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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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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). |
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INTRODUCTION |
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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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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
 |
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
 |
 | (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
 |
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
 |
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).
|
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.
|
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).
|
The locus of the onset of synchrony is also greatly influenced by the
strength and balance of excitation and inhibition, according to Eq.
C5 of APPENDIX 3. For the parameters of
Fig. 10, without excitation
(the ratio IAMPA/IGABA = 0), synchrony
appears at GABA = 7.5 ms, independent of AMPA.
An increasingly large excitation/inhibition ratio shifts the bifurcation curve
close to the diagonal line, and enlarges the oscillatory regime (at small
AMPA) dominated by the pyramid-interneuron loop
(Fig. 10A). The
population frequency is lower with a larger excitation/inhibition ratio
(Fig. 10B),
consistent with Fig. 9.
Finally, with a fixed IAMPA/IGABA but
enhancing the absolute connection strength (both gAMPA and
gGABA are increased while their ratio is preserved), the
parameter region for the asynchronous dynamics shrinks dramatically, and
coherent oscillation becomes prevalent
(Fig. 10C). Therefore
powerful feedback synaptic interactions generally are destabilizing and favor
large-amplitude coherent oscillations.
|
|
DISCUSSION |
|---|
|
In this paper, we have extended a theoretical framework for understanding
the mechanisms of fast network oscillations that are characterized by noisy
and intermittent spike firing of constituent single cells. Brunel and Hakim
(1999), using simple
(instantaneous) synaptic interactions, showed that when the network is
dominated by noise, coherent oscillation with irregular and sparse neuronal
firing can be produced with strong recurrent inhibition and strong external
excitation. In this paper, we extended this approach to a network of LIF
neurons coupled by realistic synapses. Our analysis predicts the population
frequency for noise-dominated network rhythms, given synaptic temporal
parameters, and the balance between the different types of (I-I, E-I, E-E)
feedback loops present in the network. This analysis based on a phase
condition bears some similarity with a recent study of the emergence of
rhythmic activity in a network of excitatory cells with spike frequency
adaptation (Fuhrmann et al.
2002).
In an interneuron network, ultrafast oscillations (200 Hz) with the
phenomenology of hippocampal ripples during sharp waves can be realized with
synaptic time constants similar to those observed in slice studies. The
oscillation frequency depends much more on the shortest synaptic time
constants (synaptic delay and rise time) than on the synaptic decay time.
In a network composed of both interneurons and excitatory cells, the
oscillation is a compromise between two rhythmogenesis scenarios, the purely
interneuronal scenario and the pyramidal-to-interneuron loop scenario. The
pyramidal-to-interneuron loop scenario involves a much lower frequency (in the
gamma range) than the purely interneuronal scenario. Recurrent excitation
(both in the E-I and E-E loops) reduces network frequency. The type of
oscillation that is generated depends on the relative time scales
(E and I) of excitatory and inhibitory
synaptic currents and on the balance between the E-I and I-I loops. Networks
without pyramidal-to-pyramidal connections can sustain 200-Hz
oscillations when receiving strong external inputs. Interneurons typically lag
behind pyramidal cells by 90° in such networks. Networks with
pyramidal-to-pyramidal connections are able to sustain oscillations that are
typically in the gamma range (30–100 Hz, depending on the time constants
of synapses). When the excitation-inhibition balance is roughly the same in
pyramidal cells and interneurons, we predict that there is zero phase
difference between the spike discharges of pyramidal and interneuron
populations.
