A, B). $v_{ip3r}$ reduction did not affect sigh burst magnitude as indicated by the $\int a$ in Fig. 15A. Decreasing $v_{ip3r}$ ≥50% raises the left knee of the $c$ nullcline high enough to cross the vertical $c_{tot}$ nullcline, which now intersects the $c$ nullcline on its positively sloped left branch, which causes oscillations to cease via a Hopf bifurcation (Fig. 15B, C).

Figure 15.

Role of IP₃ receptors in sigh rhythm. A, attenuation of IP₃ receptors in the model slows or stops sigh rhythms. Filled circles on the $\int a$ axis show magnitude (area) of $a$ for each burst. B, model behavior in the $(c,c_{tot})$ phase plane for corresponding values of $v_{ip3r}$. Magenta shows the $c_{tot}$ nullcline; cyan shows the $c$ nullcline. Shaded areas indicate limit cycles (direction clockwise); filled circle indicates a stable steady state (no oscillations). C, bifurcation diagrams for $c_{tot},c$, and cycle period (T) versus the parameter $v_{ip3r}$ plotted logarithmically in descending order from 25 to 5 s⁻¹. HB indicates a Hopf bifurcation point. $v_{\mathit{\text{serca}}}$ values matching the time series (A) and phase planes (B) are indicated on the abscissa. D, XII nerve recordings before and after application of 1 μM xestospongin. Sighs are denoted by σ. E, mean sigh frequency for each slice tested in control, 1 μM xestospongin, and washout. The rightmost plots show the effect size, which includes the mean change (Δ means on the y-axis) for each individual slice (gray) and for the group (red) with 95% confidence intervals (CIs) (N = 5). Note: the 95% CIs do not extend beyond the boundaries for the mean values in control and experimental conditions (1 μM xestospongin), which indicate that the statistical test is significant at $\alpha <0.05$, in this case p = 0.0359.

We tested this model prediction in slices by attenuating IP₃ receptors using xestospongin (Gafni et al., 1997; De Smet et al., 1999), injected bilaterally into the preBötC. It was impracticable to calibrate xestospongin dose to mimic 40% vs. 50% attenuation of $v_{ip3r}$. Xestospongin (1 μM) decreased sigh frequency from 1.22 ± 0.456 min⁻¹ (0.0203 ± 0.00760 Hz) to 0.35 ± 0.21 min⁻¹ (0.00583 ± 0.0035 Hz) (N = 5) or stopped it altogether (N = 1). The effect was reversible; sigh frequency returned to 0.87 ± 0.40 min⁻¹ (0.00145 ± 0.00667 Hz) (Fig. 15E). Xestospongin increased inspiratory frequency from 0.208 ± 0.133 Hz to 0.251 ± 0.142 Hz (p = 0.00992, N = 6), which recovered to 0.232 ± 0.120 Hz in washout. The xestospongin-induced acceleration of the inspiratory frequency was not predicted by the model.

These results (Figs. 13 and 15) reinforce the validity of our hypothesized role for SERCA pumps and IP₃ receptors in generating a sigh rhythm via intracellular Ca²⁺ storage and release mechanisms.

DISCUSSION

Eupnea and sigh rhythms both come from the preBötC but from which cell population(s)? Furthermore, could the slower sigh rhythm, lacking voltage dependence, depend on glia? How are the rhythms coupled? Here we elucidate these questions via modeling and tests of model predictions, which together constitute evidence that both rhythms emanate from an excitatory neuronal population in the preBötC postnatally.

We formulated a minimal model of inspiratory rhythmogenesis based on emergent network properties (Del Negro et al., 2002b, 2005; Carroll & Ramirez, 2013; Kam et al., 2013b; Kallurkar et al., 2020; Ashhad & Feldman, 2020; Ashhad et al., 2023; da Silva et al., 2023). That model is consistent with contemporary views of preBötC function, namely that it is fundamentally a network oscillator (Grillner, 2006; Grillner & El Manira, 2020), which additionally features a subset of neurons with bursting-pacemaker properties that can augment recurrent excitation leading to burst generation (Smith et al., 2000; Ausborn et al., 2018; Phillips et al., 2019, 2022) particularly in the context of development (da Silva et al., 2023). Our minimal activity model is coarse-grained regarding cellular and synaptic properties; it does not explicitly model cellular pacemakers or any other type of intrinsic membrane property. Rather, the state variable $a$ aggregates recurrent excitation into a single dynamic process, which is the same whether constituent neurons possess intrinsic burst-generating currents, or they do not. Neither can the activity model address ‘burstlet’ theory, wherein a subset of preBötC neurons progressively synchronize and generate miniature bursts, dubbed ‘burstlets’, which can amplify to become full bursts and produce motor output via recruitment of a pattern-related subset of preBötC neurons. When preBötC excitability is sufficiently high, it almost exclusively produces bursts, and the activity of a burstlet is subsumed in the preinspiratory phase of the cycle (Kam et al., 2013a; Kallurkar et al., 2020; Ashhad & Feldman, 2020; Ashhad et al., 2023). Those burstlet-to-burst dynamics are fully incorporated in the preinspiratory phase of recurrent excitation dynamics of state variable $a$. By exploiting a simplified activity model of inspiratory rhythmicity, we were better able to focus on how a single preBötC population generates two rhythms with disparate time scales (the second rhythm underlying sighs).

