Granule cell excitability regulates gamma and beta oscillations in a model of the olfactory bulb dendrodendritic microcircuit

Gamma (40–100 Hz) and beta (15–30 Hz) oscillations in mammalian olfactory bulbs represent differential involvement of higher order brain areas and different cognitive networks. In a computational model of the olfactory bulb, we find that changes in granule cell excitability can gate fast transitions from gamma to beta during odor sampling. Increased granule cell excitability, produced in many ways, releases stronger graded inhibition, which supports beta as a very stable state in disparate circumstances.

Abstract

Odors evoke gamma (40–100 Hz) and beta (20–30 Hz) oscillations in the local field potential (LFP) of the mammalian olfactory bulb (OB). Gamma (and possibly beta) oscillations arise from interactions in the dendrodendritic microcircuit between excitatory mitral cells (MCs) and inhibitory granule cells (GCs). When cortical descending inputs to the OB are blocked, beta oscillations are extinguished whereas gamma oscillations become larger. Much of this centrifugal input targets inhibitory interneurons in the GC layer and regulates the excitability of GCs, which suggests a causal link between the emergence of beta oscillations and GC excitability. We investigate the effect that GC excitability has on network oscillations in a computational model of the MC-GC dendrodendritic network with Ca²⁺-dependent graded inhibition. Results from our model suggest that when GC excitability is low, the graded inhibitory current mediated by NMDA channels and voltage-dependent Ca²⁺ channels (VDCCs) is also low, allowing MC populations to fire in the gamma frequency range. When GC excitability is increased, the activation of NMDA receptors and other VDCCs is also increased, allowing the slow decay time constants of these channels to sustain beta-frequency oscillations. Our model argues that Ca²⁺ flow through VDCCs alone could sustain beta oscillations and that the switch between gamma and beta oscillations can be triggered by an increase in the excitability state of a subpopulation of GCs.

NEW & NOTEWORTHY

Gamma (40–100 Hz) and beta (15–30 Hz) oscillations in mammalian olfactory bulbs represent differential involvement of higher order brain areas and different cognitive networks. In a computational model of the olfactory bulb, we find that changes in granule cell excitability can gate fast transitions from gamma to beta during odor sampling. Increased granule cell excitability, produced in many ways, releases stronger graded inhibition, which supports beta as a very stable state in disparate circumstances.

local field potential (LFP) oscillations in the mammalian olfactory bulb (OB) represent coordinated neural activity that is dynamically regulated during olfactory processing. These oscillations are classified into three bands: theta (1–12 Hz), low (40–60 Hz) and high (60–100 Hz) gamma (Kay 2003; Lepousez and Lledo 2013), and beta (15–30 Hz). The theta rhythm is coupled to respiratory and sniffing frequencies (Rojas-Líbano et al. 2014). The gamma and beta rhythms strongly depend on behavioral context and odor quality (for a review, see Kay et al. 2009 and Kay 2014). OB gamma and insect gamma-like odor-evoked antennal lobe oscillations are functionally related to discrimination of overlapping odor input patterns (Beshel et al. 2007; Lepousez and Lledo 2013; Nusser et al. 2001; Stopfer et al. 1997). Beta oscillation power scales with odor volatility (Lowry and Kay 2007) and increases with odor discrimination learning (Martin et al. 2006; Ravel et al. 2003). These oscillations may coordinate mitral cell (MC) spikes to drive spike-timing dependent learning in piriform cortex (PC) (de Almeida et al. 2013) and may bind neural assemblies to facilitate communication between regions (Martin et al. 2007; Mori et al. 2013).

Beta oscillations follow gamma oscillations during odor sampling (Martin et al. 2006; Martin and Ravel 2014; see Fig. 1). In anesthetized rats, beta oscillations occur during exhalation (Buonviso et al. 2003; Cenier et al. 2009; David et al. 2015; Fourcaud-Trocmé et al. 2011). In waking rats, beta oscillations can persist across many sniff cycles and also can occur during periods of low respiratory drive in late odor sampling (Martin et al. 2007; Rojas-Líbano and Kay 2012). The transition from gamma to beta can be very fast (Fig. 1A), indicating a state-like transition in a system variable. They also can be very large in amplitude (Fig. 1B), suggesting the presence of highly synchronized synaptic currents.

Fig. 1.

Examples of long-lasting beta oscillations in 2 different rats. Each plot shows the raw LFP bandpass filtered during recording at 1–300 Hz, later filtered for low-gamma (Low γ; 40–60 Hz), high-gamma (High γ; 60–100 Hz), and beta (β; 15–30 Hz) frequency bands. A: a rat trained in a 2-alternative choice task with high-volatility odors diluted in vapor phase (octanone and heptanone) pokes into the odor port at time t = 0 (black vertical line). Exit from the odor port is marked by the dashed black vertical line. In the trial shown, the nose poke was immediately followed by 4 gamma bursts corresponding to 4 sniffs (marked by gray horizontal bars; inhalation is down), a beta oscillation near 24 Hz (marked by black horizontal bar), and a late low-frequency gamma burst. The onset of beta roughly coincides with the end of the last gamma burst, indicating a very fast transition. Unlike the gamma oscillations, beta appears to persist over several respiratory cycles. B: same markings as in A. A freely behaving rat was presented with a cotton swab soaked in an undiluted high-volatility odor (ethyl butyrate). This odor evokes a large and long-lasting beta oscillation near 18 Hz that dominates the LFP signal. The beta oscillation appears ∼100 ms after the second gamma, slower than the transition in A. The appearance of what looks like 2 tapered beta oscillations may in fact correspond to 2 slower inhalations, but respiration was not measured in this rat. The large amplitude of this beta oscillation suggests the presence of highly synchronized currents. Surgical procedures and behavioral methods for obtaining these data are described elsewhere (Frederick et al. 2011; Lowry and Kay 2007).

The mechanism of OB gamma oscillations has been studied extensively (Bathellier et al. 2005; Brea et al. 2009; Freeman 1975; Friedman and Strowbridge 2003; Lagier et al. 2004; Lepousez and Lledo 2013; Nusser et al. 2001; Rall and Shepherd 1968; Rojas-Líbano and Kay 2008). It is generally agreed that they arise from ionic currents flowing between reciprocal dendrodendritic synapses formed between excitatory glutamatergic MC secondary dendrites and inhibitory GABAergic granule cell (GC) distal dendritic spines (Lagier et al. 2004). GABA can be released onto MC dendrites in a graded fashion dependent on Ca²⁺ flow through NMDA receptors (NMDARs) and other voltage-dependent Ca²⁺-dependent channels (VDCCs) expressed on GC spines (Chen et al. 2000; Isaacson and Strowbridge 1998; Schoppa et al. 1998) (but see Schoppa 2006 for evidence of AMPA-mediated dendrodendritic inhibition, DDI). One modeling study has shown that gamma oscillations can be sustained by graded inhibition via subthreshold oscillations in GC dendrites (Brea et al. 2009). Another has shown that gamma oscillations can be generated by graded asynchronous release of GABA superimposed on the slow NMDAR closing time (Bathellier et al. 2005).

Beta oscillation mechanisms have received considerably less attention. Current source density analysis in anesthetized, tracheotomized rats showed that gamma and beta oscillations never occur simultaneously, and both have dipoles centered in the OB external plexiform layer (EPL) (Neville and Haberly 2003). More recent analysis has revealed that gamma and beta oscillations may arise from distinct EPL sublaminae (Fourcaud-Trocmé et al. 2014). Nonetheless, it is unclear how DDI between MCs and GCs can support both gamma and beta frequencies.

An important clue to the origins of beta oscillations comes from lesion experiments. When centrifugal input to the OB is blocked, odor-evoked beta oscillations are extinguished or reduced, whereas gamma oscillations persist and can be enhanced (Gray and Skinner 1988; Martin et al. 2006; Neville and Haberly 2003). Analogous to cortical feedback control of thalamic reticular nucleus neurons, feedback from MC cortical projection areas primarily targets the GC layer with excitatory synapses onto GABAergic GCs and other interneurons (Balu et al. 2007; Boyd et al. 2012; Kay ad Sherman 2007). Gamma oscillations have been shown to persist even after GC somata are surgically disconnected from their distal dendrites (Lagier et al. 2004). Thus there is strong evidence that centrifugal inputs near GC somata may be critical for beta but not gamma generation.

Under certain conditions, GCs enter periods of increased excitability following a somatic spike. For example, activation of M1 muscarinic receptors in GCs transforms the afterhyperpolarization (AHP) following GC spikes into an afterdepolarization (ADP) lasting several hundred milliseconds (Pressler et al. 2007), which may trigger quasipersistent firing modes lasting several seconds (Inoue and Strowbridge 2008). The GC ADP propagates faithfully into distal dendrites, leading to increased Ca²⁺ influx (Egger et al. 2003). GCs may also undergo a long-lasting depolarization (LLD) that can last well over 1 s when a GC somatic spike is triggered by strong glomerular input (Egger 2008). We propose that a convergence of sensory, cortical, and neuromodulatory inputs onto GCs may help explain the observed dependence of beta oscillations on odorant characteristics and centrifugal feedback.

To test this hypothesis, we developed a model of the MC-GC reciprocal dendrodendritic synaptic network with graded inhibition dependent on NMDA and N-type Ca²⁺ currents. In our model, we summarize the cortical, local inhibitory/excitatory, and neuromodulatory sources of GC excitability control by a single parameter, V_rest,GC, the granule cell dendritic (GCD) resting membrane potential. To investigate the individual contributions of NMDA and N-type currents, we also create two additional models, one with only NMDA and the other with only N-type channels included in the GCDs. Our model predicts that a sudden depolarization of the membrane potential of a subpopulation of GCs can drive an increase of Ca²⁺-dependent graded inhibition that will switch the frequency from a gamma to a beta regime. The model also argues that high-power beta oscillations observed in vivo (Fig. 1) are primarily mediated by N-type currents.

METHODS

Model Architecture

In the rat OB, excitatory MCs form reciprocal dendrodendritic connections between their lateral dendrites and the spines on GCDs. MC lateral dendrites extend broadly across the OB and support bidirectional action potential propagation, allowing MCs and GCs on opposite sides of the OB to be synaptically connected. Our model represents a subset of this network, with 45 MCs and 720 GCDs. Each MC is reciprocally connected to 30% of the GCD population (Fig. 2), similar to previous models (de Almeida et al. 2013; Linster et al. 2009), yielding a total of 9,720 MC-GC dendrodendritic synapses. Our model MCs do not represent individual cells, but rather the population of MCs associated with a particular glomerulus. Although the density of Na⁺ channels is known to vary slightly along the length of MC lateral dendrites (Migliore and Shepherd 2002), in our model all excitatory synaptic weights from MCs to GCs are equal.

