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1Centre National de la Recherche Scientifique-Neurophysique et Physiologie du Système Moteur-Université Paris René Descartes, 75270 Paris Cedex 06, France; and 2Volen Center, Brandeis University, Waltham, Massachusetts 02454
Submitted 6 December 2002; accepted in final form 11 February 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX 1. NETWORKS WITH...APPENDIX 2. NETWORKS WITH...APPENDIX 3. NETWORKS WITH...ACKNOWLEDGMENTSREFERENCES |
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40-Hz gamma or 200-Hz sharp-wave ripples), the spike
trains of constituent neurons are typically irregular and sparse. The
dichotomy between rhythmic local field and stochastic spike trains presents a
challenge to the theory of brain rhythms in the framework of coupled
oscillators. Previous studies have shown that when noise is large and
recurrent inhibition is strong, a coherent network rhythm can be generated
while single neurons fire intermittently at low rates compared to the
frequency of the oscillation. However, these studies used too simplified
synaptic kinetics to allow quantitative predictions of the population rhythmic
frequency. Here we show how to derive quantitatively the coherent oscillation
frequency for a randomly connected network of leaky integrate-and-fire neurons
with realistic synaptic parameters. In a noise-dominated interneuronal
network, the oscillation frequency depends much more on the shortest synaptic
time constants (delay and rise time) than on the longer synaptic decay time,
and 200-Hz frequency can be realized with synaptic time constants taken
from slice data. In a network composed of both interneurons and excitatory
cells, the rhythmogenesis is a compromise between two scenarios: the fast
purely interneuronal mechanism, and the slower feedback mechanism (relying on
the excitatory-inhibitory loop). The properties of the rhythm are determined
essentially by the ratio of time scales of excitatory and inhibitory currents
and by the balance between the mean recurrent excitation and inhibition.
Faster excitation than inhibition, or a higher excitation/inhibition ratio,
favors the feedback loop and a much slower oscillation (typically in the gamma
range). |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX 1. NETWORKS WITH...APPENDIX 2. NETWORKS WITH...APPENDIX 3. NETWORKS WITH...ACKNOWLEDGMENTSREFERENCES |
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;
Buzsaki et al. 1992;
Csicsvari et al. 1999b;
Siapas and Wilson 1998).
Single-cell discharge rates is typically much lower than the LFP oscillation
frequency, especially in pyramidal cells but also in interneurons (Csicsvari
et al. 1998,
1999b). Indeed rhythmicity is
usually not apparent in the raw spike trains of individual cells and becomes
visible only after data processing of spike trains from multiple single units.
Thus single-cell behavior differs markedly from the population activity during
fast network oscillations. Similarly, physiological studies of primates
indicate that even when the LFP signal contains a clear rhythmic component,
simultaneously recorded single-unit spike trains usually appear irregular and
devoid of a clear-cut oscillation (Fries
et al. 2001; Logothetis et al.
2001).
Recently, oscillations have been observed in vitro
(Buhl et al. 1998;
Fellous and Sejnowski 2000;
Fisahn et al. 1998) that
resemble these characteristics: strong gamma (30–40 Hz) oscillation of
the LFP, together with low (<2 Hz) and irregular firing in pyramidal cells.
This means that a single pyramidal cell fires only once in every 15–20
cycles of the population rhythm. Fast rhythmic ripples at 100–200 Hz
have also been produced in hippocampal slices, again with intermittent
principal cell firing (Draguhn et al.
1998). The observations in the slices of rhythmic activity
patterns at high frequencies have opened a promising venue to study the
underlying cellular and circuit mechanisms.
Computational models of networks of spiking neurons have shown how
synchrony could emerge in recurrent networks of interneurons. However, in
models with weak synaptic disorder and weak noise, neurons behave typically as
oscillators and fire at network frequency (see e.g.,
Abbott and van Vreeswijk 1993;
Gerstner 1995;
Gerstner et al. 1996;
Hansel et al. 1995;
Kopell and Ermentrout 1986;
Kuramoto 1984;
Marder 1998;
Traub et al. 1996;
Treves 1993;
Wang and Buzsáki 1996).
In some cases, modes of synchrony called "clustering" occur in
which the network breaks in a small number of fixed clusters of neurons. In
these cases, the network frequency is higher than the frequency of single
cells, being equal to the number of clusters times the frequency of single
cells, but single cells still fire in a regular fashion
(Golomb and Rinzel 1994;
Kopell and LeMasson 1994;
Wang et al. 1995).
