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1Laboratory of Perception and Memory and 2Laboratory of Receptors and Cognition, Pasteur Institute, Centre National de la Recherche Scientifique Unité de Recherche Associée 2182, Paris, France
Submitted 28 October 2005; accepted in final form 21 December 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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; Lledo et al. 2005; Shepherd et al. 2004). After the pioneering studies of Lord Adrian (Adrian 1942, 1950), oscillations in the local field potential (LFP) of the OB are known to be a defining characteristic of sensory coding. From these studies, it has been suggested that synchronous fast oscillations in central sensory systems allow the formation of neural assemblies that represent sensory objects. In experiments performed on the olfactory processing circuitry, odor-triggered oscillations at a frequency of 10–40 Hz in nonmammalian vertebrates and at both the beta band (15–30 Hz) and the low- and high-gamma bands (respectively 40–60 and 60–100 Hz) in mammals, are always seen. Although odor-evoked, temporally structured activity in the OB is well known to reflect odor coding and discrimination (Kay 2003; Laurent et al. 2001; Li and Hopfield 1989; Linster and Hasselmo 1997; Martin et al. 2004; Nusser et al. 2001; Urban and Sakmann 2002), the nature of the bulbar microcircuit elements that support these fast rhythms are still unclear. Computational models of the mammalian OB were previously designed (Freeman 1975 1987; Li and Hopfield 1989; Linster and Hasselmo 1997) but were limited to the processing and coding of odorant information rather than the mechanisms of generating oscillations.
As in many cortical areas, there is a population of local interneurons in the OB, which is critical for shaping the dynamics of relay neurons, the mitral/tufted cells (MCs). These then convey information to higher integrative centers. MCs make various synaptic and nonsynaptic connections in the OB that mostly occur between dendrites (Schoppa and Urban 2003). These connections extend through multiple layers and integrate distinct microcircuits located either in glomeruli or in the external plexiform layer. Widespread interactions with the granule cells (GCs), which are local GABAergic interneurons, occur along the lateral dendrites of MCs (Lledo et al. 2005; Shepherd et al. 2004; Shipley and Ennis 1996). These dendrodendritic connections, which constitute the largest synaptic population of the OB, support both recurrent and lateral interactions (Schoppa and Urban 2003). Models have suggested for a long time that bulbar activity becomes rhythmically synchronized as a result of feedback interactions between MCs and GCs (Freeman 1975; Rall and Shepherd 1968). This is further supported by experimental studies demonstrating that local interneurons play a critical role in generating oscillatory activity in the antennal lobe of honeybees (Stopfer et al. 1997) and in the procerebral lobe of molluscs (Teyke and Gelperin 1999). Recent studies carried out in vertebrates showed that local inhibitory circuits are key elements to generating evoked gamma oscillations (Kay 2003; Lagier et al. 2004; Neville and Haberly 2003; Nusser et al. 2001). On the other hand, both theoretical (Migliore et al. 2005) and experimental studies (Christie et al. 2005; Schoppa and Westbrook 2002) recently suggested that direct excitation between MCs might be sufficient to synchronize them in the gamma range. Here, we have used a physiologically based network model of the OB combined with an experimental approach to investigate the precise synaptic organization and properties that drive the temporal patterns in OB activity. We propose that lateral inhibition plays a key role in generating sustained gamma oscillations and population synchrony.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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We used a single compartment that includes voltage-dependent currents described by Hodgkin–Huxley kinetics to model MC. These currents were fast transient (INa) and persistent (INaP) sodium currents, a delayed rectifier potassium current (IKfast), and two transient potassium currents (fast inactivating IKa, slow inactivating IKs). Synaptic (Isyn), input (Iinput), and noise (Inoise) currents were added to the voltage-dependent currents (Eq. A1 in the APPENDIX). Values for passive membrane properties (Cm = 0.01 F/m2 and Rm = 10 
m2) and the leak potential (EL = –66.5 mV) were chosen from Bhalla and Bower (1993). Values for Na (ENa = 45 mV, gNa = 500 S/m2) and Kfast (EKfast = –70 mV, gKfast = 500 S/m2) currents were also taken from Bhalla and Bower (1993) except that their activation and inactivation curves were shifted by –8 mV to lower the spike threshold in accordance with previous results (Desmaisons et al. 1999). The Ka current (EKa = –70 mV, gKa = 100 S/m2) was adapted from Wang et al. (1996) and its kinetics were taken from the Senselab databank (http://senselab.med.yale.edu). Ks (EKs = –70 mV, gKs = 310 S/m2) and NaP (ENaP = 45 mV, gNaP = 1.1 S/m2) currents were adapted from Wang (1993). Each cell received Gaussian white noise with a variance equal to 0.12 A/m2. All voltage-dependent current amplitudes were varied by ±20% among the neurons in our models. Further details are presented in the APPENDIX and in Supplementary Fig. 1. 1
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Conductance kinetics were modeled by a delayed difference of two exponential functions of time t
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r and d are the rise time and decay time of synaptic currents, respectively. Mr,d is a factor of r and d, such that the maximum of g(t) (with respect to time) is equal to 1. The resulting current is given by
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l, ranging from 0 to 3 ms (Figs. 1 to 4). Throughout this study, except for the result presented in Fig. 8A, lateral inhibition was modeled with d = 20 ms, r = 3 ms, and l = 2 ms; recurrent inhibition with d = 50 ms, r = 1 ms, and l = 1 ms; and lateral excitation with d = 10 ms, r = 0.5 ms, and l = 1.4 ms (see also Fig. 8 and Table 1).