Fig. 2.

Schematic of the reciprocal dendrodendritic MC-GC model. There are 45 MCs and 720 GCDs. Each MC represents the population of MCs associated with a glomerulus (GLO). The GLO and GC soma (open circles) are not explicitly modeled. Each MC is randomly connected to 216 GCDs (30% of the population). MCs only express GABARs, whereas GCs express AMPARs, NMDARs, and N-type channels. Ca²⁺ flow through NMDA and N-type receptors drives release of GABA vesicles, modeled as a graded inhibitory current. NMDA and AMPA currents have a spike-triggered activation, which represents the binding of glutamate released from MCs following a spike.

Our model architecture is motivated by five key experimental findings: 1) gamma and beta oscillations both have dipoles centered in the EPL, where MC-GC dendrodendritic synapses are formed (Neville and Haberly 2003); 2) GCs can release GABA in a graded (spike independent) fashion dependent on NMDA and VDCC Ca²⁺ currents (Isaacson 2001; Isaacson and Strowbridge 1998; Schoppa et al. 1998); 3) GCs spike at very low rates in awake animals compared with cortical interneurons (Cazakoff et al. 2014); 4) beta oscillations require intact centrifugal projections to the OB (Martin et al. 2006; Neville and Haberly 2003), many of which target the GC layer and regulate GC excitability; and 5) GCs can undergo periods of increased excitability with sustained elevated membrane potentials propagating faithfully into their distal dendrites (Egger 2008; Pressler et al. 2007). We represent all these sources of excitability control with a single parameter, the GCD resting potential, V_rest,GC. Because we have direct control over the excitability of the GCDs, we do not explicitly model the GC soma, whose synaptic integration and bidirectional signal propagation properties are complex and not yet fully understood (Balu et al. 2007; Egger et al. 2005; Inoue and Strowbridge 2008). Instead, we only model the entire GC dendritic tree as a single unit, which receives inputs from multiple MCs (Fig. 2). In the real system, GC dendritic spines can act independently of one another and influence each other via low-threshold Ca²⁺ spikes (Egger 2008; Egger et al. 2005), but this scenario is not captured by our model. Inhibitory coupling between GCs is also omitted for simplicity. All simulations were implemented in MATLAB R2014b, with a forward Euler integration time step of 0.1 ms. Simulations with forward and backward Euler methods were found to give identical results. The code for has been made available through the ModelDB under accession no. 185464 (Hines et al. 2004).

Neuron and Synapse Equations

MCs and GCDs are modeled as single compartments whose membrane potentials evolve in time according to

$$\tau \dot{V}=-V+W_{chan}\cdot I_{chan}+V_{rest},$$ (1)

where τ is the membrane time constant, W_chan is a dimensionless synaptic weight representing the ratio of maximum conductance of a particular channel to the membrane leak conductance (from here on referred to as synaptic weight), I_chan is the synaptic current through a particular channel, and V_rest is the resting membrane potential. We have chosen units such that resistance is equal to 1, and therefore the currents and membrane voltages are in the same units but are not in SI units. The critical parameter in this model is V_rest,GC, the resting potential of the GC dendritic tree. The MCs obey leaky integrate and fire dynamics with a hard threshold 7 mV above resting potential. Probabilistic firing is achieved by introducing noise into the MC external inputs. Although recent experiments have detected regenerative Na²⁺ spikes in GC dendrites (Bywalez et al. 2015), the GCDs do not fire action potentials in this model.

All synaptic currents in the model are conductance-based currents that obey the general form

$$I=\alpha \left(t,V,\left[Ca\right]\right)\left[E-V\right],$$ (2)

where α(t, V, [Ca]) is the product of normalized channel activation and inactivation variables (may depend on time, membrane voltage, and internal Ca²⁺ concentration, [Ca²⁺]), and E is the Nernst potential of the ionic species flowing through the channel.

Each MC receives two currents: 1) I_ext, an external sensory input, and 2) I_GABA, a graded inhibitory GABA (Cl⁻) current from GCs. Each GCD receives three currents: 1) I_AMPA, an AMPA (Na²⁺) current with MC spike-triggered conductance; 2) I_NMDA, an NMDA (Ca²⁺) current with MC spike-triggered conductance and voltage-dependent Mg²⁺ block (Baszczak and Kasicki 2005; Jahr and Stevens 1990); and 3) I_N, an N-type (Ca²⁺) current with voltage-dependent activation and Ca²⁺-dependent inactivation (Amini et al. 1999; Zeng et al. 2009). The mathematical details of the currents are summarized in Table 1.

Table 1.

MC and GCD synaptic current equations

Equation	Activation/Inactivation
MC
$I_{ext,i}=W_{ext,i}\left[1+{\sigma }_{ext}{\eta }_{ext,i}\right]\hspace{0.17em}{W}_{ext,i}={\sigma }_w{\eta }_{w,i}+W_{\min ,ext}$	Continuous
$I_{GABA,i}={\displaystyle {\sum }_j}{P}_{release,j}\left[E_{Cl}-V_{MC,i}\right]\hspace{0.17em}{P}_{release,j}=({\left[Ca\right]}_j-{\left[Ca\right]}_{baseline,j})\text{/}({\left[Ca\right]}_{th}-{\left[Ca\right]}_{baseline,j})$, [0, 1]$\hspace{0.17em}{\tau }_{Ca}\left[\dot{C}a\right]=-{\left[Ca\right]}_j+{\rho }_{Ca}\left(I_{NMDA,j}+I_{N,j}\right)$	Prelease, graded activation
GCD
$I_{AMPA,j}={\displaystyle {\sum }_i}{s}_{j,i}\left(t\right)\left[E_{AMPA}-V_{GCD,j}\right]\hspace{0.17em}{s}_{j,i}\left(t\right)=\frac{1}{s_{norm}}\left[e^{-\left(\frac{t-t_{spike,i}}{{\tau }_{decay}}\right)}-e^{-\left(\frac{t-t_{spike,i}}{{\tau }_{rise}}\right)}\right]$	s(t), MC spiking activation
$I_{NMDA,j}={\displaystyle {\sum }_i}B\left(V_{GCD,j}\right)s_{j,i}\left(t\right)\left[E_{NMDA}-V_{GCD,j}\right]B\left(V_{GCD,j}\right)=1.0\text{/}(1.0+0.28\left[Mg\right]e^{-0.062\left(V_{GCD,j}-V_{Mg}\right)})$	s(t), MC spiking activation B(VGCD), voltage-dependent activation
$I_{N,j}=m_{N,j}{h}_{N,j}\left[E_{Ca\left(N\right),j}-V_{GCD,j}\right]{\tau }_{m_{N,j}}{\dot{m}}_{N,j}={\overline{m}}_{N,j}-m_{N,j}{\tau }_{m_{N,j}}=18.0e^{-{\left(\left(V_{GCD,j}+70\right)/25\right)}^2}+0.3{\overline{m}}_{N,j}=1.0\text{/}(1.0+e^{-\left(V_{GCD,j}+45\right)/7}),h_{N,j}={10}^{-4}\text{/}({10}^{-4}+{\left[Ca\right]}_j)$	mN, voltage-dependent activation hN, [Ca2+]-dependent inactivation

The index i always refers to mitral cells (MCs) and index j always refers to granule cell dendrites (GCDs). [Ca²⁺] represents GCD internal Ca²⁺ concentration.

External excitatory input to MCs.

Following de Almeida et al. (2013), the external input I_ext to each MC is modeled by a continuous variable that represents the average instantaneous firing probability of the population of olfactory sensory neurons innervating a given glomerulus. We do not model respiration because the inhibitory circuits mediating gamma/beta oscillations are dissociable from those mediating theta oscillation (Fukunaga et al. 2014) and because we are interested in investigating the effect that GC excitability alone has on MC synchronization, which to our knowledge has not yet been done. To make MC spike firing probabilistic, we include an input noise σ_extη_ext,i, where σ_ext is a scalar and η_ext,i is drawn from the standard normal distribution for the i-th MC on each time step. Each MC represents the average activities of all the MCs innervating a particular glomerulus, and therefore this model does not capture variations in the activity of individual MCs innervating the same glomerulus. Periglomerular cells, which normally gate sensory inputs to MCs, are not included in the model because we are interested only in the modulation of MC activity by GCDs. Instead, continuous sensory input is fed directly into MCs. The input weight to the i-th MC, W_ext,i (Table 1) is determined by a scalar σ_w, a number η_w,i chosen randomly from a uniform distribution on [0 1], and a minimum weight, W_min,ext. For most simulations we set σ_ext = 0.001 and W_min,ext = 0.013, resulting in uniformly distributed, free-running (uninhibited) MC firing rates between 130 and 150 Hz. These uninhibited firing rates are much higher than experimentally recorded firing rates of individual MCs when inhibition is blocked (Lepousez and Lledo 2013). However, each of our model MCs represents a population of MCs associated with an individual glomerulus, and not a single MC. Thus high, unsynchronized firing rates are expected. Furthermore, our model aims to simulate the beta oscillations induced by high-volatility odors, which may bind olfactory sensory neurons quickly and uniformly, causing stronger convergent inputs onto MCs. We find that the network requires this strong external drive to generate the full gamma to beta range as GC excitability is varied.

Graded inhibitory input to MCs.

We assume a graded form of inhibition from GCDs onto MCs, which is proportional to the probability of GABA vesicle release, P_release. The release probability is only dependent on intracellular [Ca²⁺] (Table 1). Thus the AMPA current can only drive graded inhibition indirectly by activating NMDA and N-type channels. Much like MC inhibitory postsynaptic currents (IPSCs) recorded in slice preparations (Schoppa et al. 1998), the time course of the model MC IPSCs follows the slow exponential decay of NMDA and N-type channels. The proportionality ρ_Ca between Ca²⁺ current and Ca²⁺ concentration is chosen such that [Ca²⁺] is typically between 0.1 and 1 μM, a physiologically realistic range for [Ca²⁺] in a dendritic spine (Higley and Sabatini 2012). The threshold for maximum Ca²⁺ release, [Ca]_th, is chosen so that the maximum P_release is near 1 in the high-excitability condition. Experiments have shown that DDI is largely unaffected by intracellularly injected Ca²⁺ chelators, suggesting that the GABA release machinery in GC dendrites is tightly coupled to the Ca²⁺ influx following MC spikes (Isaacson 2001). Because the model N-type current admits a constant Ca²⁺ influx due to nonzero voltage-dependent activation, even in the absence of MC spikes there is a constant internal baseline [Ca²⁺], which we call [Ca]_baseline (see appendix for derivation of [Ca]_baseline). We subtract [Ca]_baseline in the calculation of P_release to ensure that tonic inhibition is not released onto MCs in the absence of MC spikes (Fig. 3D).