Heterogeneities tend to disrupt synchrony; but in parameter ranges for which
synchrony is present, the network oscillation is not qualitatively affected by
heterogeneity: neurons keep firing in a regular fashion and the network
frequency is close to the average frequency of the cells of the network (or an
integer multiple in case of clustering)
(Bartos et al. 2001;
Golomb and Hansel 2000;
Hansel and Mato 2001;
Wang and Buzsáki 1996;
White et al. 1998). In
contrast to the framework of coupled oscillators, several studies
(Brunel and Hakim 1999;
Brunel 2000;
Tiesinga and Jose 2000) have
shown that a network oscillation can be produced with low and intermittent
spike discharges in pyramidal cells and interneurons under conditions of
strong noise (in external inputs and/or due to disorder in recurrent
connectivity) and strong recurrent inhibition. However, Brunel and Hakim
(1999) and Brunel
(2000) used too-simplified
synaptic currents to draw quantitative conclusions about oscillation
frequencies in real networks, and Tiesinga and Jose
(2000) used a purely numerical
approach, making it difficult to identify the crucial parameters controlling
network frequency. More recently, several studies
(Lewis and Rinzel 2000;
Schmitz et al. 2001;
Traub et al. 1999;
Traub and Bibbig 2000) have
suggested that 200-Hz oscillations with sparse pyramidal firing could be
realized by gap junctions between axons of pyramidal cells. However, 200-Hz
oscillations in a slice preparation are not affected in transgenetic mice with
knockout of the gap-junction protein connexin 36
(Buhl et al 2003;
Hormuzdi et al. 2001). The
possibility remains that other subtypes of gap-junction proteins different
from connexin 36 play the hypothesized role in rhythmogenesis
(Schmitz et al. 2001).
Connexin 36-deficient mice shows gamma oscillations at the same frequency as
the control but with a reduced level of population synchrony
(Buhl et al 2003;
Hormuzdi et al. 2001).
Therefore it is still unclear whether the frequency of fast network
oscillations in hippocampus critically depends on the gap junctions.
When single neurons do not fire in a periodic fashion but rather fire stochastically at low rates, several questions remain unanswered: what determines the frequency of fast oscillations with sparsely firing neurons in networks with realistic neuronal and synaptic properties? Can such high frequencies as are observed in vivo be generated by chemical synapses and in the absence of gap junctions? To shed light onto these questions, we have analyzed coherent population oscillations, characterized by sparse and irregular firing of single cells, in a recurrent network model with realistic synaptic time courses. In this paper, we present an analytical approach to predict the population oscillation frequency from synaptic and network parameters in such a recurrent neural network. This approach allows to identify the requirements on the synaptic circuitry under which fast gamma and ripple rhythmicities occur in the sparsely firing regime.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX 1. NETWORKS WITH...APPENDIX 2. NETWORKS WITH...APPENDIX 3. NETWORKS WITH...ACKNOWLEDGMENTSREFERENCES |
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Both interneurons and pyramidal cells are described as leaky
integrate-and-fire (LIF) neurons (see e.g.,
Tuckwell 1988), with membrane
time constants 
m = 20 ms (pyramids) and 10 ms (interneurons).
The leak (resting) membrane potential is –70 mV, the spike threshold is
–52 mV, and the reset potential is –59 mV. The absolute refractory
period is 2 ms (pyramids) and 1 ms (interneurons).
Networks
The network architecture is random and sparse, with a given connection
probability. We used three types of networks: networks of
NI interneurons only, with random interneuron-interneuron
connections; networks of NI interneurons and
NE pyramidal cells, with random interneuron-interneuron,
interneuron-pyramid, and pyramid-interneuron connections (i.e., without
pyramid-pyramid connections); and networks of NI
interneurons and NE pyramidal cells, with all four
possible connections, drawn randomly with the same connection probability. In
simulations, we used typically NI = 1,000,
NE = 4,000. The connection probability p between
any pair of cells was typically 20%. Thus each cell in the network received
200 synaptic contacts from other interneurons and 800 from pyramidal
cells.
Synaptic currents
Three types of synaptic currents were used, modelling GABAergic
(inhibitory), AMPA-type (fast excitatory) and
N-methyl-D-aspartate (NMDA)-type (slow excitatory)
synaptic inputs. The synaptic currents were described as
Isyn(t) = gsyn(V
– Vsyn)s(t) where
gsyn is the synaptic conductance, Vsyn
the corresponding reversal potential, and s(t) is a function
describing the time course of synaptic currents. We used a delayed difference
of exponentials: if a presynaptic spike occurs at time 0, then after a latency
l, s(t) is updated as
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m. This normalization was chosen so that varying the synaptic
time constant does not affect the time integral of a postsynaptic current
(PSC). The peak of the function s is
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l, rise time r, and decay time
d. The reversal potential of excitatory (inhibitory) synaptic
currents is 0 mV (–70 mV). Synaptic conductances were calibrated such as
the amplitude of PSCs was in the range of 0.2–2 mV at holding potential
of –55 mV, i.e., just below threshold, in accordance with slice data
(Buhl et al. 1997;
Markram et al. 1997; Tamas et
al. 1997,
1998;
Vida et al. 1998). They
yielded peak conductances, for AMPA receptors, 1 nS; for NMDA receptors,
0.01 nS; for GABA receptors, 6 nS, compatible with experimentally
inferred values (Bartos et al.