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d = 10 ms and r = 0.5 ms, taken from Castillo et al. (1999)]. In the absence of spikes, the probability had a constant value (p0), which produced spontaneous synaptic activity. In response to a spike, a function was added to p(t) to transiently increase the number of incoming synaptic events. Two different decay times were used to describe the probability for recurrent (d = 150 ms) and lateral inhibition (d = 50 ms). The rise time and latency were both set to 0.5 ms. Finally, excitatory olfactory nerve (ON) inputs into glomeruli were modeled either by a difference of two exponentials (d = 300 ms and r = 50 ms) or by a step function (constant inputs).
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The network of MCs used for the simulations was a matrix of 10 x 10 cells. The coupling amplitude (either the magnitude of the conductance change or the probability of inhibitory events) between two MCs was dependent on their distance d (expressed as cell number). The spatial profile of connectivity was Gaussian
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Simulated local field potential
We constructed a simulated LFP (sLFP) by filtering the mean membrane potentials across all 100 cells. We used this as a parameter to assess network oscillations and synchrony, from which we could extract the frequencies of rhythmic activities. Filtering was carried out numerically using a band-pass Bessel filter (10–100 Hz) and was done twice (forward and backward) to remove artificial phase lags.
Simulation procedure
All network simulations were performed using a fourth-order Runge–Kutta integration method. The time steps were either 10, 20, or 40 µs. The results were not affected by changing this parameter. The input onset was 200 ms after the start of the simulation to allow the simulated cells to stabilize. The C++ source code was compiled on an Intel PC using a DEV C++ compiler (version 4.9.7.0 [EC] ). A simulation of 1 s of network activity took between 7 to 22 min of CPU time depending on the number of synaptic connections. Visualization and analysis of the results were carried out with Elphy software (G Sadoc, CNRS, Gif-sur-Yvette, France).
Synchronization and oscillation indices
The synchronization index is equal to
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i denotes the phase of each spike in the network with respect to local maxima in the experimental/simulated LFP (Fig. 2F). It corresponds to the length of the phase distribution vector and ranges from 0 to 1. The oscillation index (ranging from 0 to 1) corresponds to the amplitude of the secondary peak of the LFP autocorrelation, normalized by the mean and variance of the signal (Fig. 2E). The position of the secondary peak of the autocorrelograms was used to determine the oscillation frequency. In the simulated network, when inputs were step functions, the analysis was performed over a 700-ms time range, and when the inputs varied with time (as in slice recordings), the analysis was performed over a 200-ms time range.
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OB slices were prepared from 4- to 6-wk-old C57Bl/6 mice and electrophysiological recordings were carried out as previously described (Lagier et al. 2004). For extracellular single-unit recordings, thick-walled glass pipettes (Clark Electromedical Instruments, Reading, UK) were pulled with a custom vertical puller and filled with 2 M NaCl (DC resistance 5–15 M). Single-unit recordings were amplified using an Iso-DAM8 differential amplifier (World Precision Instruments, Hertfordshire, UK), filtered (0.3–3 kHz), and sampled at 20 kHz. MCs were identified by their peristimulus time histogram (PSTH) profile and their phase locking to LFP oscillations (see Fig. 5 and Lagier et al. 2004).