Fig. 3.

A: the AMPA, NMDA, and N-type currents of a single GCD connected to 14 MCs are plotted. A 5-ms-long current pulse (I_ext; amplitude not to scale) was initiated at t = 50 ms, causing all 14 MCs to fire a single action potential almost simultaneously. The latency of the currents from onset of I_ext corresponds to the integration time of the MCs before spikes. Simulations were performed under low (V_rest,GC = −70 mV; blue) and high (V_rest,GC = −60 mV; red) GC excitability conditions. The N-type current was simulated with (solid lines) and without (dashed lines) Ca²⁺-dependent inactivation (CDI). B: the N-type activation (m) and inactivation (h) variables and their product (m*h) for the N-type currents shown in A are plotted. CDI is prevented by evaluating h at [Ca]_baseline, leaving it constant. The deflections near t = 0 are transients, which can be ignored. CDI greatly reduces the magnitude of the deflections from the baseline in m*h but does not change their overall shape. C: the internal Ca²⁺ concentration ([Ca]) traces resulting from the NMDA and N-type currents shown in A are plotted. The dotted lines indicate [Ca]_baseline for low- and high-excitability conditions. D: the P_release traces derived from the internal [Ca] traces shown in C are plotted. Subtraction of [Ca]_baseline from [Ca] ensures that P_release is 0 until a MC spike is fired. The presence of CDI significantly decreases P_release but does not change the temporal structure of GABA release.

Excitatory and Ca²⁺ currents in GCDs.

The model GCDs include AMPA, NMDA, and N-type currents. The synaptic time course s(t) of the AMPA and NMDA currents is modeled as a difference of exponentials with rise and decay times that represent the opening and closing of the channels following glutamate binding (Brunel and Wang 2003). The NMDA current I_NMDA also contains an additional voltage-dependent activation, B(V_dGC) (Table 1), which represents the Mg²⁺ block as determined by Jahr and Stevens (1990). Because the connections are probabilistic, some GCDs may be connected to as few as 4 MCs, whereas others may be connected to as many as 22. The synaptic weight of the AMPA current, W_AMPA,GC, is chosen such that the magnitude of membrane depolarization for a GCD connected to an average number of 14 MCs is near 7 mV, which is comparable to the experimentally measured average size of AMPA-mediated depolarization of GC dendrites (Cang and Isaacson 2003). The experimentally recorded amplitude of NMDA current is usually smaller than the AMPA current, although this is not always the case (Schoppa et al. 1998). For our simulations, W_NMDA,GC is chosen such that the amplitude of I_NMDA is approximately one-fourth that of I_AMPA (Fig. 3A). The NMDA decay time constant is 75 ms in most of our simulations, close to the value reported in physiological concentrations of Mg²⁺ (Isaacson 2001; Schoppa et al. 1998). With these parameters, GCD membrane depolarization closely resembles that which is seen in experiments, with fast-decaying AMPA-mediated and slow-decaying NMDA-mediated components.

The N-type current I_N includes a voltage-dependent activation variable, m_N, and a Ca²⁺-dependent inactivation (CDI) variable, h_N. The activation curves of N-type channels reported in the literature are not the same across all neuronal cell types, with half-activation voltages ranging from −45 (Amini et al. 1999; Shuai et al. 2009) to −3 mV (Evans et al. 2013). To our knowledge the activation curve of N-type channels in OB GCs has not yet been measured. We choose to use a model that begins to activate at relatively low potentials so that I_N is sensitive to increases in GC excitability, which are at most only about 25 mV above the resting potential. This ensures that I_N alone can drive slowly decaying dendrodendritic inhibition when excitability is increased, as has been shown in slice preparations (Isaacson 2001). For most of our simulations, W_N,GC is chosen so that the N-type current is approximately one-third the magnitude of the NMDA current (Fig. 3A), a choice that is motivated by Ca²⁺ imaging studies, which have shown that although NMDA channels mediate the majority of Ca²⁺ entry, VDCCs still mediate a sizeable portion (Bywalez et al. 2015; Egger 2008). W_N,GC is so much greater than W_NMDA,GC because the product of the N-type activation and inactivation terms is on the order of 10⁻⁵, whereas the NMDA activation term is on the order of 1. This model does not include voltage-dependent inactivation. Because in most of our simulations the GC membrane potential does not exceed −40 mV and inactivation only becomes significant above −20 mV (Evans et al. 2013; Johnston and Wu 1995), voltage-dependent inactivation would not make a large contribution here even if it were included.

The effect of CDI is to drive the N-type inactivation variable in the opposite direction of the activation variable, as shown in Fig. 3B, causing a net negative deflection of I_N shown in Fig. 3A. Without CDI, both I_NMDA and I_N fully contribute to the internal [Ca²⁺] buildup, resulting in a high-amplitude GABA release profile that saturates in the high-excitability condition (Fig. 3D). When CDI is present, I_NMDA becomes the dominant source of Ca²⁺ for driving graded inhibition of MCs. Even though CDI significantly reduces the [Ca²⁺] amplitude and reverses the direction of the N-type current deflection, it does not change the temporal structure of graded inhibition (Fig. 3, C and D). Therefore, the model is capable of generating the full gamma-to-beta range with and without CDI as long as the excitatory and inhibitory weights are adjusted accordingly. CDI is included in all subsequent simulations.

Because of low internal Ca²⁺ concentrations, the Nernst potential for Ca²⁺ is sensitive to changes in intracellular Ca²⁺ and is calculated on each step of the simulation by the Nernst equation

$$E_{Ca}=\frac{RT}{zF}\mathit{ln}\frac{{\left[Ca\right]}_{out}}{{\left[Ca\right]}_{in}},$$ (3)

where R = 8.31 J/(mol·K) is the ideal gas constant, T = 300 K is the temperature, z = 2 is the valence of the Ca²⁺ ion, and F = 96,485 C/mol is Faraday's constant. Assuming an extracellular Ca²⁺ concentration [Ca]_out = 1.5 mM (Higley and Sabatini 2012), E_Ca is typically near 100 mV.

Simulation and spectral analysis of LFP.

The model is simulated for 700 ms. The LFP is generated by smoothing and averaging the MC IPSCs (ILFP) or MC membrane voltages (VLFP) using MATLAB's built-in smoothts function with a box filter width of 5 ms. The ILFP and VLFP have nearly identical frequencies but differ notably in phase and power, which we address in Figs. 4 and 5. Only simulation data after 100 ms are used to avoid transients due to initial conditions of the simulation. For most of our simulations we only present the ILFP. Recent work has shown that in vivo LFPs correlate more strongly with IPSCs and excitatory postsynaptic currents (EPSCs) than with membrane potentials (Atallah and Scanziani 2009; Mazzoni et al. 2015). Unless otherwise stated, the power spectra are computed by MATLAB's fft function on the mean-subtracted LFP, and we report the LFP frequency as the peak frequency of the power spectrum between 7 and 100 Hz. For reference, all power plots indicate the maximum power of the noise floor, which we define as the maximum power of the residual oscillation due to common inputs to MCs when inhibition is removed. As shown in Fig. 6, we perform a continuous Morlet wavelet transform with MATLAB's cwt function to display the instantaneous LFP frequency in response to sudden changes in GC excitability. We use a frequency range of f_range = 5–80 Hz and the standard frequency-scale relation f_c·sf/f_range (where f_c is central frequency of the wavelet and sf is sampling frequency) to define the scale range.

Fig. 4.

A: the MC-GC dendrodendritic network activity was simulated under low (i), moderate (ii), and high (iii) GC excitability conditions in response to constant olfactory receptor neuron (ORN) input current for 500 ms. The first, second, and third rows show the AMPA, NMDA, and N-type currents of all 720 GCDs for each condition, respectively. The AMPA current is scaled by half so that all axes fit on the same scale. The N-type current has a constant offset that increases with V_rest,GC due to its voltage-dependent activation. The fourth row shows the GABA release probabilities (P_release) for each of the 720 GCDs resulting from the inward Ca²⁺ currents (I_NMDA and I_N). The fifth row shows MC raster plots with cells ordered from bottom to top by increasing input strength. The sixth row shows the LFP calculated from the average MC IPSCS (ILFP; black) and MC membrane voltages (VLFP; gray). B: for each of the 3 conditions (i–iii), power spectra were calculated from the ILFP generated by the global population as well as from the 15 MCs receiving the strongest external input, the 15 MCs receiving the weakest external input, and the VLFP of the global population. The y-axes of the 3 plots are the same, but the x-axes differ. The horizontal dashed black line indicates the maximum power of the noise floor.

Fig. 5.

Ai: the simulated ILFP (pink) and VLFP (black) frequencies of the full model decrease continuously with increased GC excitability, represented by V_rest,GC. Shaded regions denote the SD from the mean of 10 simulations. Approximate frequency ranges for high/low-gamma and beta states are indicated at left. Aii: the voltage-dependent activation curves of N-type and NMDA channels are plotted. A schematic showing the average AMPA depolarization from the GCD resting potential (7 mV) is shown for low-excitability (blue) and high-excitability (red) conditions. In the excited state, the depolarization is high enough to significantly activate both NMDA and N-type currents (where the vertical dotted red line crosses the activation curves). Aiii: the power of the ILFP and VLFP from Ai is shown. The horizontal dashed line represents the power of the noise floor, where inhibition can no longer sustain oscillations. B: the means of the spike-frequency deviation (SFD; dashed line, left axis) and average maximum spike-field coherence (SFC; dotted curve, right axis) for the simulations in A are shown. SD are omitted for clarity. The minimum of the SFD is marked by the vertical dashed line and labeled V_bal. To the left of V_bal some MCs are overexcited (Over exc.), whereas to the right of V_bal some MCs are overinhibited (Over inh.). The maximum of the SFC spectrum between each MC spike train and the global ILFP was averaged across all 45 MCs to obtain each point of the plot. Ci: the excitability of a randomly chosen subpopulation of GCDs is varied while the remainder are holding at a resting potential of −75 mV. Each of the curves converges to the point where the entire GCD population is in the same state at −75 mV. The parameters for this simulation are shown in Table 2. Each of the 7 curves corresponds to a randomly selected subpopulation ranging from 30% (light gray) to 100% (black). Dotted lines indicate the borders of the beta regime. Cii: the peak power of curves in Ci is plotted. The black dashed line indicates the maximum power of the noise floor. Inset shows the cross section of power at V_rest,GC = −63 mV. The power is highest when only 30% of the GCDs are in a high-excitability state.