Firing-rate synchrony versus spike-to-spike synchrony
The regime of synchronous activity studied in this paper should be
contrasted with "spike-to-spike synchrony," which occurs usually
without or with weak noise. In networks with spike-to-spike synchrony, the
network rhythm arises from synchronization of spikes among neurons that behave
as periodic oscillators. Single cells fire a spike at each population cycle or
once every few cycles. This kind of synchronization has been studied
extensively in the theoretical framework of coupled oscillators
(Kopell and Ermentrout 1986;
Kuramoto 1984). Many studies
have focused on an inhibitory neural network because experimental and
theoretical evidence implicates interneurons in the emergence of oscillatory
synchrony patterns. In particular, it was found that synchronization depends
critically on the synaptic decay time constant
(Hansel et al. 1995;
Traub et al. 1996;
van Vreeswijk et al. 1994;
Wang and Buzsáki 1996;
Wang and Rinzel 1992;
White et al. 1998). Terman et
al. (1998) showed that the
synaptic rise time also plays a significant role in the synchronization of two
mutually inhibitory neurons. There have been some analysis of how
synchronization between coupled oscillators depends on the heterogeneity of
neuronal properties (Bartos et al.
2001; Hansel and Mato
2003; Neltner et al.
2000; Tiesinga and Jose
2000; Wang and Buzsáki
1996; White et al.
1998) or the sparse random network connectivity
(Golomb and Hansel 2000;
Wang and Buzsáki 1996).
More recently, Hansel and Mato
(2001,
2003) considered a
two-population model of excitatory and inhibitory neurons as coupled
oscillators, where the network is fully connected and without noise. This kind
of synchrony is likely to be applicable to networks of pacemaker neurons
(regular oscillators), such as in the central pattern generator systems
(Marder 1998).
The second type of network synchrony is "firing rate
synchrony," which occurs when strong noise and strong inhibitory
feedback is present (Brunel
2000; Brunel and Hakim
1999). Here, when a network displays prominent coherent
oscillation, spike discharges of single neurons tend to be extremely irregular
and intermittent, like a Poisson process with the firing rate much lower than
the population oscillation frequency. Fast oscillations in the cortex appears
to belong to this category of synchronous activity because spike trains of
single cells are usually random and sparse even though the local field
potential is rhythmic (Csicsvari et al.
1998; Fries et al
2001). In fact, this type of synchrony is already present in the
firing rate models, which were derived under the assumption that single
neurons fire spikes stochastically (Wilson and Cowan
1972,
1973).
In a purely interneuronal network, the two types of synchrony produces very
different frequency ranges. Networks with spike-to-spike synchrony are
typically in the gamma range (Traub et al.
1996; Wang and Buzsáki
1996). We have shown here that networks with firing rate synchrony
oscillate at much higher frequencies. Dependency on different synaptic time
scales is another major feature that distinguishes between the two classes of
synchronization in an interneuronal network: for spike-to-spike synchrony in
oscillator networks, synchrony can appear without any synaptic latency, and
the frequency strongly depends on synaptic decay time. For firing-rate
synchrony, synchrony depends on a non-zero latency and depends much more
weakly on the synaptic decay time. For IPSCs with very short latency and rise
times (1 ms) and longer decay times (5–10 ms), we find that an
interneuronal network is able to oscillate at a frequency determined by the
fastest time scales (latency and rise times). In this type of oscillation, the
decay time affect only weakly the frequency because it is longer than the
period of the oscillation.
In a two-population network, the differences between both types of
synchrony are less clear cut. Networks tend to oscillate in both cases at
frequencies in the gamma range. Phase lags between pyramidal cells and
interneurons occur in similar conditions for both types of synchrony
(Hansel and Mato 2003). At
gamma frequencies, the oscillation becomes sensitively dependent on decay
times of both AMPA and GABA currents. Hence, all synaptic time constants
potentially play a role in the determination of network frequency.
Comparison with hippocampal oscillations data
In vivo observations indicate that 200 Hz ripples during sharp waves are
most prominent in CA1 area, and less reliable in CA3
(Buzsaki et al. 1992;
Csicsvari et al. 1999a).
Presumably, during sharp waves, CA3 network provides a large excitatory drive,
which induces 200-Hz ripples in CA1. On the other hand, 40-Hz (gamma)
oscillations appear to be generated within CA3 and propagate to CA1
(Fisahn et al. 1998).
