QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

HOME  |   SEARCH  |   ARCHIVE  |   SUBSCRIBE  |   CONTACT  |   HELP

This Article Abstract Full Text (PDF) A correction has been published Submit an eLetter Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (375) Citing Articles via Google Scholar Google Scholar Articles by Wang, X.-J. Articles by Buzsáki, G. Search for Related Content PubMed PubMed Citation Articles by Wang, X.-J. Articles by Buzsáki, G.

 Previous Article  |  Next Article 

Volume 16, Number 20, Issue of October 15, 1996 pp. 6402-6413
Copyright ©1996 Society for Neuroscience

Gamma Oscillation by Synaptic Inhibition in a Hippocampal Interneuronal Network Model

Xiao-Jing Wang¹ and György Buzsáki²

¹ Physics Department and Center for Complex Systems, Brandeis University, Waltham, Massachusetts 02254, and ² Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, New Jersey 07102

ABSTRACT

INTRODUCTION

MATERIALS AND METHODS

RESULTS

DISCUSSION

FOOTNOTES

REFERENCES

---

ABSTRACT

Fast neuronal oscillations (gamma, 20-80 Hz) have been observed in the neocortex and hippocampus during behavioral arousal. Using computer simulations, we investigated the hypothesis that such rhythmic activity can emerge in a random network of interconnected GABAergic fast-spiking interneurons. Specific conditions for the population synchronization, on properties of single cells and the circuit, were identified. These include the following: (1) that the amplitude of spike afterhyperpolarization be above the GABA_A synaptic reversal potential; (2) that the ratio between the synaptic decay time constant and the oscillation period be sufficiently large; (3) that the effects of heterogeneities be modest because of a steep frequency-current relationship of fast-spiking neurons. Furthermore, using a population coherence measure, based on coincident firings of neural pairs, it is demonstrated that large-scale network synchronization requires a critical (minimal) average number of synaptic contacts per cell, which is not sensitive to the network size.

By changing the GABA_A synaptic maximal conductance, synaptic decay time constant, or the mean external excitatory drive to the network, the neuronal firing frequencies were gradually and monotonically varied. By contrast, the network synchronization was found to be high only within a frequency band coinciding with the gamma (20-80 Hz) range. We conclude that the GABA_A synaptic transmission provides a suitable mechanism for synchronized gamma oscillations in a sparsely connected network of fast-spiking interneurons. In turn, the interneuronal network can presumably maintain subthreshold oscillations in principal cell populations and serve to synchronize discharges of spatially distributed neurons.

Key words: gamma rhythm; hippocampus; interneurons; GABA_A; synchronization; computer model

---

INTRODUCTION

Although fast gamma cortical oscillation has been the subject of active investigation in recent years (cf. Singer and Gray, 1995), its underlying neuronal mechanisms remain elusive. Two major issues are the cellular origin of rhythmicity (Llinás et al., 1991; McCormick et al., 1993; Wang, 1993) and the mechanism(s) of large-scale population synchronicity (Freeman, 1975; Bush and Douglas 1991; Engel et al., 1991; Hansel and Sompolinsky, 1996). Traditionally, recurrent excitation between principal (pyramidal) neurons is viewed as a major source of rhythmogenesis as well as neuronal synchronization. However, in model studies in which quantitative data about the synaptic time course were incorporated, it was found that glutamatergic synaptic excitation of the AMPA type usually desynchronizes rather than synchronizes repetitive spike firings of mutually coupled neurons (Hansel et al., 1995; van Vreeswijk et al., 1995). Therefore, recurrent connections between pyramidal cells alone do not seem to account for the network coherence during cortical gamma oscillations. It was suggested that pyramidal cell populations may be entrained by synchronous rhythmic inhibition originating from fast-spiking interneurons (Buzsáki et al., 1983; Lytton and Sejnowski, 1991). During field gamma oscillations, intracellular recordings from pyramidal cells revealed both EPSPs and IPSPs phase-locked to the field oscillation frequencies (Jagadeesh et al., 1992; Chen and Fetz, 1993; Soltész and Deschênes, 1993).

In this paper, we address the question whether, in the hippocampus, an interneuronal network can generate a coherent oscillatory output to the pyramidal neurons, thereby providing a substrate for the synaptic organization of coherent gamma population oscillations. In the behaving rat, physiologically identified interneurons were shown to fire spikes in the gamma frequency range and phase-locked to the local field waves (Bragin et al., 1995). Intracellular studies and immunochemical staining demonstrated that these interneurons are interconnected via GABAergic synapses (Lacaille et al., 1987; Sik et al., 1995; Gulyás et al., 1996). Theoretical studies suggest that these GABAergic interconnections may synchronize an interneuronal network when appropriate conditions on the time course of synaptic transmission are satisfied (Wang and Rinzel, 1992, 1993; van Vreeswijk et al., 1995). Moreover, in a recent in vitro experiment (Whittington et al., 1995; Traub et al., 1996), the excitatory glutamate AMPA and NMDA synaptic transmissions were blocked in the hippocampal slice. When metabotropic glutamate receptors were activated, transient oscillatory IPSPs in the 40 Hz frequency range were observed in pyramidal cells. These IPSPs were assumed to originate from the firing activities of fast-spiking interneurons synchronized by their interconnections. Computer simulations (Whittington et al., 1995; Traub et al., 1996) lend further support to this hypothesis.

To assess whether an interneuronal network can subserve an adequate basis for the gamma frequency population rhythm in the hippocampus, it is necessary to identify its specific requirements on the cellular properties and network connectivities, as well as to determine whether these conditions are satisfied by particular interneuronal subtypes. The present study is devoted to investigate such requirements using computer simulations. We found that synaptic transmission via GABA_A receptors in a sparsely connected network of model interneurons can provide a mechanism for gamma frequency oscillations, and we compared the modeling results with the anatomical and electrophysiological data from hippocampal fast spiking interneurons.

---

MATERIALS AND METHODS

Model neuron. Each interneuron is described by a single compartment and obeys the current balance equation:

(2)

where C_m = 1 µF/cm² and I_app is the injected current (in µA/cm²). The leak current I_L = g_L(V  E_L) has a conductance g_L = 0.1 mS/cm², so that the passive time constant ₀ = C_m/g_L = 10 msec; E_L = 65 mV.

The spike-generating Na⁺ and K⁺ voltage-dependent ion currents (I_Na and I_K) are of the Hodgkin-Huxley type (Hodgkin and Huxley, 1952). The transient sodium current I_Na = g_Nam³h(V  E_Na), where the activation variable m is assumed fast and substituted by its steady-state function m = _m/(_m + _m); _m(V) = 0.1(V + 35)/(exp(0.1(V + 35))  1), _m(V) = 4exp((V + 60)/18). The inactivation variable h obeys a first-order kinetics:

(2)

where _h(V) = 0.07 exp((V + 58)/20) and _h(V) = 1/(exp(0.1(V + 28)) + 1). g_Na = 35 mS/cm²; E_Na = 55 mV,  = 5.

The delayed rectifier I_K = g_Kn⁴ (V  E_K), where the activation variable n obeys the following equation:

(2)

with _n(V) = 0.01(V + 34)/(exp(0.1(V + 34))  1) and _n(V) = 0.125 exp((V + 44)/80); g_K = 9 mS/cm², and E_K = 90 mV.

These kinetics and maximal conductances are modified from Hodgkin and Huxley (1952), so that our neuron model displays two salient properties of hippocampal and neocortical fast-spiking interneurons. First, the action potential in these cells is followed by a brief afterhyperpolarization (AHP) of about 15 mV measured from the spike threshold of approximately 55 mV (McCormick et al., 1985; Lacaille and Williams, 1990; Morin et al., 1995; Zhang and McBain, 1995). Thus, during the spike repolarization the membrane potential reaches a minimum of about 70 mV, rather than being close to the reversal potential of the K⁺ current, E_K = 90 mV. This is accomplished in the model by relatively small maximal conductance g_K and fast gating process of I_K so that it deactivates quickly during spike repolarization.

Second, these interneurons have the ability to fire repetitive spikes at high frequencies (with the frequency-current slope up to 200-400 Hz/nA) (McCormick et al., 1985; Lacaille and Williams, 1990; Zhang and McBain, 1995). With fast kinetics of the inactivation (h) of I_Na, the activation (n) of I_K, and the relatively high threshold of I_K, the model interneuron displays a large range of repetitive spiking frequencies in response to a constant injected current I_app (Fig. 1A, left). It has a small current threshold (the rheobase I_app  0.2 µA/cm²), and the firing rate is as high as 400 Hz for I_app  20 mA/cm². Similar to cortical interneurons (McCormick et al., 1985; Lacaille and Williams, 1990), the whole frequency-current curve is not linear, and the frequency-current slope is larger at smaller I_app values (lower frequencies) (Fig. 1A, right). As a consequence, the neural population is more sensitive to input heterogeneities at smaller I_app values. This is demonstrated in Figure 1B, where a Gaussian distribution of I_app is applied to a population of uncoupled neurons (N = 100), with a mean I_µ and standard deviation I. Given a fixed and small I = 0.03, the mean drive I_µ is varied systematically, and the resulting dispersion in the neuronal firing frequencies, f/f_µ (standard deviation of the firing rate/mean firing rate) is shown as function of I_µ (Fig. 1B, top). When plotted versus f_µ, it is evident that with the same amount of dispersion in applied current (I) the dispersion in firing rates f/f_µ is dramatically increased for f_µ < 20 Hz (Fig. 1B, bottom). This feature has important implications for the frequency-dependent network behaviors (see Results).

---

Fig. 1. Model of single neuron and synapse. A, Left, Firing frequency versus applied current intensity (f  I_app curve) of the model neuron. The firing rate can be as high as 400 Hz. Right, The derivative df/dI_app shows that the f/I_app slope is much larger at smaller I_app (lower f) values. B, Dispersion in firing rates caused by heterogeneity in input current. A Gaussian distribution for input currents, with standard deviation I = 0.03, is applied to a population of uncoupled neurons. The dispersion in firing rates was computed as the ratio between the standard deviation and the mean of firing rates (f/f_µ). This ratio is much larger for smaller mean current amplitude I_µ (top). Plotting f/f_µ versus f_µ shows that the dispersion in firing rates is dramatically increased for f_µ < 20 Hz (bottom). C, A brief current pulse applied to a presynaptic cell generates a single action potential, which elicits an inhibitory postsynaptic current (I_syn) and membrane potential change in a postsynaptic cell (g_syn = 0.1 mS/cm²).
[View Larger Version of this Image (15K GIF file)]

---

Model synapse. The synaptic current I_syn = g_syns(V  E_syn), where g_syn is the maximal synaptic conductance and E_syn is the reversal potential. Typically, we set g_syn = 0.1 mS/cm² and E_syn = 75 mV (Buhl et al., 1995). The gating variable s represents the fraction of open synaptic ion channels. We assume that s obeys a first-order kinetics (Perkel et al., 1981; Wang and Rinzel 1993):

(2)

where the normalized concentration of the postsynaptic transmitter-receptor complex, F(V_pre), is assumed to be an instantaneous and sigmoid function of the presynaptic membrane potential, F(V_pre) = 1/(1 + exp((V_pre  syn)/2)), where _syn (set to 0 mV) is high enough so that the transmitter release occurs only when the presynaptic cell emits a spike. The channel opening rate  = 12 msec¹ assures a fast rise of the I_syn, and the channel closing rate  is the inverse of the decay time constant of the I_syn; typically, we set  = 0.1 msec¹ (_syn = 10 msec). An example of I_syn and IPSP elicited by a single presynaptic spike is illustrated in Figure 1C.