Fig. 6.

A: the frequency (i) and power (ii) of the ILFP is plotted for varying strengths of MC excitation (W_min,ext). The pink curves correspond to parameters used in Figs. 4 and 5. B: same as in A, but the strength of MC inhibition (W_GABA,MC) is varied. W_GABA,MC is varied over the same range as W_min,ext (0.007), so the plots in A and B are visually comparable. C: the ILFP frequency (left) and power (right) are plotted with V_rest,GC = −70 mV (top) and V_rest,GC = −60 mV (bottom) as the MC inhibitory weight W_GABA,MC and minimum excitatory weight W_min,ext are varied. Red lines mark the borders of the beta regime (20–30 Hz). Regions where ILFP power is less than or equal to the baseline power are white. Pink circles indicate the parameter values used for the simulations in Figs. 4 and 5. D: vertical cross sections of the plots in C fixed at W_min,ext = 0.0134 (pink dot location) are shown for low (V_rest,GC = −70 mV; blue connected stars)- and high (V_rest,GC = −60 mV; red connected circles)-excitability conditions. The gray shaded region in Di marks the region where network oscillations are considered part of the noise floor. The horizontal black dashed line in Dii indicates the maximum power of the noise floor.

Using the frequency obtained from the fast Fourier transform of the LFP (LFP_fq), we also define a simply spike synchrony measure, which we call the spike-frequency deviation (SFD), as

$$SFD=\left|N_{spikes}-N_{MC}\left(0.6LF{P}_{fq}\right)\right|,$$ (4)

where N_spikes is the actual number of spikes fired in the last 0.6 s of simulation and the second term is the number of spikes that would be fired in 0.6 s if each MC fired at exactly the LFP frequency LFP_fq. The deviation is 0 when each MC fires exactly once per LFP oscillation cycle and is greater than 0, due to the absolute value, when MCs fire more often or less often than once per cycle.

We also compute the spike-field coherence (SFC) using the coherencycpt function included in the Chronux version 2.11 toolbox for MATLAB (Bokil et al. 2010). We used a time half-bandwidth of 5 with 9 tapers over a frequency range of 5 to 120 Hz. Finite sampling corrections were included, although they did not significantly alter the results. The SFC produces a coherence spectrum between each MC and the LFP, so we report the peak coherence averaged over the 45 MCs. The SFD and SFC measures are both used in Fig. 5.

RESULTS

Waking rats and mice show stereotypic transitions from gamma to beta oscillations after a few trials sniffing highly volatile odorants or after learning to correctly discriminate odors in operant tasks (Fig. 1). Both oscillations occur in a single sampling bout, and the switch can be very fast (10–100 ms). This has led us to propose that a fast change in a parameter value, such as GC excitability, could produce a fast change in temporal properties. We propose a mechanism by which the same circuit can generate both gamma and beta oscillations with a fast change in GC excitability.

GC Excitability Controls LFP Frequency Through Activation of NMDA and N-Type Currents

We show the activity of the full model under low, moderate, and high GC excitability conditions in Fig. 4. The model parameters for this simulation are presented in Table 2. There is a large variation in synaptic current amplitude among the 720 GCDs (Fig. 4A, first to third rows). This variation is due to differences in the number of MCs connected to each GCD, with the fewest numbering near 4 and the most near 22. GCDs connected to higher numbers of MCs produce higher values of P_release and contribute more to the MC IPSCs than GCDs with fewer connections (Fig. 4A, fourth row). For the choices of [Ca]_th and ρ_Ca used in this simulation (see Table 2), the maximum GABA release nearly saturates in the high-excitability regime (P_release = 1; Fig. 4Aiii). The N-type current (Fig. 4A, third row) has a nonzero baseline that increases with V_rest,GC due to voltage-dependent activation and causes tonically elevated internal [Ca²⁺]. This [Ca²⁺] baseline is subtracted for each GCD individually so that nonphysiological tonic inhibition is prevented (see appendix).

Table 2.

Parameters for the full model simulation presented in Fig. 4

Parameter	Value	Description	Reference
Input
Wext	See methods	Input current weights	N/A
σext	5 mV/R	Input current variability	N/A
MC
τMC	5 ms−1	MC membrane time constant	de Almeida et al. (2013)
τAMPA1,MC	1 ms−1	MC AMPA rise time	de Almeida et al. (2013)
τAMPA2,MC	2 ms−1	MC AMPA decay time	de Almeida et al. (2013)
EAMPA	0 mV	MC AMPA reversal potential	Brunel and Wang (2003)
τMC	0.5 ms−1	MC GABA rise time	Brunel and Wang (2003)
τMC	5 ms−1	MC GABA decay time	Brunel and Wang (2003)
EGABA	−80 mV	GABA reversal potential	Cang and Isaacson (2003)
Vrest,MC	−70 mV	MC resting potential	de Almeida et al. (2013)*
Vth,MC	−63 mV	MC firing threshold	Cang and Isaacson (2003)
Vhyper	−80 mV	MC/GCD hyperpolarization potential	de Almeida et al. (2013)*
WGABA,MC	0.0125	MC GABA inhibitory weight	N/A (varied in Figs. 6 and 8)
GCD
τGC	5 ms−1	GCD membrane time constant	de Almeida et al. (2013)
τAMPA1,GC	1 ms−1	GCD AMPA rise time	de Almeida et al. (2013)
τAMPA2,GC	2 ms−1	GCD AMPA decay time	de Almeida et al. (2013)
τNMDA1,GC	2 ms−1	GCD NMDA rise time	N/A (varied in Fig. 9)
τNMDA2,GC	75 ms−1	GCD NMDA decay time	N/A (varied in Fig. 9)
τNmax,GC	18 ms−1	GCD N-type activation time constant	Amini et al. (1999)
EAMPA	0 mV	GC AMPA reversal potential	Brunel and Wang (2003)
ENMDA	0 mV	GC NMDA reversal potential	de Almeida et al. (2013)
ECa(N)	∼ 120 mV	GC N-type reversal potential	Calculated each time step
[Ca]out	1,500 μM	External Ca2+ concentration	Zeng et al. (2009)
[Ca]th	1.5 μM	Ca2+ threshold for maximum GABA release	N/A
ρCa	100	Proportionality between ICa and [Ca2+]	N/A
Vrest,GC	Varied	GCD resting potential	N/A (varied in most figures)
WAMPA,GC	0.03	GCD AMPA excitatory weight	Cang and Isaacson (2003)
WNMDA,GC	0.04	GCD NMDA excitatory weight	N/A (varied in Fig. 8)
WN,GC	250	GCD N-type excitatory weight	N/A (varied in Fig. 8)

^*Voltages are shifted down by −70 mV from de Almeida et al. (2013) but remain the same relative distance from each other. N/A, not applicable.

Spike rasters line up with LFP fluctuations in each frequency range (Fig. 4A, fifth row). Field potential can be simulated either from current or voltage (ILFP or VLFP); the two oscillate at the same frequency but are phase-shifted by 180° (Fig. 4A, sixth row). We can understand this phase shift by considering what happens after a MC spike. Immediately following a spike, the MC membrane voltage undergoes hyperpolarization, producing a trough in the VLFP. At the same time, GCDs experience a strong inward Ca²⁺ current that in turn triggers a strong MC IPSC, producing a peak in the ILFP.

GC excitability alone can control the frequency of network oscillations through activation of NMDA and N-Type currents. Under low excitability conditions (V_rest,GC = −74 mV; Fig. 4Aii), the NMDA and N-type currents are not strongly activated, so inhibition is low and MCs are not fully synchronized across the population. Physiological MCs are known to skip cycles of the gamma oscillation (Bathellier et al. 2005; Lagier et al. 2004 2007); however, our model MCs represent populations of MCs associated with particular glomeruli and thus are expected to fire on each cycle. The MCs receiving the weakest external inputs, toward the bottom of the raster plot in Fig. 4Ai, are able to synchronize, but those receiving stronger inputs are overexcited and fire multiple spikes per cycle. This produces low-amplitude oscillations falling within the high-gamma band as shown in the power spectrum (Fig. 4Bi). When V_rest,GC is increased to −68 mV, the network oscillation frequency falls into the low-gamma band. The increased inhibition synchronizes all the MCs (representing populations associated with individual glomeruli), which leads to higher power LFPs (Fig. 4Aii, fifth row, and 4Bii). Finally, when V_rest,GC is further increased to −60 mV, the network oscillates at beta frequencies. The power of this beta state is lower than the low-gamma state (Fig. 4Bii), because the MCs receiving the weakest external excitation become overinhibited and cease firing spikes.

Shifts in the balance of excitation and inhibition to MCs explain the differences in power. When GC excitability, and hence MC inhibition, is low, the least excited MCs generate the largest ILFP signal (Fig. 4Bi). In contrast, during high GC excitability, the most excited MCs generate the largest ILFP signal (Fig. 4Biii). This is because when GC excitability is low, the MCs are overexcited, firing on average more than one spike per LFP cycle, but when GC excitability is high, the MCs are overinhibited, firing on average less than one spike per LFP cycle. Under moderate GC excitability conditions, excitation and inhibition are balanced such that nearly all the MCs become synchronized, and therefore the simulated LFP power of any subset of cells is very close to the global LFP. The maximum synchronization at low-gamma frequencies in our model agrees with recordings in awake mice which showed that long-range LFP coherency and MC spike synchronization occurs at low-gamma, but not high-gamma, frequencies (Lepousez and Lledo 2013). We will show later that the balance of excitation and inhibition causing peak power in the low-gamma band is primarily a property of the NMDA currents, whereas N-type currents show a balance of excitation and inhibition in the beta band (see Relative Contributions of NMDAR and N-Type Currents).