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We analyzed the frequency distributions using the Kolmogorov–Smirnov test, the pharmacological experiments using Student's t-test, and the spike-phase distributions using three different circular statistical tests (Rayleigh, Watson, and Kuiper) (Lagier et al. 2004). The level of significance was set to P = 0.05. All results are expressed as means ± SE.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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From the literature, we selected a minimal set of parameters (Bhalla and Bower 1993; Davison et al. 2000; Wang 1993) required to reproduce the characteristic properties of MCs, that is, a delayed onset of firing (e.g., Balu et al. 2004; Chen and Shepherd 1997) and repetitive firing of clustered action potentials separated by periods of subthreshold oscillations in response to DC current injection (e.g., Balu et al. 2004; Chen and Shepherd 1997; Desmaisons et al. 1999). These characteristics arise from intrinsic membrane properties because they persist in the presence of ionotropic glutamate and 
-aminobutyric acid (GABA) receptor blockers (Balu et al. 2004; Desmaisons et al. 1999). Furthermore, the presence of A-type potassium (Ka; Wang et al. 1996) and slow-inactivating potassium currents (KS; Balu et al. 2004) in MCs can explain both delayed firing onset and spike clustering (Fig. 1A).
As indicated by their sensitivity to tetrodotoxin (TTX) (Desmaisons et al. 1999), subthreshold oscillations could be generated in MCs by the interaction between a persistent sodium current (NaP), KS, and intrinsic noise. The combination of NaP and KS is known to produce a resonance of the membrane voltage in some neurons (Hutcheon and Yarom 2000). This resonance property, combined with intrinsic noise, generates irregular membrane subthreshold oscillations (Fig. 1A). The peak of resonance, and therefore the frequency of subthreshold oscillations, depends on the activation time constant of Ks (Richardson et al. 2003), which we set here to 10 ms.
The suppression of all Na+ currents abolishes the subthreshold oscillations (Desmaisons et al. 1999). We found the same in our model, with either all Na+ currents being suppressed or the NaP current being selectively suppressed (Fig. 1B). By contrast, when we changed the K+ driving force in our model (Fig. 1B) the subthreshold oscillations were not affected, which has also been seen experimentally (Desmaisons et al. 1999). We also found that the frequency of oscillations and the instantaneous intraburst spike frequency were highly correlated in our model (Fig. 1C), which has also been reported experimentally (Desmaisons et al. 1999). However, in our model, the frequency of the membrane potential oscillations was 10% faster than the firing frequency, whereas the experimental oscillation frequency was tightly linked with the firing frequency (Fig. 1D). Nevertheless, the good agreement between our simulation and experimental data suggested that our simple model cell was capturing the main features of MCs recorded in slices.
Generating oscillations in a simplified olfactory bulb circuit
We designed a model of OB local circuitry containing 100 heterogeneous MCs (see METHODS) and receiving ON inputs. This network model includes the main synaptic interactions (lateral excitation and inhibition), and recurrent action (autoexcitation and recurrent inhibition). In a first set of simulations (Figs. 2–6), all synaptic currents were approximated as their averaged time course (smooth functions) using a difference of two exponentials. This is appropriate for lateral excitation and we derived our kinetic constants from their excitatory postsynaptic potential (EPSP) time courses (Urban and Sakmann 2002). However, for lateral inhibition, this approximation masks two different aspects of the coupling. First, lateral inhibition between MCs is the sum of small asynchronous inhibitory postsynaptic currents (IPSCs) rather than being a simple exponentially decaying event. Second, it is not a direct coupling (like lateral excitation) but is mediated by GABAergic interneurons through dendrodendritic synapses. We omitted interneurons from our model circuit (but kept their resulting synaptic interactions) because the integration properties of bulbar interneurons are still unclear despite numerous recent studies (Egger et al. 2005; Hall and Delaney 2002; Isaacson and Strowbridge 1998; Murphy et al. 2005; Pinato and Midtgaard 2005; Schoppa and Westbrook 1999). A consequence of such a simplification is that the inhibitory coupling parameters are not derived from a real biophysical system. Thus our approach holds only as long as the conclusions drawn do not depend on a particular set of values chosen for these parameters. Therefore to be consistent, we analyzed systematic variations of amplitudes and time constants of lateral and recurrent inhibition and we draw our conclusions only from general observations. We also systematically varied the lateral- and autoexcitation parameters to verify that their initial values were not biasing the simulation outputs.
