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Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations With Irregular Neuronal Discharges

¹Physics Department and Volen Center for Complex Systems, Brandeis University, Waltham, Massachusetts; ²Center for Molecular and Behavioral Neuroscience, Rutgers University, Newark, New Jersey; and ³Laboratory of Neurophysics and Physiology, Unité Mixte de Recherche 8119, Centre National de la Recherche Scientifique, Université Paris René Descartes, Paris, France

Submitted 16 May 2005; accepted in final form 3 August 2005

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
During fast oscillations in the local field potential (40–100 Hz gamma, 100–200 Hz sharp-wave ripples) single cortical neurons typically fire irregularly at rates that are much lower than the oscillation frequency. Recent computational studies have provided a mathematical description of such fast oscillations, using the leaky integrate-and-fire (LIF) neuron model. Here, we extend this theoretical framework to populations of more realistic Hodgkin–Huxley-type conductance-based neurons. In a noisy network of GABAergic neurons that are connected randomly and sparsely by chemical synapses, coherent oscillations emerge with a frequency that depends sensitively on the single cell's membrane dynamics. The population frequency can be predicted analytically from the synaptic time constants and the preferred phase of discharge during the oscillatory cycle of a single cell subjected to noisy sinusoidal input. The latter depends significantly on the single cell's membrane properties and can be understood in the context of the simplified exponential integrate-and-fire (EIF) neuron. We find that 200-Hz oscillations can be generated, provided the effective input conductance of single cells is large, so that the single neuron's phase shift is sufficiently small. In a two-population network of excitatory pyramidal cells and inhibitory neurons, recurrent excitation can either decrease or increase the population rhythmic frequency, depending on whether in a neuron the excitatory synaptic current follows or precedes the inhibitory synaptic current in an oscillatory cycle. Detailed single-cell properties have a substantial impact on population oscillations, even though rhythmicity does not originate from pacemaker neurons and is an emergent network phenomenon.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
Spike trains of cortical neurons are usually very irregular and close to a Poisson process, even when the local field potential recordings exhibit coherent fast oscillations such as gamma oscillations (40–80 Hz) (Destexhe et al. 1999; Fries et al. 2001b; Pesaran et al. 2002) and sharp-wave ripples (100–200 Hz) (Buzsáki et al. 1992; Csicsvari et al. 1998, 1999a). In such oscillatory episodes, single-cell discharge rates are typically much lower than the oscillation frequency of the field potential.

Theoretical studies have demonstrated that such population rhythms appear in randomly connected networks of leaky integrate-and-fire (LIF) neurons, when the synaptic inhibitory feedback is strong and noise is sufficiently large (Brunel 2000; Brunel and Hakim 1999; Brunel and Wang 2003). Brunel and Wang (2003) showed how the population rhythmic frequency of networks of inhibitory LIF neurons depends on the time constants of the recurrent synaptic currents. With physiologically reasonable time constants, the population frequency is >100 Hz and can be as high as 300 Hz, whereas single cells fire irregularly and at a much lower rate than the population frequency. It was then shown that in a two-population network of inhibitory and excitatory LIF neurons the population frequency depends both on the time constants of excitatory and inhibitory currents and on the relative strength of recurrent excitation and inhibition: the population frequency is decreased by the synaptic excitation. In the absence of recurrent excitation among pyramidal cells the population can oscillate at 200 Hz (as observed in the CA1 area of the rat hippocampus); if the recurrent excitation is sufficiently strong, the network frequency is decreased to 100 Hz.

In a network of LIF neurons, the frequency of coherent oscillations is essentially independent of the intrinsic single-cell properties because the spiking response of an LIF model to sinusoidal input in the presence of temporally correlated noise depends only weakly on the input oscillation frequency (Brunel et al. 2001). In particular, the phase shift of the instantaneous firing rate with respect to the periodic input is very small at any input frequency. Thus through the static current–frequency relationship, single-cell properties affect the degree of synchronization of the network but not the frequency of the network oscillation.

In this paper, we examine the generality of this conclusion and show that this no longer holds true for Hodgkin–Huxley-type conductance-based neurons. The LIF neuron integrates the synaptic inputs linearly until the membrane potential reaches a threshold and a spike is triggered instantaneously. This rigid threshold behavior of LIF neurons is only a rough approximation for the actual spike-generating mechanism. Real neurons do not have a unique spiking threshold (Azouz and Gray 2000). Even if one defines a spike threshold empirically, subthreshold membrane dynamics is highly nonlinear, unlike that in the LIF model. Furthermore, after crossing the threshold the depolarization takes about 0.2–1.5 ms to reach the voltage maximum (Buhl et al. 1994; Connors et al. 1982; Lacaille and Williams 1990; Nowak et al. 2003; Zhang and McBain 1995), in contrast to the LIF model that has no spike time to peak. The precise shape of the action potential is determined by the detailed kinetic properties of the spike-generating sodium and potassium currents (Lien et al. 2002; Martina and Jonas 1997; Martina et al. 1998). These biophysical properties of action potential generation give rise to strong frequency modulation of the single neuron responsiveness to a noisy sinusoidal input (see below and Fourcaud-Trocmé et al. 2003). Consequently, for the conductance-based models, single-neuron properties can be expected to play an important role in determining the frequency of network-generated fast coherent oscillations. The purpose of the present paper is to understand how single-neuron properties affect collective oscillations in the strong noise regime.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
Model neurons

INTERNEURON. Unless stated otherwise, the interneuron model used in the simulations is a conductance-based model that is slightly modified from Wang and Buzsáki (1996). It is a one-compartment model with a total surface area of 0.02 mm². The current balance equation obeys

(1)

The capacitance of the membrane is C_m = 0.2 nF. The leak current I_L = g_L(V – V_L) has the conductance g_L = 0.02 µS and reversal potential V_L = –67 mV. I_syn is the synaptic input current and I_ext is an external applied current. The spike-generating ion currents I_Na = g_Nam³h(V – V_Na) and I_K = g_Kn⁴(V – V_K) are of the Hodgkin–Huxley type (Hodgkin and Huxley 1952). The voltage-dependent gating variables h and n are time dependent dx/dt = _x[_x(V)(1 – x) – _x(V)x] with x = {h, n}, the voltage is measured in mV, and the rate functions _x(V) and _x(V) are in ms^–1; _h = 0.07 exp[–0.05(V + 58)], _h = 1.0/{exp[–0.1(V + 28)] + 1} and _n(V) = –0.01(V + 34)/{exp[–0.1(V + 34)] – 1}, _n(V) = 0.125 exp[–0.0125(V + 44)]. The activation variable m is assumed to be fast and is substituted by its steady state m = _m/(_m + _m); _m(V) = –0.1(V + 35)/{exp[–0.1(V + 35)] – 1} and _m(V) = 4 exp[–(V + 60)/18]. The maximal conductances are g_Na = 14 µS and g_K = 1.8 µS. The reversal potentials are V_Na = 55 mV and V_K = –90 mV. The temperature factors are _n = _h = 5.

INTERNEURON WITH A-TYPE CURRENT. In hippocampal interneurons a large variety of ion channels have been found including A-type potassium currents that are activated at subthreshold voltage (Erisir et al. 1999; Lien et al. 2002; Martina et al. 1998). This finding has motivated investigations of a neuronal model containing an A-type potassium current. It is a one-compartment model with a total surface area of 0.02 mm². The current balance equation obeys

(2)

The capacitance of the membrane is C_m = 0.2 nF. The dynamics of the leak current I_L and the spike generating currents I_Na and I_K are the same as those given for the interneuron except _h = 0.07 exp[–0.05(V + 48)], _h = 1.0/{exp[–0.1(V + 18)] + 1} and _n(V) = –0.01(V + 45.7)/{exp[–0.1(V + 45.7)] – 1}, _n(V) = 0.125 exp(–0.0125(V + 55.7)), _m(V) = –0.1(V + 29.7)/{exp[–0.1(V + 29.7)] – 1} and _m(V) = 4 exp[–(V + 54.7)/18]. The maximal conductances are g_L = 0.06 µS, g_Na = 24 µS and g_K = 4 µS. The reversal potentials are V_L = –17 mV, V_Na = 55 mV, and V_K = –72 mV. The temperature factors are _n = _h = 3.8. The kinetics of the A-type potassium current I_A = g_AA³B(V – V_A) is the same as described in Connor et al. (1977), with dB/dt = (B – B)/_B, where A = 0.0761exp[(V + 94.22)/31.84]/{1 + exp[V + 1.17)/28.93]}^(1/3), B = 1/{1 + exp[(V + 53.3)/14.54]}⁴, _B = 1.24 + 2.678/{1 + exp[(V + 50)/16.027]}. The maximal conductance is g_A = 9.54 µS and the reversal potential is V_A = –75 mV. The steady-state value of the conductance g_AA³B is nonzero over a large voltage range and the current I_A contributes a significant outward current above its reversal potential. The high reversal potential of the leak current V_L = –17 mV is chosen such that the resting potential of the model neuron is –68 mV.

PYRAMIDAL CELL. In contrast with fast-spiking interneurons, pyramidal cells are characterized by pronounced spike-frequency adaptation. A two compartment model with a total surface area of 0.05 mm² (the surface area for soma and dendrite is 0.025 mm² each) accounts for adaptation properties of pyramidal cells (Wang 1998). The voltage balance equations for the soma and dendrites are, respectively

(3)

The capacitance of the membrane is C_m = 0.25 nF. The dynamics of the leak current I_L and the spike generating currents I_Na and I_K are the same as those given for the interneuron except _h = 0.07 exp[–0.1(V + 50)], _h = 1.0/{exp[–0.1(V + 20)] + 1} and _m(V) = –0.1(V + 33)/{exp[–0.1(V + 33)] – 1} and _m(V) = 4 exp[–(V + 58)/12]. The maximal conductances are g_L = 0.025 µS, g_Na = 11.25 µS and g_K = 4.5 µS. The reversal potentials are V_L = –65 mV, V_Na = 55 mV and V_K = –80 mV. The temperature factors are _n = _h = 4. The high-threshold calcium current in the dendrite I_Ca = g_Cam²(V – V_Ca), where m is assumed fast and is replaced by its steady state m = 1/{1 + exp[–(V + 20)/9]}. The maximal conductances are g_Ca = 0.25 µS and the reversal potential is V_Ca = 120 mV. The voltage-dependent, calcium-activated potassium current I_AHP = g_AHP[Ca²⁺]/([Ca²⁺] + K_d)(V – V_K) with K_d = 30 µM. The intracellular calcium follows [Ca²⁺]/dt = –I_Ca –[Ca²⁺]/_Ca, where = 4 µM/(ms · µA) and _Ca = 80 ms. The maximal conductance g_AHP = 1.25 µS.