We conclude that glia and gliotransmission are not mandatory for sigh rhythmogenesis because attenuating purinergic signaling, the dominant means by which astrocytes interact with preBötC neurons (Huxtable et al., 2010; Okada et al., 2012; Rajani et al., 2016), did not preclude or modify sigh rhythmogenesis. In another experimental context, blocking P2Y₁ receptors stopped the sigh rhythm consistently in vitro but it did not stop sigh breathing in more than half of the adult mice directly injected with a high (500 μM) dose of MRS2279 in the preBötC (Severs et al., 2023). Our conclusion is not incompatible – purinergic gliotransmission does not appear to be necessary for sigh behavior – but our data in vitro are incongruous. The disparity may be attributable to which populations were monitored. We recorded Dbx1 preBötC neurons from the rostral surface of slices. In contrast, Severs and colleagues (Severs et al., 2023) performed field recordings from the caudal surface of slices, which may preferentially reflect the caudal ventral respiratory group (rather than preBötC) containing phrenic premotor neurons (Ellenberger & Feldman, 1988, 1990; Dobbins & Feldman, 1994; Wu et al., 2017). Therefore, cessation of sigh rhythm from the caudal surface of their slices suggests that P2/P2Y₁ receptor-mediated signaling might be critical for premotor transmission of sigh bursts while remaining dispensable from the standpoint of sigh rhythmogenesis.

Our conclusion that glia are not sigh rhythmogenic assumes that their signaling is purely purinergic. However, astrocytes in the trigeminal system facilitate oromotor rhythmogenesis via paracrine transmission (Morquette et al., 2015). In that system, S100$\beta$ secretion by glia neither generates nor synchronizes oscillations but rather modulates I_NaP-mediated bursting-pacemaker properties in rhythmogenic trigeminal interneurons. That mechanism is unlikely to apply to sigh rhythmogenesis because neurons with I_NaP bursting-pacemaker properties in the preBötC oscillate much faster than the sigh rhythm (Del Negro et al., 2002b, 2005; Peña et al., 2004; Koizumi & Smith, 2008).

Neuroglial interactions potently regulate sigh activity (Severs et al., 2023), even though they do not appear to be at the core of the sigh rhythmogenic mechanism. Given that sighs mediate hypoxic arousal (Severs et al., 2022), it makes sense that sigh activity can be regulated by astrocytes that play a key role centrally in oxygen and carbon dioxide detection and communicate via purinergic transmission (Gourine et al., 2010; Angelova et al., 2015; Rajani et al., 2017; Gourine & Dale, 2022).

Regarding sigh rhythmogenic neurons, the consensus is that sigh events emerge from neurons that belong to the same, or extensively overlapping, preBötC network(s) (Lieske et al., 2000; Lieske & Ramirez, 2006b, 2006a; Ruangkittisakul et al., 2008; Tryba et al., 2008; Viemari et al., 2013). Our present data support the consensus because we imaged neurons that comprise the inspiratory core oscillator (Bouvier et al., 2010; Gray et al., 2010; Wang et al., 2014; Vann et al., 2016, 2018; Koizumi et al., 2016; Cui et al., 2016), which were universally active during inspiratory and sigh bursts.

Another model-based proposal for sigh rhythmogenesis posits that raising excitability among heterogeneous voltage-dependent pacemaker-like neurons in the preBötC gives rise to multiperiodic solutions characterized by large sigh-like population bursts interspersed with sequences of small-amplitude population bursts (Bacak et al., 2016). We reject this explanation because: 1) it does not involve intracellular Ca²⁺ whose dynamics and signaling mechanisms are linked to sigh generation and regulation (Lieske et al., 2000; Lieske & Ramirez, 2006b, 2006a; Toporikova et al., 2015; Ramos-Álvarez et al., 2015; Li et al., 2016; Morgado-Valle et al., 2022); 2) sigh frequencies in the model were on the order of 0.5–0.2 Hz, which is more than an order of magnitude faster than actual sigh rhythms recorded in vitro or in vivo; and 3) putative sighs in the Bacak et al. model require elevated excitability but we, and others, e.g. (Ruangkittisakul et al., 2008), demonstrate sigh rhythmogenesis at low levels of preBötC excitability in vitro.

