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Volume 16, Number 20,
Issue of October 15, 1996
pp. 6402-6413
Copyright ©1996 Society for Neuroscience
Gamma Oscillation by Synaptic Inhibition in a Hippocampal
Interneuronal Network Model
Xiao-Jing Wang1 and
György Buzsáki2
1 Physics Department and Center for Complex Systems,
Brandeis University, Waltham, Massachusetts 02254, and
2 Center for Molecular and Behavioral Neuroscience, Rutgers
University, Newark, New Jersey 07102
 ABSTRACT
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- FOOTNOTES
- REFERENCES
ABSTRACT 
Fast neuronal oscillations (gamma, 20-80 Hz) have been
observed in the neocortex and hippocampus during behavioral arousal.
Using computer simulations, we investigated the hypothesis that such
rhythmic activity can emerge in a random network of interconnected
GABAergic fast-spiking interneurons. Specific conditions for the
population synchronization, on properties of single cells and the
circuit, were identified. These include the following: (1) that the
amplitude of spike afterhyperpolarization be above the
GABAA synaptic reversal potential; (2) that the ratio
between the synaptic decay time constant and the oscillation period be
sufficiently large; (3) that the effects of heterogeneities be modest
because of a steep frequency-current relationship of fast-spiking
neurons. Furthermore, using a population coherence measure,
based on coincident firings of neural pairs, it is demonstrated that
large-scale network synchronization requires a critical (minimal)
average number of synaptic contacts per cell, which is not sensitive to
the network size.
By changing the GABAA synaptic maximal conductance,
synaptic decay time constant, or the mean external excitatory drive to
the network, the neuronal firing frequencies were gradually and
monotonically varied. By contrast, the network synchronization was
found to be high only within a frequency band coinciding with the gamma
(20-80 Hz) range. We conclude that the GABAA synaptic
transmission provides a suitable mechanism for synchronized gamma
oscillations in a sparsely connected network of fast-spiking
interneurons. In turn, the interneuronal network can presumably
maintain subthreshold oscillations in principal cell populations and
serve to synchronize discharges of spatially distributed neurons.
Key words:
gamma rhythm;
hippocampus;
interneurons;
GABAA;
synchronization;
computer model
INTRODUCTION
Although fast gamma cortical oscillation has been
the subject of active investigation in recent years (cf. Singer and
Gray, 1995 ), its underlying neuronal mechanisms remain elusive. Two
major issues are the cellular origin of rhythmicity (Llinás et
al., 1991; McCormick et al., 1993; Wang, 1993) and the mechanism(s) of
large-scale population synchronicity (Freeman, 1975; Bush and Douglas
1991; Engel et al., 1991; Hansel and Sompolinsky, 1996). Traditionally,
recurrent excitation between principal (pyramidal) neurons is viewed as
a major source of rhythmogenesis as well as neuronal synchronization.
However, in model studies in which quantitative data about the synaptic
time course were incorporated, it was found that glutamatergic synaptic
excitation of the AMPA type usually desynchronizes rather than
synchronizes repetitive spike firings of mutually coupled neurons
(Hansel et al., 1995; van Vreeswijk et al., 1995). Therefore, recurrent
connections between pyramidal cells alone do not seem to account for
the network coherence during cortical gamma oscillations. It was
suggested that pyramidal cell populations may be entrained by
synchronous rhythmic inhibition originating from fast-spiking
interneurons (Buzsáki et al., 1983; Lytton and Sejnowski, 1991).
During field gamma oscillations, intracellular recordings from
pyramidal cells revealed both EPSPs and IPSPs phase-locked to the field
oscillation frequencies (Jagadeesh et al., 1992; Chen and Fetz, 1993;
Soltész and Deschênes, 1993).
In this paper, we address the question whether, in the hippocampus, an
interneuronal network can generate a coherent oscillatory output to the
pyramidal neurons, thereby providing a substrate for the synaptic
organization of coherent gamma population oscillations. In the behaving
rat, physiologically identified interneurons were shown to fire spikes
in the gamma frequency range and phase-locked to the local field waves
(Bragin et al., 1995). Intracellular studies and immunochemical
staining demonstrated that these interneurons are interconnected via
GABAergic synapses (Lacaille et al., 1987; Sik et al., 1995;
Gulyás et al., 1996). Theoretical studies suggest that these
GABAergic interconnections may synchronize an interneuronal network
when appropriate conditions on the time course of synaptic transmission
are satisfied (Wang and Rinzel, 1992, 1993; van Vreeswijk et al.,
1995). Moreover, in a recent in vitro experiment
(Whittington et al., 1995; Traub et al., 1996), the excitatory
glutamate AMPA and NMDA synaptic transmissions were blocked in the
hippocampal slice. When metabotropic glutamate receptors were
activated, transient oscillatory IPSPs in the 40 Hz frequency range
were observed in pyramidal cells. These IPSPs were assumed to originate
from the firing activities of fast-spiking interneurons synchronized by
their interconnections. Computer simulations (Whittington et al., 1995;
Traub et al., 1996) lend further support to this hypothesis.
To assess whether an interneuronal network can subserve an adequate
basis for the gamma frequency population rhythm in the hippocampus, it
is necessary to identify its specific requirements on the cellular
properties and network connectivities, as well as to determine whether
these conditions are satisfied by particular interneuronal subtypes.
The present study is devoted to investigate such requirements using
computer simulations. We found that synaptic transmission via
GABAA receptors in a sparsely connected network of model
interneurons can provide a mechanism for gamma frequency oscillations,
and we compared the modeling results with the anatomical and
electrophysiological data from hippocampal fast spiking interneurons.
MATERIALS AND METHODS
Model neuron. Each interneuron is described by a
single compartment and obeys the current balance equation:
|
(2)
|
where Cm = 1 µF/cm2 and
Iapp is the injected current (in
µA/cm2). The leak current IL = gL(V  EL) has a
conductance gL = 0.1 mS/cm2, so that
the passive time constant  0 = Cm/gL = 10 msec;
EL = 65 mV.