EXPONENTIAL INTEGRATE-AND-FIRE MODEL. Fourcaud-Trocmé et al. (2003) recently showed that a simplified model, the exponential integrate-and-fire (EIF) model, can accurately reproduce the dynamics of the Wang and Buzsáki (1996) model. The advantage of this model is that its response to oscillatory input at high frequencies can be computed analytically (Fourcaud-Trocmé et al. 2003). The dynamics of the model is described by

(4)

where C_m = 0.2 nF, the leak conductance g_L = 0.02 µS, the resting potential V_L = –67 mV and I_syn is the synaptic current and I_ext is an external applied current. The exponential term represents a simplified "fast sodium current" responsible for spike initiation. The parameters of this current are chosen so that its firing rate–current relationship is identical to that of the conductance-based interneuron model (Wang and Buzsáki 1996) for currents close to the current threshold. This gives V_th = –62.45 mV and _T = 3.48 mV. The reset potential V_reset = –70.2 mV and the refractory period of 1.4 ms are chosen so that the firing rate–current relationship also matches closely for large input currents.

Single-cell studies

INPUT CURRENT. Firing rate responses of single cells are computed following Brunel et al. (2001). A conductance-based single neuron receives an input current I(t), which mimics the synaptic plus external input [I_syn(t) + I_ext(t)]. The current I(t) = I₀ + I₁ cos (2ft) + I_noise, such that the current I(t) oscillates around a mean I₀ with frequency f and amplitude I₁. The noise current is modeled as low-pass-filtered Gaussian white noise dI_noise/dt=[(t) – I_noise]/_noise where (t) is a Gaussian white-noise random variable with zero mean and SD _noise, chosen so that the SD of the subthreshold membrane potential is _v = 5 mV, which is comparable with physiological data (Destexhe and Paré 1999). The time constant _noise = 0–40 ms. This noisy input yields a highly irregular single neuron spike train, in all cases investigated in this paper. The instantaneous firing rate of the neuron (instantaneous probability of emitting a spike per unit time) is averaged over 3,000 trials and the function r(t) = r₀ + r₁(f) cos [2 ft + _cell(f)] is fitted to it using a least-square fit, where r₀ is the mean firing rate, r₁(f) is the amplitude of the modulation, f is the frequency of the input current, and _cell(f) is the phase shift between the input current and the output firing rate. The mean (I₀) and amplitude (I₁) of the input current are chosen so that r₁(f)/r₀ = 0.9 at f = 1 Hz. The length of each trial (2 s) allows a fit over at least two periods of the oscillatory input. With I₀ and I₁ fixed, the normalized amplitude r₁(f)/[r₁(f = 1 Hz)] and the phase shift _cell(f) of the instantaneous firing rate are computed for different frequencies f.

EFFECTIVE MEMBRANE TIME CONSTANT. Synaptic input modulates the membrane conductance and therefore the membrane time constant. The membrane time constant determines how fast the membrane can integrate synaptic input, and it can be used as a measure to characterize the membrane dynamics. The effective membrane time constant is defined as the inverse of the total conductance of the cell _m–eff(t) = C_m/[g_L + g_ion(t) + g_syn(t)], where g_ion is the total conductance including all ionic currents and g_syn is the sum over all synaptic conductances. In the noise-dominated regime, when the oscillation amplitude of the conductance is small, we approximate the effective membrane time constant by its time average. We exclude the spikes by excluding the conductances of the spike-generating sodium and potassium currents, which contribute to the total conductance significantly only during the spike and are small otherwise compared to the synaptic conductance

(5)

where _ion is the sum of all ionic conductances excluding the spike-generating sodium and potassium conductances and · _t denotes the time average. In the case of single-cell simulations where we add a shunting conductance to mimic synaptic input, the effective membrane time constant for the conductance-based interneuron used here is simply

(6)

Network simulations

NETWORK ARCHITECTURE. Network simulations are carried out with either one population of N_I = 1,000 interneurons, or two populations of N_I = 1,000 interneurons and N_P = 4,000 pyramidal cells. The architecture of the network is that of a sparsely and randomly directed graph: for each neuron pair, the connection probability is 10% in either direction, except in Fig. 7 where the connectivity is 5%. Thus on average, with a connectivity of p = 10% a given cell receives M_F = pN_I = 100 inhibitory synapses, and (in a two-population network) M_P = pN_P = 400 excitatory synapses.

View larger version (22K): [in this window] [in a new window]   FIG. 7. Fast (>100 Hz) oscillations in a network of inhibitory neurons. A: single cells fire randomly and sparsely at a much lower frequency than the population activity. Network oscillations are apparent in the population activity rather than on the single-cell level. Network oscillates at 125 Hz, whereas the mean firing rate of single cells is 40 Hz: (a) spike raster of 10 cells; (b) membrane potential of a single cell; (c) instantaneous population firing rate; (d) distribution of single-cell firing rates across the population; (e) power spectrum of the instantaneous population firing rate. External excitatory input is 5 kHz. B: oscillations are not detectable with the power spectrum of single-cell spike trains (the single-cell spectrum is averaged over 10 cells). C: power spectrum of "multiunit" activity (combined spike trains from 10 cells) shows a clear peak at 125 Hz. D: autocorrelation of a single spike train does not show an oscillatory pattern. E: spike-triggered average (STA) of the population rate shows oscillations at 125 Hz. Spike-triggered population rate is averaged over the spikes of one neuron (42 spikes). Length of spike trains is 2 s. Parameters: connection probability 0.05; synaptic time constants for -aminobutyric acid (GABA) Il = 0.5 ms, Ir = 0.5 ms, Id = 5.0 ms.

(7)

where the time t = 0 corresponds to the voltage maximum of the presynaptic spike. The peak conductance is given by _syn = g_syn(_r/_d)^_r/(_d–_r)(1–_r/_d). The synaptic conductances are chosen such that the postsynaptic potential has an amplitude of 1 mV at a holding potential of –60 mV for pyramidal cells and –63 mV for interneurons just below threshold (Buhl et al. 1997; Markram et al. 1997; Tamas et al. 1997, 1998; Vida et al. 1998) and such that the ratio of the peak conductance _GABAA/_AMPA 7.5 (Bartos et al. 2001, 2002; Gupta et al. 2000; Markram et al. 1997). The time constants for AMPA are _Pl = 1.5 ms, _Pr = 0.5 ms, and _Pd = 2 ms (Angulo et al. 1999; Zhou and Hablitz 1998). For -aminobutyric acid type A (GABA_A) they are _Il = 0.5 ms, _Ir = 0.5 ms and _Id = 5 ms (Bartos et al. 2001; Gupta et al. 2000; Kraushaar and Jonas 2000; Xiang et al. 1998). The reversal potential of AMPA is V_syn,AMPA = 0 mV and of GABA_A V_syn,GABAA = –75 mV. The peak conductances are _AMPAP = 1.3 nS, _GABAP = 8.75 nS, _AMPAI = 0.93 nS, and _{G ABAI} = 6.2 nS.

EXTERNAL INPUTS. Each neuron receives external synaptic input, modeled as a high-frequency Poisson spike train with a rate . The external input is mediated by AMPA synapses with conductances of 15.8 nS in pyramidal cells and 1.5 nS in interneurons. In Fig. 10 we used additional external inhibitory GABAergic input to interneurons with a conductance of 8.8 nS.

View larger version (33K): [in this window] [in a new window]   FIG. 10. A: population frequency depends on the background synaptic conductance: neurons receive strong external excitatory (40 kHz) and inhibitory (6.5 kHz) synaptic input; the network oscillates at 180 Hz, whereas single cells fire on average at 40 Hz: (a) spike raster of 10 cells; (b) membrane potential of a single cell; (c) instantaneous population firing rate; (d) distribution of single cell firing rates across the population; (e) power spectrum of the instantaneous population firing rate. B and C: each cell receives external excitatory and inhibitory Poisson inputs. Inhibitory input is adjusted so that the single-cell firing rate remains constant. B: population frequency increases with increasing input, whereas the single-cell firing rate stays constant. C: as the external inhibitory and excitatory conductances increase, the effective membrane time constant m–eff decreases. Parameters: connection probability 0.1; synaptic time constants as in Fig. 7.

MEASURE OF SYNCHRONY. To characterize the synchrony in the network we compute the autocorrelation function of the instantaneous population firing rate, normalized by the square of the average rate. In all cases described herein, the autocorrelation function is well described by a damped cosine function, with a narrow peak at the zero time lag bin, which is ascribed to the finite size of the network. To remove this finite size effect, we fit the autocorrelation function with a damped cosine, excluding the zero time lag bin. The measure of synchrony is the value of the damped cosine that best fits the data at zero time lag. This measure quantifies how much spike trains of different neurons are correlated.

Single spike train

SPECTRUM AND AUTOCORRELATION FUNCTION. To analyze the rhythmicity of a single spike train during network oscillations we calculate the power spectrum and autocorrelation function of one single spike train and of the combined spikes of a group of cells (Gabbiani and Koch 1998). The representative spike trains are selected randomly from the neural population.

SPIKE-TRIGGERED AVERAGE (STA) OF GLOBAL ACTIVITY. The spike triggered average is the cross-correlation between the spike train of a single neuron and the global activity. The instantaneous firing rate of a population during a 300-ms time window surrounding a spike (150 ms before and 150 ms after the spike time) is averaged over all spikes of the spike train of a single neuron.