Network Response to Continuous Changes in GC Excitability

A continuous change in GC excitability is not physiological, but it gives us a qualitative understanding of the relationship between network frequency and GC excitability. As V_rest,GC sweeps from −75 to −55 mV, LFP frequency changes continuously (Fig. 5Ai). The ILFP and VLFP frequencies span the full range from high-gamma to beta (approximate frequency ranges indicated at left) and have nearly identical dependence on V_rest,GC, differing only in the high-excitability condition where overinhibition tends to make the VLFP frequency slightly lower than the ILFP. The curves are steeper in the gamma than the beta regime, indicating that beta frequencies are more stable with respect to small changes in V_rest,GC. The decrease in LFP frequency in response to an increase in V_rest,GC is caused by activation of NMDA and N-type currents (Fig. 5Aii). The average AMPA-mediated membrane depolarization (7 mV) from V_rest,GC = −73 mV (Fig. 5Aii, blue curve) barely activates the currents (see vertical blue dotted line). However, the same depolarization from V_rest,GC = −60 mV (red curve) significantly activates the currents (see vertical red dotted line). Depending on how many MCs are connected to a particular GCD, the AMPA-mediated membrane depolarization can range from ∼2 to 18 mV, thus activating NMDA and N-type currents to varying degrees.

The ILFP and VLFP power plots overlap much less than their frequencies (Fig. 5Aiii). The ILFP and VLFP power values are nearly identical in the high-gamma regime, increasing dramatically with V_rest,GC as increased inhibition synchronizes the entire MC population. The power peaks in the low-gamma regime, when there is a balance of excitation and inhibition onto MCs such that all MCs are synchronized (as described in Fig. 4). Finally, as the system approaches the beta regime, some MCs become overinhibited and ILFP and VLFP power drops off, with the VLFP falling more sharply than the ILFP. Not all of the oscillations generated by the model are physiological, because we explore a wider parameter space than what is available to the real system; such high power in the low-gamma regime is rarely seen in vivo. However, this parameter exploration allows us to characterize properties of the system that otherwise may not be understood. Below we show the model response to a more physiologically realistic fast change in GC excitability (see Network Response to Fast Change in GC Excitability).

To show that the power peak in the low-gamma band is indeed due to a balance of excitation and inhibition, where the MC population fires close to one spike per cycle, we devise a simple measure that we call the spike-frequency deviation (SFD; see methods, Eq. 4). The SFD measures the deviation from the number of spikes expected if all MCs fire exactly once per LFP cycle (SFD = 0 if all MCs fire exactly 1 spike per cycle). In addition, we also calculate the spike-field coherence (SFC), which measures the phase locking between spikes and LFPs (see methods). We use the ILFP in calculating the SFD and SFC measures, although using the VLFP produces nearly indistinguishable curves.

The SFC and SFD measure spiking coherence with the LFP and deviations in spike rate, respectively, associated with changing GC excitability (Fig. 5B). The population averaged SFC is at a maximum when SFD is at a minimum and MCs are closest to the balanced condition (V_bal ∼ −71 mV). MCs are overexcited when V_rest,GC < V_bal and overinhibited when V_rest,GC > V_bal. When the number of MC spikes deviates from the number expected at maximum synchrony, the VLFP is decreased, and therefore the SFD looks almost like a mirror reflection of the VLFP. The SFC, on the other hand, has a much broader peak similar to the ILFP, because it only measures synchrony, not the number of firing MCs. The peaks of ILFP and VLFP power in Fig. 5Aiii both correspond to maximally synchronous conditions, but by construction the VLFP is more sensitive to MC spiking, whereas the ILFP is more sensitive to synchronized current flow. Overall, the ILFP and VLFP are quite similar. However, physiological LFP signals have been shown to follow synaptic currents more closely than membrane voltages (Atallah and Scanziani 2009; Mazzoni et al. 2015), and so we choose to report the frequency and power of the maximum peak in the ILFP spectrum for the remainder of this article, unless otherwise stated.

So far we have modulated the excitability of the entire population of GCDs. Such a broad modulation of excitability could potentially be mediated by diffusion of acetylcholine released from cholinergic fibers (Ma and Luo 2012; Pressler et al. 2007) or other neuromodulators. However, odorants excite only a fraction of downstream cortical neurons, which in turn feed back onto a subpopulation of GCs (Mouret et al. 2009; Poo and Isaacson 2009); thus the excitability of only a subpopulation of GCs may be modulated. We therefore continuously vary the excitability of randomly chosen GCD subpopulations of decreasing size while holding the remainder at a resting potential of −75 mV as shown in Fig. 5C. Modulating the excitability of subpopulations as small as 40% produces LFP oscillations that still span the full gamma-to-beta range (Fig. 5Ci). Beta oscillation power is actually higher when the size of the excited GCD population is reduced (Fig. 5Cii, inset). This is because MCs in the beta state are overinhibited, and reducing the population of excited GCDs reduces inhibition, allowing more MCs to participate in the oscillation. On the other hand, the high- and low-gamma power is lower for smaller excited GCD populations, because now MCs are not receiving enough inhibition. In the real system, beta power tends to be higher than gamma. Therefore, not only can modulation of the excitability of a GCDs subpopulation generate oscillations that span the high-gamma to beta range, but the simulated oscillations appear more physiological.

Beta Frequency is Highly Stable with Respect to Changes in MC Excitation and Inhibition

In Fig. 5 we showed that the ILFP power of the full model peaks in the low-gamma band, which corresponds to a state where excitation and inhibition are balanced such that all the MCs are synchronized. We next explore how network oscillations respond to changes in excitatory-inhibitory balance by directly manipulating MC excitatory and inhibitory weights. Increasing MC excitation, by increasing the minimum excitatory weight W_min,ext, raises the LFP frequency in the low-excitability regime (Fig. 6Ai) and dramatically increases the maximum power peak while shifting the peak toward higher excitabilities (Fig. 6Aii). The pink curve corresponds to W_min,ext = 0.0134 (used in Figs. 4 and 5). Note that the shift is such that the frequency at maximum power is still in the low-gamma band, because the balance of excitation is still maintained at low-gamma frequencies. Interestingly, the frequency curves converge at beta frequencies because beta frequencies are stable with respect to MC excitation. However, beta power is very sensitive to W_min,ext. We routinely observe beta oscillations with drastically different amplitudes in awake animals. In particular, low-volatility odors produce lower power beta oscillations than high-volatility odors (Lowry and Kay 2007). In our model, differences in MC excitation are sufficient to explain this phenomenon.

If we hold MC excitation constant and instead vary MC inhibition (Fig. 6B), we find that increasing W_GABA,MC reduces the ILFP frequency in the low-excitability regime and shifts the power peak toward lower excitability (Fig. 6Bi). The shift toward lower excitability is such that the peak power is in the low-gamma band, much like before. However, unlike changes in W_min,ext, changes in W_GABA,MC do not significantly alter the maximum power amplitude. Excitatory modulation essentially adds energy to the system, setting the overall scale for how high the power can be. Inhibition, on the other hand, gates how much of this power is accessible at a particular frequency. The frequencies in Fig. 6Bi also converge to beta frequencies in the high-excitability regime, indicating that beta frequency is also stable with respect to inhibition.

To further explore the extent of beta regime stability, we covaried MC excitation and inhibition under low and high GC excitability conditions (Fig. 6C). The beta frequency is quite stable over a wide parameter range under both low- and high-excitability conditions. The gamma regime (blue-green-orange colors on the frequency scale) occupies a sizeable region of parameter space under low excitability but becomes very narrow under high excitability. Under high excitability, the lower boundary of the beta regime becomes more horizontal, indicating that beta frequency becomes less sensitive to changes in W_min,ext. The power (Fig. 6C, right) increases with W_min,ext and peaks just outside of the beta regime, in the low-gamma regime. These plots show us that, overall, beta is less sensitive to changes in excitation and inhibition than gamma. This agrees qualitatively with in vivo recordings, which have shown beta oscillations to occupy a narrower frequency band than gamma (Kay 2014; Neville and Haberly 2003)

In this model, inhibition-mediated gamma frequency oscillations only occur below ∼80 Hz, because higher frequencies (gray shaded region in Fig. 6Di) are so low in power that they are essentially indistinguishable from the noise floor (dashed line in Fig. 6Dii). Under high excitability (red connected circles), the beta frequency is remarkably stable and insensitive to changes in W_GABA,MC. However, if W_GABA,MC is too low, the network frequency sharply rises and quickly becomes part of the noise floor, which is why the gamma regime is so narrow in the bottom panels of Fig. 6C. Under low excitability, as W_GABA,MC increases, the frequency falls more gradually through the gamma range until eventually a stable beta frequency is reached (Fig. 6Di, blue connected stars).

The power (Fig. 6Dii) exhibits the same low-gamma frequency peak as described earlier (Figs. 4 and 5). Pharmacological manipulations in awake mice have shown that gamma oscillation power is reduced by high concentrations of picrotoxin (a GABA antagonist) but is increased by low concentrations (Lepousez and Lledo 2013). The peak in the simulated LFP power vs. W_GABA,MC curve for low excitability in Fig. 6Dii (blue connected stars) can account for such a concentration-dependent effect. If the initial degree of inhibition puts the system just to the right of the peak power (for example, W_GABA,MC = 0.018 in Fig. 6Dii), then a small concentration of picrotoxin would only slightly decrease W_GABA,MC, thus increasing the power, but a high concentration would put W_GABA,MC to the left of the peak, resulting in lower gamma power. It is interesting to note that the concentration-dependent power modulation was restricted primarily to low-gamma frequencies in the (Lepousez and Lledo 2013) study, and in our model the peak power is in the low-gamma band. However, the same study also showed that high concentrations of picrotoxin (represented in our model as a decrease in W_GABA,MC) reduced gamma frequency, but in our model a decrease in W_GABA,MC only increases gamma frequency. Indeed, there are other factors that our model does not include, such as asynchronous GABA release (Bathellier et al. 2005) and subthreshold resonance (Brea et al. 2009), which likely play an important role in gamma generation.

Network Response to Fast Change in GC Excitability

In the living system, changes in the excitability of GC dendritic spines leading to activation of dendritic Ca²⁺ channels are not gradual as in Fig. 5 but occur very rapidly (Egger 2008; Pressler, et al. 2007). Therefore, we explore how the computational model reacts to such a fast transition. Because the exact cellular dynamics driving excitability changes in GC dendritic spines are highly complex and are still being studied (Bywalez et al. 2015; Egger 2008; Egger et al. 2005), they are beyond the scope of this model. Instead, we simulate a rapid change in the excitability of the entire GC dendritic arbor by forcing V_rest,GC to vary from −74 to −60 mV following a sigmoid defined as V_rest,GC = −74 mV + 14 mV/[1 + exp(αt)], where α defines the steepness of the transition and t is the time over which the transition occurs.