Interestingly, CA3 is endowed with an abundance of strong recurrent excitatory
connections (Miles and Wong
1986), whereas local collaterals between pyramidal cells are
relatively rare and weak in CA1 (Deuchars
and Thomson 1996). These observations raise the possibility that
the network oscillation frequency (200 vs. 40 Hz) could depend on the strength
of intrinsic pyramid-to-pyramid connections. The results shown in this paper
are consistent with such a possibility. Interestingly, during sharp wave
ripples, CA3 area typically shows variable 100-Hz oscillations
(Csicsvari et al. 1999a). The
difference in the ripple frequency (100 vs. 200 Hz) between CA3 and CA1 may
again be related to the preponderance in CA3, and paucity in CA1, of recurrent
excitatory connections.
We have shown that the same network can exhibit different population frequencies, depending on the balance between GABA and AMPA currents. In principle, this balance between excitation and inhibition in the hippocampus can be modulated depending on the behavioral states, for example, by varying differentially the external inputs targetting excitatory and inhibitory cells.
Comparison with neocortical data
Very fast oscillations (from 80 to 600 Hz) have been recently observed in
neocortex of anesthetized rats (Jones et
al. 2000; Jones and Barth
2002; Kandel and
Buzsáki 1997) and cats
(Grenier et al. 2001). In
these oscillations, single cells typically fire at much lower rates, as in
fast oscillations in the hippocampus. Oscillations with frequency <200 Hz,
as observed in the anesthetized cat
(Grenier et al. 2001), seem to
be compatible with the scenario discussed in the present paper, in which
synaptic inhibition plays a major role. A further indication in favor of the
role of inhibition is that neurons recorded with chloride-filled pipettes
change their phase relationship with the ripple. On the other hand, the faster
(300–500 Hz) oscillations seen by Kandel and Buzsáki
(1997) and Jones et al.
(2000) seem difficult to
reconcile with the present scenario, unless inhibitory time courses are
extremely short, and in particular latencies of IPSCs on interneurons are much
less than 1 ms (see Fig. 4).
Furthermore, Jones and Barth
(2002) showed that application
of bicuculline does not result in a change of oscillation amplitude or
frequency, suggesting a different mechanism is at work in such ultra-fast
oscillations.
Model predictions
Our theory has a number of predictions that can be tested experimentally.
First, in a noise-dominated inhibitory neural network, the emergence and
nature of stochastic fast oscillations depend critically on the synaptic delay
of GABAergic connections on interneurons, and in contrast to the coupled
oscillator regime, the oscillation frequency is only weakly sensitive to the
decay time constant of synaptic inhibition. Second, fast oscillations can be
produced either by the interneuronal network or the feedback inhibition in a
pyramid-interneuron reciprocal loop. Synchrony in networks with
excitatory-inhibitory loops does not necessarily imply a large phase
difference in the neural activities of the two cell populations. Pyramidal
cells and interneurons can have zero phase difference if recurrent excitatory
and inhibitory inputs are balanced in the same way in both cell types. This
prediction can be checked in the slice preparation of Fisahn et al.
(1998) by measuring the mean
excitatory and inhibitory currents in both pyramidal cells and interneurons
and comparing the ratios in the two cell types. Third, somewhat contrarily to
intuition, inhibition tends to favor very-high-frequency oscillatory patterns,
while excitation tends to favor slower oscillatory patterns. Manipulations
that alter the balance between recurrent excitation and inhibition can
therefore drastically affect network frequency.
Can very fast (200 Hz) oscillations be produced by chemical
synapses?
Our modeling results suggest that, in principle, very fast oscillations are
possible with purely chemical synaptic couplings and no gap junctions. We
would like to caution that the present study focused on the effect of synaptic
parameters on network frequency and used a simplified (the integrate-and-fire)
model of single neurons. This approach allowed us to explore systematically
the network behavior in the parameter space. Integrate-and-fire neurons are
characterized by a zero neuronal phase lag, for realistic noise models, as was
shown by Brunel et al. (2001).