Numerical methods

All equations are computed using a scheme based on the Runge–Kutta algorithm (fourth order for the network and second order for the single cell simulations) to solve the coupled differential equations (Press et al. 1992). Integration time step is t = 0.02 ms.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
Response of a single cell to noisy sinusoidal current

The transformation of the incoming synaptic inputs into an output spike train by single neurons is classically described in terms of a current–frequency relationship. However, the prediction of the collective response of a neuronal population to time-varying inputs requires a more detailed characterization of the firing properties of single cells. A standard procedure when dealing with nonstationary inputs is to characterize neurons by the linear firing rate response, i.e., the instantaneous firing rate response to noisy inputs with a weak sinusoidal modulation at an arbitrary frequency f (see e.g. Brunel et al. 2001; Gerstner 2000; Knight 1972). The idea is that during a network oscillation at frequency f, the combined external and recurrent synaptic current in a neuron can be modeled as a high-frequency inhomogeneous Poisson process, which is approximately described as

(8)

where I₀ is a constant mean current and I₁ is the amplitude of the modulation at frequency f = /2 (see METHODS). Fluctuations arising from random arrival of spikes can be well approximated by low-pass-filtered Gaussian white noise with a time constant _noise (corresponding to the synaptic decay time constant). The neuron's response to such a current can be characterized by its instantaneous firing rate r(t), obtained by an average of the response over many trials (Fig. 1A). The instantaneous firing rate follows the current with a phase shift _cell()

(9)

This linear approximation is valid for small enough I₁ and, in particular, for the values of I₁ used in Fig. 1, A and B, as shown in Fig. 2 for several representative frequencies. The amplitude of the modulation r₁() and the phase _cell() depend on the frequency f of the input current. We compute the normalized amplitude r₁()/[r₁(f = 1 Hz)] and the phase _cell() for different values of the frequency f, whereas all other parameters in the input current remain unchanged. Here, we use the convention that a negative phase corresponds to a late firing in the oscillatory cycle. In simulations, the mean current I₀ is adjusted such that the mean firing rate is fixed (r₀ = 40 Hz), so that the rate r₀ does not depend on the input frequencies f. The amplitude I₁ is chosen so that r₁(f)/r₀ = 0.9 at f = 1 Hz input frequency and r(t) is always nonzero.

View larger version (21K): [in this window] [in a new window]   FIG. 1. Firing properties of a single model neuron in response to a noisy sinusoidal input current. A: response of a single cell to 100-Hz oscillations: noisy sinusoidal current (the smooth curve is the deterministic part of the current) occasionally induces spikes, as shown in the membrane potential V(t) trace. Instantaneous firing rate r(t), averaged over 3,000 trials, oscillates at 100 Hz but is phase shifted by cell [the smooth curve is the least-square fit to the function r(t) = r0 + r1 cos (t + cell)]. Top: spike raster of 10 trials (time constant of the noise is noise = 5 ms, the average firing rate is 40 Hz, I0 = 0.12 pA, and I1 = 0.175 pA). B: phase and normalized amplitude of the trial-averaged firing rate, as a function of input frequency for conductance-based neurons. Phase and normalized amplitude depend only weakly on the time constant of the noise (noise). Normalized amplitude of the rate r1()/r1(f = 1 Hz) (left) and the phase shift cell (right) decrease with increasing frequency of the input current. C: phase and normalized amplitude as a function of frequency for leaky integrate-and-fire (LIF) neurons. Phase and normalized amplitude depend on the time constant of the noise (noise). When noise is sufficiently large, the LIF model neurons respond to the noisy sinusoidal input current without attenuation (left) and without a phase lag (right). Note that a negative phase corresponds to late firing during the oscillatory cycle.

View larger version (14K): [in this window] [in a new window]   FIG. 2. Power spectrum of the averaged firing rate r(t) is clearly dominated by a sharp peak at the input frequency f. Instantaneous firing rate of a neuron responding to a noisy sinusoidal current with frequency f = 10, 50, 150, and 250 Hz, respectively, is averaged over 3,000 trials. Mean firing rate of the neuron is 40 Hz. I0 and I1 are the same as in Fig. 1.

In sharp contrast to the LIF neuron, we found that a conductance-based neuron responds to the noisy oscillating current with a phase lag that depends very weakly on the time constant of the noise (Fig. 1B). The phase and amplitude behave essentially in the same way whether the input is Gaussian white noise (_noise = 0 ms) or filtered with a large time constant (_noise = 40 ms) (Fig. 1B). When the input current varies slowly so that f is well below the average firing rate r₀, the response of the cell follows the modulation without a phase lag and the oscillation amplitude stays constant. For larger frequencies f > r₀, the firing rate lags behind the current with a phase lag _cell() and the amplitude of the modulation r₁() decreases. For example, at f = 100 Hz the instantaneous firing rate r(t) lags behind the current by about _cell = –90°, whereas the averaged firing rate r₀ = 40 Hz is constant for all input frequencies f (Fig. 1B). The neuron acts as a low-pass filter.

A more quantitative description of the phase shift _cell() can be achieved by fitting a function to the simulation data that captures the important features of the response. We find that the phase shift _cell() as a function of the frequency of the input current f = /2 can be well described by a function of the form (see Fig. 3A)

(10)

The first term on the right-hand side of Eq. 10 is a constant delay arising from the finite spike time to peak (see Fig. 3B). After the membrane potential has reached a certain depolarization threshold (about –45 mV), the membrane dynamics is dominated by the Na⁺ and K⁺ currents and is largely independent of the fluctuations in the input current. Thus the shape of the spike is independent of the inputs. The constant delay _spike corresponds to the time to peak of the action potential, that is, the time it takes from the point when the spike is already well initiated to the voltage maximum where the spike time is defined. The second term is related to the voltage dynamics near spike initiation and below the depolarization threshold. It can be well described by a linear filter with time constant _filter, and is best understood in the context of the simplified EIF neuron (see below). In the example given in Fig. 3 we found that _spike = 0.24 ms and filter time constant _filter = 4 ms.

View larger version (11K): [in this window] [in a new window]   FIG. 3. Phase shift curve fitted by a simple delayed filter. A: single-cell simulations (full squares) and the fitted function (solid curve). Function fitted to the data is –360fspike – (180/) atan (2ffilter), with spike = 0.24 ms and filter = 4.0 ms. Single-cell firing rate is 40 Hz; the synaptic time constant noise = 5 ms; SD of the noise is noise = 5 mV. B: interpretation of fixed delay spike: spike time to peak is the time from threshold to the maximum of the spike. For the conductance-based interneuron (see METHODS) spike = 0.24 ms.

MEAN FIRING RATE. We computed the phase shift _cell() as a function of the input frequency f for different mean firing rates r₀. Different single-cell firing rates are achieved by adjusting the mean input current I₀. Simulations show that the phase shift depends substantially on the average firing rate of the single cells (Fig. 4A). The filter time constant _filter decreases with increasing mean firing rate, whereas _spike is independent of r₀ (Fig. 4B). The instantaneous firing rate follows the input current with a smaller phase lag for larger average firing rates.

View larger version (28K): [in this window] [in a new window]   FIG. 4. Dependency of the phase shift on membrane time constant and mean firing rate. A: single-cell phase shift cell() decreases with increasing single-cell mean firing rate r0 at high frequencies. Firing rate is increased by increasing the mean current I0. B: intrinsic time constant filter decreases with increasing r0. Filled triangles are filter calculated from the asymptotic behavior of the exponential integrate-and-fire (EIF) neuron and the gain of the r0–I curve filter = (CmTdr0/dI)/r0. Delay spike is not affected by r0. Effective membrane time constant is m–eff = 10 ms. C: single-cell firing rate phase shift cell() decreases with increasing shunting conductance at high frequencies. D: intrinsic time constant filter (obtained by fitting the phase data in A with Eq. 10) decreases with m–eff = Cm/(gL + gshunt). Filled triangles are filter calculated from the asymptotic behavior of the EIF neuron and the gain of the r0–I curve filter = (CmTdr0/dI)/r0. Delay spike is not affected by m–eff. The slope dr0/dI is computed numerically. Single-cell firing rate is r0 = 40 Hz.

(11)

For different values of g_shunt we adjust the SD of _noise so that the fluctuations in the membrane potential are kept at about _V = 5 mV. As shown in Fig. 4C the addition of a shunting conductance g_shunt leads to a reduction in the cellular phase lag _cell() in the firing response to a noisy sinusoidal input. The time constant characterizing the spike time to peak _spike is unaffected by changes in g_shunt, and the changes in _cell() are attributed entirely to changes in _filter (Fig. 4D). A small effective membrane time constant leads to a smaller phase lag and allows the neuron to follow high-frequency inputs better.

The exponential integrate-and-fire neuron

Are the results presented in the previous section specific to models with Hodgkin–Huxley mechanisms or can they be obtained with a simpler integrate-and-fire–like model? The exponential integrate-and-fire (EIF) model (see METHODS) was recently introduced to incorporate the dynamics of spike initiation in the LIF model (Fourcaud-Trocmé et al. 2003). In the EIF model, the activation kinetics of the fast sodium current is assumed instantaneous and the voltage-dependent activation voltage dependency is assumed to be exponential (controlled by the parameter _T, Eq. 4). The EIF model does not include the repolarizing mechanism of the potassium current but, instead, the voltage is set to a reset potential after reaching a set peak value. The parameters of the EIF model neuron are chosen such that the firing rate–current relationships of the EIF and the conductance-based model neurons are almost indistinguishable (Fig. 5B), as are the voltage traces in response to noisy currents, except for a short interval after spike initiation (Fig. 5A and Fourcaud-Trocmé et al. 2003).

View larger version (32K): [in this window] [in a new window]   FIG. 5. EIF model reproduces quantitatively the behavior of a conductance-based model neuron. A: subthreshold potentials of conductance-based and EIF neurons match well with each other, except for a short time interval right after the spike discharge. B: parameters of the EIF neuron can be chosen such that the r0–I curves of the EIF model neuron and conductance-based neuron are comparable. C: phase of the single cell's response to noisy sinusoidal current input depends on the effective membrane time constant m–eff and can be fitted by a linear filter. Smooth curve is –atan (2ffilter), where filter = 4 ms. D: time constant filter can be estimated from the cutoff frequency fc, defined as the intersection between the low- and high-frequency limits of the response amplitude (also see text).