2001,
2002;
Gupta et al. 2000;
Markram et al. 1997). See
Table 1 for more details on
parameters.
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All synaptic time scales were systematically varied, but typical
"reference" parameters were, for GABAergic currents,
l = 1 ms, r = 0.5 ms, and d =
5 ms (Bartos et al. 2001;
Gupta et al. 2000;
Kraushaar and Jonas 2000;
Xiang et al. 1998); for AMPA
currents, l = 1 ms, r = 0.5 ms, and
d = 2 ms (Angulo et al.
1999; Zhou and Hablitz
1998); for NMDA currents, l = 1 ms,
r = 2 ms, and d = 100 ms
(Hestrin et al. 1990). NMDA
conductances could be removed from all simulations without affecting any of
the results.
The equivalence between g parameters, peak conductances, amplitude of PSCs at –55 mV, and amplitude of PSPs at –55 mV are given (for the reference synaptic time scales) in the table.
External inputs
External inputs were assumed to arise from 800 external synapses of the AMPA type, with conductance 0.25 nS (on pyramids), 0.4 nS (on interneurons), and the same kinetics as recurrent AMPA synapses. The synapses are activated by random Poisson spike trains, with a given rate. In RESULTS, we mention the total input Poisson rate for each simulation shown.
Numerical methods
Simulations were done using a finite difference integration scheme based on
the second-order Runge Kutta algorithm
(Hansel et al. 1998;
Press et al. 1992;
Shelley and Tao 2001) with
time step dt = 0.05 ms. Shorter time steps did not change the results
in any significant way. Typical simulation times were carried out for 10 s of
real time. Simulations were run on workstations with alpha architecture and
lasted of the order of one hour. We used two types of synchrony indices.
SPIKE TRAIN SYNCHRONY (STS) INDEX. We compute the autocor-relation of total network activity, computed in bins of 1 ms. The autocorrelation is normalized by the square of the average firing rate of cells in the network. The spike train synchrony index is defined as the autocorrelation at zero time. Its intuitive interpretation is the following: if the index is one, it means the chance that two randomly selected neurons fire together in a 1-ms bin is 100% higher than if these neurons were firing in an uncorrelated way. The frequency of the oscillation was determined from the peak of the power spectrum of the global activity.
MEMBRANE POTENTIAL SYNCHRONY (MPS) INDEX. Average correlation
between the membrane potentials of two neurons in the network, normalized to 1
when all membrane potentials have the same time course
(Hansel and Sompolinsky
1996).
The advantage of the first index is that it is directly related to measurable quantities in vivo such as cross-correlations between spike trains (it is equal to the CC at 0 time, averaged over pairs). The advantage of the second is that it is bounded between 0 and 1. In all simulations series, we found, unsurprisingly, that both indices behave qualitatively in a very similar way.
Synchrony indices are always nonzero in simulated networks due to finite
size effects (Brunel and Hakim
1999; Hansel and Sompolinsky
1996; Wang and Buzsáki
1996). To determine whether the network is in an asynchronous or
synchronous state, we performed simulations with varying network sizes,
keeping the number of connections and the synaptic conductance fixed so as to
keep unchanged the temporal average and fluctuations of the synaptic currents
as network size was varied. In an asynchronous state, the synchrony indices
strongly decrease and go to zero with increasing N. In a synchronous
state, the synchrony index decreases only mildly and tends to a finite value
in the large N limit. An alternative strategy for finite size scaling
has been proposed by Golomb and Hansel
(2000). Both approaches are
expected to give the same results in the limit in which connection probability
becomes small.
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RESULTS |
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Figure 1 shows the behavior
of a simulated interneuronal network. A pronounced population activity
oscillation is clearly visible at a frequency of 180 Hz. On the other
hand, the single cell activity reveals a much lower activity (average of
20 spikes/s), with a wide range of individual firing rates (from 0 to
100), and the spiking process is highly irregular. Thus in any cycle of the
oscillation, only 10% of the interneurons actually fire. The intuitive
explanation for the oscillatory phenomenon is the following: single neurons
receive a strong inhibitory drive due to powerful recurrent inhibition. Thus
they fire at low rates, even though they receive strong external excitatory
inputs. The firing is irregular because the average total (external excitatory
plus recurrent inhibitory) current is subthreshold, and firing is triggered by
fluctuations due to noise in external and recurrent inputs. On the other hand,
the oscillation is stable because of the repetitive succession of the
following events: at the peak of a cycle, there is strong inhibitory firing.