This fact allows the network to oscillate at frequencies that are limited only
by the speed of synaptic transmission. However, real neurons are endowed with
a variety of voltaged-gated ion channels and might differ significantly from
the integrate-and-fire model. At least two features of real neurons are not
present in integrate-and-fire neurons: subthreshold resonance (see e.g.,
Hutcheon and Yarom 2000) and
dynamics of ionic channels leading to spike generation. Subthreshold
resonances recorded in most neurons appear at rather low frequencies
(typically theta), so they should not affect the network dynamics at the high
frequencies considered here. On the other hand, dynamics of ionic channels
leading to spike generation could in principle affect the high-frequency
neuronal response. In a companion paper (C. Geisler, N. Brunel, and X.-J.
Wang, unpublished observations), the contribution of intrinsic cellular
dynamics to the oscillation frequency and network synchronization is
investigated extensively. In particular, we find that with Hodgkin-Huxley-type
conductance-based neurons, instead of the integrate-and-fire neurons,
very-high-frequency (200 Hz) oscillations can be achieved (Geisler, et al.,
unpublished observations) provided that certain assumptions are satisfied, for
example, when the effective membrane time constants of the neurons are small
enough.
Furthermore, even if the chemical synapse mechanism is sufficient to
produce 200-Hz oscillations under some conditions, it does not rule out other
scenarios for the generation of very fast oscillations, such as axo-axonic gap
junctions between pyramidal cells (Lewis
and Rinzel 2000; Schmitz et
al. 2001; Traub et al.
1999; Traub and Bibbig
2000). In some systems, very fast oscillations are not suppressed
by GABAergic antagonists (Draguhn et al.
1998; Jones and Barth
2002), leaving gap junctions as the leading alternative mechanism.
In other systems, such that the hippocampus of a freely moving rat, whether
chemical, electrical, or both, connections are needed remains an open
question.
Further experimental and theoretical work will be necessary to obtain a
general framework for understanding synchronous network rhythms characterized
by irregular neuronal spike discharges and help to resolve the well-known
dichotomy between oscillatory local field potential of a cortical circuit and
the highly stochastic spike trains of its constituent neurons
(Engel et al. 1992;
Fries et al. 2001).
|
|
APPENDIX 1. NETWORKS WITH E-I CONNECTIONONLY |
|---|
|
Step 1
In the excitatory-inhibitory network, we look for solutions of network
activity of the form
 | (A1) |
 | (A2) |
E,0 and I,0 are the average firing rates,
E and I are the relative strengths of the
deviations to the stationary firing rates (including a possible phase lag
between the two populations). Step 2
The GABAergic synaptic variable is given by Eq. 6. The AMPA
synapses are described by
 | (A3) |
 |
 |
The recurrent synaptic currents are
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
Step 3
The firing rate of pyramidal cells and interneurons are
 |
 |
E,0/E,0
E,0(IsynE,0)/IsynE,0. Step 4
Steps 3 and 1 imply that the oscillatory components in pyramidal cell
activity and interneuron oscillatory activity are related by
 | (A8) |
 | (A9) |
 |
(or f =
/(2)). Note that, by our convention, interneurons lag excitatory
cells by E(). |
|
APPENDIX 2. NETWORKS WITH E-I AND I-I LOOPS |
|---|
|
|
| (B1) |
 | (B2) |
 | (B3) |
 |
 |
 | (B4) |
 | (B5) |
 | (B6) |
E,0,
IAMPAI,0, and
IGABAI,0.
Equalizing the rates in Eq. B4 and in Eq. A1, it is clear
that the oscillatory components in pyramidal cell activity and interneuron
oscillatory activity are related by
 | (B7) |
 | (B8) |
 | (B9) |
The phase lag between interneurons and pyramidal cells is
 | (B10) |
|
|
APPENDIX 3. NETWORKS WITH E-E, E-I, AND I-I LOOPS |
|---|
|
 | (C1) |
|
| (C2) |
 |
 | (C3) |
E() = 0 whose solution is = 0.
When the balance between excitation and inhibition is the same in pyramidal
cells and interneurons
 |
 | (C4) |
 | (C5) |
 | (C6) |
E, I,
SE, and SI and on the balance between
recurrent excitation and inhibition,
IAMPA/IGABA. Combining Eq. C4
with Eq. C2, we find that E = I. In
other words, there is zero phase lag between pyramidal cells and
interneurons.