A simple analytical estimate of _filter for the EIF model neuron can be obtained from the low- and high-frequency limits investigated by Fourcaud-Trocmé et al. (2003) (see also Fig. 5D). In the low-frequency range, the linear response amplitude is essentially constant and proportional to the gain of the r₀–I curve at the corresponding mean frequency, i.e., dr₀/dI. In the high-frequency range, the amplitude decays as r₀/(C_mT2f). This behavior is reminiscent of a simple low-pass-filter rate model of the type dr/dt = –r + [I(t)]. For such a rate model, in response to a sinusoidal input, the amplitude of the firing rate modulation decays with f as , and the phase shift of the rate modulation is atan (f/f_c), where the cutoff frequency is related to the time constant of the rate model by f_c = 1/(2). For such a rate model, the cutoff frequency corresponds to the frequency at which the asymptotic expressions for r₁(f) in the low (r₁ 1) and high (r₁ f_c/f) frequency limits are equal. We can define the cutoff frequency in a similar way for the EIF model, which gives f_c = r₀/(2C_mTdr₀/dI) (Fig. 5D). This in turn gives an estimate of the filter time constant, _filter = 1/(2f_c), or _filter = (C_mTdr₀/dI)/r₀. This estimate turns out to be very close to the one obtained by the fitting procedure, as shown in Figs. 4, B and D. Note that one deduces directly from the analytical formula that the filter time constant decreases when the firing rate increases. Furthermore, it shows that the filter time constant is also proportional to the slope (gain) of the r₀–I curve, and to the spike activation parameter _T. The input conductance affects _filter through its effect on the slope of the r₀–I curve (not shown) (Chance et al. 2002). In the high-noise regime considered here, increasing input conductance decreases the gain of the r₀–I curve, when the mean firing rate is maintained constant. As a result, the filter time constant is decreased.

To summarize the single cell results, we found that, unlike the LIF model, the firing-rate response of a conductance-based model neuron to synaptically filtered noisy sinusoidal input is highly dependent on the input frequency. The phase shift in the firing rate can be described as the sum of two terms: the near-threshold voltage dynamics give rise to a filter term with _filter, and the threshold-to-peak spike width leads to an intrinsic latency _spike. The time constant _filter is highly dependent on the effective membrane time constant and mean firing rate, whereas _spike is not. Unlike LIF neurons, which can follow fast transients, the response of conductance-based neurons is strongly dependent on the single cell firing rate and the input conductance. The response properties of the conductance-based neuron can be well captured by the EIF neuron.

Theoretical determination of the population frequency of a network of inhibitory neurons

We now incorporate the response properties of single neurons in a theoretical framework that allows us to determine quantitatively the frequency of network oscillations. This represents an extension of the analysis of Brunel and Wang (2003), which assumed that single neurons respond instantaneously to inputs at all frequencies. To start with, we assume that during collective oscillatory population activity, the averaged instantaneous firing rate r_I(t) can be roughly described as a sinusoidal function. Single cells fire irregularly in time with a discharge probability equal to this sinusoidal function. Thus, the activity of each neuron in the population can be described by

(12)

where r_I,0 is the mean firing rate of the cell, = r_I,1/r_I,0 is the relative amplitude of the sinusoidal modulation in the firing rate, and = f/2, where f is the frequency of the sinusoidal modulation and corresponds to the population frequency (Figs. 1A and 6), yet to be calculated.

View larger version (21K): [in this window] [in a new window]   FIG. 6. Theoretical prediction of the rhythmic frequency of an interneuronal population. A: illustration of the analytical method. An inhibitory neuron has an average firing rate rI(t). Because of synaptic filtering, the fraction of open channels sI(t) lags behind the firing rate with a phase I,syn(f). Synaptic current II(t) is antiphasic to sI(t) because it is an inhibitory current. The neuron lags behind the synaptic current by I,cell(f). For network oscillations to emerge the phase must satisfy – = I,syn(f) + I,cell(f), where f is the population frequency. B: total phase shift vs. frequency: population frequency is the intersection of the total phase shift [dashed line, synaptic phase I,syn(f); solid line, synaptic + single-cell phase I,syn(f) + I,cell(f)] with the horizontal line at –180°. In the presence of synaptic filtering only, the population frequency is almost 300 Hz. Single-cell filtering reduces the population frequency to 125 Hz.

(13)

Note that a negative phase shift corresponds to a phase lag. The synaptic current is the product of the fraction of open channels at GABAergic synapses multiplied by the driving force. Assuming that the temporal variations in the driving force are small compared to the temporal variations in the fraction of open channels, the phase shift of the GABAergic current is given by _I,syn() – , where the factor comes from the inhibitory nature of GABAergic currents.

The next step is to determine the time course of the firing rate of a postsynaptic neuron that receives an oscillatory current with a phase shift _I,syn() – . We have seen in the previous section that such a postsynaptic neuron will respond to a noisy sinusoidal current with a phase shift _I,cell(), which also depends on the input frequency f. Thus the total phase shift of the postsynaptic firing rate with respect to the presynaptic one is _I,syn() – + _I,cell(). Because the instantaneous firing rate of pre- and postsynaptic neurons must be in phase for network oscillations to emerge, this total phase shift must be equal to –2, i.e.

(14)

Equation 14 gives the predicted population frequency. It shows that long synaptic time constants increase the phase and decrease the population frequency (see also Brunel and Wang 2003). When the phase shift arising from the single-cell response is negligible as in the case of LIF neurons, _I,cell() = 0, the population frequency can be computed by solving the equation = _Il + atan (_Ir) + atan (_Id) (Brunel and Wang 2003). An additional phase lag [_I,cell() < 0] reduces the population frequency as shown in Fig. 6B.

To obtain a prediction of the network frequency, we insert the phenomenological description of the phase shift _I,cell() of the single cell's response to noisy sinusoidal current (Eq. 10) in Eq. 14. The condition for the total phase in an oscillating network can then be written

(15)

Thus, the population frequency of a network of inhibitory coupled neurons can be calculated by knowing the inhibitory synaptic time constants and the two cellular time constants _I,spike and _I,filter. For example, when the phase shift is negligible for all input frequencies [_I,cell(f) 0, corresponding to LIF neurons], Eq. 15 predicts a population frequency of almost f = 300 Hz, assuming the standard synaptic time constants (_Il = 0.5 ms, _Ir = 0.5 ms, _Id = 5.0 ms). A delay _I,spike = 0.24 ms corresponding to the spike time to peak lowers the population frequency to about f = 230 Hz. The single-cell filtering time constant _I,filter has a much stronger effect on the population frequency. For a firing rate of 40 Hz and an effective membrane time constant of 10 ms, we have seen in the previous section that _I,filter = 4 ms. This leads to a decrease of the population frequency to f = 95 Hz. However, the effective membrane time constant is significantly reduced during network activity as a result of the synaptic conductance change and we expect a higher population frequency. For example, if the effective membrane time constant changes tenfold (from _m–eff = 10 ms to _m–eff = 1 ms) and the average firing rate of the neurons is r_I,0 = 40 Hz we predict a population frequency f 140 Hz.

Emergent oscillations in a population of conductance-based inhibitory interneurons

How does the analysis compare with simulation results? Simulations of a network composed of a single population of mutually connected inhibitory GABAergic Hodgkin–Huxley-type conductance-based cells show a prominent fast rhythm when the external drive is sufficiently large and recurrent inhibition is strong (Fig. 7A). A similar result was shown for a network of LIF neurons (Brunel and Wang 2003). During network activity single cells are subjected to synaptic currents, external excitation (Poisson rate of the AMPA-receptor–mediated synaptic inputs), and recurrent inhibition. The recurrent inhibition has an oscillatory component stemming from the emerging network oscillations. The synaptic inputs to each cell of the network induce a noisy subthreshold oscillation in the membrane voltage; fluctuations around the average subthreshold time course of the voltage occasionally cause a cell to spike. The spike pattern differs from cell to cell and the single-cell firing rates are heterogeneous across the network because external drive is Poisson and recurrent connections are random and sparse. The power spectrum of a single spike train does not indicate any rhythmicity (Fig. 7B), even though "multiunit" activity averaged over a group of 10 cells during the same time interval shows a peak in the power spectrum at 125 Hz (Fig. 7C). This narrow peak in the power spectrum indicates that the network activity is indeed dominated by a single frequency component. Similarly, the autocorrelation of a single spike train does not show an oscillatory pattern (Fig. 7D); thus oscillations are hardly apparent in a single cell's spike train, because the probability of firing in any given cycle is small. On the other hand, the population firing rate, averaged over a large number of cells, clearly reveals the network oscillation at a much higher frequency than the averaged single cell firing rate (Fig. 7A). Although single cells do not discharge rhythmically, they fire at a preferred phase of the population oscillation, as can be seen in the cross-correlation between the spike train of a single cell and the population activity (spike-triggered average of the population firing rate, Fig. 7E).

The oscillation frequency of 125 Hz (Fig. 7A) can be predicted from the single-cell analysis in the following way. First, given the synaptic time constants the synaptic phase shift can be computed as a function of the population frequency _I,syn() (Eq. 13). The next step is to determine the single-cell phase shift _I,cell(), which depends on the average single cell firing rate (40 Hz) and the effective membrane time constant. Each neuron receives on average an inhibitory input of 2 kHz [average single cell firing rate (40 Hz) multiplied by the number of connections (50)] and an excitatory input of 5 kHz. Considering the conductances of recurrent inhibition (_GABA1 = 6.2 nS) and external excitation (_extI = 1.5 nS) and the time constants of each synapse, the total synaptic conductance can be computed to be 0.11 µS. The neuron has therefore an effective membrane time constant of 1.5 ms. Using a single cell with a shunt conductance of 0.11 µS, the single-cell phase shift can be determined. The time constants _I,filter = 1.6 ms and _I,spike = 0.24 ms can be obtained with a fit to the phase shift. The last step is to evaluate the self-consistent solution (Eq. 15), which leads to a population frequency of 127 Hz, remarkably close to what we observe (125 Hz) in network simulations.

How does the network frequency depend on the external excitation? The increase in external excitation leads to an increase in the single cell's firing rate and a decrease of the effective membrane time constant (resulting from increased external and recurrent synaptic conductance). Brunel and Wang (2003) showed that the oscillation frequency of a population of LIF neurons is independent of the external excitatory input because the response of the single LIF neuron to noisy oscillating synaptic current does not depend on the single-cell properties. In contrast, Fig. 8 shows that the conductance-based neuron's response to noisy oscillating synaptic current does depend on the single-cell properties. Indeed, when the external drive is varied gradually, the single-cell firing rate as well as the population frequency increase almost linearly. The population can oscillate at frequencies ranging from <100 to >200 Hz. This frequency range is significantly lower than the population frequency of a network of LIF model neurons (300 Hz), when the comparison is made with the same synaptic parameters (latency _Il = 0.5 ms, rise time _Ir = 0.5 ms, and decay time _Id = 5.0 ms). Because the synaptic conductance affects the single-cell properties, and thus the phase shift with which the conductance-based neuron responds to the synaptic current, the population frequency of conductance-based neurons can be modulated by the synaptic afferents.