Random network connectivity. The network model consists of N cells. The coupling between neurons is randomly assigned, with a fixed average number of synaptic inputs per neuron, M_syn. The probability that a pair of neurons are connected in either direction is p = M_syn/N. For comparison, we also used fully coupled (all-to-all) connectivity (M_syn = N). In the model, the maximal synaptic conductance g_syn is divided by M_syn, so that when the number of synapses M_syn is varied, the total synaptic drive per cell in average remains the same.

M_syn is the convergence/divergence factor of the neural coupling in the network. Experimentally, an estimate of this important quantity has been obtained for an interneuronal network of the CA1 hippocampus (Sik et al., 1995). A parvalbumin-positive (PV⁺) basket interneuron was stained intracellularly by biocytin in vivo. Its axonal arborization was largely confined in the striatum pyramidale (Fig. 2A). Other PV⁺ interneurons were stained immunochemically, and the contacts made by the biocytin-filled cell with other PV⁺ cells were counted (Sik et al., 1995). It was concluded that a single PV⁺ basket cell makes synaptic contacts with at least 60 other PV⁺ cells (a majority of which are basket cells) within a spatial region of the volume up to 0.1-0.2 mm³ (Sik et al., 1995) (Fig. 2B). This tissue volume contains as many as 500-1000 other PV⁺ cells, because the PV⁺ neurons in the pyramidal layer have a cell density of 5.4 × 10³ cells/mm³ (Aika et al., 1994). Hence, for the CA1 network of basket cells, the experimentally estimated M_syn is ~60. The probability of postsynaptic contacts, however, decreases with the distance between the cell pair (Fig. 2C).

---

Fig. 2. In vivo double staining of parvalbumin-positive interneuron in the hippocampus. A, The axonal arbor of an intracellularly labeled basket cell, largely confined in the pyramidal layer, is overlayed with immunochemically stained other parvalbumin-positive cells. The two-dimensional distribution of the interneuron-interneuron contacts is shown in B and then collapsed to a one-dimensional distribution (along the septo-temporal axis) in C. Overall, 99 boutons in contact with 64 parvalbumin-positive cells were counted (adapted from Sik et al., 1995).
[View Larger Version of this Image (34K GIF file)]

---

Heterogeneous input. In the model, single neurons are not identical. Each receives a depolarizing current I_app of different intensity and, hence, has a different firing rate. The bias current I_app has a Gaussian distribution with a mean I_µ and a standard deviation I. The parameter I_µ determines the mean excitation by the external drive; I measures the degree of the heterogeneity in the cell population.

A measure of network coherence. To quantify the synchronization of neuronal firings in the network, we introduce a coherence measure based on the normalized cross-correlations of neuronal pairs in the network (Gerstein and Kiang, 1960; Welsh et al., 1995). The coherence between two neurons i and j is measured by their cross-correlation of spike trains at zero time lag within a time bin of t = . More specifically, suppose that a long time interval T is divided into small bins of  and that two spike trains are given by X(l) = 0 or 1, Y(l) = 0 or 1, l = 1, 2, ... , K (T/K = ). Thus, we define a coherence measure for the pair as:

(2)

We have also used a slightly different definition where the mean firing rates are substracted from X(l) and Y(l); the substraction did not significantly change our results reported below.

The population coherence measure () is defined by the average of _ij() over many pairs of neurons in the network. This coherence measure presents a number of useful properties. First, it is naturally based on cross-correlations and, although we use it here to describe synchronization of oscillations, it can be applied to quantify the synchrony of nonoscillatory neuronal firings. It is calculated from spike trains, requires relatively small sample sizes, and can be used for data analysis of experimental extracellular multiunit recordings. Second, () is between 0 and 1 for all . For very small , () is close to 1 (resp. 0) in the case of maximal synchrony (resp. asynchrony). Third, the degree of synchrony of the network dynamics can be quantified by how () behaves as function of . In the case of full synchrony, () is 1 for all nonzero  values; whereas in the case of total asynchrony, () is a linearly increasing function of  (see below). Finally, the distribution of _ij() over neural pairs provides detailed information about the interneuronal interactions and synchronization.

In Results, the population coherence measure () is calculated by averaging over all neural pairs in the network of size N. Typically N = 100.

Numerical methods. The network model was integrated using the fourth-order Runge-Kutta method, with a time step of 0.05 msec. For random network connectivity and heterogeneity, each set of simulations was run with three to five random realizations of the network connections and applied current distribution. As initial conditions, the membrane potential is uniformly distributed between 70 and 50 mV and the other channel-gating variables are set at their corresponding steady-state values. Coherence was calculated after 1000 msec transients. Simulations were performed on a SUN Sparc Station or a Y-MP Cray Supercomputer.

---

RESULTS

Spike afterhyperpolarization, inhibition, and synchronization

We start by considering a simple case in which all individual cells are identical (i.e., without heterogeneity) and are coupled in an all-to-all fashion (i.e., without randomness in connectivity). As shown in Figure 3A (left), cells starting at random and asynchronous initial conditions quickly become synchronized and within 5-6 oscillatory cycles their spiking times are perfectly in-phase. The spike AHP amplitude is 15 mV measured from the spike threshold (52 mV), so the maximal AHP, V_AHP = 67 mV, stays above the reversal potential of the synaptic current (E_syn = 75 mV). The inequality V_AHP > E_syn means that during the time course of an action potential and its repolarization, the synaptic action is always hyperpolarizing. This relationship between intrinsic and synaptic properties was found to be an important condition under which the global network synchronization can be achieved (Fig. 3B,C). In the example (Fig. 3B), the speed of the I_Na inactivation and the I_K activation is slowed down ( = 3.33 instead of 5), so that repolarization becomes larger (V_AHP  73 mV, close to E_syn). In this case, the network synchronization takes much longer time to realize. With  = 2 (Fig. 3C), V_AHP is 78 mV, which is below E_syn, and global network synchrony is lost. Instead, the network is dynamically broken into two clusters: within each cluster the cells fire spikes simultaneously, and the two clusters alternate in time. Such clustering dynamics is a commonly observed behavior of interneuronal networks (Golomb et al., 1994). Hence, synaptic inhibition of the GABA_A-type provides a mechanism by which a macroscopic coherence of the network (global synchrony or clustering) can be realized.

---

Fig. 3. Synchronization by GABA_A synapses. In these simulations, neurons are identical and coupled in an all-to-all fashion. Left panels, Rastergrams; right panels, membrane potentials of two cells (dotted line, 52 mV). The synchrony is realized when the spike AHP of the cells does not fall below the synaptic reversal potential E_syn = 75 mV (dot-dashed line on the right panels). From A to C,  = 5, 3.33, and 2 respectively; I_app = 1, 1.2, and 1.4 µA/cm² accordingly to preserve a similar oscillation frequency. With smaller  values, I_K is slower and the AHP amplitude (V_AHP) is more negative. When V_AHP < E_syn, the full synchrony is lost (C).
[View Larger Version of this Image (56K GIF file)]

---

To investigate further the dependence of the network synchronization on the inhibitory nature of synaptic interactions, the intrinsic cell properties were kept unchanged, while the reversal potential E_syn was gradually varied (Fig. 4). In Figure 4A the global coherence index  (compare Eq. 2.5), plotted versus E_syn, remains at 1 (perfect synchrony) for E_syn < V_AHP. It displays an abrupt drop for E_syn > V_AHP and is close to 0 for E_syn > 60 mV. The oscillation frequency dramatically increases as the synaptic effect becomes depolarizing (not shown). Unlike the two cluster dynamics of Figure 3C with the coherence index  = 0.5, in the regime characterized by  ~ 0 the relative timing of neuronal firings is essentially arbitrary. This happens in our network model when the synaptic interactions are excitatory. In the example given in Figure 4, B and C, E_syn = 0 mV and _syn = 2 msec, so that the synaptic current mimic that of the glutamate AMPA type. Because recurrent excitation considerably enhances the neural discharge rates, weaker external drive (I_µ = 0.1 instead of 1) was used so as to obtain a similar (~40 Hz) firing frequency with excitatory rather than inhibitory interactions. In this case, although all neurons have very similar rhythmic frequencies, their relative firing phases are almost uniformly distributed between zero and the common oscillation period T (Fig. 4B). Hence, the probability of coincident firing within a time bin  between two cells increases proportionally with , and the network coherence function () grows linearly from zero at  = 0 to its maximal value of 1 at  = T msec (Fig. 4C).

---

Fig. 4. Dependence of the network synchrony on the synaptic reversal potential E_syn. A, The coherence index  ( = 1 msec) is plotted versus E_syn. As E_syn is varied, V_AHP remains essentially the same (vertical dashed line). There is a sudden transition from synchrony to asynchrony as E_syn is increased above V_AHP. B, An example of asynchronous behavior when cells are coupled by excitatory synapses (E_syn = 0 mV, _syn = 2 msec; I_app = 0.1). The oscillation frequency is f = 43 Hz. C, The network coherence function () increases linearly with , from 0 (at  = 0) to 1 (at  = T), showing that the relative firing time of neural pairs is almost uniformly distributed between 0 and the oscillation period T = 1/f.
[View Larger Version of this Image (22K GIF file)]

---

Heterogeneity and asynchrony

It is intuitively expected that network synchrony cannot be globally maintained if individual neurons display very different intrinsic oscillation frequencies. We studied the effects of heterogeneity, using a Gaussian distribution of the applied current intensity I_app with standard deviation I. As illustrated in Figure 5A, the network coherence deteriorates quite rapidly with increasing I. This sensitivity is related to the large frequency-current slope of fast-spiking interneurons (Fig. 1A,B), so that in a population of uncoupled cells a small current dispersion may imply a wide distribution of firing frequencies. When neurons are synaptically coupled, the distribution of firing frequencies is a product of their interactions. As shown in Figure 5B, for small dispersion in I_app (I < 0.02), minor differences in intrinsic firing rates are overcome by the coupling, and the standard deviation of firing rates f = 0. As the network coherence erodes with larger I values, f grows linearly with I (Fig. 5B). This linear frequency-current relationship in standard deviation is a network property of coupled cells. By contrast, the mean firing rate f_µ changes only slightly, illustrating the relative independence of the neural firing rates and synchronicity. The moderate decrease in f_µ, however, is related to the decreased degree of network coherence, because asynchronized cell firings result in an averaged tonic hyperpolarization that slows down the firing rate (see below).