After a time lag of 2.5 ms due to synaptic filtering, every neuron in the
network feels a massive inhibitory input and activity goes down, hence the
trough in global activity. Subsequently, 2.5 ms later, the synaptic
currents decay away, the total input becomes high due to strong external
stimulation, and there is another surge of activity. The period of the
oscillation is therefore about two times the synaptic lag, i.e., 5 ms in this
case. Qualitatively, the oscillation is as described by Brunel and Hakim
(1999). In the following text,
we present an approximate analytical approach to quantitatively predict the
population oscillation frequency.
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Without recurrent inhibitory interactions, neurons would show asynchronous
spike discharges due to external excitatory drive. This asynchronous state is
destabilized, and synchronous oscillation emerges, when inhibitory recurrent
feedback becomes sufficiently strong. The inhibitory feedback can be enhanced
in different ways: by increasing either the coupling strength (the number of
connections per neurons, the synaptic conductance) or the average inhibitory
firing rate through an increase in external excitatory currents.
Figure 2 shows how synchrony
depends on the magnitude of the external excitatory input. In the
"thermodynamical" (large N) limit, synchrony appears
above some critical level of external stimulation (10 kHz). Firing rates
of interneurons increase quasi-linearly with the external input as expected in
strongly coupled networks (Brunel
2000; van Vreeswijk and
Sompolinsky 1996). On the other hand, the frequency of the
population oscillation stays relatively constant, between 150 and 200 Hz.
Therefore the network frequency is independent of single cell firing rate and
depends only weakly on the magnitude of external drive. This dissociation
between network oscillation frequency and single neuron firing rate will be
confirmed below by analytical calculations.
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Analytical approach for predicting the network frequency
In this section, we outline our approach for one population of inhibitory
neurons. Later, we will extend the method to two populations of excitatory and
inhibitory cells. The instantaneous firing rate of the interneuronal
population 
I(t) is defined as the fraction of neurons
firing in a short interval [t, t + dt] where dt is
small, divided by dt. In an asynchronous state, the firing rate is
stationary (independent of time) apart from finite size effects. The
population firing rate I is determined by the sum of two
synaptic currents, the excitatory external drive Iext and
the feedback inhibition IGABA. IGABA
in turn depends on the population activity, hence is a function of
I itself. Given a presynaptic firing rate pre,
one can calculate the synaptic current
IGABA(pre). Then the postsynaptic firing
rate as a function F of the sum
Isyn(pre) = Iext
– IGABA(pre) can be evaluated,
post =
F[Isyn(pre)]. Finally, because
both the pre- and postsynaptic firing rates are of the same neural population,
they must be the same and equal to I. Hence, I =
F[Isyn(I)] yields a
self-consistent equation for I.
To understand whether asynchrony or synchrony is present in the network, a
linear stability analysis of the asynchronous state is performed
(Abbott and van Vreeswijk 1993;
Brunel and Hakim 1999;
Treves 1993). Small deviations
around the stationary state, in which the instantaneous firing rate is a sum
of a stationary firing rate I0 plus a small exponential
component I0
I exp(µt +
i
t), where I « 1, are
considered. When µ < 0, this corresponds to a damped oscillation with
frequency ; when µ = 0, this corresponds to a sinusoidal wave around
the stationary state; when µ > 0, this corresponds to an oscillation
that amplifies with time. Thus self-consistent solutions of network activity
with µ > 0 signal an oscillatory instability: an oscillation with a
finite amplitude develops from the asynchronous state. The onset of synchrony
is therefore signaled by the appearance of solutions with µ = 0. Here, we
investigate the conditions under which the network activity has a sinusoidal
component with µ = 0. In such a way we obtain the population frequency
close to the onset of oscillations.
Specifically, the procedure can be decomposed in the following four steps.
STEP 1. ASSUME A PRESYNAPTIC RHYTHMIC FIRING RATE. The
instantaneous population firing rate is assumed to have the form
 | (1) |
I0 is the average firing rate, I is the
relative modulation of the oscillatory deviation to the stationary firing
rate, and is the frequency of the network oscillation.
STEP 2. OBTAIN THE POST-SYNAPTIC CURRENT FROM THE PRESYNAPTIC FIRING
RATE. We next calculate the synaptic conductance produced by presynaptic
cells firing at the rate I(t). The sum of all
inhibitory synaptic variables in a given cell sI is given
by the sum of two exponentials (see METHODS) or equivalently by
 | (2) |
 | (3) |
i,j
(t
– ti,j) is
the compound spike train of all presynaptic neurons connected to the cell. In
average, a postsynaptic cell receives inputs from CI =
pNI presynaptic cells, where p is the connection
probability and NI is the total number of interneurons in
the network. The variables sI and x obey the
equations
 | (4) |
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| (5) |
(t – l) and random
fluctuations. Solving these two equations, we obtain the average synaptic
variable sI(t), which has the same form as the
firing rate I(t) but with an amplitude attenuation
factor SI() and a phase shift
I(). More precisely
 | (6) |
 | (7) |
 | (8) |
; the lag due to the rise time; and the lag due to the decay time. The
latter two lags are linear in at low frequencies and saturate at
/2 at high frequencies.