The speed of transition in V_rest,GC may account for the fast changes in frequency we see in waking animals (Fig. 1). For the slowest changes (Fig. 7, top 2 traces), the ILFP frequency changes continuously, whereas for the fastest changes (bottom 2 traces), the transition from gamma to beta frequencies is sharp but is accompanied by a large low-frequency artifact. The gamma-to-beta transitions observed in vivo (Fig. 1) are sharp (i.e., there are no intermediate frequencies) but also spectrally clean (i.e., there are no sudden artifacts). Therefore, the closest qualitative match between our model and experimental observation is found in the middle trace of Fig. 7, where the gamma-to-beta transition is sharp and no artifacts are produced. The change in excitability does not have to be instantaneous to elicit an apparent sharp gamma-to -beta transition. When the excitability switch is fast, the intermediate values of GC excitability shown in Figs. 4 and 5 are skipped, and therefore high-power oscillations in the low-gamma regime are skipped. This may explain why high-power oscillations in the low-gamma band are rarely observed in vivo (although see Kay 2003).

Fig. 7.

Left: simulated ILFP (black left y-scale) in response to changing V_rest,GC (red right y-scale) from −74 to −60 mV over 1 s with varying speeds (slowest at top, fastest at bottom). In the middle row, the dotted vertical lines indicate the duration of the excitability transition for that plot, which is roughly 80 ms. Right: a Morlet wavelet transform spectrogram (see methods) gives the instantaneous frequency (y-axis) and power (color scale) of the ILFP traces. Freq., frequency.

The simulations producing sharp and clean gamma-to-beta transitions have sigmoidal widths of ∼60–80 ms (Fig. 7, middle trace). This is slow relative to the onset of ADP and LLD in individual spines, which is on the order of 1 ms (Egger 2008; Pressler et al. 2007), but relatively fast on a behavioral timescale (i.e., <1 sniff). Thus our model predicts that the excitability transitions of a population of dendritic spines should take place over roughly 60–80 ms to drive a gamma-to-beta transition during odor sampling.

Relative Contributions of NMDA and N-Type Currents

In the full model, graded inhibition of MCs is mediated by NMDA and N-type Ca²⁺ currents (Figs. 4 and 5). Slice studies have shown that Ca²⁺ flow through NDMARs alone can drive DDI (Schoppa et al. 1998). However, with NMDAR current pharmacologically blocked, slowly decaying DDI can also be evoked solely by AMPAR activation under pharmacological conditions that increase GC excitability (Isaacson 2001), and thus the activation of N- and P/Q- type VDCCs by AMPA-mediated depolarization alone can trigger GABA release. Furthermore, pharmacological manipulations of OB activity in awake mice have shown that odor-evoked beta oscillation frequency and power are not very sensitive to NMDAR antagonists (Fig. 3 in Lepousez and Lledo 2013), providing evidence that beta oscillations may depend on Ca²⁺ flow through N- and P/Q- type VDCCs, but not through NMDARs.

To test the relative contributions of Ca²⁺ flow through NMDARs and VDCCs to the network oscillations we simulate a complete pharmacological block of NMDAR or N-type currents by setting the weight of either current equal to 0 (Fig. 8, A and B). In these simulations the [Ca²⁺] threshold for maximum GABA release ([Ca]_th), the GCD AMPA weight (W_AMPA,GC), and external inputs to MCs (W_min,ext) are identical to those of the full model, but MC inhibition (W_GABA,MC) and the GCD NMDA/N-type excitatory weights (W_NMDA,GC and W_N,GC) are varied. For both pure NMDA and pure N-type models, W_GABA,MC is varied from 0.01 to 0.07. The minimum and maximum W_NMDA,GC and W_N,GC values are chosen such that the transition to the unsynchronized state, where the power is below the noise floor, happens near the midpoint on the horizontal and vertical axes. This allows for a visual comparison of the two models, despite W_N,GC being ∼10⁵ greater than W_NMDA,GC.

Fig. 8.

Parameter exploration of pure NMDA and N-type models. A: the ILFP frequency (left) and power (right) of the pure NMDA model are plotted with V_rest,GC = −70 mV (top) and V_rest,GC = −60 mV (bottom) as the MC inhibitory W_GABA,MC and GC excitatory NMDA weights W_NMDA,GC are varied. Red lines mark the borders of the beta regime (20–30 Hz). Regions where ILFP power is less than or equal to the baseline power are white. In Ai we simulated the default NMDA model with τ_NMDA2 = 75 ms. In Aii we switched the NMDA and N-type constants so that τ_NMDA2 = 18 ms. B: same layout and color scales as in A, but for the pure N-type model with τ_NMDA2 = 18 ms (i; default N-type model), with τ_NMDA2 = 75 ms (ii; switched decay time constant with NMDA model), without CDI (iii), and with N-type Nernst potential lowered to ∼20 mV by reducing the external Ca²⁺ concentration (iv). Ci: a gradual NMDA block was simulated by reducing W_NMDA,GC with GCDs fixed at V_rest,GC = −60 mV. The maximum frequency during the gradual NMDA block is plotted for 3 values of W_N,GC, showing that beta frequency can be sustained by N-type currents as NMDA currents are reduced. Cii: the maximum power of the same 3 curves increases as W_NMDA,GC is decreased because the MCs are overinhibited when W_NMDA,GC is maximum. Di: the mean N-type activation (m_N) is shown for the same 3 curves in C. Because m_N is oscillatory, we include the shaded regions to indicate the minimum and maximum extent of the oscillations. Dii: same as in Di, but for N-type inactivation (h_N). Diii: same as in Di, but for the product (m_N*h_N).

The frequency of both pure models decreases with increased V_rest,GC, as is evident when the same points from low- and high-excitability conditions are compared, and both models generate oscillations spanning the gamma-to-beta range. The power of the pure NMDA model is highest at low-gamma frequencies (Fig. 8A, right), and the gamma regime under high excitability is very narrow (Fig. 8A, bottom left), much like in the full model (see Fig. 6D). The pure N-type model, on the other hand, generates peak power within the 20- to 30-Hz beta regime (outlined by red lines) and even for frequencies below the beta regime (Fig. 8B, bottom right). Why does the full model behave more like the pure NMDA model than the pure N-type model? One factor is the CDI of the N-type current by the NMDA Ca²⁺ current, which reduces the N-type current as NMDA current increases. Another factor is the balance of excitation and inhibition. In the pure N-type model there is a balance at beta frequencies, but in the pure NMDA model the balance is in the low-gamma regime and the MCs in the beta regime are already overinhibited. Therefore, when the two pure currents are combined, the MCs continue to be overinhibited in the beta state and only the peak at low-gamma power remains.

The maximum power of the ILFP oscillations generated by the pure NMDA model is much lower than the maximum power of the pure N-type model (Fig. 8, Ai and Bi). By systematically exploring parameters, we found that the main cause for this difference is the difference in decay time constants of the two models. The default decay times of the NMDA and N-type currents are τ_NMDA2 = 75 ms and τ_Nmax = 18 ms. If we switch the values of the time constants, we find that the pure NMDA power increases dramatically (Fig. 8Aii), whereas the pure N-type power decreases to levels close to the original pure NMDA model (Fig. 8Bii). The maximum power of the oscillations is inversely proportional to the decay time constant, because inhibitory current pulses with short decay times will decay fully each cycle, causing a high amplitude between peaks and troughs of the inhibitory current oscillation, but for longer decay times the pulses fall off more slowly and the difference between peaks and troughs is smaller. With a shorter decay time the pure NMDA model releases less GABA over time, so higher excitatory/inhibitory weights are required to sustain beta oscillations (note the shift of beta regime toward higher W_GABA,MC and W_NMDA,GC in Fig. 8Aii compared with Fig. 8Ai). Interestingly, this shift toward higher weights is accompanied by a greater stability in both gamma and beta regimes. Furthermore, the maximum power is shifted into the beta regime, as was the case for the original pure N-type model (Fig. 8Bi). Therefore, the peak power at low gamma is not a specific consequence of the NMDA model but a general consequence of a long time constant.

We explored the sensitivity of the N-type model's frequency and power landscapes to the absence of CDI. Removal of CDI results in dramatically higher graded inhibitory release probability (Fig. 3D), which causes the pure N-type model to be highly overinhibited for large W_N,GC in the high-excitability condition (Fig. 8Biii, bottom). Curiously, the absence of CDI caused a broadening of the gamma and beta regimes under low excitability (Fig. 8Biii, bottom). This shows that the N-type model has a substantial amount of self-induced CDI and that CDI prevents the inhibitory current from getting too large too quickly as excitability is increased.

We also explored the effect of changing the reversal potential on the frequency and power landscapes. Glutamate receptors such as AMPARs and NMDARs tend to have Nernst potentials near 0 mV, whereas VDCCs tend to have Nernst potentials at the Ca²⁺ reversal potential, E_Ca. In our model E_Ca oscillates with the network near 120 mV, because it is calculated on each time step, but its mean value is controlled by the extracellular Ca²⁺ concentration [Ca]_out (see Eq. 3). When [Ca]_out is reduced to 5 μM, the Nernst potential of E_Ca oscillates near ∼25 mV. This manipulation does not significantly alter the maximum power, but it shifts the beta regime of the pure N-type model toward higher weights (Fig. 8Biv). This is because a lower (positive) Ca²⁺ reversal potential causes the Ca²⁺ current to reverse more quickly so that less inhibition is released over time and higher weights are needed to sustain beta oscillations. In contrast, we increased the NMDA reversal potential from 0 to 100 mV (not shown). This also had little effect on the ILFP power, but the beta regime was now shifted toward lower weights, because higher reversal potential allow more graded inhibition to be released.

Our models show that NMDARs or N-type VDCCs alone can mediate graded DDI, confirming what has been found empirically (Isaacson 2001; Isaacson and Strowbridge 1998; Schoppa et al. 1998). Furthermore, they suggest that Ca²⁺ flow through N-type channels may be responsible for generating the high-power beta oscillations we record in experiments (Fig. 1) due to the shorter decay time of the N-type current compared with the NMDA current. The parameter exploration in Fig. 8, A and B, also shows that the gamma and beta regimes can always be reached by sufficiently increasing the excitatory/inhibitory weights. There can be multiple paths to generating beta as long as inhibitory current is sufficiently strong. This suggests that potentiation of the MC-GC synapse could lead to beta oscillations, although this is not explicitly modeled here.