---

Fig. 5. Effects of the network heterogeneity. A, The coherence index  ( = 1 msec) versus I (the standard deviation of the applied current distribution). The network becomes asynchronous for I  0.05. Examples indicated by arrows (I = 0.03 and 0.1) are illustrated in C-E and F-H, respectively. B, The mean (f_µ) and standard deviation (f) of the firing rates averaged over individual neurons are plotted versus I. Note a decrease of f_µ and a linear increase of f for I  0.05. The sensitivity to I is related to the steep frequency-current relationship of single cells. C-E, A partially synchronous state. C, The rastergram. D, The coherence function () increases with  rapidly, displays a plateau, and reaches the value of 1 near  = 1/f_µ. E, The derivative of () shows a sharp peak at  = 0. F-H, An asynchronous state, as seen by the rastergram (F). () is linear with  and reaches 1 near  = 1/f_µ (G), and its derivative is flat (H).
[View Larger Version of this Image (27K GIF file)]

---

Figure 5, C and D, illustrates a partially coherent state. In the rastergram (Fig. 5C), neurons are labeled in the increasing order of their I_app values. The cells with the smallest injected currents are not in synchrony with those firing at higher rates. The population coherence measure () increases sharply with  and reaches the value of 1 at   1/f_µ (Fig. 5D). Considered as a function of , () may be viewed as a distribution function of the neural pairs whose relative firing phase is , between 0 and the mean oscillation period (estimated as T_µ = 1/f_µ). The derivative d/d, therefore, is the corresponding distribution density. Network synchronization is manifested by a peak at zero phase of d/d (Fig. 5E), reflecting the sharp increase of  near  = 0. Note that a second peak is expected near  = T_µ (Fig. 5E) because the spiking event is periodic.

The network dynamics with I = 0.1 is illustrated in Fig. 5,F- H. By contrast to Figure 5C-E, here the rastergram is quite irregular (Fig. 5F). The linear form of the function () (Fig. 5G) and its flat derivative (Fig. 5H) are consistent with a totally desynchronized behavior, where the relative firing phase of neural pairs is uniformly distributed between 0 and T_µ.

Without the aid of the coherence function (), it would be difficult to conclude from the rastergram (F) that the network is completely disordered. Indeed, if one looks at the summated synaptic drive, s(t) = (1/N) ^N_i=1 s_i(t), oscillatory fluctuations are significant in this ``population field'' (Fig. 6B, top). This is because, if every cell oscillates, the summation of a relatively small number (e.g., N = 100) of oscillatory signals would still show some oscillations, even if cells are completely asynchronous. To assess the macroscopic coherence of the network, one can compute the temporal variance ² of s(t) for different network sizes and assess whether the fluctuations in s(t) persist in large networks. The network is asynchronous if ² of s(t) decreases inversely with the network size, ²(N) ~ 1/N (Hansel and Sompolinsky, 1996). This is shown to be the case in Figure 6A. As N increases, s(t) becomes flatter (Fig. 6B) and the peak in the power spectrum gets smaller (Fig. 6C). This example shows that global network synchrony cannot be assessed correctly by oscillatory fluctuations in the population field by the presence of a peak in the power spectrum if the network size is not sufficiently large.

---

Fig. 6. Synaptic field in a large asynchronous network. To demonstrate further the asynchronous nature of the network behavior of Figure 5F-H, the temporal variance ² (N) of the population synaptic field s(t) was calculated for different network sizes (N = 100, 200, ... , 1000). As expected for asynchronoized network states (see text), ² (N) decreases as ~1/N (A). Three examples of s(t) are shown in B, and their power spectra in C (arrow indicating increasing N). Thus, the fluctuations of s(t) vanish for large network sizes.
[View Larger Version of this Image (24K GIF file)]

---

Sparse network and minimal connectedness

Random connections among interneurons can be introduced into the model by assuming that a cell makes synaptic contact to a second cell with a probability p. Then, if N is the total number of cells, there are M_syn = pN presynaptic cells that converge to a postsynaptic cell, on average. As shown above, the network can be synchronized with all-to-all connectivity (M_syn = N, p = 1). Because synchronization is impossible without synaptic connections (M_syn = 0), we examined the dependence of the population synchrony on M_syn. To evaluate the effect of sparse connectivity separately, M_syn was gradually varied for a network of 100 identical neurons (i.e., without heterogeneity). As shown in Figure 7A, the population coherence (as measured by ) is essentially zero for M_syn below a critical value 40; above which it starts to become significant, increases rapidly with M_syn, and reaches the maximum of 1 in the all-to-all limit (M_syn = N). Thus, the dependence of the network synchrony on M_syn is highly nonlinear. There is a minimal value of M_syn and neural interconnections have to be sufficiently dense to generate global population synchronization. This critical M_syn value is not simply a required minimum for the total synaptic drive per cell, because it does not change noticeably when the maximal synaptic conductance is reduced by a factor of 2 (Fig. 7A). Also, in the presence of heterogeneity (e.g., with I = 0.03), the critical M_syn value remains the same, but the quantitative degree of network coherence for M_syn larger than the critical value was reduced (data not shown). On the other hand, this minimal connectedness depends on the probability rules in the network design and on the network dynamical state under consideration. For instance, it is much smaller if the number of synapses per cell is exactly the same number M_syn, but the actual set of presynaptic cells is chosen randomly (Fig. 7A). Or, when the oscillation frequency is increased from ~35 Hz to ~100 Hz with I_µ = 3 instead of 1, the network is synchronized only with M_syn > 75 (Fig. 7A).

---

Fig. 7. Minimal random connectivity is required for large-scale network synchronization. A, The coherence index  ( = 1 msec) is plotted versus the mean number of synaptic inputs per cell M_syn (N = 100). Filled circle, I_µ = 1 and g_syn = 0.1 (reference parameter set). In this case, the network synchrony is realized when M_syn is larger than a critical value M_crit  40. This curve remains essentially the same when the maximal synaptic conductance is reduced by 1/2 (open circle). By comparison, if the number of inputs per cell is identical to all cells, and equals M_syn, the synchrony occurs with very small values of M_syn (5; filled square). With I_µ = 3 (solid triangle), the mean oscillation frequency is increased from 35 to 100 Hz; the critical value of M_syn for the network coherence is much larger (70). B, The coherence index  ( = 1 msec) versus M_syn for different numbers of neurons N = 100, 200, 500, 1000 (with reference parameter set). The onset of network coherence occurs at a critical value of M_syn, which does not grow as a fraction of the network size.
[View Larger Version of this Image (17K GIF file)]

---

We next examined whether the required minimal connectedness increases in proportion with the network size. Simulations were performed with N = 100, 200, 500, and 1000, and it was found that the onset of network coherence corresponded to a small value of M_syn, close to M_syn = 60 for large network sizes (Fig. 7B). Therefore, the minimal number of synapses per cell that is required for the network synchronization is not a fraction of the total number N of cells. It either remains finite for large N or it could conceivably depend weakly on N.

Partial synchronization in a sparse and heterogeneous network

As stated above, the minimal connectedness for a network coherence remains the same when a moderate amount of heterogeneity is added. Figure 8 illustrates a partially synchronized network, with both sparse connections (M_syn = 60) and a dispersion in the external drive (I = 0.03). In that case, a major fraction of neurons (group I) displays similar firing rates (close to 39 Hz), although their intrinsic firing rates are different. The remaining neurons (group II) have lower firing rates (below 34 Hz) that are scattered in the diagram. The membrane potential trace of a representive cell is shown in Figure 8B, together with its synaptic drive s_syn (sum of the synaptic gating variables to that cell) and the population synaptic field, both displaying fairly regular oscillations at the same frequency. Cross-correlations of membrane potentials between the representative cell (a in Fig. 8C) and three other cells (b, c, and d in Fig. 8C) show that some pairs (ac and ad) are synchronized with near-zero phase shift, but a significant phase difference may be present for other pairs (e.g., ab; Fig. 8D).

---

Fig. 8. A partially synchronous state with sparse connectivity (M_syn = 60) and heterogeneity (I = 0.03). A, The firing rates of neurons in the network (filled circle) are lower than those when the neurons are uncoupled (open circle). The bias current varies from 0.91 to 1.09, the firing rate from 55 to 63 Hz when g_syn = 0. With g_syn = 0.1, a large fraction of neurons have firing rates close to 39 Hz, the remaining cells have lower firing rates. B, Time traces of the population synaptic field s(t), the membrane potential V(t) of one cell and its summated synaptic drive s_syn(t). C, The rastergram, with the cell shown in B indicated by a. D, The cross-correlations between this cell with three other cells b, c, d indicated in C. E, The coherence index _ij ( = 1 msec) for each of all the pairs in the network, plotted versus the difference in the firing rates |f_i  f_j| of the pair. Top, Pairs with oscillation frequencies above 34 Hz; bottom, remaining pairs. Only pairs in the top panel show a high degree of zero-phase synchrony. F, The histograms of _ij in three groups of pairs: those not monosynaptically coupled (top), those coupled in one direction (middle), and those coupled in both directions (bottom). The population averaged () function shows a steep rise for small  values (G), and its derivative has a peak at  = 0 (H).
[View Larger Version of this Image (25K GIF file)]

---

The normalized cross-correlation of spike trains at zero phase lag, defined as our coherence index _ij for the pair (i, j), was calculated for all neural pairs in the network and plotted against the difference in the firing rates of the pair |f_i  f_j| (Fig. 8E). Those pairs with both firing rates above 34 Hz (group I) are plotted in the top panel, and the other pairs are plotted in the bottom panel. Comparison of the two panels reveals that high zero-phase synchronization (large _ij) occurs only in pairs of group I neurons, and with similar firing frequencies (small |f_i  f_j|). For those pairs in the bottom panel, _ij is small even for pairs with almost identical firing frequencies. Therefore, the network is subdivided into two groups of neurons, and only group I neurons are synchronized with near zero-phase shift. It is not immediately clear why all neurons in the asynchronous group have lower, but not higher, firing frequencies than the synchronous group.

On the other hand, the neural pairs can be classified into three categories, according to whether they are monosynaptically uncoupled, coupled in one direction, or coupled in both directions. Histograms of _ij, however, do not show conspicious difference between such categories (Fig. 8F), indicating that the degree of synchronization between a pair is not primarily determined by their direct monosynaptic contacts. We conclude, therefore, that interneuronal coherence emerges as a network phenomenon. The global character of network synchrony is quantified by , the average of _ij over all pairs. The population coherence function () increases rapidly with the time bin  and reaches the value of 1 for   T_µ, where again T_µ is the average oscillation period (Fig. 8G). Its derivative shows a clear peak at  = 0 (Fig. 8H).

The partially synchronous dynamics cannot be maintained if the interneuronal connections are too sparse (Fig. 7). Indeed, when M_syn is decreased from 60 to 30, with all of the other parameters remaining the same, the network becomes asynchronous (Fig. 9). In this case, neurons are not locked to a same firing frequency (Fig. 9A), the rastergram looks irregular (Fig. 9C), the population synaptic field is almost constant in time (although the synaptic drive to a single cell still shows residual oscillatory fluctuations) (Fig. 9B), and the cross-correlations between cell pairs are flat (Fig. 9D). Corroboratively, the zero-phase synchronization is very low for all neural pairs (Fig. 9E,F), in contrast with Figure 8, E and F. The global coherence function () is linear in  (Fig. 9G), and its derivative is flat (Fig. 9H), as expected for a fully asynchronous neural network.

---

Fig. 9. Desynchronization with reduced network connectivity. Same as Figure 8, except that M_syn = 30 instead of 60. The network dynamics is asynchronous, as seen by the scattered distribution of firing rates (A), the disordered rastergram (C), the small fluctuations of the synaptic field (B), and flat cross-correlations (D). This asynchronous state is further characterized by small values of _ij (E, F), the linear function of the network averaged () (G), and its flat derivative (H).
[View Larger Version of this Image (23K GIF file)]

---

Dependence on the synaptic time constant

It was shown that the 40 Hz oscillations in hippocampal interneurons are sensitive to the decay time constant _syn of the GABA_A synapse (Whittington et al., 1995; Traub et al., 1996). We have confirmed their result that an increased _syn induces a decrease in the oscillation frequency (Fig. 10A, left). This frequency reduction occurs largely because the slowly decaying synaptic inhibition accumulates in time, resulting in a tonic level of hyperpolarization that counteracts the external depolarization. This is shown in Figure 10B, in which the synaptic drive to a representative cell is displayed for three different values of _syn. The time average is indicated by a horizontal line, which is higher (0.5, 0.64, 0.7) for larger _syn values (10, 20, and 40 msec, respectively).