The spike-generating Na+ and K+
voltage-dependent ion currents (INa and
IK) are of the Hodgkin-Huxley type (Hodgkin and
Huxley, 1952). The transient sodium current INa = gNam3 h(V ENa), where the activation variable m is
assumed fast and substituted by its steady-state function
m =  m/(m +  m); m(V) = 0.1(V + 35)/(exp(0.1(V + 35)) 1),
m(V) = 4exp((V + 60)/18).
The inactivation variable h obeys a first-order
kinetics:
|
(2)
|
where h(V) = 0.07 exp((V + 58)/20) and
h(V) = 1/(exp(0.1(V + 28)) + 1). gNa = 35 mS/cm2;
ENa = 55 mV,  = 5.
The delayed rectifier IK = gKn4 (V EK), where the activation variable n obeys
the following equation:
|
(2)
|
with n(V) = 0.01(V + 34)/(exp(0.1(V + 34)) 1) and
n(V) = 0.125 exp((V + 44)/80); gK = 9 mS/cm2, and
EK = 90 mV.
These kinetics and maximal conductances are modified from Hodgkin and
Huxley (1952), so that our neuron model displays two salient properties
of hippocampal and neocortical fast-spiking interneurons. First, the
action potential in these cells is followed by a brief
afterhyperpolarization (AHP) of about 15 mV measured from the spike
threshold of approximately 55 mV (McCormick et al., 1985; Lacaille
and Williams, 1990; Morin et al., 1995; Zhang and McBain, 1995). Thus,
during the spike repolarization the membrane potential reaches a
minimum of about 70 mV, rather than being close to the reversal
potential of the K+ current, EK = 90 mV. This is accomplished in the model by relatively small maximal
conductance gK and fast gating process of
IK so that it deactivates quickly during spike
repolarization.
Second, these interneurons have the ability to fire repetitive spikes
at high frequencies (with the frequency-current slope up to 200-400
Hz/nA) (McCormick et al., 1985; Lacaille and Williams, 1990; Zhang and
McBain, 1995). With fast kinetics of the inactivation (h) of
INa, the activation (n) of
IK, and the relatively high threshold of
IK, the model interneuron displays a large range
of repetitive spiking frequencies in response to a constant injected
current Iapp (Fig.
1A, left). It has a
small current threshold (the rheobase Iapp  0.2 µA/cm2), and the firing rate is as high as 400 Hz for
Iapp 20 mA/cm2. Similar to
cortical interneurons (McCormick et al., 1985; Lacaille and Williams,
1990), the whole frequency-current curve is not linear, and the
frequency-current slope is larger at smaller
Iapp values (lower frequencies) (Fig.
1A, right). As a consequence, the
neural population is more sensitive to input heterogeneities at smaller
Iapp values. This is demonstrated in Figure
1B, where a Gaussian distribution of
Iapp is applied to a population of uncoupled
neurons (N = 100), with a mean
Iµ and standard deviation
I . Given a fixed and small
I = 0.03, the mean drive
Iµ is varied systematically, and the resulting
dispersion in the neuronal firing frequencies,
f/fµ (standard
deviation of the firing rate/mean firing rate) is shown as function of
Iµ (Fig. 1B,
top). When plotted versus fµ, it is
evident that with the same amount of dispersion in applied current
(I) the dispersion in firing rates
f/fµ is dramatically
increased for fµ < 20 Hz (Fig.
1B, bottom). This feature has important
implications for the frequency-dependent network behaviors (see
Results).
Fig. 1.
Model of single neuron and synapse. A,
Left, Firing frequency versus applied current intensity
(f Iapp curve) of the
model neuron. The firing rate can be as high as 400 Hz.
Right, The derivative df/dIapp shows
that the f/Iapp slope is much larger at smaller
Iapp (lower f) values.
B, Dispersion in firing rates caused by heterogeneity in
input current. A Gaussian distribution for input currents, with
standard deviation I = 0.03, is applied to a
population of uncoupled neurons. The dispersion in firing rates was
computed as the ratio between the standard deviation and the mean of
firing rates
(f/fµ).
This ratio is much larger for smaller mean current amplitude
Iµ (top). Plotting
f/fµ versus
fµ shows that the dispersion in firing rates
is dramatically increased for fµ < 20 Hz
(bottom). C, A brief current pulse applied to
a presynaptic cell generates a single action potential, which
elicits an inhibitory postsynaptic current
(Isyn) and membrane potential change in a
postsynaptic cell (gsyn = 0.1 mS/cm2).
[View Larger Version of this Image (15K GIF file)]
Model synapse. The synaptic current
Isyn = gsyns(V Esyn), where gsyn is the
maximal synaptic conductance and Esyn is the
reversal potential. Typically, we set gsyn = 0.1 mS/cm2 and Esyn = 75 mV (Buhl et
al., 1995). The gating variable s represents the fraction of
open synaptic ion channels. We assume that s obeys a
first-order kinetics (Perkel et al., 1981; Wang and Rinzel 1993):
|
(2)
|
where the normalized concentration of the postsynaptic
transmitter-receptor complex, F(Vpre), is
assumed to be an instantaneous and sigmoid function of the presynaptic
membrane potential, F(Vpre) = 1/(1 + exp((Vpre  syn)/2)), where
syn (set to 0 mV) is high enough so that the transmitter
release occurs only when the presynaptic cell emits a spike. The
channel opening rate = 12 msec 1 assures a fast rise
of the Isyn, and the channel closing rate is
the inverse of the decay time constant of the
Isyn; typically, we set = 0.1 msec1 (syn = 10 msec). An example of
Isyn and IPSP elicited by a single presynaptic
spike is illustrated in Figure 1C.
Random network connectivity. The network model consists of
N cells. The coupling between neurons is randomly assigned,
with a fixed average number of synaptic inputs per neuron,
Msyn. The probability that a pair of neurons are
connected in either direction is p = Msyn/N. For comparison, we also used fully
coupled (all-to-all) connectivity (Msyn = N). In the model, the maximal synaptic conductance
gsyn is divided by Msyn,
so that when the number of synapses Msyn is
varied, the total synaptic drive per cell in average remains the
same.