Solving numerically Eq. C6, we find that when excitatory time
constants are shorter or equal to inhibitory time constants, excitation
decreases monotonically the oscillation frequency. The condition for
this to happen is sin(E(I)) > 0, where
I is the network frequency for a purely inhibitory network.
When excitation time constants are long enough, resulting in
sin(E(I)) < 0, excitation increases the
oscillation frequency for small values of the ratio
IAMPA/IGABA. This phenomenon is
investigated in more detail in Geisler (unpublished observations).
|
|
ACKNOWLEDGMENTS |
|---|
|
X. J. Wang was supported by the National Institute of Mental Health Grant MH-62349, the Alfred P. Sloan Foundation, and the Swartz Foundation.
|
|
FOOTNOTES |
|---|
Address for reprint requests: N. Brunel, CNRS-NPSM-Université Paris René Descartes, 45 rue des Saints Pères 75270 Paris Cedex 06, France (E-mail: brunel{at}biomedicale.univ-paris5.fr).
|
|
REFERENCES |
|---|
|
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.
Bartos M, Vida
I, Frotscher M, Geiger JRP, and Jonas P. Rapid signaling at inhibitory
synapses in a dendate gyrus interneuron network. J
Neurosci 21:
2687–2698, 2001.
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.
Bragin A, Jando G, Nadasdy Z, Hetke J, Wise K, and Buzsáki G. Gamma (40–100 Hz) oscillation in the hippocampus of the behaving rat. J Neurosci 15: 47–60, 1995.[Abstract]
Brunel N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8: 183–208, 2000.[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.[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.[Web of Science][Medline]
Buhl DL, Harris
KD, Hormuzd SG, Monyer H, and Buzsáki G. Selective impairment of
hippocampal gamma oscillations in connexin-36 knock-out mouse in vivo.
J. Neurosci 23:
1013–1018, 2003
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.
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 interneurons
of cat visual cortex. J Physiol
500: 689–713,
1997.
Buzsaki G,
Urioste R, Hetke J, and Wise K. High-frequency network oscillation in the
hippocampus. Science 256:
1025–1027, 1992.
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.[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.
Deuchars J and Thomson AM. CA1 pyramid-pyramid connections in rat hippocampus in vitro: dual intracellular recordings with biocytin filling. Neuroscience 74: 1009–1018, 1996.[Web of Science][Medline]
Draguhn A, Traub RD, Schmitz D, and Jefferys JGR. Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro. Nature 394: 189–193, 1998.[Medline]
Engel AK, Konig TBSP, Kreiter AK, and Singer WR. Temporal coding in the visual cortex: new vistas on integration in the nervous system. Trends Neurosci 15: 218–226, 1992.[Web of Science][Medline]
Fellous J and Sejnowski TJ. Cholinergic induction of oscillations in the hippocampal slice in the slow (0.5–2 Hz), theta (5–12 Hz) and gamma bands. Hippocampus 10: 187–197, 2000.[Web of Science][Medline]
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.[Medline]
Fourcaud N and Brunel N. Dynamics of firing probability of noisy integrate-and-fire neurons. Neural Comput 14: 2057–2110, 2002.[Web of Science][Medline]
Freeman WJ. Mass Action in the Nervous System. New York: Academic Press, 1975.
Fries P,
Reynolds JH, Rorie AE, and Desimone R. Modulation of oscillatory neuronal
synchronization by selective visual attention. Science
291: 1560–1563,
2001.
Fuhrmann G,
Markram H, and Tsodyks M. Spike frequency adaptation and neocortical
rhythms. J Neurophysiol 88:
761–770, 2002.
Gerstner W. Time structure of the activity in neural network models. Phys Rev E 51: 738–758, 1995.
Gerstner W, van Hemmen L, and Cowan J. What matters in neuronal locking? Neural Comput 8: 1653–1676, 1996.[Web of Science][Medline]
Golomb D and Hansel D. The number of synaptic inputs and the synchrony of large sparse neuronal networks. Neural Comput 12: 1095–1139, 2000.[Web of Science][Medline]
Golomb D and Rinzel J. Clustering in globally coupled inhibitory neurons. Physica D 72: 259–282, 1994.