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Inhibition- versus excitation-coupled network models
We assessed the stability of the synchrony for variations in the recurrent inhibition strength (which controls MC excitability) and lateral coupling strength (inhibition for the I-network in Fig. 3A, left, and excitation for the E-network in Fig. 3A, right). In the I-network, we found that a wide range of parameters (lateral and recurrent inhibition between 1 and 32 S/m2) led to high synchronization indices (S.I. >0.4; Fig. 3A, left, predominance of warm colors). By contrast, in the E-network, high levels of synchronization (S.I. >0.4) were limited to a small set of parameters (Fig. 3A, right, predominance of cold colors) that corresponded to a combination of strong recurrent inhibition (>32 S/m2) and strong lateral excitation (>0.4 S/m2, Fig. 3A, right). We observed a similar stability with oscillations: the E-network was more sensitive than the I-network to changes in both lateral and recurrent inhibition amplitudes (Fig. 3B).
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Frequency of oscillations and firing rates
We analyzed the firing pattern of MCs to differentiate the oscillatory states in the I- and E-networks. We found that the oscillatory activity in the E-network always had mean firing rates equal to or higher than the oscillation frequency (Fig. 4B1). The MCs fired at least once during each oscillation cycle (see Fig. 4B2) and the network oscillations arose from synchronization of all MCs. This spiking behavior was observed for all the MC and network parameters we tested. In particular, even after reducing MC excitability by a drastic reduction in sodium conductance [from 500 S/m2 taken from a theoretical model in Bhalla and Bower (1993) to 90 S/m2 given by Bischofberger and Jonas (1997) who effectively measured sodium conductance in the primary dendrite of MCs], no cycle skipping of the MC spike was observed (see Supplementary Fig. 1). Therefore we conclude that only the lateral inhibitory model reproduces the skips in MC firing. Similarly, changes in the time constants of the lateral excitation did not lead to a state where MCs were firing at a lower rate than the oscillation frequency (see Supplementary Fig. 2). In all cases, individual spiking frequency came from alternating phases of inhibition (caused by a spike in the cell) and excitation (coming from ON input and from surrounding activity). This required an excitatory coupling strong enough to make all neurons fire at the same time (synchronization of the individual oscillators; see for example Van Vreeswijk et al. 1994), and an inhibition strong enough to silence each neuron afterward. Therefore the balance between excitation (including input strength) and inhibition strictly determined the oscillation frequency (Fig. 3, C and D). By contrast, in the I-network, the individual neurons fired at a lower rate than the global oscillation frequency (Fig. 4A1), with the individual neurons not firing on each oscillation cycle (Fig. 4A2). Therefore in this model, the coupling periodically modulates the instantaneous firing rate of the network rather than synchronizing the oscillating neurons. Thus oscillations in the I- and E-networks seemed to occur through two different mechanisms.
Some of the dynamic properties of these mechanisms could be studied using electrophysiological experiments. In particular, our first set of simulations yielded one simple prediction. A low individual firing rate in a network (e.g., Fig. 4A2) suggests that lateral inhibition dominates lateral excitation. By contrast, a high individual firing rate, equal to the oscillation frequency (e.g., Fig. 4B2), suggests that lateral excitation dominates for both oscillation and synchronization. Therefore we studied electrophysiological recordings in OB slices to investigate which coupling type predominates for generating gamma oscillations. First, we recorded ON-induced LFP oscillations using simultaneous MC single-unit recordings (Fig. 4C). The instantaneous firing frequencies (Fig. 4D, white columns, n = 245) almost never overlapped LFP oscillation frequencies (black columns, n = 50; Kolmogorov–Smirnov test, P < 0.001). On average, the MC firing frequency was 35.9 ± 3.2 Hz, whereas the mean LFP oscillation was 63.1 ± 1.0 Hz (Fig. 4E; n = 16; Kolmogorov–Smirnov test, P < 0.001). All neurons used in this analysis showed a sustained firing rate and a phase locking to the LFP (Fig. 5, for an individual cell in A and for all the recorded cells in B). These results showed that lateral inhibition was the major coupling required to generate gamma oscillations in the OB network. Therefore we focused our analysis on the I-network.