View larger version (17K): [in this window] [in a new window]   FIG. 8. Network oscillation frequency and neuronal firing rate as function of the input amplitude. A: population frequency increases with increasing input strength. Each cell receives an external excitatory Poisson input with rate . With increasing the single-cell frequency (filled diamonds; averaged over all cells) and the population frequency (triangles) increase. B: network synchrony increases rapidly when the input rate crosses a threshold (5 kHz) and saturates for larger input. Parameters as in Fig. 7.

An important conclusion from the single-cell study is that the effective membrane time constant of the single cell has a strong influence on the phase shift and therefore on the oscillation frequencies. We predict that fast oscillations can be obtained by a small single-cell effective membrane time constant. This raises the question as to whether physiologically observed frequencies of very fast oscillations (around 200 Hz) are realizable. We now discuss two possible scenarios by which a small effective membrane time constant might be obtained and, consequently, fast network oscillations might emerge. In the first scenario an intrinsic current activated in the subthreshold range is used, whereas in the second scenario large synaptic inputs lead to a decrease in the effective membrane time constant.

The A-type potassium current is an example of a membrane current that is activated below threshold and contributes to the effective membrane time constant (Connor et al. 1977) (see METHODS). When the voltage is close to threshold, an activated A-channel conductance increases the total membrane conductance, thus decreasing the effective membrane time constant _m–eff. We find that an inhibitory network of conductance-based neurons endowed with an A-type potassium current (Connor et al. 1977) can oscillate at very high frequencies (Fig. 9). With the same synaptic time constants and the same average single-cell firing rate (40 Hz) as in Fig. 7 A, the population oscillates at 220 Hz instead of 125 Hz. This dramatic increase of the population frequency compared to the network of cells without an A-type current (Fig. 7A) is explained by the fact that the effective membrane time constant is very small, _m–eff = 0.15 ms in this model.

View larger version (27K): [in this window] [in a new window]   FIG. 9. A population of conductance-based neurons with a small effective membrane time constant can oscillate at 220 Hz. Single cells contain an A-type potassium current that contributes to reducing the effective membrane time constant at subthreshold potentials. During network activity, the membrane time constant m–eff = 0.15 ms: (a) spike raster of 10 cells; (b) membrane potential of a single cell; (c) instantaneous population firing rate; (d) distribution of firing rates across neurons; (e) power spectrum of the instantaneous population firing rate. External excitatory input is 9 kHz. Parameters: connection probability 0.1; synaptic time constants as in Fig. 7.

Is the change of _m–eff sufficient to account for the increase in the population frequency in these two scenarios? This question can be addressed by comparing network simulations with the theoretical prediction (Fig. 11). To predict the population frequency we first need to know the synaptic phase shift _I,syn(f), which can be calculated from the synaptic time constants. Second, the single-cell phase shift _I,cell(f) depends on the time to peak of the spike (_I,spike = 0.24 ms) and the filter time constant _I,filter. The latter depends on the effective membrane time constant _m–eff and on the single cell firing rate _I,0 (here _I,0 = 40 Hz) (Fig. 4). Using the relationship between _m–eff and _I,filter obtained from the EIF neuron model (Fig. 4D) we can predict the population rhythmic frequency of a neural network, as a function of _m–eff (Fig. 11, solid curve). The theoretical prediction agrees well with results from direct network simulations (Fig. 10B), in which the synaptic conductance was varied so that the single cell's effective membrane time constant changed from _m–eff = 1.5 ms (Fig. 7) to _m–eff = 0.3 ms (corresponding to large synaptic bombardment, Fig. 10). The agreement is also good for a simulated network of neurons endowed with an A-current (_m–eff = 0.15 ms, population frequency of 220 Hz).

View larger version (17K): [in this window] [in a new window]   FIG. 11. Population frequency increases with decreasing effective membrane time constant. Solid curve: analytical prediction from Eq. 14 using the EIF model and the relationship between filter and the gain of the r0–I curve I,filter = (CmTdrI,0/dI)/rI,0 (see text and Fig. 4D); we assume the standard synaptic time constants (Il = 0.5 ms, Il = 0.5 ms, Id = 5.0 ms) and a finite spike width (I,spike = 0.24 ms). Open triangle: simulations of networks with additional external excitatory and inhibitory synaptic input (Fig. 10B). Star: simulations of a network of interneurons with A-type potassium current (Fig. 9).

Oscillations in a two-population network of excitatory and inhibitory neurons

We have shown that an interneuronal network can by itself generate very fast rhythms. In cortical networks, interneurons are reciprocally connected to pyramidal cells. How do the interactions with pyramidal cells affect the patterns and, in particular, the frequency of such fast oscillations? To address this question we investigate synchronous oscillations in a two-population network of inhibitory interneurons and excitatory pyramidal cells. With this type of architecture, two possible scenarios for rhythmogenesis are present: the interneuronal network and the loop between interneurons and pyramidal cells. Brunel and Wang (2003) showed that in a two-population network of LIF neurons, the oscillation frequency depends on the synaptic time constants, the relative strength between the excitatory and inhibitory synaptic currents and the connectivity among pyramidal cells. Increasing the relative strength of excitation versus inhibition typically decreases the population frequency. Including recurrent connections among pyramidal cells reduces the population frequency further.

Do these results hold for conductance-based neurons as well? Based on the single-cell simulations shown in Figs. 1–4, we expect that in a two-population network of conductance-based cells, the population frequency should also depend on the phase response of single interneurons and pyramidal cells to the synaptic current. We expect the population frequency to be lower than that of a network of LIF neurons. To understand in more detail how the oscillation frequency depends on single-cell and synaptic parameters, the population frequency can be computed from both synaptic parameters and single-cell phase-shift curves, using calculations similar to those presented for a one-population network (see Fig. 12 and Appendix A). The calculation proceeds according to the following steps:

• The population firing rates of interneurons and of pyramidal cells are approximated by sinusoidal functions. These functions are characterized by the mean firing rates of interneurons (_I,0) and pyramidal cells (r_P,0), the relative amplitude of the sinusoidal modulation (_I for interneurons and _P for pyramidal cells), and the population frequency f = /2. The firing rates of pyramidal cells and interneurons are not necessarily in phase.
  
• The fraction of open channels at the postsynaptic membrane follows the presynaptic firing probability with a phase delay as a result of synaptic filtering. The phase delay of the inhibitory (excitatory) synaptic currents _I,syn() – [_P,syn()] depends on the latency, rise, and decay time constants of the synaptic currents.
  
• Each neuron receives a total current that is the sum of inhibitory and excitatory synaptic currents. For the sake of simplicity, we consider here the special case when the relative strength of excitatory and inhibitory synaptic current (I_AMPA/I_GABA ratio) is the same for interneurons and pyramidal cells. The phase of the total current depends on this relative strength and on the phase difference between excitation and inhibition.
  
• In general, a neuron responds to the total synaptic current with a phase shift, which depends on the single-cell properties, such as firing rate and effective membrane time constant. The phase shifts of interneurons _I,cell() and pyramidal cells _P,cell() depend on the time constants _I,spike, _I,filter and _P,spike, _P,filter, respectively, and can be determined from single cell simulations.
  
• Self-consistency between the presynaptic and postsynaptic firing rates, which are both the source and recipient of the recurrent synaptic input, leads to equations for the phases of excitatory and inhibitory neurons. These two equations give the frequency f of the global oscillation and the phase shift between the two populations.
  

View larger version (23K): [in this window] [in a new window]   FIG. 12. Predicting the population frequency in a two-population network of interneurons and pyramidal cells: the synaptic current II(t) of an oscillating averaged instantaneous firing rate of interneurons rI(t) is phase delayed by I,syn(f) – . Synaptic current IP(t) of an oscillating averaged instantaneous firing rate of pyramidal cells rP(t) is phase delayed by P,syn(f). Cells respond to the total synaptic current II(t) + IP(t) with a phase lag I,cell(f) (interneurons) and P,cell(f) (pyramidal cells).

We found that a wide range of frequencies can be realized in a network with the same set of synaptic time constants and the same architecture. Critical determinants of the network oscillation frequency are the level of balance between synaptic excitation and inhibition (the time-averaged I_AMPA/I_GABA ratio, which was imposed to be the same in pyramidal cells and interneurons in our simulations) and the phase shift between the excitatory and inhibitory synaptic currents. When the I_AMPA/I_GABA ratio is low, which can be achieved by very low firing rates of pyramidal cells compared to interneurons (e.g., interneurons fire at 35 Hz, whereas the firing rates in pyramidal cells is 1 Hz), the population frequency is mainly determined by the interneuronal network. The pyramidal cells are basically paced by the rhythmic inhibition and the population frequency is >110 Hz (Fig. 13A). On the other hand, with a high I_AMPA/I_GABA ratio, the network displays synchronous gamma frequency (40 Hz) when the firing rates of pyramidal cells are high compared to interneurons (e.g., interneurons fire on average at 8 Hz and pyramidal cells on average at 3 Hz, and see Fig. 14 ). In that case, the oscillations are mainly maintained by the loop between interneurons and pyramidal cells.

View larger version (33K): [in this window] [in a new window]   FIG. 13. A two-population network can oscillate at different frequencies depending on the balance of excitatory and inhibitory synaptic currents. In both cases are shown the membrane voltage of a single pyramidal cell (a) and interneuron (b); the instantaneous population firing rate of pyramidal cells (gray) and interneurons (black) (c); the distribution of the single cell firing rates across the population (d); and the power spectrum of the instantaneous population firing rate of pyramidal cells (e). A: high-frequency oscillations (125 Hz). IAMPA/IGABA ratio is essentially zero and the population frequency is mainly determined by the interneuronal network. Average single-cell firing rates: pyramidal cells 1 Hz, interneurons 35 Hz. External input to pyramidal cells 3 kHz, to interneurons 4.5 kHz. B: high-frequency oscillations (190 Hz). When the IAMPA/IGABA ratio is not zero but low, the population frequency is determined by the interneuronal network and the interneuron–pyramidal cell loop. Average single-cell firing rate: pyramidal cells 15 Hz and interneurons 40 Hz. External input to pyramidal cells 5.06 kHz and to interneurons 5.6 kHz. Synaptic time constants: GABA Il = 0.5 ms, Ir = 0.5 ms, Id = 5.0 ms; AMPA Pl = 1.5 ms, Pr = 0.5 ms, Pd = 2.0 ms; connection probability 0.1.