Msyn is the convergence/divergence factor of the
neural coupling in the network. Experimentally, an estimate of this
important quantity has been obtained for an interneuronal network of
the CA1 hippocampus (Sik et al., 1995). A parvalbumin-positive
(PV+) basket interneuron was stained intracellularly by
biocytin in vivo. Its axonal arborization was largely
confined in the striatum pyramidale (Fig.
2A). Other PV+
interneurons were stained immunochemically, and the contacts made by
the biocytin-filled cell with other PV+ cells were counted
(Sik et al., 1995). It was concluded that a single PV+
basket cell makes synaptic contacts with at least 60 other
PV+ cells (a majority of which are basket cells) within a
spatial region of the volume up to 0.1-0.2 mm3 (Sik et
al., 1995) (Fig. 2B). This tissue volume
contains as many as 500-1000 other PV+ cells, because the
PV+ neurons in the pyramidal layer have a cell density of
5.4 × 103 cells/mm3 (Aika et al., 1994).
Hence, for the CA1 network of basket cells, the experimentally
estimated Msyn is ~60. The probability of
postsynaptic contacts, however, decreases with the distance between the
cell pair (Fig. 2C).
Fig. 2.
In vivo double staining of parvalbumin-positive
interneuron in the hippocampus. A, The axonal arbor of an
intracellularly labeled basket cell, largely confined in the pyramidal
layer, is overlayed with immunochemically stained other
parvalbumin-positive cells. The two-dimensional distribution of the
interneuron-interneuron contacts is shown in B and then
collapsed to a one-dimensional distribution (along the septo-temporal
axis) in C. Overall, 99 boutons in contact with 64 parvalbumin-positive cells were counted (adapted from Sik et al.,
1995).
[View Larger Version of this Image (34K GIF file)]
Heterogeneous input. In the model, single neurons are not
identical. Each receives a depolarizing current
Iapp of different intensity and, hence, has a
different firing rate. The bias current Iapp has
a Gaussian distribution with a mean Iµ and a
standard deviation I. The parameter
Iµ determines the mean excitation by the
external drive; I measures the degree of the
heterogeneity in the cell population.
A measure of network coherence. To quantify the
synchronization of neuronal firings in the network, we introduce a
coherence measure based on the normalized cross-correlations
of neuronal pairs in the network (Gerstein and Kiang, 1960; Welsh et
al., 1995). The coherence between two neurons i and
j is measured by their cross-correlation of spike trains at
zero time lag within a time bin of  t = . More
specifically, suppose that a long time interval T is divided
into small bins of and that two spike trains are given by
X(l) = 0 or 1, Y(l) = 0 or 1, l = 1, 2, ... , K (T/K = ). Thus, we define a coherence
measure for the pair as:
|
(2)
|
We have also used a slightly different definition where the mean
firing rates are substracted from X(l) and
Y(l); the substraction did not significantly change
our results reported below.
The population coherence measure  () is defined by the
average of ij() over many pairs of neurons
in the network. This coherence measure presents a number of useful
properties. First, it is naturally based on cross-correlations and,
although we use it here to describe synchronization of oscillations, it
can be applied to quantify the synchrony of nonoscillatory neuronal
firings. It is calculated from spike trains, requires relatively small
sample sizes, and can be used for data analysis of experimental
extracellular multiunit recordings. Second, () is between 0 and 1 for all . For very small , () is close to 1 (resp. 0) in
the case of maximal synchrony (resp. asynchrony). Third, the degree of
synchrony of the network dynamics can be quantified by how ()
behaves as function of . In the case of full synchrony,
() is 1 for all nonzero values; whereas in the case of total
asynchrony, () is a linearly increasing function of (see
below). Finally, the distribution of
ij() over neural pairs provides detailed
information about the interneuronal interactions and
synchronization.
In Results, the population coherence measure () is
calculated by averaging over all neural pairs in the network of size
N. Typically N = 100.
Numerical methods. The network model was integrated using
the fourth-order Runge-Kutta method, with a time step of 0.05 msec.
For random network connectivity and heterogeneity, each set of
simulations was run with three to five random realizations of the
network connections and applied current distribution. As initial
conditions, the membrane potential is uniformly distributed between
70 and 50 mV and the other channel-gating variables are set at
their corresponding steady-state values. Coherence was calculated after
1000 msec transients. Simulations were performed on a SUN Sparc Station
or a Y-MP Cray Supercomputer.
RESULTS
Spike afterhyperpolarization, inhibition, and synchronization
We start by considering a simple case in which all individual
cells are identical (i.e., without heterogeneity) and are coupled in an
all-to-all fashion (i.e., without randomness in connectivity). As shown
in Figure 3A (left), cells
starting at random and asynchronous initial conditions quickly become
synchronized and within 5-6 oscillatory cycles their spiking times are
perfectly in-phase. The spike AHP amplitude is 15 mV measured from the
spike threshold (52 mV), so the maximal AHP,
VAHP = 67 mV, stays above the reversal
potential of the synaptic current (Esyn = 75
mV). The inequality VAHP > Esyn means that during the time course of an action
potential and its repolarization, the synaptic action is always
hyperpolarizing. This relationship between intrinsic and synaptic
properties was found to be an important condition under which the
global network synchronization can be achieved (Fig.
3B,C). In the example (Fig. 3B),
the speed of the INa inactivation and the
IK activation is slowed down ( = 3.33 instead
of 5), so that repolarization becomes larger
(VAHP 73 mV, close to
Esyn). In this case, the network synchronization
takes much longer time to realize. With = 2 (Fig.
3C), VAHP is 78 mV, which
is below Esyn, and global network synchrony is
lost. Instead, the network is dynamically broken into two clusters:
within each cluster the cells fire spikes simultaneously, and the two
clusters alternate in time. Such clustering dynamics is a commonly
observed behavior of interneuronal networks (Golomb et al., 1994).
Hence, synaptic inhibition of the GABAA-type provides a
mechanism by which a macroscopic coherence of the network (global
synchrony or clustering) can be realized.
Fig. 3.