---

Fig. 10. Synaptic time constant can modulate the oscillation frequency and population synchrony. A, Slowing down the synaptic current decay (with increasing _syn) decreases the mean oscillation frequency f_µ (left), whereas the network coherence shows a peaked region centered at _syn  7 msec (solid circle, right). Here, to take into account the change in firing rates, the coherence index  was calculated with  = 0.1/f_µ. For large _syn values, the network coherence decreases because of the heterogeneities in the connectivity and external drive, as can be seen by isolating each of the two effects (dotted and dashed lines, respectively). In the absence of both, the coherence is maximal ( = 1) for _syn > 4 msec (dash-dotted line). B, Decreased firing frequency is caused partly by the fact that with larger _syn, the summated synaptic drive has a greater time average (horizontal lines) and shows less oscillatory fluctuations, thus providing an enhanced level of tonic hyperpolarization, which counteracts the depolarizing drive of the cells. C, Rastergram of an asynchronous network with _syn = 2 msec (top). The globally coupled network of identical cells shows a two-cluster dynamics; thus, it is only partially synchronous (bottom).
[View Larger Version of this Image (35K GIF file)]

---

We also considered how the network synchronization depends on _syn. Within a fixed time window, a pair of cells has a higher chance to fire simultaneously if their firing frequencies are higher. Hence, to make a meaningful comparison, the coherence measure  should be calculated with different time bin  when _syn is varied. In Figure 10A (right), we chose to use  equal to one-tenth the mean oscillation period. It was found that the coherence index displays a peaked region around _syn = 7 msec; the network synchronization is lost for both smaller and larger _syn values.

The decrease in the coherence index  with larger _syn is a consequence of network heterogeneities. Indeed, as shown in Figure 10A, if the dispersion in applied current is absent (I = 0), or if the connectivity is not random (M_syn = 100),  is larger but still decreases with _syn. But if both I = 0 and M_syn = 100,  (hence, the network synchrony) is now maximal for all larger _syn values. Note that the heterogeneity in either the applied current or the connectivity produces stronger effects for larger _syn values, because in the lower frequency range the network is more sensitive to input heterogeneities (see Fig. 1B).

On the other hand, the decrease in the network synchrony for small _syn is presumably attributable to a synaptic (dynamical) effect: inhibition should not be too fast compared to the oscillation period, so as to synchronize the network (Wang and Rinzel 1992, 1993; van Vreeswijk et al., 1995; Ermentrout, 1996). This is illustrated in Figure 10C with _syn = 2 msec, where the rastergram is quite irregular (top). This is probably related to the fact that with global connectivity (M_syn = 100) and without heterogeneity (I = 0), the network displays two-clusters (bottom, Fig. 10D), similar to Figure 3C. In that case, the global network synchrony (one-cluster) was also observed with different initial conditions, but it was very sensitive to the network heterogeneity (data not shown).

It follows from the left and right panels of Figure 10A that a peaked region for  versus _syn implies that the network coherence is high only within a limited range of the mean oscillation frequencies. This interesting observation was confirmed with different levels of the external drive I_µ (Fig. 11A). When I_µ is larger, the oscillation frequencies are higher (left). Moreover, the maximum of  is located at a smaller value of _syn (right). The network synchrony does not depend simply on _syn, but on the ratio between _syn and the mean oscillation period T_µ, which is an increasing function of _syn (Fig. 11B). When the coherence index is plotted versus _syn/T_µ, it is small for small _syn/T_µ ratios and peaks at _syn/T_µ  0.2 for all three external excitation levels (I_app = 1, 2, 3; Fig. 11C).

---

Fig. 11. Dependence of the network coherence on the synaptic time constant _syn. A, The mean oscillation frequency as function of _syn is shown with three levels of the network drive I_µ (left). The coherence index displays a peak which is shifted to smaller _syn value with larger I_µ (right). B, The ratio between the synaptic time constant _syn and the oscillation period T_µ increases with _syn. C, The coherence index versus the ratio _syn/T_µ. For all three I_µ values, the coherence index peaks at the same _syn/T_µ (0.2) and decreases at smaller ratio values (M_syn = 60, I = 0.03; the coherence index  was calculated with  = 0.1/f_µ).
[View Larger Version of this Image (28K GIF file)]

---

Optimal synchronization in the gamma frequency range

That network synchronization is highest in a limited frequency range appears to be a robust finding in our simulations. For instance, the phenomenon was also observed when the mean input current (I_µ) was varied continuously (Fig. 12A). With stronger external drive, the mean oscillation frequency increases monotonically (Fig. 12A, top), but the network coherence displays high values only for an intermediate I_µ range (Fig. 12A, bottom). In an all-to-all network of identical neurons, the network coherence was found to remain maximal ( = 1) for the entire I_µ range (data not shown). Furthermore, at small I_µ, heterogeneity in an all-to-all network reduces the synchrony dramatically (with I_µ = 0.4,  = 1, and 0.1 for I = 0 and 0.03, respectively). Thus, the decrease of the network synchrony at smaller I_µ values is attributable to the high network sensitivity to heterogeneities at lower frequencies (see Fig. 1B) in addition to a decreased _syn/T_µ ratio (dynamical effect). On the other hand, at larger I_µ (higher frequencies), the network coherence requires tighter connectedness and our fixed M_syn = 60 may no longer be sufficient (see Fig. 7A).

---

Fig. 12. High network coherence in the gamma oscillation frequency range. A, With increasing mean drive I_µ, the average oscillation frequency monotonically increases (top), but the coherence index  is large only for intermediate I_µ values (bottom). B, Similarly, as the maximal synaptic conductance g_syn is increased, with stronger inhibitory interactions the average oscillation frequency f_µ monotonically decreases (top). An increase in the external drive shifts the curve upwards. On the other hand, the coherence index  displays a pronounced peak for I_µ = 1, which flattens for larger I_µ values (bottom). C, The coherence index is plotted as function of f_µ for all four curves from (A, B). The macroscopic network coherence is observed only in the gamma range of the oscillation frequencies (20-80 Hz) (M_syn = 60, I = 0.03; the coherence index  was calculated with  = 0.1/f_µ).
[View Larger Version of this Image (19K GIF file)]

---

We also varied the coupling strength g_syn systematically, with three different I_µ values. With stronger synaptic inhibition, the oscillation frequency decreases monotonically (Fig. 12B, top). The network coherence, on the other hand, shows a peaked region at intermediate g_syn values (Fig. 12B, bottom). Again, without input heterogeneity and coupling sparseness, the network coherence is maximal ( = 1) for the entire g_syn range (data not shown). Therefore, similar to the case of I_µ variation, the decrease of  is presumably caused by a reduced stability of the network synchrony against input heterogeneity at low frequencies (large g_syn) and by a lack of sufficiently tight connectivity (combined with a weakened coupling strength) at high frequencies (small g_syn).

When the network coherence index is plotted against the mean frequency, for all four curves in Figure 12, A and B, it is clear that the network synchronization is realized only in a relatively narrow range of oscillation frequencies (30-80 Hz; Fig. 12C). As shown by the three curves from Figure 12B, this frequency range of high network synchrony can be shifted by the level of external drive. With I_µ = 2 or 3 (instead of 1), the network is more excited and the coherence peak is located at higher frequencies, as expected, and the peak is somewhat enlarged. However, the amplitude of the peak is dramatically reduced (compare I_µ = 1 and 3). This can be explained, again, by the fact that at higher frequencies, the network synchronization requires denser connections (see Fig. 7A), whereas here M_syn = 60 is kept constant. Figure 12C demonstrates that, although the optimal frequency range for synchronization is not precisely defined and does depend quantitatively on network parameters and external drive, the high network coherence is robustly limited to a frequency band that coincides roughly with the gamma (20-80 Hz) frequency range.

---

DISCUSSION

We examined the emergence of synchronous gamma oscillations in an interneuronal network model. The following conditions were identified for the synchronizing mechanism by GABA_A synaptic inhibition. (1) The spike afterhyperpolarization of single neurons should be above the synaptic reversal potential, so that the effect of synaptic inputs is always hyperpolarizing. (2) The synaptic current decay should be relatively slow, such that the ratio between the decay time constant and the oscillation period is not small. (3) Heterogeneities should be sufficiently small. (4) The random network connectedness should be higher than a well defined minimum, which is not sensitive to the network size. When the four conditions are fulfilled, a large-scale network coherence was observed preferentially in the gamma (20-80 Hz) frequency range, although uncoupled neurons are potentially capable of discharging in a wide range of frequencies (0-400 Hz).

Synchronization by synaptic inhibition

Rhythmogenesis in many biological central pattern generators is based on circuits of coupled inhibitory neurons exhibiting postinhibitory rebound excitation (Selverston and Moulins, 1985). In such systems, however, rhythmic patterns are usually slower than the kinetic time constants of the inhibitory synapses, and reciprocally connected neurons typically fire out-of-phase (Perkel and Mulloney, 1974). More recently, it was recognized that neural oscillations can be synchronized by mutual inhibition at zero phase shift, provided that the synaptic kinetics is sufficiently slow as compared to the oscillation period (Wang and Rinzel, 1992, 1993). Intuitively, slow synaptic decay offers the possibility for neurons to ``escape'' synchronously as the synaptic inhibition wanes below a certain threshold, thus fire at the same time. A general conclusion from this scenario is that synapses with given temporal characteristics may be suitable to synchronizing large neural populations in a particular oscillatory frequency range; a major determining factor is the ratio between the synaptic time constant and the oscillation period. Other computational works have since shown that the mechanism of synchronization by inhibition may be quite general (van Vreeswijk et al., 1995; Whittington et al., 1995; Ermentrout, 1996; Traub et al., 1996). Synchronization by GABA_A synapses is facilitated if the synaptic reversal potential is more negative than the maximum spike afterhyperpolarization. This, however, is necessary only for the perfect global, but not for partial, synchronization. Although fast-spiking interneurons typically display larger AHP amplitudes than pyramidal cells, their measured maximal AHP is usually not below 70 mV (McCormick et al., 1985; Lacaille and Williams, 1990; Morin et al., 1995; Sik et al., 1995), hence, above the reversal potential of 75 mV for GABA_A synapses (Buhl et al., 1995). On the other hand, the time constant _syn of the synaptic current determines the range of the oscillation frequencies where the synchronization can be realized by mutual inhibition. For gamma oscillations (~40 Hz), the _syn should be larger than 3 msec, compatible with the estimated _syn (Otis and Mody, 1992). Therefore, these requirements seem to be fulfilled by cortical fast-spiking interneurons, especially the basket cells, interconnected by GABA_A synapses. On the other hand, the synchronization mechanism does not require interneurons to be endowed with a postinhibitory rebound property.