View larger version (24K): [in this window] [in a new window]   FIG. 14. A two-population network oscillates at gamma frequency (40 Hz). A: membrane voltage of a single pyramidal cell (a) and interneuron (b), the instantaneous population firing rate of pyramidal cells (gray) and interneurons (black) (c), the distribution of the single cell firing rates across the population (d), and the power spectrum of the instantaneous population firing rate of pyramidal cells (e). Average single-cell firing rates: pyramidal cells 3 Hz, interneurons 8 Hz. External input to pyramidal cells 0.82 kHz, to interneurons 0.4 kHz. B: oscillations are not detectable in the power spectrum of single-cell spike trains. C: power spectrum of multiunit activity averaged over 50 spike trains shows a clear peak at 40 Hz. D: autocorrelation of a single spike train (averaged over 50 cells) does not show a clear oscillatory pattern. E: STA of the population rate shows oscillations at gamma frequency. Spike-triggered population rate is averaged over the spikes of one neuron (52 spikes). Length of spike train: 10 s. Synaptic parameters as in Fig. 13.

(16)

The phase shift _current() (Eq. 16) depends on the synaptic time constants through the synaptic phase shifts of excitation _P,syn() and inhibition _I,syn() – , and on the single-cell properties of pyramidal cells and interneurons through _P,cell() and _I,cell(), respectively.

In Fig. 15A, we illustrate different scenarios by plotting the relative locations of the peaks of AMPA and GABA currents in an oscillatory cycle of the total synaptic current, when the I_AMPA/I_GABA ratio is <1. If in such a cycle |_P,cell + _P,syn| < |_I,cell + _I,syn| (respectively |_P,cell + _P,syn| > |_I,cell + _I,syn|) I_AMPA follows (respectively precedes) I_GABA. As a first example, consider the case in which _P,cell() = _I,cell(), and excitatory time constants are shorter than inhibitory time constants, |_P,syn()| < |_I,syn()| [Fig. 15A(i)]. Then _current() > 180°, excitation follows inhibition, and therefore an increase in excitation strength will decrease the oscillation frequency. This is the scenario that was considered in Brunel and Wang (2003). On the other hand, if the single-cell phase shift of pyramidal cells is sufficiently larger than that of interneurons, due to a larger _m–eff and/or _P,spike, then _current() can become <180°, and excitation now precedes inhibition [Fig. 15A(ii)]. Thus, somewhat counterintuitively, a larger neuronal phase shift of pyramidal cells would favor the regime where an increased I_AMPA/I_GABA ratio accelerates the network oscillation.

View larger version (22K): [in this window] [in a new window]   FIG. 15. Population frequency depends on the balance (IAMPA/IGABA) and on the relative phase shift between the excitatory and inhibitory synaptic currents current = P,syn(f) + P,cell(f) – I,syn(f) – I,cell(f) + . A: illustration of different scenarios that can lead to excitatory current IAMPA following (i) and (iv) or preceding (ii) and (iii) inhibitory current IGABA, respectively (see text). B, left: population frequency decreases with increasing recurrent excitation when excitation follows inhibition. Right: phase shift current is >180° for all values of IAMPA/IGABA. Excitatory synaptic current follows the inhibitory synaptic current, as illustrated in the top trace (AMPA–latency = 0.5 ms; all other synaptic parameters as in Fig. 13). C, left: population frequency increases with increasing recurrent excitation when excitation precedes inhibition. Right: for small values of IAMPA/IGABA the excitatory synaptic current precedes the inhibitory synaptic current, as illustrated in the top trace. There is a critical IAMPA/IGABA ratio, above which a second stable solution becomes possible and the trend is reversed: the oscillation frequency shows a discrete drop from ripple frequency to gamma frequency range (solid line: stable solution; dotted line: unstable solution). (AMPA–latency = 1.5 ms; all other synaptic parameters as in Fig. 13). Solid/dotted curves: analytical prediction; open circle: network simulations.

The phase shift between the firing rates of interneurons and pyramidal cells depends only on single-cell characteristics

(17)

when the I_AMPA/I_GABA ratio is the same in interneurons and pyramidal cells (APPENDIX B). When _rate > 0 interneurons follow pyramidal cells. For LIF neurons _rate = 0, because interneurons and pyramidal cells respond to oscillating synaptic current with negligible phase lag and, as a consequence, both populations oscillate in phase. On the other hand, conductance-based neurons generally respond to oscillating synaptic currents with a phase lag that is larger for pyramidal cells than that for interneurons. When the I_AMPA/I_GABA ratio is the same in pyramidal cells and interneurons, pyramidal cells tend to follow interneurons. However, when recurrent connections among pyramidal cells are absent, or more generally when the ratio of excitation to inhibition is weaker in pyramidal cells than in interneurons, pyramidal cells tend to precede interneurons (Brunel and Wang 2003).

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
During certain behavioral states, neocortex and hippocampus display fast (40–200 Hz) oscillations that are detected in the local field potential, but not easily in spike trains of individual neurons that are typically very irregular (Engel et al. 1992; Fries et al. 2001a,b; Logothetis et al. 2001). This is at odds with most theoretical studies of network synchronization, in which neurons are coupled oscillators (Borgers and Kopell 2003; Ermentrout and Rinzel 1984; Gerstner et al. 1996; Hansel et al. 1995; Kopell and LeMasson 1994; Marder 1998; Nomura et al. 2003; Traub et al. 1996; van Vreeswijk et al. 1994; Wang and Buzsáki 1996). On the other hand, rhythmic phenomena that mimic the phenomenology of fast oscillations occur in networks in which single neurons are subject to large amounts of noise, which can be attributed either to external inputs or to intrinsic network dynamics in networks with sparse and random connectivity (Brunel 2000; Brunel and Hakim 1999; Brunel and Wang 2003). The present study represents a further step along this line of research, and extends the theoretical framework to networks of Hodgkin–Huxley-type conductance-based models. We showed that the frequency of network oscillations is determined by the interplay between synaptic kinetics and the single cell's spiking properties. We extended a theoretical framework for predicting the population frequency, on the basis of quantitative synaptic kinetic properties and the response properties of single cells.

Collective network rhythmic frequency depends on single-cell spiking properties

In sharp contrast to the LIF model, the firing response of a single conductance-based model neuron to a noisy sinusoidal input strongly depends on the stimulus frequency (see also Fourcaud-Trocmé et al. 2003). This cellular response property has a major impact on the frequency of synchronous network oscillations. In interneuronal networks, with the same network connectivity, same synaptic time constants, and same average firing rates of single neurons, varying the single-cell properties alone can change the network rhythmic frequency by a significant amount (e.g., from 100 to 200 Hz). The present study shows that, when single neurons fire irregularly and fast coherent oscillation is an emergent network phenomenon, synchronization properties (in particular the frequency) depend critically on the membrane dynamics that control the upstroke of action potentials. This is explicitly demonstrated by the exponential integrate-and-fire model that, with just one nonlinear voltage dependence, is shown to adequately account for the single cell's impact on the network oscillation frequency.

Moreover, we show that the membrane time constant of single cells greatly affects the frequency of network oscillations. An increased total membrane conductance, described either to spontaneous synaptic activity (Borg-Graham et al. 1998; Chance et al. 2002; Destexhe and Paré 1999; Häusser et al. 2001) or to intrinsic ion channels (Connor et al. 1977; Softky 1994) leads to a smaller effective membrane time constant, which favors a higher population rhythmic frequency. Thus the general conclusions of Brunel and Wang (2003) that an interneuronal network of inhibitory cells can give rise to >100 Hz coherent oscillations with irregular neural discharges still holds. However, the effective membrane time constant needs to be <0.5 ms in the models we considered here to achieve a 200-Hz oscillation. It is not known whether such a short effective membrane time constant is realized in real neurons in vivo. Another parameter that strongly influences the phase lag of single cells at high frequency is the sharpness of spike initiation, as measured by the parameter _T of the EIF model (Fourcaud-Trocmé et al. 2003). Thus if real neurons have significantly sharper spike initiation than the Hodgkin–Huxley-type models considered here, 200-Hz oscillations could be sustained with larger effective membrane time constants.

To predict the frequency of weakly synchronous rhythms in a noise-dominated network, it is necessary to quantitatively characterize the responsiveness of single cells to a noisy sinusoidal input (see also Fuhrmann et al. 2002). We showed that a Hodgkin–Huxley-like conductance-based neuron has a smaller response amplitude and larger phase lag with increasingly higher input frequency. This modulation is independent of the synaptic time constant, in contrast to the LIF model for which the frequency dependency becomes negligible when the synaptic time constant becomes comparable to or larger than the neuronal membrane time constant (Brunel et al. 2001). The phase lag can be approximately described by the sum of a linear filter, related to the membrane dynamics for the upstroke leading to a spike threshold; and a constant phase shift related to the spike time to peak. The time constant for the linear filter is shorter with smaller effective membrane time constant and higher single-cell firing rate. This leads to a smaller phase lag of single cells, which implies faster population frequency in the network. Note that our approach with a filter is only a phenomenological description. Other single-cell conductance-based models display phase advance at low frequencies arising from negative feedback mechanisms (Fuhrmann et al. 2002; Richardson et al. 2003; Shriki et al. 2003). However, these phase-advance phenomena are generally observed at much lower frequencies than the network frequencies investigated in this paper, and they should not interfere with the mechanisms giving rise to the fast network oscillation, although they could modulate such an oscillation slowly.