Our model and our framework for analysis places the sigh-generating mechanism at the signaling interface of the ER and the plasma membrane: the ER-dominated Ca²⁺ oscillation periodically releases Ca²⁺ to the cytosol, which activates I_CAN at plasma membrane to produce the sigh burst. Another model-based viewpoint posits that ER Ca²⁺ dynamics are instead involved in turning subthreshold burstlets into full magnitude inspiratory bursts via ER Ca²⁺ release that also periodically activates I_CAN (Phillips & Rubin, 2022). Given the experimental support for recurrent synaptic excitation and emergent network properties that underlie burstlets reaching threshold to evoke bursts (Kam et al., 2013a; Kallurkar et al., 2020; Ashhad & Feldman, 2020; Ashhad et al., 2023) we favor the view that ER Ca²⁺ dynamics underlie sigh bursts rather than burstlet-to-burst transitions.

Both experiment and modeling suggest that inspiratory and sigh rhythms can coexist because they use separate cellular tool kits. A canonical network oscillator produces inspiratory bursts with a cycle period on the order of seconds. The activity variable $a$ captures recurrent excitation dynamics (Tabak & Rinzel, 2005) that lead to burst generation (Carroll & Ramirez, 2013; Kam et al., 2013b, 2013a; Kallurkar et al., 2020; Ashhad & Feldman, 2020). Refractory variable $s$ encapsulates presynaptic factors like synaptic depression (Guerrier et al., 2015; Kottick & Del Negro, 2015) and postsynaptic factors like Na/K ATPase pumps (which produce electrogenic outward current), ATP-dependent K⁺ current, Na⁺-dependent K⁺ current, which all activate in concert with neural activity, as well as I_NaP inactivation during inspiratory bursts (Ballanyi, 2004; Del Negro et al., 2009; Krey et al., 2010). $s$ terminates the burst and governs the refractory period that coincides with postinspiration (note: refractoriness does not define postinspiration), but $s$ recovers completely prior to the next inspiratory burst. Another variable $\theta$, which we dub cellular adaptation, ultimately sets the timing of the inspiratory burst. $\theta$, too, represents pre- and postsynaptic factors. One possibility is that $\theta$ represents the slowly inactivating transient outward current (I_A), which acts as ‘clamp’ of network excitability by acting specifically in the dendrites of rhythmogenic preBötC neurons (Hayes et al., 2008; Phillips et al., 2018). A second possibility is that $\theta$ reflects synchronization of constituent rhythmogenic neurons, which takes place predominantly during the preinspiratory phase of the cycle (Carroll & Ramirez, 2013; Ashhad & Feldman, 2020; Ashhad et al., 2023), long after refractoriness ($s$) has recovered. In both cases, the declining influence of outward current I_A in conjunction with network synchronization constitutes necessary factors that govern inspiratory burst generation. Those factors are abstractly captured in the activity-based model framework in one state variable $\theta$, which cannot by its nature elucidate the mechanisms of synchronization or the role of synaptic topology in the preBötC and their detailed contribution to inspiratory rhythm (Kam et al., 2013a; Ashhad & Feldman, 2020; Ashhad et al., 2023). Although lacking explanatory power with regard to network properties, the strength of the simplified activity model is to show that two mechanisms (for sigh and inspiration) can comfortably coexist in preBötC neurons because a network oscillator and a biochemical oscillator are fundamentally different and can be separately regulated by manipulating either membrane excitability (inspiration) (Del Negro et al., 2009; Doi & Ramirez, 2010) or Gq-mediated intracellular signaling (sigh) (Li et al., 2016; Morgado-Valle et al., 2022).

Each constituent neuron hosts an intracellular signaling system linked to plasma membrane Ca²⁺ flux as well as ER Ca²⁺ storage and release mechanisms. That system can produce biochemical oscillations on the order of minutes, much slower than inspiration, and trigger neural bursts substantially larger than inspiration via Ca²⁺-induced Ca²⁺ release that evokes the burst-generating inward current, I_CAN. The slow $\left(c,c_{tot}\right)$ subsystem of the activity model counterintuitively predicted that partial SERCA blockade would increase sigh frequency whereas partial blockade of IP₃ receptors would decrease it. Experimental confirmation of these predictions validates both the preexisting Ca²⁺ hypothesis of sigh rhythmogenesis (Jasinski et al., 2013; Toporikova et al., 2015) as well as the specific oscillatory mechanisms introduced here and demonstrated in the model. Specifically, the model explains how SERCA or IP₃ receptor blockade influences sigh frequency and amplitude by changing the $c_{tot}$ (i.e., total Ca²⁺) threshold that compels ER Ca²⁺ release as it refills during the inter-sigh interval. The inspiratory subsystem $\left(a, s, \theta \right)$ explains the dynamics of inspiratory rhythm generation in an abstract form that subsumes recurrent excitation, burstlet-to-burst transitions, and the role of intrinsic cellular burst-generating currents.