Grenier F,
Timofeev I, and Steriade M. Focal synchronization of ripples (80–200
Hz) in neocortex and their neuronal correlates. J
Neurophysiol 86:
1884–1898, 2001.
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.
Hansel D and Mato G. Existence and stability of persistent states in large neuronal networks. Phys Rev Lett 10: 4175–4178, 2001.
Hansel D and Mato G. Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons. Neural Comput 15: 1–56, 2003.[Web of Science][Medline]
Hansel D, Mato G, and Meunier C. Synchrony in excitatory neural networks. Neural Comput 7: 307–337, 1995.[Web of Science][Medline]
Hansel D, Mato G, Meunier C, and Neltner L. On numerical simulations of integrate-and-fire neural networks. Neural Comput 10: 467–483, 1998.[Web of Science][Medline]
Hansel D and Sompolinsky H. Chaos and synchrony in a model of a hypercolumn in visual cortex. J Comput Neurosci 3: 7–34, 1996.[Web of Science][Medline]
Hestrin S, Sah P, and Nicoll R. Mechanisms generating the time course of dual component excitatory synaptic currents recorded in hippocampal slices. Neuron 5: 247–253, 1990.[Web of Science][Medline]
Hormuzdi SG, Pais I, LeBeau FE, Towers SK, Rozov A, Buhl EH, Whittington MA, and Monyer H. Impaired electrical signaling disrupts gamma frequency oscillations in connexin 36-deficient mice. Neuron 31: 487–495, 2001.[Web of Science][Medline]
Hutcheon B and Yarom Y. Resonance, oscillation and the intrinsic frequency preferences of neurons. Trends Neurosci 23: 216–222, 2000.[Web of Science][Medline]
Jefferys JG, Traub RT, and Whittington MA. Neuronal networks for induced "40-Hz" rhythms. Trends Neurosci 19: 202–208, 1996.[Web of Science][Medline]
Jones MS and
Barth DS. Effects of bicuculline methiodide on fast (>200 Hz)
electrical oscillations in rat somatosensory cortex. J
Neurophysiol 88:
1016–1025, 2002.
Jones MS,
MacDonald KD, Chol B, Dudek E, and Barth DS. Intracellular correlates of
fast (>200 Hz) electrical oscillations in rat somatosensory cortex.
J Neurophysiol 84:
1505–1518, 2000.
Kandel A and
Buzsáki G. Cellular-synaptic generation of sleep spindles,
spike-and-wave discharges, and evoked thalamocortical responses in the
neocortex of the rat. J Neurosci
17: 6783–6797,
1997.
Kopell N and Ermentrout GB. Symmetry and phaselocking in chains of weakly coupled oscillators. Comm Pure Appl Math 39: 623–660, 1986.
Kopell N and
LeMasson G. Rhythmogenesis, amplitude modulation, and multiplexing in a
cortical architecture. Proc Natl Acad Sci USA
91: 10586–10590,
1994.
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.
Kuramoto Y. Chemical Oscillations, Waves and Turbulence. New York: Springer-Verlag, 1984.
Leung LS.
Nonlinear feedback model of neuronal populations in hippocampal CA1 region.
J Neurophysiol 47:
845–868, 1982.
Lewis TJ and Rinzel J. Self-organized synchronous oscillations in a network of excitable cells coupled by gap junctions. Network 11: 299–320, 2000.[Web of Science][Medline]
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.[Medline]
Marder E. From biophysics to models of network function. Annu Rev Neurosci 21: 25–45, 1998.[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 (London)
500: 409–440,
1997.
Miles R and
Wong RK. Excitatory synaptic interactions between ca3 neurones in the
guinea pig hippocampus. J Physiol
373: 397–418,
1986.