Synchronization by GABAA synapses. In
these simulations, neurons are identical and coupled in an all-to-all
fashion. Left panels, Rastergrams; right panels,
membrane potentials of two cells (dotted line, 52 mV). The
synchrony is realized when the spike AHP of the cells does not fall
below the synaptic reversal potential Esyn = 75 mV (dot-dashed line on the right panels).
From A to C, = 5, 3.33, and 2 respectively;
Iapp = 1, 1.2, and 1.4 µA/cm2
accordingly to preserve a similar oscillation frequency. With smaller
values, IK is slower and the AHP amplitude
(VAHP) is more negative. When
VAHP < Esyn, the full
synchrony is lost (C).
[View Larger Version of this Image (56K GIF file)]
To investigate further the dependence of the network synchronization on
the inhibitory nature of synaptic interactions, the intrinsic cell
properties were kept unchanged, while the reversal potential
Esyn was gradually varied (Fig.
4). In Figure 4A the global coherence
index (compare Eq. 2.5), plotted versus
Esyn, remains at 1 (perfect synchrony) for
Esyn < VAHP. It displays
an abrupt drop for Esyn > VAHP and is close to 0 for Esyn > 60 mV. The oscillation frequency dramatically increases as the
synaptic effect becomes depolarizing (not shown). Unlike the two
cluster dynamics of Figure 3C with the coherence index = 0.5, in the regime characterized by ~ 0 the relative timing of
neuronal firings is essentially arbitrary. This happens in our network
model when the synaptic interactions are excitatory. In the
example given in Figure 4, B and C,
Esyn = 0 mV and syn = 2 msec, so
that the synaptic current mimic that of the glutamate AMPA type.
Because recurrent excitation considerably enhances the neural discharge
rates, weaker external drive (Iµ = 0.1 instead
of 1) was used so as to obtain a similar (~40 Hz) firing frequency
with excitatory rather than inhibitory interactions. In this case,
although all neurons have very similar rhythmic frequencies, their
relative firing phases are almost uniformly distributed between zero
and the common oscillation period T (Fig.
4B). Hence, the probability of coincident
firing within a time bin between two cells increases proportionally
with , and the network coherence function () grows linearly
from zero at = 0 to its maximal value of 1 at = T
msec (Fig. 4C).
Fig. 4.
Dependence of the network synchrony on the
synaptic reversal potential Esyn. A,
The coherence index ( = 1 msec) is plotted versus
Esyn. As Esyn is varied,
VAHP remains essentially the same
(vertical dashed line). There is a sudden transition from
synchrony to asynchrony as Esyn is increased
above VAHP. B, An example of
asynchronous behavior when cells are coupled by excitatory synapses
(Esyn = 0 mV, syn = 2 msec;
Iapp = 0.1). The oscillation frequency is
f = 43 Hz. C, The network coherence function
() increases linearly with , from 0 (at = 0) to 1 (at = T), showing that the relative firing time of neural
pairs is almost uniformly distributed between 0 and the oscillation
period T = 1/f.
[View Larger Version of this Image (22K GIF file)]
Heterogeneity and asynchrony
It is intuitively expected that network synchrony cannot be
globally maintained if individual neurons display very different
intrinsic oscillation frequencies. We studied the effects of
heterogeneity, using a Gaussian distribution of the applied current
intensity Iapp with standard deviation
I. As illustrated in Figure
5A, the network coherence deteriorates quite
rapidly with increasing I. This sensitivity
is related to the large frequency-current slope of fast-spiking
interneurons (Fig. 1A,B), so that in a
population of uncoupled cells a small current dispersion may imply a
wide distribution of firing frequencies. When neurons are synaptically
coupled, the distribution of firing frequencies is a product of their
interactions. As shown in Figure 5B, for small dispersion in
Iapp (I < 0.02),
minor differences in intrinsic firing rates are overcome by the
coupling, and the standard deviation of firing rates
f = 0. As the network coherence erodes with
larger I values, f
grows linearly with I (Fig.
5B). This linear frequency-current relationship
in standard deviation is a network property of coupled
cells. By contrast, the mean firing rate fµ
changes only slightly, illustrating the relative independence of the
neural firing rates and synchronicity. The moderate decrease in
fµ, however, is related to the
decreased degree of network coherence, because asynchronized cell
firings result in an averaged tonic hyperpolarization that slows down
the firing rate (see below).
Fig. 5.
Effects of the network heterogeneity.
A, The coherence index ( = 1 msec) versus
I (the standard deviation of the applied
current distribution). The network becomes asynchronous for
I  0.05. Examples indicated by
arrows (I = 0.03 and 0.1) are
illustrated in C-E and F-H, respectively.
B, The mean (fµ) and
standard deviation (f) of the firing
rates averaged over individual neurons are plotted versus
I. Note a decrease of
fµ and a linear increase of
f for I 0.05. The sensitivity to I is related to the steep
frequency-current relationship of single cells. C-E, A
partially synchronous state. C, The rastergram.
D, The coherence function () increases with rapidly, displays a plateau, and reaches the value of 1 near = 1/fµ. E, The derivative of ()
shows a sharp peak at = 0. F-H, An asynchronous state,
as seen by the rastergram (F). () is
linear with and reaches 1 near = 1/fµ
(G), and its derivative is flat
(H).
[View Larger Version of this Image (27K GIF file)]
Figure 5, C and D, illustrates a partially
coherent state. In the rastergram (Fig. 5C), neurons
are labeled in the increasing order of their Iapp
values. The cells with the smallest injected currents are not in
synchrony with those firing at higher rates. The population coherence
measure () increases sharply with and reaches the value of 1 at 1/fµ (Fig. 5D).
Considered as a function of , () may be viewed as a
distribution function of the neural pairs whose relative firing phase
is , between 0 and the mean oscillation period (estimated as
Tµ = 1/fµ). The derivative
d/d, therefore, is the corresponding distribution
density. Network synchronization is manifested by a peak at zero phase
of d/d (Fig. 5E), reflecting the
sharp increase of near = 0. Note that a second peak is expected
near = Tµ (Fig. 5E)
because the spiking event is periodic.