Neglecting temporal variations in the driving force, the GABAergic synaptic
current is simply sI multiplied by a constant factor
 | (9) |
gGABACI.
The GABAergic current experienced by the neuron is therefore the sum of
three terms: an average drive IGABA,0 due to the average
firing rate I0 of inhibitory cells; an oscillatory component
due to the global oscillation; and a noisy component due to the random arrival
of spikes, after filtering by the synaptic kinetics. The total synaptic
current is
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 | (10) |
in the
phase appears because of the minus sign introduced by inhibitory
interactions.
STEP 3. OBTAIN THE POSTSYNAPTIC FIRING RATE FROM THE POSTSYNAPTIC
CURRENT. We now calculate the postsynaptic firing rate
I(t) in response to the synaptic current
Isyn (Eq. 10). For a synaptic input
Isyn(t) that varies periodically in time, the
response I(t) is expected in general to depend on the
frequency of the oscillatory input. For example, one might expect both
amplitude change and phase shift between the oscillatory components of
Isyn and I at high frequencies .
Therefore, in general, the input-output relationship between
Isyn(t) and I(t) is
expected to depend explicitly on . This subject has been analytically
investigated in (Brunel et al.
2001; Fourcaud and Brunel
2002) for the LIF neuron model. It was found that when synaptic
time constants are very fast compared to the membrane time constant,
I(t) shows a phase lag with respect to
Isyn(t) and the amplitude of the modulation is
attenuated at high frequencies. On the other hand, with a sufficient amount of
noise filtered by synaptic time constants that are of the order of the
membrane time constant, the postsynaptic firing rate follows instantaneously
the variations in input currents. In other words, the response of the neuron
to oscillatory currents at frequency has an amplitude that is nearly
independent of the frequency and has no phase lag. The
specific conditions for this to be true are: single neurons are described by
the LIF model; synaptic noise is of large amplitude and with a decay times are
comparable to the membrane time constant; and the variations in input currents
are such that the firing rate remains strictly positive.
Under these conditions, the dynamics of our network can be described by
firing rate dynamics that are purely determined by the synaptic time
constants. Specifically, the firing rate is simply a function of the total
synaptic current
 | (11) |
I is small, we can expand this function as
F(Isyn(t)) 
F(Itot,0) +
F'(Itot,0)(Isyn(t)
– Itot,0), where F' is the derivative
of F with respect to the input current. Combining Eq. 10
with Eq. 11, the firing rate of a cell is approximately given by
 | (12) |
I,0 = F(Itot,0) and
AI =
F'(Itot,0)Itot,0/I,0
is the relative variation in firing rate due to a relative variation in input
current. AI is proportional to the slope of the
f-I curve F, normalized in such a way as to be
dimensionless. For example, if the firing rate increases by 10% when the input
current is increased by 10%, then AI = 1.
STEP 4. SELF-CONSISTENT EQUATION FOR THE FIRING RATE. The last
step is to equate the postsynaptic firing rate (Eq. 12) with the
presynaptic firing rate (Eq. 1), yielding a self-consistent equation
for the firing rate of neurons in the network
 | (13) |
For the left- and right-hand sides to be equal, two relations have to be
satisfied, one for the amplitude
 | (14) |
 | (15) |
The phase condition, Eq. 15, allows to determine the frequency of
the network oscillation in terms of the synaptic temporal parameters.
Using the frequency given by Eq. 15, Eq. 14 can be solved to
determine the value of a particular network parameter for which the onset of
the oscillation occurs. For example, an increase in the synaptic connection
strength increases the ratio
IGABA0/Itot,0. The value of the
synaptic connection strength beyond which synchronized oscillations occur can
therefore be obtained from Eq. 14 once SI and
AI are known.
How network frequency depends on the synaptic time constants
The phase Eq. 15 indicates that the frequency of the
population oscillation at the onset of oscillations is purely determined by
the synaptic parameters lI, rI, and
dI. This is illustrated graphically in
Fig. 3B, for the same
model parameters as the network simulation of
Fig. 1. The function
I( = 2f) is plotted against the frequency
f, the intersection of this curve with the horizontal line =
occurs at the population frequency fpop, according to
Eq. 15. For the parameters of Fig.
1, the theoretically predicted frequency of
fpop 180 Hz is very close to that of the simulated
network oscillation (Fig.
1).