Neltner L, Hansel D, Mato G, and Meunier C. Synchrony in heterogeneous neural networks. Neural Comp 12: 1607–1641, 2000.[Web of Science][Medline]
Press WH, Teukolsky SA, Vetterling WT, and Flannery BP. Numerical Recipes in C. Cambridge, UK: Cambridge Univ. Press, 1992.
Salin PA and
Prince DA. Spontaneous GABAA receptor mediated inhibitory
currents in adult rat somatosensory cortex. J
Neurophysiol 75:
1573–1588, 1996.
Schmitz D, Schuchmann S, Fisahn A, Draguhn A, Buhl EH, Petrasch-Parwez W, Dermietzel R, Heinemann U, and Traub RD. Axo-axonal coupling a novel mechanism for ultrafast neuronal communication. Neuron 31: 831–840, 2001.[Web of Science][Medline]
Shelley MJ and Tao L. Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks. J Comput Neurosci 11: 111–119, 2001.[Web of Science][Medline]
Siapas AG and Wilson MA. Coordinated interactions between hippocampal ripples and cortical spindles during slow-wave sleep. Neuron 21: 1123–1128, 1998.[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 500:
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.
Tegnér J, Compte A, and Wang X-J. Dynamical stability of hebbian reverberatory neural networks. Biol Cybern 87: 471–481, 2002.[Web of Science][Medline]
Terman D, Kopell N, and Bose A. Dynamics of two mutually coupled slow inhibitory neurons. Physica D 117: 241–275, 1998.
Tiesinga PH and Jose JV. Robust gamma oscillations in networks of inhibitory hippocampal interneurons. Network 11: 1–23, 2000.[Web of Science][Medline]
Traub RD and
Bibbig A. A model of high-frequency ripples in the hippocampus based on
synaptic coupling plus axon-axon gap junctions between pyramidal neurons.
J Neurosci 20:
2086–2093, 2000.
Traub RD, Schmitz D, Jefferys JGR, and Draguhn A. High-frequency population oscillations are predicted to occur in hippocampal pyramidal neuronal networks interconnected by axoaxonal gap junctions. Neuroscience 92: 407–426, 1999.[Web of Science][Medline]
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.
Treves A. Mean-field analysis of neuronal spike dynamics. Network 4: 259–284, 1993.
Tsodyks MV,
Skaggs WE, Sejnowski TJ, and McNaughton BL. Paradoxical effects of
external modulation of inhibitory interneurons. J
Neurosci 17:
4382–4388, 1997.
Tuckwell HC. Introduction to Theoretical Neurobiology. Cambridge, UK: Cambridge Univ. Press, 1988.
van Vreeswijk C, Abbott L, and Ermentrout GB. When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1: 313–321, 1994.[Medline]
van Vreeswijk C and Sompolinsky H. Chaos in neuronal networks with balanced excitatory and
inhibitory activity. Science
274: 1724–1726,
1996.
Vida I, Halasy
K, Szinyei C, Somogyi P, and Buhl EH. Unitary IPSPs evoked by interneurons
at the stratum radiatum-stratum lacunosum-molecolare border in the CA1 area of
the rat hippocampus in vitro. J Physiol
506: 755–773,
1998.
Wang X-J.
Synaptic basis of cortical persistent activity: the importance of NMDA
receptors to working memory. J Neurosci
19: 9587–9603,
1999.
Wang X-J and
Buzsáki G. Gamma oscillation by synaptic inhibition in a
hippocampal interneuronal network model. J Neurosci
16: 6402–6413,
1996.
Wang X-J,
Golomb D, and Rinzel J. Emergent spindle oscillations and intermittent
burst firing in a thalamic model: specific neuronal mechanisms.
Proc Natl Acad Sci USA 92:
5577–5581, 1995.
Wang X-J and Rinzel J. Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 4: 84–97, 1992.
White JA, Chow CC, Soto-Treviño C, and Kopell N. Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. J Comput Neurosci 5: 5–16, 1998.[Web of Science][Medline]
Wilson HR and Cowan JD. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12: 1–24, 1972.
Wilson HR and Cowan JD. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13: 55–80, 1973.[Web of Science][Medline]
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.
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.[Web of Science][Medline]
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