Fast oscillations in two-population network of pyramidal cells and interneurons

In this paper, we examined rhythmogenesis both in a one-population network of inhibitory interneurons and in a two-population network of interneurons and pyramidal cells. The oscillation frequency in a two-population network of conductance-based neurons depends strongly on the current balance (I_AMPA/I_GABA ratio) and time constants of excitatory and inhibitory synaptic interactions, as has been shown in the network of LIF neurons (Brunel and Wang 2003). It has been shown that strong recurrent excitation typically reduces the oscillation frequency in a two-population network of LIF neurons, compared to the purely interneuronal network (Brunel and Wang 2003). We observed that intrinsic and/or synaptic dynamics of excitatory neurons, which are slower than those of inhibitory interneurons, can lead to faster rhythmic frequencies in the two-population network, compared to the purely interneuronal network. Intuitively, this happens when the combined (synaptic and cellular) phase lag for excitation exceeds that for inhibition by >180°, so that excitation appears to be in advance of inhibition. Under this condition, 200-Hz oscillations can be realized even with reasonable effective membrane time constants (1.2 ms), unlike the purely interneuronal network. Moreover, because the population frequency is larger with higher single-cell firing rates, it is easier to realize 200-Hz network rhythms with increased neural activity. This is in consonance with the experimental observation that single-cell firing rates increase significantly during 200-Hz sharp-wave ripples compared to non–sharp-wave episodes (Csicsvari et al. 1999b). In the scenario in which recurrent excitation increases the population frequency at low I_AMPA/I_GABA ratios, one can distinguish two well-separated frequency bands: a high-frequency band (120–250 Hz) at low I_AMPA/I_GABA ratios; and the gamma frequency band (40–80 Hz) at larger I_AMPA/I_GABA ratios.

Our analysis of rhythmogenesis in a recurrent network of noisy neurons requires knowledge of synaptic kinetics and strength, as well as of how a single cell responds to noisy sinusoidal inputs. Thus it is crucial to examine experimentally how cortical neurons (both pyramidal cells and interneurons) respond to a weak oscillatory input in the presence of a large amount of noise, and especially how the response amplitude and phase depend on the input frequency. The present study also highlights the importance of measuring the ratio of the mean excitatory and inhibitory currents (Anderson et al. 2000; Borg-Graham et al. 1996; Compte et al. 2003; Shu et al. 2003) and, in particular, examining whether this ratio is roughly the same in interneurons and pyramidal cells.

If the I_AMPA/I_GABA ratio is the same in two cell types, the phase lag of firing rates between the two populations is solely determined by the single cell's properties. Recently, it was found that pyramidal cells precede fast-spiking interneurons by 90° during fast oscillations (Csicsvari et al. 1999b; Klausberger et al. 2003). In a network in which the excitation–inhibition balance is the same in two cell types, such an experimental finding could be accommodated only if interneurons have a larger cellular phase shift than that of pyramidal cells. On the other hand, such a phase shift between firing rates can be accounted for in a network in which the excitation–inhibition balance is lower in pyramidal cells than that in interneurons (Brunel and Wang 2003). In this scenario, the high network frequency could be compatible only with very small single-cell filter time constants, which could be obtained by a massive increase of input conductance, and/or a very sharp spike initiation.

To conclude, we have developed a theoretical framework for predicting the rhythmic frequency and relative phase relationship between cell populations for a noisy neural network, in terms of cellular and synaptic biophysical properties. This work helps to reconcile the apparent dichotomy between oscillatory local field potentials and almost Poisson-like stochastic spike discharges of single neurons, a characteristic of fast coherent oscillations observed in the neocortex of awake behaving animals (Averbeck and Lee 2004; Baker et al. 2001; Fries et al. 2001b).

	Appendix

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
A. Oscillation frequency of a two-population network

The presynaptic firing rates of interneurons and pyramidal cells are

(A1)

where r_P,0 and r_I,0 are the mean rates of interneurons and pyramidal cells, respectively, and v_P and v_I are relative deviations from the mean. The fraction of open channels follows the firing rate with a phase lag as a result of the synaptic filtering

(A2)

where the attenuation in the amplitude of the oscillation induced by synaptic filtering

(A3)

and the phase introduced by synaptic filtering

(A4)

depend on the synaptic time constants: latency _Pl, rise _Pr, and decay _Pd time for excitation; and latency _Il, rise _Ir, and decay _Id time for inhibition. Neglecting fluctuations in the driving force, the synaptic current can be written as

(A5)

The factor – comes from the fact that I_GABA(t) is an inhibitory current and is therefore phase-reversed compared to the fraction of open channels s_GABA(t). The total synaptic current for interneurons and pyramidal cells is a superposition of excitatory and inhibitory current

(A6)

where

(A7)

The postsynaptic firing rate follows the synaptic current, but with an additional phase shift

(A8)

where _P,cell() and _I,cell() are the phase shifts attributed to intrinsic cell properties. A_P() and A_I() are normalized oscillation amplitudes of the firing rates, and _P and _I are the gains of the r₀–I curves at the frequency r₀.

To find the self-consistent solution we equate the pre- and postsynaptic firing rates (Eqs. A1 and A8, respectively) of pyramidal cells and of interneurons

(A9)

(A10)

which leads to

(A11)

where the amplitude X() and phase () are given as

(A12)

with , = {P, I}. In general the ratio of excitation and inhibition is not the same for interneurons and pyramidal cells but might differ by a factor

(A13)

Note that this is equivalent to X_PP()/X_IP() = X_PI()/X_II(). We can now write down the condition for the phase that determines the population frequency

(A14)

When the ratio of excitation and inhibition is the same for interneurons and pyramidal cells ( = 1), the population frequency is simply given by

(A15)

In this case, the population frequency depends only on the ratio of excitatory and inhibitory current (I_PP/I_II) and the synaptic and single-cell properties of interneurons and pyramidal cells.

	Appendix

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
B. The relative phase between excitatory and inhibitory neurons

The postsynaptic firing rate (Eq. A8) can be written in the form

(B1)

which allows a direct comparison of the two firing rate probabilities. The phase difference _rate() and the amplitude can be calculated from Eqs. A9–A11. The amplitude is

(B2)

and the phase is given by

(B3)

The phase shift between excitatory and inhibitory currents can be derived from

(B4)

and is then

(B5)

In the balanced case, when = 1, the phase differences reduce to

(E6)

(B7)

	GRANTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
This work was supported in part by National Institute of Mental Health Grant MH-62349 and the Swartz Foundation to C. Geisler and X.-J. Wang.

	FOOTNOTES

 
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Address for reprint requests and other correspondence: X.-J. Wang, Volen Center for Complex Systems, Brandeis University, Waltham, MA 02454 (E-mail: xjwang{at}brandeis.edu)

	REFERENCES

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION Appendix Appendix GRANTS REFERENCES

 
Anderson JS, Carandini M, and Ferster D. Orientation tuning of input conductance, excitation, and inhibition in cat primary visual cortex. J Neurophysiol 84: 909–926, 2000.[Abstract/Free Full Text]

Angulo MC, Rossier J, and Audinat E. Postsynaptic glutamate receptors and integrative properties of fast-spiking interneurons in the rat neocortex. J Neurophysiol 82: 1295–1302, 1999.[Abstract/Free Full Text]

Averbeck BB and Lee D. Coding and transmission of information by neural ensembles. Trends Neurosci 27: 225–230, 2004.[CrossRef][Web of Science][Medline]

Azouz R and Gray CM. Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo. Proc Natl Acad Sci USA 97: 8110–8115, 2000.[Abstract/Free Full Text]

Baker SN, Spinks R, Jackson A, and Lemon RN. Synchronization in monkey motor cortex during a precision grip task. I. Task-dependent modulation in single-unit synchrony. J Neurophysiol 85: 869–885, 2001.[Abstract/Free Full Text]

Bartos M, Vida I, Frotscher M, Geiger JRP, and Jonas P. Rapid signaling at inhibitory synapses in a dentate gyrus interneuron network. J Neurosci 21: 2687–2698, 2001.[Abstract/Free Full Text]

Bartos M, Vida I, Frotscher M, Meyer A, Monyer H, Geiger JRP, and Jonas P. Fast synaptic inhibition promotes synchronized gamma oscillations in hippocampal interneuron networks. Proc Natl Acad Sci USA 99: 13222–13227, 2002.[Abstract/Free Full Text]

Borg-Graham L, Monier C, and Frégnac Y. Voltage-clamp measurement of visually-evoked conductances with whole-cell patch recordings in primary visual cortex. J Physiol (Paris) 90: 185–188, 1996.[CrossRef][Web of Science][Medline]

Borg-Graham LJ, Monier C, and Frégnac Y. Visual input evokes transient and strong shunting inhibition in visual cortical neurons. Nat Neurosci 393: 369–373, 1998.

Borgers C and Kopell N. Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Comput 15: 509–538, 2003.[CrossRef][Web of Science][Medline]

Brunel N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8: 183–208, 2000.[CrossRef][Web of Science][Medline]

Brunel N, Chance F, Fourcaud N, and Abbott L. Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86: 2186–2189, 2001.[CrossRef][Web of Science][Medline]

Brunel N and Hakim V. Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput 11: 1621–1671, 1999.[CrossRef][Web of Science][Medline]

Brunel N and Wang X-J. What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation–inhibition balance. J Neurophysiol 90: 415–430, 2003.[Abstract/Free Full Text]

Buhl EH, Han ZS, Lorinczi Z, Stezhka VV, Karnup SV, and Somogyi P. Physiological properties of anatomically identified axo-axonic cells in the rat hippocampus. J Neurophysiol 71: 1289–1307, 1994.[Abstract/Free Full Text]

Buhl EH, Tamas G, and Fisahn A. Cholinergic activation and tonic excitation induce persistent gamma oscillations in mouse somatosensory cortex in vitro. J Physiol 513: 117–126, 1998.[Abstract/Free Full Text]

Buhl EH, Tamas G, Szilagyi T, Stricker C, Paulsen O, and Somogyi P. Effect, number and location of synapses made by single pyramidal cells onto aspiny interneurones of cat visual cortex. J Physiol 500: 689–713, 1997.[Abstract/Free Full Text]

Buzsáki G, Urioste R, Hetke J, and Wise K. High frequency network oscillation in the hippocampus. Science 256: 1025–1027, 1992.[Abstract/Free Full Text]

Chance FS, Abbott LF, and Reyes AD. Gain modulation from background synaptic input. Neuron 35: 773–782, 2002.[CrossRef][Web of Science][Medline]

Compte A, Sanchez-Vives MV, McCormick DA, and Wang X-J. Cellular and network mechanisms of slow oscillatory activity (<1 Hz) and wave propagations in a cortical network model. J Neurophysiol 89: 2707–2725, 2003.[Abstract/Free Full Text]

Connor JA, Walter D, and McKown R. Neural repetitive firing—modifications of the Hodgkin–Huxley axon suggested by experimental results from crustacean axons. Biophys J 18: 81–102, 1977.[Web of Science][Medline]

Connors BW, Gutnick MJ, and Prince DA. Electrophysiological properties of neocortical neurons in vitro. J Neurophysiol 48: 1302–1320, 1982.[Abstract/Free Full Text]

Csicsvari J, Hirase H, Czurko A, and Buzsáki G. Reliability and state dependence of pyramidal cell–interneuron synapses in the hippocampus: an ensemble approach in the behaving rat. Neuron 21: 179–189, 1998.[CrossRef][Web of Science][Medline]

Csicsvari J, Hirase H, Czurko A, Mamiya A, and Buzsáki G. Fast network oscillations in the hippocampal CA1 region of the behaving rat. J Neurosci 19: RC20, 1999a.