Determining the oscillation frequency in the I-network
We investigated the properties of oscillations in the I-network by independently varying the inhibitory coupling event time constants (around the reference values given in METHODS) while keeping the total charge of the event constant to minimize changes in the mean network firing rate. We always kept the global onset of the event (i.e., latency + rise time) >2–3 ms to maintain the oscillation frequency in the gamma range. For shorter onset times, the frequencies were >80 Hz (data not shown; see Brunel and Wang 2003). Rise time constant and latency both had the same independent effect on oscillation frequency while preserving oscillations and MC synchrony. Indeed, increasing rise time (from 0.2 to 5 ms) caused the oscillation frequency to continually decrease (from 73.4 to 46.1 Hz; Fig. 6A, left), without greatly affecting synchrony and oscillation indices (O.I. ranged from 0.76 to 0.58; S.I. from 0.52 to 0.69; Fig. 6A, right). Similarly, increasing the latency of the coupling event (from 0.2 to 5 ms) caused a steady decrease in frequency (from 79.0 to 33.5 Hz; Fig. 6B, left), again without greatly affecting the synchrony and oscillation indices (O.I. ranging from 0.62 to 0.66; S.I. from 0.49 to 0.71; Fig. 6B, right). By contrast, changing the decay time did not greatly affect the oscillation frequency but did affect the synchrony and oscillation indices. For decays >5 ms, the oscillation frequency was unchanged (between 50 and 55 Hz, Fig. 6C, left) whereas the MC synchrony decreased for decays >40 ms (S.I. from 0.65 to 0.24; Fig. 6C, right). Likewise, the oscillation quality was reduced with both short (<20 ms) and long decays (>80 ms; O.I. <0.35 in both cases; Fig. 6C, right). These results demonstrate the high sensitivity of the I-network oscillatory dynamic to the temporal features of lateral inhibitory coupling. Furthermore, the frequency is crucially determined by the onset (i.e., latency and rise time) of the coupling rather than by its duration (i.e., decay time). Given that the rise time of lateral inhibition coupling is strongly dependent on the modalities of GABA release, we developed a simple model to check the influence of the sum of multiple individual IPSCs on the time course of global IPSCs.
Modeling the temporal features of inhibitory synaptic events in mitral/tufted cells
Lateral and recurrent inhibition can be considered as a long-lasting global increase in the frequency of vesicular GABA release. Thus in our second model, IPSCs received by MCs were considered as the sum of unitary IPSCs (Fig. 7, A–C, black traces). In a given MC, activity transiently changed the probability of receiving unitary IPSCs (Fig. 7, A–C, red traces). We tuned parameters to the particular properties of GC to MC inhibition [i.e., decay of unitary IPSCs of 10 ms, see Castillo et al. (1999); ratio of lateral to reciprocal inhibition of 0.1, see Isaacson and Strowbridge (1998)]. We focused on this type of synapse because it is the main inhibitory connection involved in generating gamma oscillations (Lagier et al. 2004).
We observed that the rise time of the global coupling event (Fig. 7C, black trace) was large (around 8–10 ms) as a result of the gradual summing of IPSCs despite the fast rise time of the individual IPSCs (0.5 ms) and of the release probability (0.5 ms). Thus the asynchronous GABA release time constants were in the range of parameters giving rise to oscillations in the gamma frequencies. We then simulated a voltage-clamp experiment for pairs of MCs to further validate this model (Fig. 7D). We observed rise times for IPSPs very similar to those observed by Urban and Sakmann (2002) but we found that the latencies were different. This difference can be explained by the rise in global calcium levels in GCs (leading probably to lateral GABA release) needing sufficient presynaptic activation (Egger et al. 2005). Further experiments on this class of bulbar interneurons are required for a better understanding of their temporal integration.
Sensitivity of oscillation and synchronization to changes in the degree of inhibition
Because our results suggested that inhibition was the main key player involved in gamma oscillation, we built a network having asynchronous recurrent inhibition and asynchronous lateral inhibition to verify our general conclusions in a more realistic framework. We validated our second prediction (i.e., stability) by studying the effect on stability of global changes in inhibition strength. We subjected the model to proportional changes of both lateral and recurrent inhibition (Fig. 8A). With asynchronous GABA release, the oscillations were slightly reduced compared with smooth IPSCs (compare Fig. 3 with Fig. 8) but were closer to experimental data. This can be partly explained by a large amount of synaptic noise in MCs (colored noise as opposed to the white noise that was injected into the modeled cells) being caused by the asynchronicity of the release that was not taken into account in our first approach.