The network dynamics with I = 0.1 is
illustrated in Fig. 5,F- H. By contrast to Figure
5C-E, here the rastergram is quite irregular (Fig.
5F). The linear form of the function () (Fig.
5G) and its flat derivative (Fig.
5H) are consistent with a totally desynchronized
behavior, where the relative firing phase of neural pairs is
uniformly distributed between 0 and
Tµ.
Without the aid of the coherence function (), it would be
difficult to conclude from the rastergram (F) that the
network is completely disordered. Indeed, if one looks at the summated
synaptic drive, s(t) = (1/N)  Ni=1
si(t), oscillatory fluctuations are significant in
this ``population field'' (Fig. 6B,
top). This is because, if every cell oscillates, the
summation of a relatively small number (e.g., N = 100)
of oscillatory signals would still show some oscillations, even if
cells are completely asynchronous. To assess the macroscopic coherence
of the network, one can compute the temporal variance
 2 of s(t) for different network sizes
and assess whether the fluctuations in s(t) persist in large
networks. The network is asynchronous if 2 of
s(t) decreases inversely with the network size,
2(N) ~ 1/N (Hansel and Sompolinsky,
1996). This is shown to be the case in Figure 6A. As
N increases, s(t) becomes flatter (Fig.
6B) and the peak in the power spectrum gets
smaller (Fig. 6C). This example shows that global
network synchrony cannot be assessed correctly by oscillatory
fluctuations in the population field by the presence of a peak in the
power spectrum if the network size is not sufficiently large.
Fig. 6.
Synaptic field in a large asynchronous network. To
demonstrate further the asynchronous nature of the network behavior of
Figure 5F-H, the temporal variance 2
(N) of the population synaptic field s(t)
was calculated for different network sizes (N = 100, 200, ... , 1000). As expected for asynchronoized network states
(see text), 2 (N) decreases as ~1/N
(A). Three examples of s(t) are shown in
B, and their power spectra in C (arrow
indicating increasing N). Thus, the fluctuations of
s(t) vanish for large network sizes.
[View Larger Version of this Image (24K GIF file)]
Sparse network and minimal connectedness
Random connections among interneurons can be introduced into the
model by assuming that a cell makes synaptic contact to a second cell
with a probability p. Then, if N is the total
number of cells, there are Msyn = pN
presynaptic cells that converge to a postsynaptic cell, on
average. As shown above, the network can be synchronized with
all-to-all connectivity (Msyn = N, p = 1). Because synchronization is impossible without synaptic
connections (Msyn = 0), we examined the
dependence of the population synchrony on Msyn.
To evaluate the effect of sparse connectivity separately,
Msyn was gradually varied for a network of 100 identical neurons (i.e., without heterogeneity). As shown in Figure
7A, the population coherence (as measured by
) is essentially zero for Msyn below a
critical value 40; above which it starts to become significant,
increases rapidly with Msyn, and reaches the
maximum of 1 in the all-to-all limit (Msyn = N). Thus, the dependence of the network synchrony on
Msyn is highly nonlinear. There is a minimal
value of Msyn and neural interconnections have
to be sufficiently dense to generate global population synchronization.
This critical Msyn value is not simply a
required minimum for the total synaptic drive per cell, because it does
not change noticeably when the maximal synaptic conductance is reduced
by a factor of 2 (Fig. 7A). Also, in the presence of
heterogeneity (e.g., with I = 0.03), the
critical Msyn value remains the same, but the
quantitative degree of network coherence for
Msyn larger than the critical value was reduced
(data not shown). On the other hand, this minimal connectedness depends
on the probability rules in the network design and on the network
dynamical state under consideration. For instance, it is much smaller
if the number of synapses per cell is exactly the same number
Msyn, but the actual set of presynaptic cells is
chosen randomly (Fig. 7A). Or, when the oscillation
frequency is increased from ~35 Hz to ~100 Hz with
Iµ = 3 instead of 1, the network is
synchronized only with Msyn > 75 (Fig.
7A).
Fig. 7.
Minimal random connectivity is required for
large-scale network synchronization. A, The coherence index
( = 1 msec) is plotted versus the mean number of synaptic inputs
per cell Msyn (N = 100).
Filled circle, Iµ = 1 and
gsyn = 0.1 (reference parameter set). In this
case, the network synchrony is realized when
Msyn is larger than a critical value
Mcrit 40. This curve remains essentially the
same when the maximal synaptic conductance is reduced by 1/2
(open circle). By comparison, if the number of inputs per
cell is identical to all cells, and equals Msyn,
the synchrony occurs with very small values of
Msyn (5; filled square). With
Iµ = 3 (solid triangle), the mean
oscillation frequency is increased from 35 to 100 Hz; the critical
value of Msyn for the network coherence is much
larger (70). B, The coherence index ( = 1 msec)
versus Msyn for different numbers of neurons
N = 100, 200, 500, 1000 (with reference parameter set).
The onset of network coherence occurs at a critical value of
Msyn, which does not grow as a fraction of the
network size.
[View Larger Version of this Image (17K GIF file)]
We next examined whether the required minimal connectedness increases
in proportion with the network size. Simulations were performed with
N = 100, 200, 500, and 1000, and it was found that the
onset of network coherence corresponded to a small value of
Msyn, close to Msyn = 60 for large network sizes (Fig. 7B). Therefore, the
minimal number of synapses per cell that is required for the network
synchronization is not a fraction of the total number
N of cells. It either remains finite for large N
or it could conceivably depend weakly on N.
Partial synchronization in a sparse and heterogeneous network
As stated above, the minimal connectedness for a network coherence
remains the same when a moderate amount of heterogeneity is added.
Figure 8 illustrates a partially synchronized network,
with both sparse connections (Msyn = 60) and a
dispersion in the external drive (I = 0.03).