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The dependency of the network frequency on the synaptic parameters is shown
in Fig. 4. It is apparent that
the network frequency is more sensitive to relative variations of the shortest
time scales (the latency and the rise time) than to variations in the longest
time scale (the decay time). To understand these observations theoretically,
let us re-write Eq. 15 with = 2f
 | (16) |
/2, the right-hand side
of Eq. 16 can be equal to 
only with a strictly positive
latency lI. Therefore the latency of synaptic transmission is
critical for the emergence of coherent oscillations in this model.
Furthermore, simple bounds for the population frequency can be obtained using
the inequalities atan(x) < x, atan (x) <
/2 and atan (x) > /2 – 1/x
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rI is much
shorter than the decay time dI. Indeed, voltage-clamp
measurements of GABAA receptor-mediated IPSCs show that the latency
and rise time are of the order of ≤1 ms, while the decay time is longer, of
order 5–10 ms (Bartos et al.
2001,
2002;
Gupta et al. 2000;
Kraushaar and Jonas 2000;
Salin and Prince 1996;
Xiang et al. 1998). When the
decay time is much slower than the rise time, dI >>
rI, the bound becomes
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lI
and rI) because the bounds are independent of
dI. They provide a simple way to estimate the order of
magnitude of the network frequency. For example, if lI =
rI = 1 ms, we obtain 125 Hz < fpop <
159 Hz. For the parameters of Fig.
1 (lI = 1 ms, rI = 0.5 ms), 167 Hz
< fpop < 225 Hz in agreement of the observed
frequency of 180 Hz.
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Figure 4 shows how the
degree of network synchronization, or the oscillation amplitude, depends on
the three synaptic time constants. The simulation results can be qualitatively
understood with the help of our theoretical analysis. The synaptic time
constants affect the degree of synchrony in two ways: through the dependency
of SI(), as described by Eq. 7, and
through a change of oscillation frequency, as governed by Eq. 15. The
attenuation factor SI does not depend on the latency
lI explicitly. An increase in lI affects the
degree of synchrony only indirectly through a decrease in the population
frequency. The attenuation due to synaptic filtering is smaller at lower
frequencies, hence the network oscillation is amplified with a longer latency
lI. Changes in the rise time rI have two
effects: an increase of rI decreases SI
through the
factor in Eq. 7; but it also reduces the population frequency
, for the same reason as for the latency. These two opposing effects
tend to counterbalance each other. As long as the rise time is of the same
order as the latency and shorter than the decay time, the decreased
dominates over the increased rI, so that the product
rI is smaller, and the network synchrony is higher with
a larger rI. For longer rise times, the attenuation factor has
a stronger influence and becomes predominant, thus the oscillation amplitude
decreases. Finally, a longer decay time decreases the amplitude of the
oscillation. This is because the decay time has little effect on the
frequency, but on the other hand, it increases the attenuation by synaptic
filtering due to the factor
in Eq. 7.
In summary, recurrent synaptic inhibition with a large latency and very short rise and decay times, i.e., close to a delayed delta function, lead to pronounced oscillations. On the contrary, IPSCs with negligible latency lead to asynchrony in a network of leaky integrate-and-fire neurons. This feature is crucially dependent on the fact that the neuronal firing rate follows instantaneously (i.e., without phase lag) inputs at any frequency. If a neuronal phase lag is present, as expected for Hodgkin-Huxley conductance-based neurons, then network synchrony can be obtained even in absence of a latency. However, in general we expect that large synaptic latency tends to facilitate synchrony.
Above the onset of oscillations, the oscillation amplitude becomes large,
and our theoretical analysis is no longer valid. Numerical simulations show
that the network frequency decreases and network coherence increases, similar
to what happens in the simplified network of Brunel and Hakim
(1999). For very high external
inputs, the firing rate of single neurons becomes comparable to the network
frequency. The network reaches an almost fully synchronized state with
regularly firing neurons and therefore enters the regime of coupled
oscillators.
Two population networks
OSCILLATIONS DUE TO PYRAMIDAL-INTERNEURON FEEDBACK LOOP. An
alternative to the interneuronal network model of fast oscillations is the
feedback inhibition model: pyramidal neurons excite interneurons, which in
turn send inhibition back onto pyramidal cells
(Freeman 1975;
Jefferys et al. 1996;
Leung 1982). In a recent slice
experiment (Fisahn et al.
1998), spontaneously occuring 40-Hz oscillations have been shown
to depend both on the excitatory and inhibitory synaptic transmissions. Both
types of loops (pyr 
int pyr and int int) are present in a
cortical network. The two preceding mentioned mechanisms are not necessarily
mutually exclusive and may cooperate in the generation of a coherent network
rhythm.
To understand how the pyramidal-interneuron loop is involved in the generation of population synchrony, it is useful to consider first the feedback inhibition scenario in isolation, in which only pyramidal-to-interneuron and interneuron-to-pyramidal connections are present (no pyramidal-to-pyramidal and no interneuron-to-interneuron connections).