Csicsvari J, Hirase H, Czurko A, Mamiya A, and Buzsáki G. Oscillatory coupling of hippocampal pyramidal cells and interneurons in the behaving rat. J Neurosci 19: 274–287, 1999b.[Abstract/Free Full Text]

Destexhe A, Contreras D, and Steriade M. Spatiotemporal analysis of local field potentials and unit discharges in cat cerebral cortex during natural wake and sleep states. J Neurosci 19: 4595–4608, 1999.[Abstract/Free Full Text]

Destexhe A and Paré D. Impact of network activity on the integrative properties of neocortical pyramidal neurons in vivo. J Neurophysiol 81: 1531–1547, 1999.[Abstract/Free Full Text]

Engel AK, Konig P, Kreiter AK, Schillen TB, and Singer W. Temporal coding in the visual cortex: new vistas on integration in the nervous system. Trends Neurosci 15: 218–226, 1992.[CrossRef][Web of Science][Medline]

Erisir A, Lau D, Rudy B, and Leonard CS. Function of specific K⁺ channels in sustained high-frequency firing of fast-spiking cortical interneurons. J Neurophysiol 82: 2476–2489, 1999.[Abstract/Free Full Text]

Ermentrout GB and Rinzel J. Beyond a pacemaker's entrainment limit: phase walk-through. Am J Physiol 246: 102–106, 1984.

Fisahn A, Pike FG, Buhl EH, and Paulsen O. Cholinergic induction of network oscillations at 40 Hz in the hippocampus in vitro. Nature 394: 186–189, 1998.[CrossRef][Medline]

Fourcaud-Trocmé N, Hansel D, van Vreeswijk C, and Brunel N. How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci 23: 11628–11640, 2003.[Abstract/Free Full Text]

Fries P, Neuenschwander S, Engel AK, Goebel R, and Singer W. Rapid feature selective neuronal synchronization through correlated latency shifting. Nat Neurosci 4: 194–200, 2001a.[CrossRef][Web of Science][Medline]

Fries P, Reynolds JH, Rorie AE, and Desimone R. Modulation of oscillatory neuronal synchronization by selective visual attention. Science 291: 1560–1563, 2001b.[Abstract/Free Full Text]

Fuhrmann G, Markram H, and Tsodyks M. Spike frequency adaptation and neocortical rhythms. J Neurophysiol 88: 761–770, 2002.[Abstract/Free Full Text]

Gabbiani F and Koch C. Principles of spike train analysis. In: Methods in Neuronal Modeling, edited by Koch C and Segev I. Cambridge, MA: MIT Press, 1998, p. 313–360.

Gerstner W. Population dynamics of spiking neurons: fast transients, asynchronous states, and locking. Neural Comput 12: 43–89, 2000.[CrossRef][Web of Science][Medline]

Gerstner W, van Hemmen L, and Cowan J. What matters in neuronal locking? Neural Comput 8: 1653–1676, 1996.[Web of Science][Medline]

Gupta A, Wang Y, and Markram H. Organizing principles for a diversity of GABAergic interneurons and synapses in the neocortex. Science 287: 273–278, 2000.[Abstract/Free Full Text]

Hansel D, Mato G, and Meunier C. Synchrony in excitatory neural networks. Neural Comput 7: 307–337, 1995.[Web of Science][Medline]

Häusser M, Major G, and Stuart GJ. Differential shunting of EPSPs by action potentials. Science 291: 138–141, 2001.[Abstract/Free Full Text]

Hodgkin AL and Huxley AF. A quantitative description of membrane current and its application to conductance and excitation in nerve. J Physiol 117: 500–544, 1952.[Free Full Text]

Klausberger T, Magill PJ, Marton LF, Roberts JDB, Cobden PM, Buzsáki G, and Somogyi P. Brain-state- and cell-type-specific firing of hippocampal interneurons in vivo. Nature 421: 844–848, 2003.[CrossRef][Medline]

Knight BW. Dynamics of encoding in a population of neurons. J Gen Physiol 59: 734–766, 1972.[Abstract/Free Full Text]

Kopell N and LeMasson G. Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture. Proc Natl Acad Sci USA 91: 10586–10590, 1994.[Abstract/Free Full Text]

Kraushaar U and Jonas P. Efficacy and stability of quantal GABA release at a hippocampal interneuron–principal neuron synapse. J Neurosci 20: 5594–5607, 2000.[Abstract/Free Full Text]

Lacaille J and Williams S. Membrane properties of interneurons in stratum oriens-alveus of the CA1 region of rat hippocampus in vitro. Neuroscience 36: 349–359, 1990.[CrossRef][Web of Science][Medline]

Lien C-C, Martina M, Schultz JH, Ehmke H, and Jonas P. Gating, modulation and subunit composition of voltage-gated K⁺ channels in dendritic inhibitory interneurons of rat hippocampus. J Physiol 538: 405–419, 2002.[Abstract/Free Full Text]

Logothetis NK, Pauls J, Augath MA, Trinath T, and Oeltermann A. Neurophysiological investigation of the basis of the fMRI signal. Nature 412: 150–157, 2001.[CrossRef][Medline]

Marder E. From biophysics to models of network function. Annu Rev Neurosci 21: 25–45, 1998.[CrossRef][Web of Science][Medline]

Markram H, Lubke J, Frotscher M, Roth A, and Sakmann B. Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. J Physiol 500: 409–440, 1997.[Abstract/Free Full Text]

Martina M and Jonas P. Functional differences in Na⁺ channel gating between fast-spiking interneurones and principal neurons of rat hippocampus. J Physiol 505: 593–603, 1997.[Abstract/Free Full Text]

Martina M, Schultz JH, Ehmke H, Monyer H, and Jonas P. Functional and molecular differences between voltage-gated K⁺ channels of fast-spiking interneurons and pyramidal neurons of rat hippocampus. J Neurosci 15: 8111–8125, 1998.

Nomura M, Fukai T, and Aoyagi T. Synchrony of fast-spiking interneurons interconnected by GABAergic and electrical synapses. Neural Comput 15: 2179–2198, 2003.[CrossRef][Web of Science][Medline]

Nowak LG, Azouz R, Sanchez-Vives MV, Gray CM, and McCormick DA. Electrophysiological classes of cat primary visual cortical neurons in vivo as revealed by quantitative analyses. J Neurophysiol 89: 1541–1566, 2003.[Abstract/Free Full Text]

Pesaran B, Pezaris JS, Sahani M, Mitra PP, and Andersen RA. Temporal structure in neuronal activity during working memory in macaque parietal cortex. Nat Neurosci 5: 805–811, 2002.[CrossRef][Web of Science][Medline]

Press WH, Teukolsky SA, Vetterling WT, and Flannery BP. Numerical Recipes in C. Cambridge, UK: Cambridge Univ. Press, 1992.

Richardson MJ, Brunel N, and Hakim V. From subthreshold to firing rate resonance. J Neurophysiol 89: 2538–2554, 2003.[Abstract/Free Full Text]

Shriki O, Hansel D, and Sompolinsky H. Rate models for conductance-based cortical neuronal networks. Neural Comput 15: 1809–1841, 2003.[CrossRef][Web of Science][Medline]

Shu Y, Hasenstaub A, and McCormick DA. Turning on and off recurrent balanced cortical activity. Nature 423: 288–293, 2003.[CrossRef][Medline]

Softky W. Sub-millisecond coincidence detection in active dendritic trees. Neuroscience 58: 13–41, 1994.[CrossRef][Web of Science][Medline]

Tamas G, Buhl EH, and Somogyi P. Fast IPSPs elicited via multiple synaptic release sites by different types of GABAergic neurons in the cat visual cortex. J Physiol 5002: 715–738, 1997.

Tamas G, Somogyi P, and Buhl EH. Differentially interconnected networks of GABAergic interneurons in the visual cortex of the cat. J Neurosci 18: 4255–4270, 1998.[Abstract/Free Full Text]

Traub RD, Whittington MA, Collins SB, Buzsáki G, and Jefferys JGR. Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. J Physiol 493: 471–484, 1996.[Abstract/Free Full Text]

van Vreeswijk C, Abbott L, and Ermentrout GB. When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1: 313–321, 1994.[CrossRef][Medline]

Vida I, Halasy K, Szinyei C, Somogyi P, and Buhl EH. Unitary IPSPs evoked by interneurons at the stratum radiatum–stratum lacunosum moleculare border in the CA1 area of the rat hippocampus in vitro. J Physiol 506: 755–773, 1998.[Abstract/Free Full Text]

Wang X-J. Calcium coding and adaptive temporal computation in cortical pyramidal neurons. J Neurophysiol 79: 1549–1566, 1998.[Abstract/Free Full Text]

Wang X-J and Buzsáki G. Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J Neurosci 16: 6402–6413, 1996.[Abstract/Free Full Text]

Xiang Z, Huguenard JR, and Prince DA. GABAA receptor mediated currents in interneurons and pyramidal cells of rat visual cortex. J Physiol 506: 715–730, 1998.[Abstract/Free Full Text]

Zhang K and McBain CJ. Potassium conductances underlying repolarization and after-hyperpolarization in rat CA1 hippocampal interneurones. J Physiol 488: 647–660, 1995.[Abstract/Free Full Text]

Zhou F-M and Hablitz JJ. AMPA receptor–mediated EPSCs in rat neocortical layer II/III interneurons have rapid kinetics. Brain Res 780: 166–169, 1998.[CrossRef][Web of Science][Medline]

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