In that case, a major fraction of neurons (group I) displays similar
firing rates (close to 39 Hz), although their intrinsic firing rates
are different. The remaining neurons (group II) have lower firing rates
(below 34 Hz) that are scattered in the diagram. The membrane potential
trace of a representive cell is shown in Figure 8B,
together with its synaptic drive ssyn (sum of
the synaptic gating variables to that cell) and the population synaptic
field, both displaying fairly regular oscillations at the same
frequency. Cross-correlations of membrane potentials between the
representative cell (a in Fig. 8C) and
three other cells (b, c, and d in Fig.
8C) show that some pairs (ac and
ad) are synchronized with near-zero phase shift, but a
significant phase difference may be present for other pairs (e.g.,
ab; Fig. 8D).
Fig. 8.
A partially synchronous state with sparse
connectivity (Msyn = 60) and heterogeneity
(I = 0.03). A, The firing rates of
neurons in the network (filled circle) are lower than
those when the neurons are uncoupled (open circle). The bias
current varies from 0.91 to 1.09, the firing rate from 55 to 63 Hz when
gsyn = 0. With gsyn = 0.1, a large fraction of neurons have firing rates close to 39 Hz, the
remaining cells have lower firing rates. B, Time traces of
the population synaptic field s(t), the membrane potential
V(t) of one cell and its summated synaptic drive
ssyn(t). C, The
rastergram, with the cell shown in B indicated by
a. D, The cross-correlations between this cell
with three other cells b, c, d indicated in C.
E, The coherence index ij ( = 1 msec) for each of all the pairs in the network, plotted versus the
difference in the firing rates |fi fj| of the pair. Top, Pairs with
oscillation frequencies above 34 Hz; bottom, remaining
pairs. Only pairs in the top panel show a high degree of
zero-phase synchrony. F, The histograms of
ij in three groups of pairs: those not
monosynaptically coupled (top), those coupled in one
direction (middle), and those coupled in both directions
(bottom). The population averaged () function shows a
steep rise for small values (G), and its derivative has
a peak at = 0 (H).
[View Larger Version of this Image (25K GIF file)]
The normalized cross-correlation of spike trains at zero phase lag,
defined as our coherence index ij for the
pair (i, j), was calculated for all neural pairs in the
network and plotted against the difference in the firing rates of the
pair |fi fj| (Fig.
8E). Those pairs with both firing rates above
34 Hz (group I) are plotted in the top panel, and the other pairs are
plotted in the bottom panel. Comparison of the two panels reveals that
high zero-phase synchronization (large ij)
occurs only in pairs of group I neurons, and with similar firing
frequencies (small |fi fj|). For those pairs in the bottom panel,
ij is small even for pairs with almost
identical firing frequencies. Therefore, the network is subdivided into
two groups of neurons, and only group I neurons are synchronized with
near zero-phase shift. It is not immediately clear why all neurons in
the asynchronous group have lower, but not higher, firing frequencies
than the synchronous group.
On the other hand, the neural pairs can be classified into
three categories, according to whether they are monosynaptically
uncoupled, coupled in one direction, or coupled in both directions.
Histograms of ij, however, do not show
conspicious difference between such categories (Fig.
8F), indicating that the degree of
synchronization between a pair is not primarily determined
by their direct monosynaptic contacts. We conclude, therefore, that
interneuronal coherence emerges as a network phenomenon. The global
character of network synchrony is quantified by , the average of
ij over all pairs. The population coherence
function () increases rapidly with the time bin and reaches
the value of 1 for Tµ, where again
Tµ is the average oscillation period (Fig.
8G). Its derivative shows a clear peak at = 0 (Fig. 8H).
The partially synchronous dynamics cannot be maintained if the
interneuronal connections are too sparse (Fig. 7). Indeed, when
Msyn is decreased from 60 to 30, with all of the
other parameters remaining the same, the network becomes asynchronous
(Fig. 9). In this case, neurons are not locked to a same
firing frequency (Fig. 9A), the rastergram looks
irregular (Fig. 9C), the population synaptic field
is almost constant in time (although the synaptic drive to a single
cell still shows residual oscillatory fluctuations) (Fig.
9B), and the cross-correlations between cell pairs
are flat (Fig. 9D). Corroboratively, the zero-phase
synchronization is very low for all neural pairs (Fig.
9E,F), in contrast with Figure 8, E and
F. The global coherence function () is linear in (Fig. 9G), and its derivative is flat (Fig.
9H), as expected for a fully asynchronous neural
network.
Fig. 9.
Desynchronization with reduced network
connectivity. Same as Figure 8, except that Msyn = 30 instead of 60. The network dynamics is asynchronous, as seen by
the scattered distribution of firing rates (A), the
disordered rastergram (C), the small fluctuations of
the synaptic field (B), and flat cross-correlations
(D). This asynchronous state is further
characterized by small values of ij (E,
F), the linear function of the network averaged ()
(G), and its flat derivative
(H).
[View Larger Version of this Image (23K GIF file)]
Dependence on the synaptic time constant
It was shown that the 40 Hz oscillations in hippocampal
interneurons are sensitive to the decay time constant
syn of the GABAA synapse (Whittington et
al., 1995; Traub et al., 1996). We have confirmed their result that an
increased syn induces a decrease in the oscillation
frequency (Fig. 10A,
left). This frequency reduction occurs largely
because the slowly decaying synaptic inhibition accumulates in time,
resulting in a tonic level of hyperpolarization that
counteracts the external depolarization. This is shown in Figure
10B, in which the synaptic drive to a representative
cell is displayed for three different values of syn. The
time average is indicated by a horizontal line, which is higher (0.5, 0.64, 0.7) for larger syn values (10, 20, and 40 msec,
respectively).
Fig. 10.
Synaptic time constant can modulate the
oscillation frequency and population synchrony. A, Slowing
down the synaptic current decay (with increasing syn)
decreases the mean oscillation frequency fµ
(left), whereas the network coherence shows a peaked region
centered at syn 7 msec (solid circle,
right). Here, to take into account the change in firing rates, the
coherence index was calculated with = 0.1/fµ. For large syn values,
the network coherence decreases because of the heterogeneities in the
connectivity and external drive, as can be seen by isolating each of
the two effects (dotted and dashed lines,
respectively). In the absence of both, the coherence is maximal ( = 1) for syn > 4 msec (dash-dotted line).