When we increased the degree of inhibition, we observed a large reduction in the MC mean firing rate, from 105 to 17 Hz, whereas the oscillation frequency was only slightly decreased (from 89 to 57 Hz with a large plateau around 60 Hz, Fig. 8A, left). This reduction in firing rate was accompanied by an increase in MC synchrony (from 0.02 to 0.34, Fig. 8A, right), whereas the oscillation stability initially increased up to a peak at around 0.02–0.03 S/m2 before decreasing with stronger inhibition. We studied this experimentally by recording induced LFP oscillations in OB slices bathed with increasing concentrations of gabazine, a GABAA receptor antagonist (from 0.5 to 20 µM). We observed a large reduction in the firing of MCs (Fig. 8B, top right, n = 5 for all concentrations; paired t-test with control conditions, P < 0.005) with decreasing gabazine concentrations. However, the oscillation frequency remained constant (Fig. 8B, top right, n = 5, paired t-test, P > 0.2), with the lowest oscillation frequency being 64.7 ± 2.7 Hz for 2 µM gabazine and the highest being 68.2 ± 4.3 Hz for 1 µM gabazine. The decrease in firing rate we observed with decreasing gabazine concentrations was accompanied by an increase in synchronization of MCs. The S.I. increased from 0.03 ± 0.01 (20 µM gabazine) to 0.31 ± 0.06 (control) (Fig. 8B, bottom right, P < 0.005 for 20 and 2 µM and P < 0.05 for 1 µM; paired t-test with control conditions). In these experiments, the O.I. was sensitive only to moderately high concentrations of gabazine (0.12 ± 0.08 and 0.39 ± 0.07 for 20 and 2 µM, respectively, compared with 0.57 ± 0.06 for control conditions; P < 0.005 and P < 0.05, respectively, with paired t-test). At low gabazine concentrations (1 and 0.5 µM), the O.I. was unchanged (0.56 ± 0.09 and 0.59 ± 0.05 compared with 0.57 ± 0.06 for 1 and 0.5 µM and control conditions, respectively; Fig. 8B, left traces and Fig. 8B, bottom right). In the presence of 20 µM of gabazine, the O.I. after stimulation was not different from that from before stimulation (respectively, 0.12 ± 0.08 and 0.22 ± 0.3), indicating that the stimulation of sensory inputs did not induce oscillations. Values for the frequency, O.I., and S.I. from the model were close to those observed experimentally. The combination of theoretical and experimental approaches clearly demonstrates that lateral inhibition reliably generates evoked gamma oscillations in the OB circuit. Regarding lateral excitation, additional experiments need to be conducted to unambiguously demonstrate whether it plays only a minor role.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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Generating gamma oscillations in a neuronal network
Various mechanisms have been proposed to account for the generation of fast oscillations in neuronal networks. Previous studies focused particularly on the role of weak coupling. Weak couplings, whether they are excitatory (Van Vreewijk et al. 1994) or inhibitory (Wang and Buzsáki 1996), always require the intrinsic neuronal properties to be relatively homogeneous. However, this was not true for the OB network because the firing patterns were heterogeneous among MCs (see Fig. 4D, individual irregular firing pattern; Fig. 4E, mean firing frequency ranging from 15 to 66 Hz). Also, cell-to-cell variability limits the synchronization mechanisms that can be based on finely tuned subthreshold properties (e.g., Lampl and Yarom 1993). Indeed, the variability of the intrinsic MC oscillatory properties causes the resonant properties of the MC ensemble to disappear. For example, when we changed the mean subthreshold properties of the MCs in our model (e.g., resonance frequency) the oscillation frequency and quality of the network were not significantly affected (data not shown). Synchronization between MCs may also be triggered by a correlated, fast-fluctuating input (Hopfield and Brody 2003; Nikonov et al. 2002), although such oscillating inputs have so far never been reported in mammalian olfactory systems.
The precise nature of lateral excitation within the OB is not yet known. Therefore we assessed the role of excitatory coupling across a wide range of parameters, including brief excitatory couplings similar to those provided by gap junctions. Independent of these variations, the outputs of our network models were identical (see Supplementary Fig. 2). We always observed irregular firing and cycle skipping in the I-network and homogeneous firing without cycle skipping in the E-network. Thus given the particular characteristics of the OB network (i.e., slow lateral inhibition, cell-to-cell variability), we found that the effects of lateral inhibitory coupling dominate those of excitatory coupling in controlling the features of induced gamma oscillations.
Mechanism of oscillations with irregular individual spiking
Gamma oscillations with a low individual spiking rate have been observed in the hippocampus of behaving rats (Bragin et al. 1995) and, more recently, in slices of rat hippocampus or somatosensory cortex (Buhl et al. 1998; Fellous and Sejnowski 2000; Fisahn et al. 1998) using pharmacological manipulations. Previous detailed models of the hippocampus successfully reproduced these oscillations (e.g., Traub et al. 2001), but the mechanism generating population rhythm with irregularly spiking neurons is still unclear. In OB slices, no pharmacological agent is required to trigger gamma oscillations, which occur after activation of sensory fibers (see Figs. 4 and 8; Lagier et al. 2004). However, the similarity in frequency and individual firing rates between these studies and the present study suggests that the mechanisms underlying these two phenomena share some general features.
