J Neurophysiol 94: 4344-4361, 2005.
First published August 10, 2005; doi:10.1152/jn.00510.2004
0022-3077/05 $8.00
Contributions of Intrinsic Membrane Dynamics to Fast Network Oscillations With Irregular Neuronal Discharges
Caroline Geisler1,2,
Nicolas Brunel3 and
Xiao-Jing Wang1
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
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ABSTRACT
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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.
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INTRODUCTION
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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.
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METHODS
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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 Cm = 0.2 nF. The leak current IL = gL(V – VL) has the conductance gL = 0.02 µS and reversal potential VL = –67 mV. Isyn is the synaptic input current and Iext is an external applied current. The spike-generating ion currents INa = gNam
3h(V – VNa) and IK = gKn4(V – VK) 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 gNa = 14 µS and gK = 1.8 µS. The reversal potentials are VNa = 55 mV and VK = –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
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The capacitance of the membrane is Cm = 0.2 nF. The dynamics of the leak current IL and the spike generating currents INa and IK 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 gL = 0.06 µS, gNa = 24 µS and gK = 4 µS. The reversal potentials are VL = –17 mV, VNa = 55 mV, and VK = –72 mV. The temperature factors are n = h = 3.8. The kinetics of the A-type potassium current IA = gAA3B(V – VA) is the same as described in Connor et al. (1977), with dB/dt = (B – B)/
B, where A = 0.0761
exp[(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 gA = 9.54 µS and the reversal potential is VA = –75 mV. The steady-state value of the conductance gAA3B is nonzero over a large voltage range and the current IA contributes a significant outward current above its reversal potential. The high reversal potential of the leak current VL = –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
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The capacitance of the membrane is Cm = 0.25 nF. The dynamics of the leak current IL and the spike generating currents INa and IK 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 gL = 0.025 µS, gNa = 11.25 µS and gK = 4.5 µS. The reversal potentials are VL = –65 mV, VNa = 55 mV and VK = –80 mV. The temperature factors are n = h = 4. The high-threshold calcium current in the dendrite ICa = gCam2(V – VCa), where m is assumed fast and is replaced by its steady state m = 1/{1 + exp[–(V + 20)/9]}. The maximal conductances are gCa = 0.25 µS and the reversal potential is VCa = 120 mV. The voltage-dependent, calcium-activated potassium current IAHP = gAHP[Ca2+]/([Ca2+] + Kd)(V – VK) with Kd = 30 µM. The intracellular calcium follows [Ca2+]/dt = –ICa –[Ca2+]/Ca, where = 4 µM/(ms · µA) and Ca = 80 ms. The maximal conductance gAHP = 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
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where Cm = 0.2 nF, the leak conductance gL = 0.02 µS, the resting potential VL = –67 mV and Isyn is the synaptic current and Iext 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 Vth = –62.45 mV and 
T = 3.48 mV. The reset potential Vreset = –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 [Isyn(t) + Iext(t)]. The current I(t) = I0 + I1 cos (2
ft) + Inoise, 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 dInoise/dt=[
(t) – Inoise]/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) = Cm/[gL + gion(t) + gsyn(t)], where gion is the total conductance including all ionic currents and gsyn 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
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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 NI = 1,000 interneurons, or two populations of NI = 1,000 interneurons and NP = 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 MF = pNI = 100 inhibitory synapses, and (in a two-population network) MP = pNP = 400 excitatory synapses.
SYNAPTIC CURRENTS.
The synaptic currents are described by Isyn = gsyns(t)(V – Vsyn), where gsyn is the synaptic conductance, s(t) is the fraction of open channels at time t, and Vsyn is the reversal potential. The time course of s(t) is faster for -amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA)–mediated excitation than for GABAergic inhibition and can be characterized by the three time constants: synaptic latency (l), rise time (r), and decay time (d)
 | (7) |
where the time t = 0 corresponds to the voltage maximum of the presynaptic spike. The peak conductance is given by syn = gsyn(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 (GABAA) 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 Vsyn,AMPA = 0 mV and of GABAA Vsyn,GABAA = –75 mV. The peak conductances are AMPA
P = 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.
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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.
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THE INSTANTANEOUS FIRING RATE.