B, Decreased firing frequency is caused partly by the fact
that with larger syn, the summated synaptic drive has a
greater time average (horizontal lines) and shows less
oscillatory fluctuations, thus providing an enhanced level of
tonic hyperpolarization, which counteracts the depolarizing
drive of the cells. C, Rastergram of an asynchronous network
with syn = 2 msec (top). The globally coupled
network of identical cells shows a two-cluster dynamics; thus, it is
only partially synchronous (bottom).
[View Larger Version of this Image (35K GIF file)]
We also considered how the network synchronization depends on
syn. Within a fixed time window, a pair of cells has a
higher chance to fire simultaneously if their firing frequencies are
higher. Hence, to make a meaningful comparison, the coherence measure
should be calculated with different time bin when
syn is varied. In Figure 10A
(right), we chose to use equal to one-tenth the
mean oscillation period. It was found that the coherence index displays
a peaked region around syn = 7 msec; the network
synchronization is lost for both smaller and larger syn
values.
The decrease in the coherence index with larger syn
is a consequence of network heterogeneities. Indeed, as shown in Figure
10A, if the dispersion in applied current is absent
(I = 0), or if the connectivity is not random
(Msyn = 100), is larger but still decreases
with syn. But if both I = 0 and Msyn = 100, (hence, the network
synchrony) is now maximal for all larger syn values.
Note that the heterogeneity in either the applied current or the
connectivity produces stronger effects for larger syn
values, because in the lower frequency range the network is more
sensitive to input heterogeneities (see Fig.
1B).
On the other hand, the decrease in the network synchrony for small
syn is presumably attributable to a synaptic (dynamical)
effect: inhibition should not be too fast compared to the oscillation
period, so as to synchronize the network (Wang and Rinzel 1992, 1993;
van Vreeswijk et al., 1995; Ermentrout, 1996). This is illustrated in
Figure 10C with syn = 2 msec, where the
rastergram is quite irregular (top). This is probably
related to the fact that with global connectivity
(Msyn = 100) and without heterogeneity
(I = 0), the network displays two-clusters
(bottom, Fig. 10D), similar to
Figure 3C. In that case, the global network synchrony
(one-cluster) was also observed with different initial conditions, but
it was very sensitive to the network heterogeneity (data not
shown).
It follows from the left and right panels of Figure
10A that a peaked region for versus
syn implies that the network coherence is high only
within a limited range of the mean oscillation frequencies. This
interesting observation was confirmed with different levels of the
external drive Iµ (Fig.
11A). When
Iµ is larger, the oscillation frequencies are
higher (left). Moreover, the maximum of is
located at a smaller value of syn
(right). The network synchrony does not depend
simply on syn, but on the ratio between
syn and the mean oscillation period
Tµ, which is an increasing function of
syn (Fig. 11B). When the
coherence index is plotted versus
syn/Tµ, it is small for small
syn/Tµ ratios and
peaks at syn/Tµ 0.2 for all three external excitation levels
(Iapp = 1, 2, 3; Fig. 11C).
Fig. 11.
Dependence of the network coherence on the
synaptic time constant syn. A, The mean
oscillation frequency as function of syn is shown with
three levels of the network drive Iµ
(left). The coherence index displays a peak which is shifted
to smaller syn value with larger
Iµ (right). B, The ratio
between the synaptic time constant syn and the
oscillation period Tµ increases with
syn. C, The coherence index versus the ratio
syn/Tµ. For all three
Iµ values, the coherence index peaks at the
same syn/Tµ (0.2) and
decreases at smaller ratio values (Msyn = 60, I = 0.03; the coherence index was
calculated with = 0.1/fµ).
[View Larger Version of this Image (28K GIF file)]
Optimal synchronization in the gamma frequency range
That network synchronization is highest in a limited frequency
range appears to be a robust finding in our simulations. For instance,
the phenomenon was also observed when the mean input current
(Iµ) was varied continuously (Fig.
12A). With stronger
external drive, the mean oscillation frequency increases monotonically
(Fig. 12A, top), but the network coherence
displays high values only for an intermediate
Iµ range (Fig. 12A,
bottom). In an all-to-all network of identical neurons, the
network coherence was found to remain maximal ( = 1) for the entire
Iµ range (data not shown). Furthermore, at
small Iµ, heterogeneity in an all-to-all
network reduces the synchrony dramatically (with
Iµ = 0.4, = 1, and 0.1 for
I = 0 and 0.03, respectively). Thus, the
decrease of the network synchrony at smaller
Iµ values is attributable to the high network
sensitivity to heterogeneities at lower frequencies (see Fig.
1B) in addition to a decreased
syn/Tµ ratio
(dynamical effect). On the other hand, at larger
Iµ (higher frequencies), the network coherence
requires tighter connectedness and our fixed
Msyn = 60 may no longer be sufficient (see Fig.
7A).
Fig. 12.
High network coherence in the gamma oscillation
frequency range. A, With increasing mean drive
Iµ, the average oscillation frequency
monotonically increases (top), but the coherence index is large only for intermediate Iµ values
(bottom). B, Similarly, as the maximal synaptic
conductance gsyn is increased, with stronger
inhibitory interactions the average oscillation frequency
fµ monotonically decreases (top).
An increase in the external drive shifts the curve upwards. On the
other hand, the coherence index displays a pronounced peak for
Iµ = 1, which flattens for larger
Iµ values (bottom). C,
The coherence index is plotted as function of
fµ for all four curves from (A,
B). The macroscopic network coherence is observed only in
the gamma range of the oscillation frequencies (20-80 Hz)
(Msyn = 60, I = 0.03;
the coherence index was calculated with = 0.1/fµ).
[View Larger Version of this Image (19K GIF file)]
We also varied the coupling strength gsyn
systematically, with three different Iµ
values. With stronger synaptic inhibition, the oscillation frequency
decreases monotonically (Fig. 12B, top).
The network coherence, on the other hand, shows a peaked region at
intermediate gsyn values (Fig.