In this scenario, it is straightforward to repeat the analysis of the
previous section (see APPENDIX 1 for details). The population
frequency is now given by
 | (17) |
 | (18) |
E() (see
Eq. A9 in APPENDIX 1). In particular, if synaptic time
scales of excitation and inhibition are identical, then
E() = I() = /2, hence
interneurons lag pyramidal cells by 90°, and the population frequency will
be more than halved compared to the frequency of the purely interneuronal
network.
As an example, we take GABA synapses with latency lI = 0.5
ms, rise time rI = 0.5 ms, decay time dI = 5
ms, and AMPA synapses with latency lI = 1 ms, rise time
rI = 0.4 ms, and decay time dI = 2 ms. In the
purely interneuronal scenario, fpop is equal to 296 Hz,
while in the E-I loop scenario, the frequency goes down to 79 Hz, with
interneurons lagging behind pyramidal cells by 104°—a drastic
reduction in population frequency.
In the presence of both types of feedback loops (E-I loop and I-I loop), a
network tends to settle in an oscillation that is a compromise between the two
scenarios with a frequency and phase lag that are intermediate between these
two extremes. The frequency and phase lag are then determined by the relative
strength of the E-I and I-I connections through Eqs. B9 and
B10 of APPENDIX 2. As an example,
Fig. 5 shows a simulation of a
two-population network without pyramid-to-pyramid connections. Note that such
a network could represent a simplified model for a CA1 network where pyramid
to pyramid connections are rare. The synaptic conductances are as indicated in
METHODS. With a small external drive, the network is essentially
asynchronous, and the power spectrum of the population firing rate is flat
(Fig. 5A). When the
external drive is sufficiently strong, coherent 200-Hz oscillations emerge in
the network. In this oscillation, the interneurons lag behind pyramidal cells
by 90°. Note that for the synaptic parameters chosen here, a
one-population interneuronal network would oscillate at 300 Hz (see
Fig. 4C1). Thus the
pyramidal-interneuron loop slows down the oscillation significantly from 300
to 200 Hz. Both pyramidal cells and interneurons fire intermittently at much
lower rates than the population rhythm, and there is a broad distribution of
firing rates across individual cells (3–150 Hz, average: 50 Hz) for
interneurons, 1–20 Hz, average: 7 Hz) for pyramidal cells. Neurons in
CA1 show similar intermittent spike activity during sharp wave ripples in vivo
(Buzsaki et al. 1992;
Csicsvari et al. 1998,
1999b).
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Effect of pyramidal-to-pyramidal connections on oscillations
To understand the effect of pyramidal-to-pyramidal connections on oscillations, it is useful to consider first a network in which only these connections are present. In such networks, it is straightforward to show that instabilities can only occur with f = 0 Hz (a rate instability). Adding pyramidal-to-pyramidal connections to a network with all other types of connections tends again to decrease network frequency because these connections tend to prolong the positive phases of each cycle of the oscillation. The observed frequency and phase lags between the two populations is now a compromise between the strength of the all the feedback loops (E-E, E-I, and I-I), see Eq. C3 of APPENDIX 3.
An example is shown in Fig.
6 of a network oscillation when the pyramid-to-pyramid excitatory
connections are included into the model. The network architecture is now
closer to that of the CA3 hippocampus, with extensive recurrent collaterals
between pyramidal cells. The oscillation frequency is dramatically reduced by
the insertion of such connections, from 200 to 110 Hz
(Fig. 6).
|
The two-population network, with all four (E-to-E, E-to-I, I-to-E and
I-to-I) types of connections, displays fast oscillations which are typically
in the frequency range of 30–110 Hz, depending on the synaptic time
constants and on the balance between the loops. In
Fig. 7 is shown a simulation
with slightly longer (latency and decay) time constants of synaptic
inhibition, compared to Fig. 6.
With slower inhibition, the network oscillation frequency is lower (50 instead
of 110 Hz). In this case, the model reproduces the salient characteristics of
40-Hz oscillations in CA3 (Fisahn et al.
1998) and neocortical (Buhl et
al. 1998) slices. During 40-Hz population rhythm, single-cell
firing rates are low, 10 Hz in interneurons and 2 Hz in pyramidal cells,
in average. Spike trains of individual neurons are irregular and intermittent.
It would be difficult to detect the oscillation from such analysis as
autocorrelation function and power spectrum of spike trains. However, a
subthreshold oscillation is apparent in the membrane potential traces. The
membrane potential hovers below and near the firing threshold. Spikes are
triggered randomly by noise fluctuations, leading to sparse and irregular
spike trains. As in the experiments, network oscillation was abolished in the
model by blockade of either AMPA-mediated excitation or GABA-mediated
inhibition but not NMDA-mediated excitation. Thus the salient observations of
these experiments (Fisahn et al.