The two oscillators have another point of intersection: plasma membrane Ca²⁺ influx. Periodic Ca²⁺ entry during inspiratory rhythm influences burst amplitude via its ability to recruit I_CAN mediated by Trpm4 ion channels, which is inward current that augments burst generation (Pace et al., 2007b; Koizumi et al., 2018; Picardo et al., 2019). Because the Ca²⁺ channel antagonist Cd²⁺ stops preBötC bursts, yet leaves burstlets unperturbed, it is reasonable to conclude that Ca²⁺ and I_CAN are involved in burstlet-to-burst transitions in addition to the sigh burst rhythm (Kam et al., 2013a). In addition to plasma membrane flux, the synaptic Ca²⁺ entry leading to I_CAN recruitment during inspiratory bursts may depend to some extent on intracellular Ca²⁺ release because it involves metabotropic glutamate receptors and dendritic wavelike propagation (Pace et al., 2007b; Pace et al., 2008; Mironov, 2008). Therefore, the extent to which intracellular Ca²⁺ storage and release via the ER contributes to inspiratory bursts (in addition to its role in sighs) remains unsettled.

Inspiratory rhythmic activity increases Ca²⁺ influx leading to a modest increase in the sigh related Ca²⁺ oscillation frequency. This confers (minimal) voltage dependence on the sigh oscillation, which is approximately 6% as sensitive as inspiratory rhythm to changes in membrane excitability (see Figs. 6D and 11D). Activity-dependent Ca²⁺ influx may also promote synchronization of intracellular Ca²⁺ oscillations across the neural population. In the model of 500 interacting neurons, ionotropic synaptic signaling synchronized intracellular Ca²⁺ oscillations underlying the sigh rhythm (Fig. 12). In the biological system, additional synchronizing mechanisms are available including metabotropic glutamatergic transmission (Pace et al., 2007b; Mironov, 2008; Ben-Mabrouk et al., 2012) as well paracrine signaling mechanisms yet to be identified.

The in vitro slice model of breathing and sighing has a long track record of revealing essential mechanisms that apply in vivo (Lieske et al., 2000; Funk & Greer, 2013). It is particularly applicable for sigh rhythms whose cycle period is on the order of 1 min^–1 both in slices and intact adult mice (Li et al., 2016; Li & Yackle, 2017).

The evolution of the lung in mammals introduced the physiological need for regularly timed large-volume breaths to reinflate collapsed or collapsing alveoli. As the breathing neural-control system evolved for terrestrial vertebrates it produced an inexorable inspiratory oscillator but not a sigh rhythm. As the physiological need arose in mammals whose lungs are composed of collapsible air sacs, rather than develop a distinct brain region for sighs, we posit that adaptive evolution utilized a previously existing intracellular Ca²⁺ signaling tool kit to produce a low frequency, large amplitude oscillation that operates quasi-independently of the inspiratory oscillator but within the same core of Dbx1 preBötC neurons.

This work demonstrates the operation of a dual-rhythmic CPG system with relevance to health and physiology. Its activity can be understood via a low-dimensional dynamical system with fast and slow time scales.

KEY POINTS.

• A simplified activity-based model of the preBötC generates inspiratory and sigh rhythms from a single neuron population

• Inspiration is attributable to a canonical excitatory network oscillator mechanism

• Sigh emerges from intracellular calcium signaling

• The model predicts that perturbations of calcium uptake and release across the endoplasmic reticulum counterintuitively accelerate and decelerate sigh rhythmicity, respectively, which was experimentally validated

• Vertebrate evolution may have adapted existing intracellular signaling mechanisms to produce slow oscillations needed to optimize pulmonary function in mammals

Biography

Daniel S. Borrus completed his Ph.D. work at William & Mary in Virginia in 2022 and is currently a postdoctoral associate with the Yale School of Medicine at Yale University in Connecticut. His primary interests involve the application of mathematical and computer science tools to understand biological phenomenon. The dramatic union between dynamical systems and cellular neuroscience presented in this work is one of his proudest achievements as a scientist. He hopes this work inspires scientists everywhere to leverage the power of dynamics to reveal the beautiful and complex architecture that underlies the systems around us.

DATA AVAILABILITY

All the equations and Matlab code are in the public repository on Model DB (Accession No. 267252).