We were curious to see how the network responds to a gradual block of the NMDA current, since infusion of pharmacological blockers in vivo likely results in only a partial block. Figure 8C shows the network response to decreasing W_NMDA,GC under high GC excitability conditions (V_rest,GC = −60 mV) for three different values of W_N,GC. For these simulations the MC inhibitory weight W_GABA,MC was increased to 0.015 to maintain a strong inhibitory current when NMDA is completely removed. We point out that models with higher W_GABA,MC still generate the full gamma-to-beta range (see Fig. 6Bi). With this arrangement, the beta frequency is quite stable with decreasing W_NMDA,GC (Fig. 8Ci). For the lowest W_N,GC, the LFP frequency rises out of the low-gamma regime as W_NMDA,GC approaches 0, but for higher W_N,GC the frequency remains in the beta regime even when the NMDA current is completely blocked. Interestingly, the gradual reduction of NMDA current actually increases beta power (Fig. 8Cii), because we are reducing the degree of MC overinhibition. We saw a similar effect when reducing the fraction of excited GCs in Fig. 5Cii.

We also investigated the effect that the gradual NMDA block has on the inactivation of N-type current through CDI. Recall that the inactivation variable, h_N, has an inverse dependence on internal [Ca²⁺] (Table 1). We expected that a reduction in NMDA current would lower internal [Ca²⁺] and therefore raise the overall N-type activation by raising h_N. As shown in Fig. 8Diii, the overall N-type activation does increase with decreased NMDA current, but not because of an increase in h_N, which is largely flat (Fig. 8Dii). Instead, the voltage-dependent activation variable m_N is responsible for the overall increased N-type activation, whereas h_N only reflects the background level of Ca²⁺ influx due to the increased excitatory weights W_N,GC. The reduction of MC overinhibition results in more MC spikes, which drives higher GCD membrane depolarization through the AMPA current, thus raising m_N. The increased internal [Ca²⁺] nearly balances the reduction of [Ca²⁺] due to blocking the NMDA current, resulting in a nearly flat h_N. These simulated results provide a possible explanation for in vivo experiments showing that NMDA blockers have little effect on beta frequency (Fig. 3 in Lepousez and Lledo 2013). Namely, blocking the NMDA current results in a higher activation of N-type current through reducing MC overinhibition, which keeps the beta frequency stable. However, there is only partial agreement with the model because these pharmacological experiments also showed the beta power to be relatively unchanged, whereas the beta power in the model is sensitive to NMDA current.

Beta Frequency Dependence on Rise and Fall Time Constants of Ca²⁺ Currents

In this model the beta frequency emerges as a consequence of strong graded inhibition when the NMDAR and N-type channels are sufficiently activated under high GC excitability conditions. In Fig. 8 we show that the beta regime shifts toward higher W when the decay time constants are decreased. In Fig. 9 we explore the dependence of the model response on the time constants alone, holding everything else fixed. The literature reports a wide range of MC IPSC decay times, from about 50 ms to several hundred ms, which may reflect varying concentrations of Mg²⁺ at synaptic clefts as well as varying proportions of NMDARs and VDCCs expressed on individual GCDs (Isaacson 2001; Isaacson and Strowbridge 1998; Schoppa et al. 1998; Urban and Sakmann 2002). As described in methods, NMDA activation is governed by two time constants, the rise time τ_NMDA1 and the decay time τ_NMDA2. For simulations in which τ_NMDA1 is varied, τ_NMDA2 is fixed at 75 ms (Fig. 9, Ai and Aii), and for simulations in which τ_NMDA2 is varied, τ_NMDA1 is fixed at 2 ms (Fig. 9, Aiii and Aiv), which are the default values used throughout this work. The NMDA rise and decay times are close to those used in other models (Bathellier et al. 2005; de Almeida et al. 2013). The N-type model only has one time constant, τ_N. Because τ_N itself is dependent on V_rest,GC, we vary its maximum value, τ_Nmax. The default value of τ_Nmax = 18 ms is taken from Amini et al. (1999) and Zeng et al. (2009).

Fig. 9.

A: dependence of the LFP frequency and power on rise time τ_NMDA1 and decay time τ_NMDA2 of the pure NMDA model. In Ai and Aii, τ_NMDA2 is fixed at 75 ms. In Aiii and Aiv, τ_NMDA2 is fixed at 2 ms. B: dependence of the LFP frequency and power on the N-type activation time constant maximum τ_Nmax of the pure N-type model. Each plot shows 20 curves calculated for uniformly spaced values of V_rest,GC ranging from −75 (blue) to −55 mV (red), as indicated by the color scale.

NMDA rise times as high as 10 ms have been reported in slice studies (Isaacson and Strowbridge 1998; Schoppa et al. 1998). When the rise time τ_NMDA1 of the pure NMDA model is increased, the LFP frequency is driven down (Fig. 9Ai). If the excitability is too low, the slower rise times generate a slower MC IPSC that fails to sufficiently inhibit the MC before a spike is elicited by the constant olfactory receptor neuron (ORN) stimulation. This prohibits MC synchronization, and the frequency shoots upward while the power falls to noise floor levels (Fig. 9Aii). For sufficiently fast rise times (<3 ms), the full gamma-to-beta range can be generated. Interestingly, longer rise times create higher power in the beta regime because the slower rise time increases the inhibition released over time. Increasing the decay time τ_NMDA2 also decreases LFP frequency, but for short decay times (<30 ms) beta oscillations cannot be sustained because the inhibition is not strong enough to sufficiently delay MC spikes (Fig. 9Aiii). As τ_NMDA2 is increased, the LFP frequency in the high excitability condition enters the beta regime. The beta frequency becomes nearly stable after τ_NMDA2 > 50 ms, because lengthening the tail of the inhibitory pulse decay does not significantly change the overall inhibition of MCs each cycle. Therefore, in the pure NMDA model, the beta frequency only emerges if the decay time is long enough, but beta is not very sensitive to the exact value.

A slightly different picture arises for the pure N-type model ILFP oscillations; the frequency changes very gradually with a nearly constant slope over the entire range of decay times (Fig. 9Bi), and for higher excitabilities the frequency is nearly constant. The NMDA model only showed such constant frequency dependence for τ_NMDA2 > 50 ms. This difference between the models is due to the absence of separate rise and fall times for the N-type model, which has a single, dynamic time constant that effectively produces rise and fall times that adapt to GC excitability conditions. Therefore, the frequency of pure N-type ILFP oscillations is much more sensitive to GC excitability and choice of inhibitory excitatory weights than to τ_Nmax. However, as was shown earlier in Fig. 8B, the power is quite sensitive to τ_Nmax (Fig. 9Bii). We find that both pure NMDA and pure N-type models are capable of generating beta oscillations when graded inhibition is sufficiently strong, but near their physiological parameter regimes the pure N-type model generates higher power beta oscillations that are more invariant to parameter changes.

DISCUSSION

We demonstrate that GC excitability could play a pivotal role in regulating OB oscillations, thus closely linking ADPs and LLDs in GCs to beta generation. We hypothesized that GC excitability could control the strength of graded inhibition through NMDA and N-type Ca²⁺ currents, allowing MC-GC dendrodendritic synapses to support both gamma and beta oscillations. Our model generates LFP oscillations ranging from high gamma to beta by varying GC excitability over a range of ∼15 mV. Gradual increases in GC excitability cause monotonic decreases in LFP frequency due to graded inhibitory release through activation of NMDA and N-type currents (Fig. 5). This contrasts with spiking inhibitory systems where increased interneuron excitability leads to increased firing rates and higher synchronization frequencies (Fisahn et al. 2004; Lakatos et al. 2005). Interestingly, beta power increased with smaller GCD populations (Fig. 5C), suggesting that high-power beta oscillations may only recruit a subpopulation of GCs in high-excitability states. Such subpopulations of excited GCs may arise from intrinsic individual GC differences in plateau currents (Egger et al. 2005) and cortical/neuromodulatory inputs to GCs associated with odor-selective MCs (Matsutani and Yamamoto 2008).

GC excitability exerts control over LFP frequency by shifting the excitation-inhibition balance such that low GC excitability induces overexcitation and high GC excitability induces overinhibition (Fig. 4B). With intermediate GC excitability all the MCs fire at low-gamma frequencies (Figs. 4 and 5), reminiscent of type-2 gamma (35–65 Hz) oscillations, which have high coherence across the OB and PC in vivo (Kay 2003; Lepousez and Lledo 2013). We found that maximum power at low-gamma frequencies was primarily due to the long NMDA decay times (Fig. 8A). Interestingly, high-power, low-frequency gamma oscillations are routinely seen in slice preparations (Bathellier et al. 2005; Friedman and Strowbridge 2003; Gire and Schoppa 2008), where Mg²⁺ concentrations are often low and NMDA currents may contribute more to DDI than they do in vivo. A more realistic fast change in GC excitability drives a transition from high gamma to beta, skipping the low-gamma regime (Fig. 7). A change in GC excitability as slow as 100 ms can produce a sudden switch from gamma to beta frequencies, rather than a gradual shift.

In a pure N-type model, the maximum power is in the beta band and is much higher than the pure NMDA due to its shorter decay time (Fig. 8B), suggesting that high-power beta oscillations (Fig. 1) triggered by high-volatility odors (Lowry and Kay 2007) and learning (Martin et al. 2006; Ravel et al. 2003) may depend on VDCCs more than NMDA currents. Starting with both NMDA and N-type currents, we showed that beta oscillations can be sustained as NMDA is blocked (Fig. 8C), in agreement with past experiments (Fig. 3 in Lepousez and Lledo 2013). Comparison of LFP power between models and experiments should be done with care, because our simple model is only a caricature of the LFP, and experimentally recorded LFPs can vary in amplitude depending on electrode position and placement. Because we assume that the same synaptic currents produce both gamma and beta oscillations, we compare the relative size of the beta and gamma oscillations in the model as we do in experiments.

In the full model we found that beta frequencies are more stable than gamma with respect to MC inhibitory and excitatory weights (Fig. 6), but beta power increased dramatically with MC excitatory weight (Fig. 6A). If high-volatility odorants represent stronger inputs, this may explain the high-power beta oscillations produced by these odors (Lowry and Kay 2007). In the pure NMDA and N-type models, the beta regime became more stable as their excitatory synaptic decay times were shortened (Fig. 8). We found that CDI is critical to maintaining beta oscillations in the high excitability of the pure N-type model, since without CDI the MCs became overinhibited. Finally, long decay times are critical to sustaining beta oscillations in the pure NMDA model but not in the pure N-type model (Fig. 9). Together, these results argue that the odor-evoked gamma to beta transition could be triggered by an increase in GC excitability, which drives an increase in VDCC-mediated graded inhibition.