1998; Buhl et al.
1998) can be reproduced and understood in this simple setting.
|
Phase shift between two populations
Recurrent excitation also tends to decrease the phase shift between
excitatory and inhibitory populations. In the absence of
pyramidal-to-pyramidal connections, interneurons can lag excitatory cells by
>90° as shown in the preceding text. When pyramidal-to-pyramidal
connections are present and the balance between inhibition and excitation is
equal in pyramidal cells and interneurons, the analysis predicts that the
phase shift becomes essentially zero (see APPENDIX 3 for details).
Figure 8 shows that the zero
phase shift is indeed observed in simulations where these conditions hold. In
hippocampal slices, where gamma oscillation appears to depend on the
pyramid-interneuron connections, Fisahn et al.
(1998) found no significant
phase shift between spike activities of pyramidal and interneuronal
populations. Significant phase lag was seen only between pyramidal cell firing
and EPSCs and IPSCs. This observation is reproduced by our model, where
pyramidal and interneuronal spiking activities are synchronized with zero
phase difference. EPSCs and IPSCs lag behind the pyramidal spiking by 2 and 5
ms, respectively (Fig. 8).
These phase lags can be simply accounted for by the time-to-peak of the
excitatory and inhibitory synaptic currents. Therefore gamma oscillation in a
pyramid-interneuron network is compatible with zero phase difference between
pyramidal cells and interneurons like in the experiment of Fisahn et al.
(1998). The simple intuitive
reason for this phenomenon is that if the balance of excitation and inhibition
is the same in pyramidal cells and interneurons, the inputs to both cell types
must be in phase, and hence the firing rates of both cell types must also be
in phase.
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Dependence of oscillation frequency on the balance and relative speeds of excitation and inhibition
To understand better the two-population network with all four (E-to-E, E-to-I, I-to-E, and I-to-I) connections, we analyzed the oscillatory behavior under the assumption that the ratio of AMPA to GABA conductances is equal in pyramidal cells and interneurons, as a reasonable working hypothesis for neocortical and CA3 networks. Hence the ratio IAMPA/IGABA is the same for both pyramidal cells and interneurons.
In such a network, the analysis predicts that the frequency strongly
depends on the balance between AMPA and GABA synaptic currents (Eq.
C6 of APPENDIX 3). This is a manifestation of the fact that
both pyramidal-to-interneuron connections (via the E-I loop) and the
pyramidal-to-pyramidal connections tend to decrease population frequency. In
the simulations shown in Fig.
9, we varied systematically the balance between
IGABA and IAMPA by varying the AMPA
conductances on both pyramidal cells and interneurons were varied from 0 to
100% of their "control" value (indicated in METHODS).
When IAMPA/IGABA is increased from
zero (without recurrent excitation, purely I-I oscillation) to 0.5 (strong
recurrent excitation), the population frequency decreases from 180 to 70
Hz (Fig. 9). The theoretical
prediction (Eq. C6, ——) fits well with direct network
simulations (
, 
, 
). For these parameters, the ratio could
not be increased beyond 0.5 in the simulation, because the network became very
strongly synchronized, and hence the synchrony regime fell outside of the
scope of the present study.
|
Unlike the one-population model where the network oscillation frequency
depends only weakly on the synaptic decay time constant, the behavior of the
two-population network also critically depends on the relative time constants
of synaptic excitation and inhibition. As shown in
Fig. 10, the parameter plane
of the decay times AMPA and GABA is separated
into two regions for the asynchronous and synchronous dynamics. The boundary
between the two regions is the locus of the onset of synchronized oscillation
(a bifurcation in the language of dynamical systems).
Figure 10 shows that different
values of the synaptic temporal parameters can favor one of the two competing
kinds of oscillatory instabilities, the purely interneuronal mechanism or the
pyramidal-interneuronal loop mechanism. This can be seen clearly along a
horizontal line in Fig.
10A (say at fixed GABA = 5 ms). The
asynchronous behavior is realized only in an intermediate range of
AMPA values. With very short AMPA, oscillation
occurs, as expected when excitation is faster than feedback inhibition
(Tegnér et al. 2002;
Tsodyks et al. 1997;
Wang 1999;
Wilson and Cowan 1973). With
these short excitatory time constants, the E-I loop strongly influences the
network oscillation, the population frequency is relatively low. On the other
hand, with sufficiently long AMPA, the excitatory synaptic
current strongly attenuates the network oscillation. Driven by tonic
excitation, the interneuronal network alone is able to generate a synchronous
oscillation, which now has the characteristics of oscillation in the
one-population model (with a very high population frequency). The population
frequency is much higher when the oscillation is dominated by the
interneuronal network (with long AMPA) than when it largely
depends on the pyramid-to-interneuron loop (with short AMPA;
Fig. 10B).
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