Brunel and Wang (2003) recently proposed that the oscillatory function might occur within the neuron population to explain fast oscillations with low individual firing rate. They propose that the oscillation occurs through instability of the firing rate (i.e., a Hopf bifurcation; see Brunel and Hakim 1999) as a result of the global negative feedback provided by lateral inhibition. The oscillation drives itself in the following way: at the peak of the oscillation a fraction of neurons (that can change randomly from cycle to cycle) fires and produces a large barrage of IPSPs within the entire population (because of the widespread connectivity). This barrage prevents most of the population firing for a while, thus creating the silent phase of the oscillation. Then, when the population is no longer laterally inhibited, a new fraction of neurons can fire and the cycle starts again. This model predicted that synaptic time constants were crucial to the oscillation frequency as a result of the dominant role of lateral couplings. In a network of sparsely firing neurons that are coupled by inhibition, the frequency of the network oscillation depends on the latency and the rise time, but not on the decay time, of the inhibitory coupling. Indeed, we observed these predictions in our I-network model (see Fig. 6).
In the present study, our model gave new results concerning oscillation stability. After analyzing the role of the coupling amplitude and coupling time constants together for determining frequency, we showed that a network similar to that described by Brunel and Wang (2003) exhibited a remarkable oscillation frequency stability with respect to changes in coupling strength. We verified this in the OB, suggesting that gamma oscillations are a population phenomenon that occurs under the influence of lateral inhibition. However, further mathematical and experimental investigations are needed to understand how similar the bulbar gamma oscillations described here are to other known fast oscillations in the brain.
Potential roles for excitatory connections
MCs projecting to the same glomerulus are coupled by excitation with high probability (Urban and Sakmann 2002). Nevertheless, the mechanism of this excitatory stimulation is still unclear. Some studies have reported a coupling through glutamate spillover (relying on N-methyl-D-aspartate receptors, see Carlson et al. 2000; or on 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors, see Urban and Sakmann 2002), and others by an interaction between glutamate spillover and electrical coupling (Schoppa and Westbrook 2002). Glutamate spillover has also been suggested to take place between MC lateral dendrites (Isaacson 1999). Also, some local interneurons located in the GC layer have been reported as being excitatory (Didier et al. 2001). Despite the diversity of mechanisms and location of the coupling, we modeled a single source of lateral excitation (Fig. 2B). Lateral excitation generally improved the synchrony of oscillations generated by a dominant lateral inhibition (Fig. 2, B and C, top traces). A similar result has been found in a network coupled by inhibitory synapses in which gap junctions were added (Kopell and Ermentrout 2004; Traub et al. 2001). Nevertheless, experimental evidence is needed to support these predictions.
We noticed that the E-network had a tendency to oscillate at low frequencies (5–20 Hz). These rhythms may correspond to slow waves in cells connected to the same glomerulus (Schoppa and Urban 2003), suggesting that lateral excitation may play a role in slower rhythm generation. As excitatory connections between MCs mainly occur within a single glomerulus (Urban and Sakmann 2002), lateral excitation makes the functional units formed by individual glomeruli more homogeneous.
Features of granule cell physiology
The present work concludes to a central role of lateral inhibition in the generation and tuning of OB network gamma oscillations. Previous works described some properties of this lateral inhibition, that is, its independence toward sodium spikes, its long delay, and its long-lasting asynchronous release (e.g., Isaacson and Strowbridge 1998; Schoppa and Westbrook 1999; Urban and Sakmann 2002; reviewed in Schoppa and Urban 2003). GCs, the largest neuronal population in the OB circuit (Shepherd et al. 2004), can provide lateral inhibition in MCs through distinct mechanisms. First, MCs integrate IPSCs over long distances because of the length of their lateral dendrites, which in turn leads to a slow, asynchronous activation of a large number of GCs. Second, individual GC spines release GABA asynchronously as a result of prolonged high calcium levels or prolonged depolarization. Thus lateral inhibition features also depend on GC integration properties. However, little is known about the physiology of GCs despite a number of recent studies (Egger et al. 2005; Hall and Delaney 2002; Isaacson and Strowbridge 1998; Murphy et al. 2005; Schoppa and Westbrook 1999). In particular, their integration properties during network gamma oscillations are still unclear. We have shown that they do not induce somatic sodium spikes phase locked to the oscillations (Lagier et al. 2004). This suggests that the processing of MC inputs, which leads to their synchrony, is independent of the somatic spiking activity of GCs. A recent study in the OB of turtles revealed that GCs could generate Na+ spikes in their dendritic arborization (Pinato and Midtgaard 2005). Rall and Shepherd (1968) suggested that lateral inhibition could be supported by electrotonic propagation of GC inputs. Calcium spikes have also been proposed as a mechanism to support signal integration and GABA release (Egger et al. 2005). Calcium dynamics appear to be a good candidate for explaining the asynchronicity of GABA release onto MCs. The threshold observed by Egger et al. (2005) in generating calcium spikes may be involved in the latency seen in the lateral inhibitory events recorded in pairs of MCs (Urban and Sakmann 2002) and may explain why only trains of MC spikes elicited lateral inhibitory events. We did not take this apparent nonlinearity in GC physiology into account in our model and therefore the long coupling latency was absent. However, MC pair recordings in magnesium-free solutions did not show any long latency (Isaacson and Strowbridge 1998). Therefore it seems reasonable to propose that during network oscillations, GCs receive massive inputs, and thus maintain high levels of calcium in the dendrites, which masks the nonlinearity of GC integration. More work on GC physiology, especially calcium dynamics, is needed to understand their contribution during network oscillations.