The network synchronization was quantified by a coherence index that was defined in terms of the zero-time cross-correlations of spike trains. This proved to be a useful and reliable measure of population synchrony. By contrast, population-averaged quantities like the ``synaptic field'' may display significant oscillatory fluctuations even when most neurons are in fact asynchronous, if the size of the probed neural population is small. In that case, however, the field fluctuations decrease with the network size and become almost flat for large networks. This observation suggests caution in the interpretation of oscillatory local field potential in experimental measurements and in small-network simulations.

We studied the dependence of the network coherence on the heterogeneity in interneuronal properties. Typically, with moderate heterogeneities, coupled oscillators break down into a synchronous and an asynchronous subpopulations. In our model, asynchronous neurons in such a partially synchronous state all have lower firing frequencies than the synchronous ones. The observed high sensitivity on the degree of heterogeneity can be attributed partly to the large frequency-current slope of these fast-spiking cells, so that a minor dispersion in the external drives may result in a wide distribution of single cells' oscillation frequencies. It would be of interest to investigate whether the network synchronization may become more robust in the presence of heterogeneities and noise, if the neural connections are structured in space (Somers and Kopell, 1995), or when an excitatory pyramidal population is included that reciprocally interacts with the interneuronal population (Kopell and LeMasson, 1994; Wang et al., 1995). Moreover, information processing in the cortex may involve a small subset of pyramidal neurons at a time. These pyramidal cells, by activating their common interneuron targets, may exert localized effects on the synchronous oscillations in a selective neural subpopulation of the cortical network.

Random connection and critical sparseness

One of our main objectives was to assess how dense synaptic interconnections must be for the maintenance of the synchronized network oscillations. The degree of network coherence depended on the average number of synaptic contacts per cell, M_syn, in a highly nonlinear fashion. A minimal M_syn value was identified, below which the network becomes totally asynchronous. For M_syn above its critical value, the degree of network coherence becomes nonzero and increases with M_syn. We demonstrated that this critical connectedness does not increase in proportion with the number of neurons, hence, is not sensitive to the network size. However, further analysis is needed to determine whether there is a very weak (e.g., logarithmic) dependence. Such a dependence is expected, for instance, if the required connectedness is related to the minimal number of links for a random network to be topologically connected at large scales (Erdös and Rényi, 1960). The existence of a small critical connectedness has also been found in other neural network models (Barkai et al., 1990; Wang et al., 1995), suggesting that this may be a general feature of sparsely connected random neural networks.

Note that the actual value of this minimal connectedness depends on the details of single-cell and network properties, as well as on the cooperative dynamics under consideration. Indeed, we found that synchronization of oscillations at higher frequencies requires tighter interneuronal connections (Fig. 7A). Further analyses are needed to provide a theoretical understanding of this simulation result. For rhythmicities in the gamma frequency range, our model network require 60 or more synaptic contacts from an interneuron to other interneurons. This number is comparable with the estimated divergence/convergence factor of CA1 basket cells in the hippocampus (Sik et al., 1995).

We also demonstrated that synchronization of oscillatory neurons usually cannot be attributed simply to the presence of monosynaptic couplings between the respective cells. The macroscopic network synchronization is thus a truly emergent phenomenon of large neuronal aggregates.

Frequency selection for synchronization

An important finding of the present study is that interneuronal networks can be synchronized by GABA_A synapses preferentially within the gamma frequency range. This happens despite the wide range (0-400 Hz) of possible single-neuron firing rates. For instance, when the synaptic time constant (_syn), external drive (I_µ), or coupling strength (g_syn) is varied gradually, the neuronal firing frequencies change monotonically and cover a wide frequency range. In all three cases, however, the degree of network synchronization shows a relatively narrow peak within the gamma frequency range (20-80 Hz).

This phenomenon of frequency selection may be qualitatively understood in terms of the following three neural and network properties: (1) the high network sensitivity to heterogeneities at low frequencies, attributable to the steep frequency/current curve of single neurons; (2) the minimal connectedness for the synchrony, which is larger at higher oscillation frequencies; (3) the ``dynamical effect,'' namely, network synchronization is impossible or fragile against heterogeneities if the ratio between _syn and the oscillation period T (_syn/T) becomes too small. Hence, given _syn  10 msec, if the oscillation frequency is lower than 20 Hz (T > 50 msec), the network coherence may be abolished by a high sensitivity to input heterogeneity (see Fig. 1B) and by a reduced _syn/T ratio. On the other hand, if the frequency is higher than 80 Hz, the synchrony may no longer be possible because the interneuronal connectivity is not sufficiently tight (see Fig. 7A). Consequently, the network can be highly synchronized only at 20-80 Hz.

We would like to emphasize that this frequency band for coherent oscillations cannot be determined in an absolute precision, and its quantitative limits do depend on the details of the model. However, our results robustly demonstrated that the synaptic time constant delimits a frequency band of coherent network oscillations, and the GABA_A-type synapse (with _syn  10 msec) seems especially suitable for the gamma rhythmicity. The limited frequency range of population gamma oscillations has been observed in the behaving rat (Bragin et al., 1995) and in hippocampal slices (Whittington et al., 1995; Traub et al., 1996). Our findings suggest that the synchronization mechanism by an interneuronal network would be especially effective if the fast oscillations are generated in the gamma frequency range by pacemaker neurons (Llinás et al., 1991; McCormick et al., 1993). On the other hand, even in the absence of pacemaker neurons, coherent field oscillations could still be observed in the gamma frequency range, not because possible firing rates of inhibitory interneurons are restricted to a narrow frequency band, but because outside this frequency range the large-scale coherence is not possible (by this particular synaptic mechanism).

Physiological implications

Recurrent excitatory connections have long been regarded as a possible synaptic substrate underlying correlated neural firings in general and massively synchronous brain rhythms in particular. Although coherent slow (epileptic) rhythmic bursting may indeed emerge in a disinhibited pyramidal cell network (Chagnac-Amitai and Connors, 1989; Traub et al., 1993), modeling studies suggest that mutual excitation via the AMPA-type synapse often cannot synchronize neural oscillations in the gamma frequency range, at least for simple repetitive spiking neurons (Hansel et al., 1995; van Vreeswijk, et al., 1995) (present work). An alternative mechanism for the fast entrainment of principal cells has emerged recently. Previous work (Bragin et al., 1995; Whittington et al., 1995; Traub et al., 1996) and the present model suggest that networks of interneurons are critically involved in the generation of coherent gamma oscillations. The advantage of such ``synchronizer'' function of interneuronal networks is the maintenance of subthreshold and coherent modulation of the large, sparsely connected pyramidal cell populations and the resulting precise timing of their action potentials (Buzsáki and Chrobak, 1995; Hopfield, 1995).

---

FOOTNOTES

Received May 5, 1996; revised June 25, 1996; accepted July 31, 1996.

  

This work was supported by the National Institute of Mental Health (MH53717-01), Office of Naval Research (N00014-95-1-0319), and the Sloan Foundation to X.J.W.; and HFSP and the National Institute of Neurological Disease and Stroke (NS34994) to G.B. and X.J.W. Simulations were partly performed at the Pittsburgh Supercomputing Center. We thank D. Golomb, D. Hansel, J.-C. Lacaille, and C. McBain for discussions, A. Sik for preparing Figure 2, and L. Abbott, J. Lisman, and R. Traub for carefully reading this manuscript.

Correspondence should be addressed to Xiao-Jing Wang, Center for Complex Systems, Brandeis University, Waltham, MA 02254.