Multiple Factors May Contribute to Odor-Evoked Gamma-to-Beta Transition

Our model shows that increased GC excitability can drive beta oscillations, but it is agnostic as to which inputs control GC excitability. Because GC excitability is regulated by centrifugal, neuromodulatory, and local inputs, beta oscillations may be supported by a convergence of inputs onto GCs, which we summarize in Fig. 10. For example, cholinergic inputs could transform the AHP current to an ADP current (Pressler et al. 2007) so that cortical inputs could trigger excitability increases in GCs and hence generate odor-evoked beta oscillations. This could potentially explain why beta oscillations are dependent on such varied circumstances as stimulus characteristics, the state of the animal, and behavioral context (Cenier et al. 2008, 2009; Lowry and Kay 2007; Martin et al. 2006). Because many factors influence the strength of inhibition, such as the inhibitory/excitatory weights and even the external Ca²⁺ concentration (Fig. 8), our model argues that there can be multiple paths to generating beta oscillations, as long as there is sufficiently strong inhibition.

Fig. 10.

A summary of the possible factors contributing to the gamma-to-beta transition. Those that explicitly control GC excitability are marked in bold with gray arrows. GC excitability can be regulated by at least 3 distinct inputs to GCs, all of which may cooperate in the generation of beta oscillations. Neuromodulatory fibers may target the soma and GC dendritic arbors and gate excitatory inputs from MCs and cortical neurons. Our model summarizes these different sources of GC excitability control by a single parameter, the GC resting potential, to show that GC excitability has a direct influence on the LFP frequency. However, our model does not include respiratory modulation, ORN adaptation, and corticobulbar loops, which may also contribute to beta generation.

Respiration may also influence beta generation (Fig. 10). Fast airflow during inhalation coincides with gamma oscillations, and decreased airflow during exhalation coincides with beta oscillations in urethane-anesthetized rats (Cenier et al. 2008, 2009; Fourcaud-Trocmé et al. 2011). Waking rats also slow their respiration during late odor sampling when beta emerges (Rojas-Líbano and Kay 2012).

The gamma-to-beta sequence recorded in rats resembles the transition from fast to slow oscillations detected in moth olfactory LFPs associated with firing rate adaptation of peripheral ORNs (Ito et al. 2009). Firing rate adaption also occurs in rat and amphibian ORNs (Kurahashi and Menini 1997; Laing and Mackay-Sim 1975; Zufall and Leinders-Zufall 2000). It is possible that ORN firing rate adaptation plays a role in the gamma to beta transition in rats (Fig. 10).

Comparison with Other Theories of Olfactory Beta Generation

Beta oscillations are highly coherent between the OB and PC, and it has been proposed that beta oscillations may be generated by long-distance action potential propagation within the loop encompassing MCs, PC pyramidal neurons, and GCs (Martin et al. 2006; Neville and Haberly 2003). However, PC pyramidal neurons also target deep short axon cells, which strongly inhibit GCs (Boyd et al. 2012). Furthermore, beta oscillations are also coherent between the OB and the entorhinal cortex and hippocampus (Gourévitch et al. 2010; Kay and Beshel 2010; Martin et al. 2007), so it is unclear which of these loops would be responsible for beta. Our model generates beta oscillations intrinsically in the OB and relies on centrifugal innervation solely for the regulation of GC excitability, reminiscent of the role that OB input to PC plays in the generating PC oscillations (Freeman 1968). The model predicts beta-band directional influence from the OB to other cortical areas, as we have shown experimentally (Gourévitch et al. 2010; Kay and Beshel 2010). Nonetheless, it is possible that a combination of GC excitability regulation and long-distance action potential propagation stabilizes the beta frequency.

To our knowledge, there are only two other computational models of OB beta (David et al. 2015, Fourcaud-Trocmé et al. 2011). Both of these models, like ours, imply that the mutual exclusivity of gamma and beta oscillations suggests a common mechanism, namely, the ionic currents at the dendrodendritic MC-GC synapses. The Fourcaud-Trocmé et al. model also uses graded inhibition but argues that the critical parameter in switching from gamma to beta is the excitatory synaptic conductance to the MCs. The David et al. model argues that the switch between gamma and beta depends on nonspiking (graded) and spiking states of the GCs respectively. Our model is closer to the David et al. model in spirit, because both models assume a switch in the state of the GCs. Although experiments have shown that the majority of DDI is mediated by Ca²⁺ currents through NMDARs and VDCCs (Chen et al. 2000; Isaacson and Strowbridge 1998; Schoppa et al. 1998), neither of the other models includes Ca²⁺ currents, which are essential to beta generation in our model.

Differences between modeling approaches are to some extent motivated by differences in experimental data. Whereas the other models aim to reproduce experimental data showing gamma-to-beta transitions within a single sniff in anesthetized rats (Cenier et al. 2009), we reproduce high-power beta epochs lasting several sniffs that occur during both learning and passive exposure to high-volatility odorants (Fig. 1). These models together suggest that beta oscillations arise from an appropriate convergence of sensory stimulation and centrifugal feedback onto GCs.

Recent current source density analysis has suggested that gamma and beta oscillations are generated by distinct sublaminar networks within the EPL (Fourcaud-Trocmé et al. 2014). Distinct sublaminar networks are compatible with our GC excitability-based hypothesis, because different sublaminae may represent distinct developmental stages of GCs with different excitability characteristics (Lepousez et al. 2013; Petreanu and Alvarez-Buylla 2002).

Limitations of the Model

Every model must trade off simplicity with accuracy. We used a standard NMDA current model (Jahr and Stevens 1990), which does not include CDI, but NMDA channels can also exhibit CDI (Legendre et al. 1993; Zhang et al. 1998), which could dynamically compete with VDCC currents. Our model also excludes the GC soma and relies on graded inhibition alone. However, somatic spikes could propagate through the entire dendritic arbor of a GC to trigger global lateral inhibition of MCs (Mouret et al. 2009). We do not include this in our model because it would require simulating bidirectional conductance along GC primary dendrites, the properties of which are still being studied (Balu et al. 2007; Egger et al. 2005). Nonetheless, our model implicitly depends on GC spikes, because spikes trigger the ADP (Pressler et al. 2007) or LLD (Egger 2008) that provides the mechanism for transition between gamma and beta oscillations. Although GC spikes were recently recorded for the first time in awake animals (Cazakoff et al. 2014), they have yet to be recorded during beta oscillations.

Our GCDs are modeled as single compartments, but recent experiments have shown that GC dendritic spines can independently support Na⁺ spike generation (Bywalez et al. 2015) and that locally produced Ca²⁺ spikes mediated by T-type VDCCs can spread activity across the entire dendritic arbor to synchronize GABA release from all the dendritic spines of a given GC cell (Egger et al. 2005). Our model does not capture this fine-grained activity because we aimed to model population activity. However, it remains an interesting and open question how this fine-grained activity influences population activity.

GRANTS

This work was supported by NIDCD Grant R01DC014367 (to L. M. Kay).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

B.L.O. and L.M.K. conception and design of research; B.L.O. performed experiments; B.L.O. analyzed data; B.L.O. and L.M.K. interpreted results of experiments; B.L.O. prepared figures; B.L.O. drafted manuscript; B.L.O. and L.M.K. edited and revised manuscript; B.L.O. and L.M.K. approved final version of manuscript.

APPENDIX

Derivation of [Ca]_baseline

Because in the absence of MC spikes I_N has a nonzero activation even for resting potentials as low as −75 mV, there is a constant Ca²⁺ current that sustains an internal [Ca], which we call [Ca]_baseline. To prevent [Ca]_baseline from driving tonic inhibition of MCs, we subtract it in the calculation of P_release, as shown in Table 1. We derive [Ca]_baseline by solving the steady-state equation for [Ca] evaluated at V_rest,dGC:

$${\tau }_{Ca}\left[\dot{C}a\right]=-\left[Ca\right]+{\rho }_{Ca}\left(I_{NMDA}+I_N\right)=0$$ (A1)

$$\left[Ca\right]={\rho }_{Ca}\cdot I_N={\rho }_{Ca}\cdot W_N{\overline{m}}_N\frac{{10}^{-4}}{{10}^{-4}+\left[Ca\right]}\left[E_{Ca\left(N\right)}-V_{rest,dGC}\right]$$

Note that in the absence of MC spikes, the steady state of I_NMDA is 0 and the steady state of m_N is m̄_N evaluated at V_rest,dGC. Now we have a quadratic expression in [Ca],

$${\left[Ca\right]}^2+{10}^{-4}\left[Ca\right]-{10}^{-4}{\rho }_{Ca}{W}_N{\overline{m}}_N\left[E_{Ca\left(N\right)}-V_{rest,dGC}\right]=0,$$ (A2)

which has the following solution:

$$\left[Ca\right]=\frac{-{10}^{-4}\pm \sqrt{{10}^{-8}-4\ast {10}^{-4}{\rho }_{Ca}{W}_N{\overline{m}}_N\left[E_{Ca\left(N\right)}-V_{rest,dGC}\right]}}{2}.$$ (A3)

$$\left[Ca\right]\approx \pm \sqrt{-{10}^{-4}{\rho }_{Ca}{W}_N{\overline{m}}_N\left[E_{Ca\left(N\right)}-V_{rest,dGC}\right]}$$

In the approximation we take advantage of the fact that the second term in the square root is of order 1 >> 10⁻⁴ >>10⁻⁸. Now we must recognize that E_Ca itself is a logarithmic function of [Ca], and thus an analytical solution here is not tractable. Instead, [Ca]_baseline is found as the value for which the left-hand side (LHS) and right-hand side (RHS) of the following equation intersect:

$$V_{rest,dGC}+\frac{{\left[Ca\right]}_{baseline}^2}{{10}^{-4}{\rho }_{Ca}{W}_N{\overline{m}}_N}=\frac{RT}{zF}\ln \frac{{\left[Ca\right]}_{out}}{{\left[Ca\right]}_{baseline}}.$$ (A4)

In practice this is achieved by finding the index of the minimum of abs(LHS − RHS), giving a value for [Ca]_baseline that agrees well with the steady-state internal [Ca] of all 720 simulated GCDs. One should note that this method breaks down if [Ca]_baseline reaches or overshoots [Ca]_th, because in that case P_release will be 0 or negative. We keep our model parameters in a range where this problem is not encountered.