Tuning inhibition for network plasticity
Our results demonstrate that even if lateral inhibitory coupling between MCs is slow, fast oscillations still occur. Slow and long-lasting lateral inhibition may synchronize projection neurons on slow timescales, as seen in insects (Lei et al. 2002) and may contribute to dynamic slow changes in odor-evoked MC activity patterns, as seen in fish (Friedrich et al. 2004). Overall, our simulation over a large range of parameters showed that the same simple network architecture could lead to several dynamic states depending on the choice of the parameters. In particular, synchrony among a population of neurons can be adjusted by changes in inhibition strength without altering the global frequency. In principle, this could happen within local populations of neurons (such as a cluster of few glomeruli). Because changes in the degree of GC inhibition are linked to plasticity and learning (Lledo et al. 2005), we need to consider how learning can be included in our current model. For example, short-term plasticity of lateral inhibition can mediate the progressive synchronization of the neurons during repeated odor stimulation (e.g., for insects see Bazhenov et al. 2005; Stopfer et al. 1997). Irrespective of whether the plasticity affects the amplitude of unitary IPSCs or the probability of release, the consequences in the network will be similar.
Making sense of spike timing
Deciphering odor signals presents a common challenge to all animals. Interestingly, there are striking similarities between species in the organization of the olfactory pathway (Ache and Young 2005), indicating that many animals share general principles of odor information processing. Currently, it is well established that in mammals and nonmammalian vertebrates and homologous insect systems, both spatial (Leon and Johnson 2003) and temporal (Friedrich 2002; Laurent 2002) components of OB activity patterns convey sensory information (Friedrich and Stopfer 2001). However, it is unclear how these two dimensions are interrelated, despite all data supporting the presence of nonstatic activity patterns in the OB after odorant presentation. Fast oscillatory synchronization may have many roles in odor coding. Oscillations may integrate synaptic inputs over time to provide temporal formatting for the readout of activity patterns by preventing action potential firing for a given period within each cycle. Oscillatory activity may also provide a clock signal for temporal coding. Alternatively, oscillations may help form functional neuronal assemblies. In insects, the information conveyed by the synchronization of projection neurons seems to be related to precise odors (Laurent 2002). However, in this model, categorization (i.e., the ability to link a specific odorant to a group or category of odorants) may not rely on spike synchrony (Sivan and Kopell 2004). By contrast, synchronous MC spikes seem to convey information about odor categories in vertebrates, the odor identity being conveyed by asynchronous spikes (Friedrich et al. 2004). Our results demonstrate that lateral inhibition mediates a regulatory function in temporal activity patterning in the OB and may therefore account for the distribution—in space and in time—of MC activity patterns in the coding space.
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A single compartment that includes voltage-dependent currents described by Hodgkin–Huxley kinetics was used to model mitral cells. Their membrane potentials were calculated with the equations
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Ka
m(V) = 25 exp[(V + 45)/13.3]/{exp[(V + 45)/10] + 1} h(V) = 55.5 exp[(V + 70)/5.1]/{exp[(V + 70)/5] + 1} 
= 1/{exp[–(V – 70)/14] + 1} = 1/{exp[(V + 47.4)/6] + 1} Ks
m(V) = 10 h(V) = 2000 + 220/{exp[–(V + 71.6)/6.85] + 1} = 1/{exp[–(V + 34)/6.5] + 1} = 1/{exp[(V + 65)/6.6] + 1} NaP
Kfast
. Na
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, not different from homogeneous distribution; #, not different from baseline (before the stimulation).