The spike times from all neurons are binned in a sliding window with t = 0.2 ms. The spike times are taken at the voltage maximum. The instantaneous firing rate at time t, r(t), is then the number of spikes in the time window [t, t + t], divided by the number of neurons and by t.
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.
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RESULTS
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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
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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.
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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.
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It has been shown for LIF neurons that the response of a neuron to a noisy sinusoidal current strongly depends on the time constant of the noise noise (Brunel et al. 2001 and Fig. 1C). The phase lag decreases and the amplitude increases for larger noise time constants. When the time constant noise is sufficiently large, the LIF model responds to sinusoidal input superimposed on synaptically filtered noise with negligible phase shift, independent of the input frequency. This salient feature of the LIF model neuron is crucial for neurons to follow fast transients and to enable a network of LIF inhibitory neurons to generate very fast (up to 300 Hz) coherent oscillations (Brunel and Wang 2003; Brunel et al. 2001).
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.
How do these two time constants filter and spike depend on the properties of the neuron? As already mentioned, spike depends exclusively on the interplay of the Hodgkin–Huxley currents leading to spike generation, independently of the inputs. On the other hand, filter does depend on the synaptic inputs. Our single-cell simulations allow us to identify two crucial parameters that control filter: the single-cell mean firing rate and the input conductance.
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.
INPUT CONDUCTANCE.
Synaptic inputs as well as intrinsic currents that are active in the subthreshold voltage range introduce changes in the input conductance of the neuron. If the fluctuations in the conductance are small, the synaptic conductance can be approximated by a constant term. Thus we introduce a shunting term into the input current that mimics the total synaptic conductance (external and recurrent) during network activity. This allows us to vary the neuron's input conductance and, equivalently, the neuron's effective membrane time constant m–eff (see also Eq. 6)
 | (11) |
For different values of gshunt 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 gshunt 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 gshunt, 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).
The linear response of the EIF model neuron was obtained in the same way as for the conductance-based model (see previous section and METHODS). Simulations show that the single cell's response to the noisy sinusoidal current is essentially independent of the time constant of the noise noise (Fourcaud-Trocmé et al. 2003), as it is for the conductance-based model. As shown analytically by Fourcaud-Trocmé et al. (2003), the phase shift cell(f) goes to 90° for large input frequencies f. Indeed, Fig. 5C shows that, in addition, the phase shift can be well fitted with an arctangent function (Eq. 10). The time constants spike and filter are taken from a fit with the function in Eq. 10 to the simulated data. The fact that the phase goes to 90° at large frequencies implies that for the EIF model spike = 0. The remaining term is the low-pass filter with time constant filter. The values of filter for different shunt conductances, or equivalently for different m–eff, are similar to those of the conductance-based model (compare Figs. 4C and 5C). Thus, all the results obtained with the conductance-based models can be reproduced quantitatively by the simpler EIF model. The use of the simpler model confirms that the phase shift in the single cell response arises from the dynamics of the spike initiation that is well captured by the exponential term of the EIF, and that the effective membrane time constant has a major influence on the single-cell phase shift.
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/(CmT2f). 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/fc), where the cutoff frequency is related to the time constant of the rate model by fc = 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 fc/f) frequency limits are equal. We can define the cutoff frequency in a similar way for the EIF model, which gives fc = r0/(2CmTdr0/dI) (Fig. 5D). This in turn gives an estimate of the filter time constant, filter = 1/(2fc), or filter = (CmTdr0/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 rI(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 rI,0 is the mean firing rate of the cell, 
= rI,1/rI,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.
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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.
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First, we consider the time course of GABAergic currents in the inhibitory neurons of the network. A sinusoidally varying presynaptic firing rate leads to the average fraction of open channels at GABAergic synapses varying as a sinusoidal function of time. However, because of temporal characteristics (latency Il, rise time Ir, and decay time Id) of GABAergic synapses, there is a phase shift of the time course of the fraction of open channels with respect to the presynaptic firing rate. For synapses described by Eq. 7 this phase shift I,syn() is given by (Brunel and Wang 2003)
 | (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 rI,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.
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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 inpu | |
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ABSTRACT
INTRODUCTION