---

REFERENCES

• Aika Y, Ren JQ, Kosaka K, Kosaka T (1994) Quantitative analysis of GABA-like-immunoreactive and parvalbumin-containing neurons in the CA1 region of the rat hippocampus using a stereological method, the disector. Exp Brain Res 99:267-276 . [Web of Science][Medline]
• Barkai E, Kanter I, Sompolinsky H (1990) Properties of sparsely connected excitatory neural networks. Phys Rev A 41:590-597.[Medline]
• Bragin A, Jandó G, Nádasdy Z, Hetke J, Wise K, Buzsáki G (1995) Gamma (40-100 Hz) oscillation in the hippocampus of the behaving rat. J Neurosci 15:47-60 . [Abstract]
• Buhl EH, Cobb SR, Halasy K, Somogyi P (1995) Properties of unitary IPSPs evoked by anatomically identified basket cells in the rat hippocampus. Eur J Neurosci 7:1989-2004 . [Web of Science][Medline]
• Bush PC, Douglas RJ (1991) Synchronization of bursting action potential discharge in a model network of neocortical neurons. Neural Comput 3:19-30.
• Buzsáki G, Chrobak JJ (1995) Temporal structure in spatially organized neuronal ensenbles: a role for interneuronal networks. Curr Opin Neurobiol 5:504-510 . [Web of Science][Medline]
• Buzsáki G, Leung L, Vanderwolf CH (1983) Cellular bases of hippocampal EEG in the behaving rat. Brain Res Rev 6:139-171.
• Chagnac-Amitai Y, Connors BW (1989) Horizontal spread of synchronized activity in neocortex and its control by GABA-mediated inhibition. J Neurophysiol 61:747-758 . [Abstract/Free Full Text]
• Chen DF, Fetz EE (1993) Effect of synchronous neural activity on synaptic transmission in primate cortex. Soc Neurosci Abstr 19:319.7.
• Engel AK, König P, Kreiter AK, Singer W (1991) Interhemispheric synchronization of oscillatory neuronal responses in cat visual cortex. Science 252:1177-1179 . [Free Full Text]
• Erdös P, Rényi A (1960) On the evolution of random graphs. MTA Mat Kut Int Közl 5:17-61.
• Ermentrout GB (1996) Type I membranes, phase resetting and synchrony. Neural Comput 8:979-1001. [Web of Science][Medline]
• Freeman WJ (1975) Mass action in the nervous system. .
• Gerstein GL, Kiang WY (1960) An approach to the quantitative analysis of electrophysiological data from single neurons. Biophys J 1:15-28.
• Golomb D, Wang X-J, Rinzel J (1994) Synchronization properties of spindle oscillations in a thalamic reticular nucleus model. J Neurophysiol 72:1109-1126 . [Abstract/Free Full Text]
• Gulyás AI, Hájos N, Freund TF (1996) Interneurons containing calretinin are specialized to control other interneurons in the rat hippocampus. J Neurosci 16:3397-3411 . [Abstract/Free Full Text]
• Hansel D, Sompolinsky H (1996) Chaos and synchrony in a model of a hypercolumn in visual cortex. J Comput Neurosci 3:7-34 . [Web of Science][Medline]
• Hansel D, Mato G, Meunier C (1995) Synchrony in excitatory neural networks. Neural Comput 7:307-335. [Web of Science][Medline]
• Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol (Lond) 117:500-544.
• Hopfield J (1995) Pattern recognition computation using action potential timing for stimulus representation. Nature 376:33-36 . [Medline]
• Jagadeesh B, Gray CM, Ferster D (1992) Visually evoked oscillations of membrane potential in cells of cat visual cortex. Science 257:552-554 . [Abstract/Free Full Text]
• Kopell N, LeMasson G (1994) Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture. Proc Natl Acad Sci USA 91:10586-10590 . [Abstract/Free Full Text]
• Lacaille J-C, Williams S (1990) Membrane properties of interneurons in stratum oriens-alveus of the CA1 region of rat hippocampus in vitro. Neuroscience 36:349-359 . [Web of Science][Medline]
• Lacaille J-C, Mueller A, Kunkel DD, Schwartzkroin PA (1987) Local circuit interactions between oriens/alveus interneurons and CA1 pyramidal cells in hippocampal slices: electrophysiology and morphology. J Neurosci 7:1979-1993 . [Abstract]
• Llinás RR, Grace AA, Yarom Y (1991) In vitro neurons in mammalian cortical layer 4 exhibit intrisic oscillatory activity in the 10-50 Hz frequency range. Proc Natl Acad Sci USA 88:897-901 . [Abstract/Free Full Text]
• Lytton WW, Sejnowski TJ (1991) Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons. J Neurophysiol 66:1059-1079 . [Abstract/Free Full Text]
• McCormick DA, Connors BW, Lighthall JW, Prince DA (1985) Comparative electrophysiology of pyramidal and sparsely spiny stellar neurons of the neocortex. J Neurophysiol 54:782-806 . [Abstract/Free Full Text]
• McCormick DA, Gray C, Wang Z (1993) Chattering cells: a new physiological subtype which may contribute to 20-60 Hz oscillations in cat visual cortex. Soc Neurosci Abstr 16:359.9.
• Morin F, Beaulieu C, Lacaille J-C (1996) Membrane properties and synaptic currents evoked in CA1 interneuron subtypes in rat hippocampal slices. J Neurophysiol, in press.
• Murthy VN, Fetz EE (1992) Coherent 25-35 Hz oscillations in the sensorimotor cortex of the awake behaving monkey. Proc Natl Acad Sci USA 89:5670-5674 . [Abstract/Free Full Text]
• Otis TS, Mody I (1992) Modulation of decay kinetics and frequency of GABA_A receptor-mediated spontaneous inhibitory postsynaptic currents in hippocampal neurons. Neuroscience 49:13-32 . [Web of Science][Medline]
• Perkel DH, Mulloney B (1974) Motor pattern production in reciprocally inhibitory neurons exhibiting postinhibitory rebound. Science 185:181-183 . [Abstract/Free Full Text]
• Perkel DH, Mulloney B, Budelli RW (1981) Quantitative methods for predicting neuronal behavior. Neuroscience 6:823-827 . [Web of Science][Medline]
• Selverston AI, Moulins M (1985) Oscillatory neural networks. Annu Rev Physiol 47:29-48 . [Web of Science][Medline]
• Sik A, Pentonnen M, Ylinen A, Buzsáki G (1995) Hippocampal CA1 interneurons: an in vivo intracellular labeling study. J Neurosci 15:6651-6665 . [Abstract/Free Full Text]
• Singer W, Gray CM (1995) Visual feature integration and the temporal correlation hypothesis. Annu Rev Neurosci 18:555-586 . [Web of Science][Medline]
• Soltész I, Deschênes M (1993) Low- and high-frequency membrane potential oscillations during theta activity in CA1 and CA3 pyramidal neurons of the rat hippocampus under ketamine-xylazine anesthesia. J Neurophysiol 70:97-116 . [Abstract/Free Full Text]
• Somers D, Kopell N (1995) Waves and synchrony in networks of oscillators of relaxation and non-relaxation type. Physica D 89:169-183.[Web of Science]
• Traub RD, Miles R, Jeffreys JGR (1993) Synaptic and intrinsic conductances shape picrotoxin-induced synchronized afterdischarges in the guinea-pig hippocampal slice. J Physiol (Lond) 93:525-547.
• Traub RD, Whittington MA, Colling SB, Buzsáki G, Jeffreys JGR (1996) Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. J Physiol (Lond), in press.
• van Vreeswijk C, Abbott LF, Ermentrout GB (1995) When inhibition, not excitation synchronizes neural firing. J Comput Neurosci 1:313-322.
• Wang X-J (1993) Ionic basis for intrinsic 40 Hz neuronal oscillations. NeuroReport 5:221-224 . [Web of Science][Medline]
• Wang X-J, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 4:84-97.
• Wang X-J, Rinzel J (1993) Spindle rhythmicity in the reticularis thalami nucleus: synchronization among mutually inhibitory neurons. Neuroscience 53:899-904 . [Web of Science][Medline]
• Wang X-J, Golomb D, Rinzel J (1995) Emergent spindle oscillations and intermittent burst firing in a thalamic model: specific neuronal mechanisms. Proc Natl Acad Sci USA 92:5577-5581 . [Abstract/Free Full Text]
• Welsh JP, Lang EJ, Sugihara I, Llinás RR (1995) Dynamic organization of motor control within the olivocerebellar system. Nature 374:453-457 . [Medline]
• Whittington MA, Traub RD, Jeffreys JGR (1995) Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature 373:612-615 . [Medline]
• Zhang L, McBain CJ (1995) Potassium conductances underlying action potential repolarization and afterhyperpolarization in rat CA1 hippocampal interneurons. J Physiol (Lond) 488:661-672 . [Abstract/Free Full Text]

---

Copyright ©1996 Society for Neuroscience   0270-6474/1996/166402-12$05.00/0

This article has been cited by other articles:

				J. Cui, C. C. Canavier, and R. J. Butera Functional Phase Response Curves: A Method for Understanding Synchronization of Adapting Neurons J Neurophysiol, July 1, 2009; 102(1): 387 - 398. [Abstract] [Full Text] [PDF]

				P. J. Uhlhaas, F. Roux, W. Singer, C. Haenschel, R. Sireteanu, and E. Rodriguez The development of neural synchrony reflects late maturation and restructuring of functional networks in humans PNAS, June 16, 2009; 106(24): 9866 - 9871. [Abstract] [Full Text] [PDF]

				E. R. Williams and S. N. Baker Renshaw Cell Recurrent Inhibition Improves Physiological Tremor by Reducing Corticomuscular Coupling at 10 Hz J. Neurosci., May 20, 2009; 29(20): 6616 - 6624. [Abstract] [Full Text] [PDF]

				S. D. Muthukumaraswamy, R. A.E. Edden, D. K. Jones, J. B. Swettenham, and K. D. Singh Resting GABA concentration predicts peak gamma frequency and fMRI amplitude in response to visual stimulation in humans PNAS, May 19, 2009; 106(20): 8356 - 8361. [Abstract] [Full Text] [PDF]

				S. Achuthan and C. C. Canavier Phase-Resetting Curves Determine Synchronization, Phase Locking, and Clustering in Networks of Neural Oscillators J. Neurosci., April 22, 2009; 29(16): 5218 - 5233. [Abstract] [Full Text] [PDF]

				A. Racz, A. A. Ponomarenko, E. C. Fuchs, and H. Monyer Augmented Hippocampal Ripple Oscillations in Mice with Reduced Fast Excitation onto Parvalbumin-Positive Cells J. Neurosci., February 25, 2009; 29(8): 2563 - 2568. [Abstract] [Full Text] [PDF]

				F. N. Karameh and S. G. Massaquoi Intracortical Augmenting Responses in Networks of Reduced Compartmental Models of Tufted Layer 5 Cells J Neurophysiol, January 1, 2009; 101(1): 207 - 233. [Abstract] [Full Text] [PDF]

				A. Compte, R. Reig, V. F. Descalzo, M. A. Harvey, G. D. Puccini, and M. V. Sanchez-Vives Spontaneous High-Frequency (10-80 Hz) Oscillations during Up States in the Cerebral Cortex In Vitro J. Neurosci., December 17, 2008; 28(51): 13828 - 13844. [Abstract] [Full Text] [PDF]

				S. A. Prescott, S. Ratte, Y. De Koninck, and T. J. Sejnowski Pyramidal Neurons Switch From Integrators In Vitro to Resonators Under In Vivo-Like Conditions J Neurophysiol, December 1, 2008; 100(6): 3030 - 3042. [Abstract] [Full Text] [PDF]

				M. A. Smith and A. Kohn Spatial and Temporal Scales of Neuronal Correlation in Primary Visual Cortex J. Neurosci., November 26, 2008; 28(48): 12591 - 12603. [Abstract] [Full Text] [PDF]

				D. Doischer, J. Aurel Hosp, Y. Yanagawa, K. Obata, P. Jonas, I. Vida, and M. Bartos Postnatal Differentiation of Basket Cells from Slow to Fast Signaling Devices J. Neurosci., November 26, 2008; 28(48): 12956 - 12968. [Abstract] [Full Text] [PDF]

				C. Borgers, S. Epstein, and N. J. Kopell Gamma oscillations mediate stimulus competition and attentional selection in a cortical network model PNAS, November 18, 2008; 105(46): 18023 - 18028. [Abstract] [Full Text] [PDF]

				M. Bazhenov, N. F. Rulkov, and I. Timofeev Effect of Synaptic Connectivity on Long-Range Synchronization of Fast Cortical Oscillations J Neurophysiol, September 1, 2008; 100(3): 1562 - 1575. [Abstract] [Full Text] [PDF]

				G. Gonzalez-Burgos and D. A. Lewis GABA Neurons and the Mechanisms of Network Oscillations: Implications for Understanding Cortical Dysfunction in Schizophrenia Schizophr Bull, September 1, 2008; 34(5): 944 - 961. [Abstract] [Full Text] [PDF]

				H. Kondgen, C. Geisler, S. Fusi, X.-J. Wang, H.-R. Luscher, and M. Giugliano The Dynamical Response Properties of Neocortical Neurons to Temporally Modulated Noisy Inputs In Vitro Cereb Cortex, September 1, 2008; 18(9): 2086 - 2097. [Abstract] [Full Text] [PDF]

				C. I. Buia and P. H. Tiesinga Role of Interneuron Diversity in the Cortical Microcircuit for Attention J Neurophysiol, May 1, 2008; 99(5): 2158 - 2182. [Abstract] [Full Text] [PDF]

				S. K. Towers and S. Hestrin D1-Like Dopamine Receptor Activation Modulates GABAergic Inhibition But Not Electrical Coupling between Neocortical Fast-Spiking Interneurons J. Neurosci., March 5, 2008; 28(10): 2633 - 2641. [Abstract] [Full Text] [PDF]

				K. Morita, R. Kalra, K. Aihara, and H. P. C. Robinson Recurrent Synaptic Input and the Timing of Gamma-Frequency-Modulated Firing of Pyramidal Cells during Neocortical "UP" States J. Neurosci., February 20, 2008; 28(8): 1871 - 1881. [Abstract] [Full Text] [PDF]

				L. Badel, S. Lefort, R. Brette, C. C. H. Petersen, W. Gerstner, and M. J. E. Richardson Dynamic I-V Curves Are Reliable Predictors of Naturalistic Pyramidal-Neuron Voltage Traces J Neurophysiol, February 1, 2008; 99(2): 656 - 666. [Abstract] [Full Text] [PDF]

				M. Goldin, J. Epsztein, I. Jorquera, A. Represa, Y. Ben-Ari, V. Crepel, and R. Cossart Synaptic Kainate Receptors Tune Oriens-Lacunosum Moleculare Interneurons to Operate at Theta Frequency J. Neurosci., September 5, 2007; 27(36): 9560 - 9572. [Abstract] [Full Text] [PDF]

				S. M. Montgomery and G. Buzsaki Gamma oscillations dynamically couple hippocampal CA3 and CA1 regions during memory task performance PNAS, September 4, 2007; 104(36): 14495 - 14500. [Abstract] [Full Text] [PDF]

				W. K. Mitchell, M. R. Baker, and S. N. Baker Muscle responses to transcranial stimulation in man depend on background oscillatory activity J. Physiol., September 1, 2007; 583(2): 567 - 579. [Abstract] [Full Text] [PDF]

				A. B. L. Tort, H. G. Rotstein, T. Dugladze, T. Gloveli, and N. J. Kopell On the formation of gamma-coherent cell assemblies by oriens lacunosum-moleculare interneurons in the hippocampus PNAS, August 14, 2007; 104(33): 13490 - 13495. [Abstract] [Full Text] [PDF]

				S. M. Barman and G. L. Gebber Role of ventrolateral medulla in generating the 10-Hz rhythm in sympathetic nerve discharge Am J Physiol Regulatory Integrative Comp Physiol, July 1, 2007; 293(1): R223 - R233. [Abstract] [Full Text] [PDF]

				C. Geisler, D. Robbe, M. Zugaro, A. Sirota, and G. Buzsaki Hippocampal place cell assemblies are speed-controlled oscillators PNAS, May 8, 2007; 104(19): 8149 - 8154. [Abstract] [Full Text] [PDF]

				C. L. Witham and S. N. Baker Network oscillations and intrinsic spiking rhythmicity do not covary in monkey sensorimotor areas J. Physiol., May 1, 2007; 580(3): 801 - 814. [Abstract] [Full Text] [PDF]

				A. Kepecs and S. Raghavachari Gating Information by Two-State Membrane Potential Fluctuations J Neurophysiol, April 1, 2007; 97(4): 3015 - 3023. [Abstract] [Full Text] [PDF]

				M. Trevino, C. Vivar, and R. Gutierrez {beta}/{gamma} Oscillatory Activity in the CA3 Hippocampal Area is Depressed by Aberrant GABAergic Transmission from the Dentate Gyrus after Seizures J. Neurosci., January 3, 2007; 27(1): 251 - 259. [Abstract] [Full Text] [PDF]

				G. Orban, T. Kiss, and P. Erdi Intrinsic and Synaptic Mechanisms Determining the Timing of Neuron Population Activity During Hippocampal Theta Oscillation J Neurophysiol, December 1, 2006; 96(6): 2889 - 2904. [Abstract] [Full Text] [PDF]

				T. G. Banke and C. J. McBain GABAergic Input onto CA3 Hippocampal Interneurons Remains Shunting throughout Development. J. Neurosci., November 8, 2006; 26(45): 11720 - 11725. [Abstract] [Full Text] [PDF]

				Y. Aponte, C.-C. Lien, E. Reisinger, and P. Jonas Hyperpolarization-activated cation channels in fast-spiking interneurons of rat hippocampus J. Physiol., July 1, 2006; 574(1): 229 - 243. [Abstract] [Full Text] [PDF]

				R. Dodla, G. Svirskis, and J. Rinzel Well-Timed, Brief Inhibition Can Promote Spiking: Postinhibitory Facilitation J Neurophysiol, April 1, 2006; 95(4): 2664 - 2677. [Abstract] [Full Text] [PDF]

				B. Bathellier, S. Lagier, P. Faure, and P.-M. Lledo Circuit Properties Generating Gamma Oscillations in a Network Model of the Olfactory Bulb J Neurophysiol, April 1, 2006; 95(4): 2678 - 2691. [Abstract] [Full Text] [PDF]

				F. Saraga, L. Ng, and F. K. Skinner Distal Gap Junctions and Active Dendrites Can Tune Network Dynamics J Neurophysiol, March 1, 2006; 95(3): 1669 - 1682. [Abstract] [Full Text] [PDF]

				D. Golomb, A. Shedmi, R. Curtu, and G. B. Ermentrout Persistent Synchronized Bursting Activity in Cortical Tissues With Low Magnesium Concentration: A Modeling Study J Neurophysiol, February 1, 2006; 95(2): 1049 - 1067. [Abstract] [Full Text] [PDF]

				V. S. Sohal and J. R. Huguenard Inhibitory coupling specifically generates emergent gamma oscillations in diverse cell types PNAS, December 20, 2005; 102(51): 18638 - 18643. [Abstract] [Full Text] [PDF]

				C. Geisler, N. Brunel, and X.-J. Wang Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations With Irregular Neuronal Discharges J Neurophysiol, December 1, 2005; 94(6): 4344 - 4361. [Abstract] [Full Text] [PDF]

				N. Lozovaya, N. Yatsenko, A. Beketov, T. Tsintsadze, and N. Burnashev Glycine Receptors in CNS Neurons as a Target for Nonretrograde Action of Cannabinoids J. Neurosci., August 17, 2005; 25(33): 7499 - 7506. [Abstract] [Full Text] [PDF]

				H. G. Rotstein, D. D. Pervouchine, C. D. Acker, M. J. Gillies, J. A. White, E. H. Buhl, M. A. Whittington, and N. Kopell Slow and Fast Inhibition and an H-Current Interact to Create a Theta Rhythm in a Model of CA1 Interneuron Network J Neurophysiol, August 1, 2005; 94(2): 1509 - 1518. [Abstract] [Full Text] [PDF]

				D. S.F. Ling and L. S. Benardo Nootropic Agents Enhance the Recruitment of Fast GABAA Inhibition in Rat Neocortex Cereb Cortex, July 1, 2005; 15(7): 921 - 928. [Abstract] [Full Text] [PDF]

				E. B. Merriam, T. I. Netoff, and M. I. Banks Bistable Network Behavior of Layer I Interneurons in Auditory Cortex J. Neurosci., June 29, 2005; 25(26): 6175 - 6186. [Abstract] [Full Text] [PDF]

				C. Borgers, S. Epstein, and N. J. Kopell Background gamma rhythmicity and attention in cortical local circuits: A computational study PNAS, May 10, 2005; 102(19): 7002 - 7007. [Abstract] [Full Text] [PDF]

				F. K. Skinner, J.Y.J. Chung, I. Ncube, P. A. Murray, and S. A. Campbell Using Heterogeneity to Predict Inhibitory Network Model Characteristics J Neurophysiol, April 1, 2005; 93(4): 1898 - 1907. [Abstract] [Full Text] [PDF]

				H. L Garner, M. A Whittington, and Z. Henderson Induction by kainate of theta frequency rhythmic activity in the rat medial septum-diagonal band complex in vitro J. Physiol., April 1, 2005; 564(1): 83 - 102. [Abstract] [Full Text] [PDF]

				V. Nimmrich, N. Maier, D. Schmitz, and A. Draguhn Induced sharp wave-ripple complexes in the absence of synaptic inhibition in mouse hippocampal slices J. Physiol., March 15, 2005; 563(3): 663 - 670. [Abstract] [Full Text] [PDF]

				J. R. Gibson, M. Beierlein, and B. W. Connors Functional Properties of Electrical Synapses Between Inhibitory Interneurons of Neocortical Layer 4 J Neurophysiol, January 1, 2005; 93(1): 467 - 480. [Abstract] [Full Text] [PDF]

				A. Fisahn Kainate receptors and rhythmic activity in neuronal networks: hippocampal gamma oscillations as a tool J. Physiol., January 1, 2005; 562(1): 65 - 72. [Abstract] [Full Text] [PDF]

				C. Foldy, J. Dyhrfjeld-Johnsen, and I. Soltesz Structure of cortical microcircuit theory J. Physiol., January 1, 2005; 562(1): 47 - 54. [Abstract] [Full Text] [PDF]

				M. Galarreta, F. Erdelyi, G. Szabo, and S. Hestrin Electrical Coupling among Irregular-Spiking GABAergic Interneurons Expressing Cannabinoid Receptors J. Neurosci., November 3, 2004; 24(44): 9770 - 9778. [Abstract] [Full Text] [PDF]

				N. Kopell and B. Ermentrout Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks PNAS, October 26, 2004; 101(43): 15482 - 15487. [Abstract] [Full Text] [PDF]

				I. Aradi, V. Santhakumar, and I. Soltesz Impact of Heterogeneous Perisomatic IPSC Populations on Pyramidal Cell Firing Rates J Neurophysiol, June 1, 2004; 91(6): 2849 - 2858. [Abstract] [Full Text] [PDF]

				T. Bem and J. Rinzel Short Duty Cycle Destabilizes a Half-Center Oscillator, But Gap Junctions Can Restabilize the Anti-Phase Pattern J Neurophysiol, February 1, 2004; 91(2): 693 - 703. [Abstract] [Full Text] [PDF]

				N. Fourcaud-Trocme, D. Hansel, C. van Vreeswijk, and N. Brunel How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating Inputs J. Neurosci., December 17, 2003; 23(37): 11628 - 11640. [Abstract] [Full Text] [PDF]

				R. Maex and E. De Schutter Resonant Synchronization in Heterogeneous Networks of Inhibitory Neurons J. Neurosci., November 19, 2003; 23(33): 10503 - 10514. [Abstract] [Full Text] [PDF]

				F Saraga, C P Wu, L Zhang, and F K Skinner Active dendrites and spike propagation in multicompartment models of oriens-lacunosum/moleculare hippocampal interneurons J. Physiol., November 1, 2003; 552(3): 673 - 689. [Abstract] [Full Text] [PDF]

				C. J. Wierenga and W. J. Wadman Excitatory Inputs to CA1 Interneurons Show Selective Synaptic Dynamics J Neurophysiol, August 1, 2003; 90(2): 811 - 821. [Abstract] [Full Text] [PDF]

				N. Maier, V. Nimmrich, and A. Draguhn Cellular and Network Mechanisms Underlying Spontaneous Sharp Wave-Ripple Complexes in Mouse Hippocampal Slices J. Physiol., August 1, 2003; 550(3): 873 - 887. [Abstract] [Full Text] [PDF]

				N. Brunel and X.-J. Wang What Determines the Frequency of Fast Network Oscillations With Irregular Neural Discharges? I. Synaptic Dynamics and Excitation-Inhibition Balance J Neurophysiol, July 1, 2003; 90(1): 415 - 430. [Abstract] [Full Text] [PDF]

				A. Compte, M. V. Sanchez-Vives, D. A. McCormick, and X.-J. Wang Cellular and Network Mechanisms of Slow Oscillatory Activity (<1 Hz) and Wave Propagations in a Cortical Network Model J Neurophysiol, May 1, 2003; 89(5): 2707 - 2725. [Abstract] [Full Text] [PDF]

				C.-C. Lien and P. Jonas Kv3 Potassium Conductance is Necessary and Kinetically Optimized for High-Frequency Action Potential Generation in Hippocampal Interneurons J. Neurosci., March 15, 2003; 23(6): 2058 - 2068. [Abstract] [Full Text] [PDF]

				D. L. Buhl, K. D. Harris, S. G. Hormuzdi, H. Monyer, and G. Buzsaki Selective Impairment of Hippocampal Gamma Oscillations in Connexin-36 Knock-Out Mouse In Vivo J. Neurosci., February 1, 2003; 23(3): 1013 - 1018. [Abstract] [Full Text] [PDF]

				M. R Baker and S. N Baker The effect of diazepam on motor cortical oscillations and corticomuscular coherence studied in man J. Physiol., February 1, 2003; 546(3): 931 - 942. [Abstract] [Full Text] [PDF]

				M. I. Banks, J. B. Hardie, and R. A. Pearce Development of GABAA Receptor-Mediated Inhibitory Postsynaptic Currents in Hippocampus J Neurophysiol, December 1, 2002; 88(6): 3097 - 3107. [Abstract] [Full Text] [PDF]

				D. C. McIntyre, B. Hutcheon, K. Schwabe, and M. O. Poulter Divergent GABAA Receptor-Mediated Synaptic Transmission in Genetically Seizure-Prone and Seizure-Resistant Rats J. Neurosci., November 15, 2002; 22(22): 9922 - 9931. [Abstract] [Full Text] [PDF]

				J. E. Mikkonen, T. Gronfors, J. J. Chrobak, and M. Penttonen Hippocampus Retains the Periodicity of Gamma Stimulation In Vivo J Neurophysiol, November 1, 2002; 88(5): 2349 - 2354. [Abstract] [Full Text] [PDF]

				P. M. Baker, P. S. Pennefather, B. A. Orser, and F. K. Skinner Disruption of Coherent Oscillations in Inhibitory Networks With Anesthetics: Role of GABAA Receptor Desensitization J Neurophysiol, November 1, 2002; 88(5): 2821 - 2833. [Abstract] [Full Text] [PDF]

				A. Kepecs, X.-J. Wang, and J. Lisman Bursting Neurons Signal Input Slope J. Neurosci., October 15, 2002; 22(20): 9053 - 9062. [Abstract] [Full Text] [PDF]

				M. Bartos, I. Vida, M. Frotscher, A. Meyer, H. Monyer, J. R. P. Geiger, and P. Jonas Fast synaptic inhibition promotes synchronized gamma oscillations in hippocampal interneuron networks PNAS, October 1, 2002; 99(20): 13222 - 13227. [Abstract] [Full Text] [PDF]

				M. Galarreta and S. Hestrin Electrical and chemical synapses among parvalbumin fast-spiking GABAergic interneurons in adult mouse neocortex PNAS, September 17, 2002; 99(19): 12438 - 12443. [Abstract] [Full Text] [PDF]

				R. C. Elson, A. I. Selverston, H. D. I. Abarbanel, and M. I. Rabinovich Inhibitory Synchronization of Bursting in Biological Neurons: Dependence on Synaptic Time Constant J Neurophysiol, September 1, 2002; 88(3): 1166 - 1176. [Abstract] [Full Text] [PDF]

				T. Klausberger, J. D. B. Roberts, and P. Somogyi Cell Type- and Input-Specific Differences in the Number and Subtypes of Synaptic GABAA Receptors in the Hippocampus J. Neurosci., April 1, 2002; 22(7): 2513 - 2521. [Abstract] [Full Text] [PDF]

				K. L. Perkins GABA Application to Hippocampal CA3 or CA1 Stratum Lacunosum-Moleculare Excites an Interneuron Network J Neurophysiol, March 1, 2002; 87(3): 1404 - 1414. [Abstract] [Full Text] [PDF]

				X.-J. Wang Pacemaker Neurons for the Theta Rhythm and Their Synchronization in the Septohippocampal Reciprocal Loop J Neurophysiol, February 1, 2002; 87(2): 889 - 900. [Abstract] [Full Text] [PDF]

				I Aradi and I Soltesz Modulation of network behaviour by changes in variance in interneuronal properties J. Physiol., January 1, 2002; 538(1): 227 - 251. [Abstract] [Full Text] [PDF]

				Z. Nusser, L. M. Kay, G. Laurent, G. E. Homanics, and I. Mody Disruption of GABAA Receptors on GABAergic Interneurons Leads to Increased Oscillatory Power in the Olfactory Bulb Network J Neurophysiol, December 1, 2001; 86(6): 2823 - 2833. [Abstract] [Full Text] [PDF]

				M. Alkondon and E. X. Albuquerque Nicotinic Acetylcholine Receptor alpha 7 and alpha 4beta 2 Subtypes Differentially Control GABAergic Input to CA1 Neurons in Rat Hippocampus J Neurophysiol, December 1, 2001; 86(6): 3043 - 3055. [Abstract] [Full Text] [PDF]

				R. D. Traub, N. Kopell, A. Bibbig, E. H. Buhl, F. E. N. LeBeau, and M. A. Whittington Gap Junctions between Interneuron Dendrites Can Enhance Synchrony of Gamma Oscillations in Distributed Networks J. Neurosci., December 1, 2001; 21(23): 9478 - 9486. [Abstract] [Full Text] [PDF]

				M.-J. Bourque and A. Kolta Properties and Interconnections of Trigeminal Interneurons of the Lateral Pontine Reticular Formation in the Rat J Neurophysiol, November 1, 2001; 86(5): 2583 - 2596. [Abstract] [Full Text] [PDF]

				E. Bracci, M. Vreugdenhil, S. P. Hack, and J. G. R. Jefferys Dynamic Modulation of Excitation and Inhibition During Stimulation at Gamma and Beta Frequencies in the CA1 Hippocampal Region J Neurophysiol, June 1, 2001; 85(6): 2412 - 2422. [Abstract] [Full Text] [PDF]

				K. Stenkamp, J. M. Palva, M. Uusisaari, S. Schuchmann, D. Schmitz, U. Heinemann, and K. Kaila Enhanced Temporal Stability of Cholinergic Hippocampal Gamma Oscillations Following Respiratory Alkalosis In Vitro J Neurophysiol, May 1, 2001; 85(5): 2063 - 2069. [Abstract] [Full Text] [PDF]

				M. Bartos, I. Vida, M. Frotscher, J. R. P. Geiger, and P. Jonas Rapid Signaling at Inhibitory Synapses in a Dentate Gyrus Interneuron Network J. Neurosci., April 15, 2001; 21(8): 2687 - 2698. [Abstract] [Full Text] [PDF]

				C. Bou-Flores and A. J. Berger Gap Junctions and Inhibitory Synapses Modulate Inspiratory Motoneuron Synchronization J Neurophysiol, April 1, 2001; 85(4): 1543 - 1551. [Abstract] [Full Text] [PDF]

				C. T. Dickson, G. Biella, and M. de Curtis Evidence for Spatial Modules Mediated by Temporal Synchronization of Carbachol-Induced Gamma Rhythm in Medial Entorhinal Cortex J. Neurosci., October 15, 2000; 20(20): 7846 - 7854. [Abstract] [Full Text] [PDF]

				L. Acsady, I. Katona, F. J. Martinez-Guijarro, G. Buzsaki, and T. F. Freund Unusual Target Selectivity of Perisomatic Inhibitory Cells in the Hilar Region of the Rat Hippocampus J. Neurosci., September 15, 2000; 20(18): 6907 - 6919. [Abstract] [Full Text] [PDF]

				S. Mahon, J.-M. Deniau, S. Charpier, and B. Delord Role of a Striatal Slowly Inactivating Potassium Current in Short-Term Facilitation of Corticostriatal Inputs: A Computer Simulation Study Learn. Mem., September 1, 2000; 7(5): 357 - 362. [Abstract] [Full Text]

				J. A. White, M. I. Banks, R. A. Pearce, and N. J. Kopell Networks of interneurons with fast and slow gamma -aminobutyric acid type A (GABAA) kinetics provide substrate for mixed gamma-theta rhythm PNAS, June 23, 2000; (2000) 100124097. [Abstract] [Full Text]

				R. D. Traub and A. Bibbig A Model of High-Frequency Ripples in the Hippocampus Based on Synaptic Coupling Plus Axon-Axon Gap Junctions between Pyramidal Neurons J. Neurosci., March 15, 2000; 20(6): 2086 - 2093. [Abstract] [Full Text] [PDF]

				J. M. Palva, K. Lamsa, S. E. Lauri, H. Rauvala, K. Kaila, and T. Taira Fast Network Oscillations in the Newborn Rat Hippocampus In Vitro J. Neurosci., February 1, 2000; 20(3): 1170 - 1178. [Abstract] [Full Text] [PDF]

				B. Hutcheon, P. Morley, and M. O Poulter Developmental change in GABAA receptor desensitization kinetics and its role in synapse function in rat cortical neurons J. Physiol., January 1, 2000; 522(1): 3 - 17. [Abstract] [Full Text] [PDF]

				R. H. Joho, C. S. Ho, and G. A. Marks Increased gamma - and Decreased delta -Oscillations in a Mouse Deficient for a Potassium Channel Expressed in Fast-Spiking Interneurons J Neurophysiol, October 1, 1999; 82(4): 1855 - 1864. [Abstract] [Full Text] [PDF]

				C. A. Chapman and J.-C. Lacaille Cholinergic Induction of Theta-Frequency Oscillations in Hippocampal Inhibitory Interneurons and Pacing of Pyramidal Cell Firing J. Neurosci., October 1, 1999; 19(19): 8637 - 8645. [Abstract] [Full Text] [PDF]

				D. D. Mott, Q. Li, M. M. Okazaki, D. A. Turner, and D. V. Lewis GABAB-Receptor-Mediated Currents in Interneurons of the Dentate-Hilus Border J Neurophysiol, September 1, 1999; 82(3): 1438 - 1450. [Abstract] [Full Text] [PDF]

				L. M. d. l. Prida and J. V. Sanchez-Andres Nonlinear Frequency-Dependent Synchronization in the Developing Hippocampus J Neurophysiol, July 1, 1999; 82(1): 202 - 208. [Abstract] [Full Text] [PDF]

				P. Mann-Metzer and Y. Yarom Electrotonic Coupling Interacts with Intrinsic Properties to Generate Synchronized Activity in Cerebellar Networks of Inhibitory Interneurons J. Neurosci., May 1, 1999; 19(9): 3298 - 3306. [Abstract] [Full Text] [PDF]

				F. K. Skinner, L. Zhang, J. L. P. Velazquez, and P. L. Carlen Bursting in Inhibitory Interneuronal Networks: A Role for Gap-Junctional Coupling J Neurophysiol, March 1, 1999; 81(3): 1274 - 1283. [Abstract] [Full Text] [PDF]

				C. A. Chapman and J.-C. Lacaille Intrinsic Theta-Frequency Membrane Potential Oscillations in Hippocampal CA1 Interneurons of Stratum Lacunosum-Moleculare J Neurophysiol, March 1, 1999; 81(3): 1296 - 1307. [Abstract] [Full Text] [PDF]

				R. D. Traub, M. A. Whittington, E. H. Buhl, J. G. R. Jefferys, and H. J. Faulkner On the Mechanism of the gamma  right-arrow beta  Frequency Shift in Neuronal Oscillations Induced in Rat Hippocampal Slices by Tetanic Stimulation J. Neurosci., February 1, 1999; 19(3): 1088 - 1105. [Abstract] [Full Text] [PDF]

				Y. Zhang, J. L. Perez Velazquez, G. F. Tian, C.-P. Wu, F. K. Skinner, P. L. Carlen, and L. Zhang Slow Oscillations (less than equal to 1 Hz) Mediated by GABAergic Interneuronal Networks in Rat Hippocampus J. Neurosci., November 15, 1998; 18(22): 9256 - 9268. [Abstract] [Full Text] [PDF]

This Article Abstract Full Text (PDF) A correction has been published Submit an eLetter Alert me when this article is cited Alert me when eLetters are posted Alert me if a correction is posted Citation Map Services Email this article to a friend Similar articles in this journal Similar articles in Web of Science Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Web of Science (375) Citing Articles via Google Scholar Google Scholar Articles by Wang, X.-J. Articles by Buzsáki, G. Search for Related Content PubMed PubMed Citation Articles by Wang, X.-J. Articles by Buzsáki, G.

Home  |   Search  |   Archive  |   Subscribe  |   Contact  |   Help