12B, bottom). Again, without input
heterogeneity and coupling sparseness, the network coherence is maximal
( = 1) for the entire gsyn range (data not
shown). Therefore, similar to the case of Iµ
variation, the decrease of is presumably caused by a reduced
stability of the network synchrony against input heterogeneity at low
frequencies (large gsyn) and by a lack of
sufficiently tight connectivity (combined with a weakened coupling
strength) at high frequencies (small gsyn).
When the network coherence index is plotted against the mean frequency,
for all four curves in Figure 12, A and B, it is
clear that the network synchronization is realized only in a relatively
narrow range of oscillation frequencies (30-80 Hz; Fig.
12C). As shown by the three curves from Figure
12B, this frequency range of high network synchrony
can be shifted by the level of external drive. With
Iµ = 2 or 3 (instead of 1), the network is
more excited and the coherence peak is located at higher frequencies,
as expected, and the peak is somewhat enlarged. However, the amplitude
of the peak is dramatically reduced (compare Iµ = 1 and 3). This can be explained, again, by the fact that at higher
frequencies, the network synchronization requires denser connections
(see Fig. 7A), whereas here
Msyn = 60 is kept constant. Figure
12C demonstrates that, although the optimal frequency range
for synchronization is not precisely defined and does depend
quantitatively on network parameters and external drive, the high
network coherence is robustly limited to a frequency band that
coincides roughly with the gamma (20-80 Hz) frequency range.
DISCUSSION
We examined the emergence of synchronous gamma oscillations in an
interneuronal network model. The following conditions were identified
for the synchronizing mechanism by GABAA synaptic
inhibition. (1) The spike afterhyperpolarization of single neurons
should be above the synaptic reversal potential, so that the effect of
synaptic inputs is always hyperpolarizing. (2) The synaptic current
decay should be relatively slow, such that the ratio between the decay
time constant and the oscillation period is not small. (3)
Heterogeneities should be sufficiently small. (4) The random network
connectedness should be higher than a well defined minimum, which is
not sensitive to the network size. When the four conditions are
fulfilled, a large-scale network coherence was observed preferentially
in the gamma (20-80 Hz) frequency range, although uncoupled neurons
are potentially capable of discharging in a wide range of frequencies
(0-400 Hz).
Synchronization by synaptic inhibition
Rhythmogenesis in many biological central pattern generators is
based on circuits of coupled inhibitory neurons exhibiting
postinhibitory rebound excitation (Selverston and Moulins, 1985). In
such systems, however, rhythmic patterns are usually slower than the
kinetic time constants of the inhibitory synapses, and reciprocally
connected neurons typically fire out-of-phase (Perkel and Mulloney,
1974). More recently, it was recognized that neural oscillations can be
synchronized by mutual inhibition at zero phase shift, provided that
the synaptic kinetics is sufficiently slow as compared to the
oscillation period (Wang and Rinzel, 1992, 1993). Intuitively, slow
synaptic decay offers the possibility for neurons to ``escape''
synchronously as the synaptic inhibition wanes below a certain
threshold, thus fire at the same time. A general conclusion from this
scenario is that synapses with given temporal characteristics may be
suitable to synchronizing large neural populations in a particular
oscillatory frequency range; a major determining factor is the ratio
between the synaptic time constant and the oscillation period. Other
computational works have since shown that the mechanism of
synchronization by inhibition may be quite general (van Vreeswijk et
al., 1995; Whittington et al., 1995; Ermentrout, 1996; Traub et al.,
1996). Synchronization by GABAA synapses is facilitated if
the synaptic reversal potential is more negative than the maximum spike
afterhyperpolarization. This, however, is necessary only for the
perfect global, but not for partial, synchronization. Although
fast-spiking interneurons typically display larger AHP amplitudes than
pyramidal cells, their measured maximal AHP is usually not below 70
mV (McCormick et al., 1985; Lacaille and Williams, 1990; Morin et al.,
1995; Sik et al., 1995), hence, above the reversal potential of 75 mV
for GABAA synapses (Buhl et al., 1995). On the other hand,
the time constant syn of the synaptic current determines
the range of the oscillation frequencies where the synchronization can
be realized by mutual inhibition. For gamma oscillations (~40 Hz),
the syn should be larger than 3 msec, compatible with
the estimated syn (Otis and Mody, 1992). Therefore,
these requirements seem to be fulfilled by cortical fast-spiking
interneurons, especially the basket cells, interconnected by
GABAA synapses. On the other hand, the synchronization
mechanism does not require interneurons to be endowed with a
postinhibitory rebound property.
The network synchronization was quantified by a coherence index that
was defined in terms of the zero-time cross-correlations of spike
trains. This proved to be a useful and reliable measure of population
synchrony. By contrast, population-averaged quantities like the
``synaptic field'' may display significant oscillatory fluctuations
even when most neurons are in fact asynchronous, if the size
of the probed neural population is small. In that case, however, the
field fluctuations decrease with the network size and become almost
flat for large networks. This observation suggests caution in the
interpretation of oscillatory local field potential in experimental
measurements and in small-network simulations.
We studied the dependence of the network coherence on the heterogeneity
in interneuronal properties. Typically, with moderate heterogeneities,
coupled oscillators break down into a synchronous and an asynchronous
subpopulations. In our model, asynchronous neurons in such a partially
synchronous state all have lower firing frequencies than the
synchronous ones. The observed high sensitivity on the degree of
heterogeneity can be attributed partly to the large frequency-current
slope of these fast-spiking cells, so that a minor dispersion in the
external drives may result in a wide distribution of single cells'
oscillation frequencies. It would be of interest to investigate whether
the network synchronization may become more robust in the presence of
heterogeneities and noise, if the neural connections are structured in
space (Somers and Kopell, 1995), or when an excitatory pyramidal
population is included that reciprocally interacts with the
interneuronal population (Kopell and LeMasson, 1994; Wang et al.,
1995). Moreover, information processing in the cortex may involve a
small subset of pyramidal neurons at a time. These pyramidal cells, by
activating their common interneuron targets, may exert localized
effects on the synchronous oscillations in a selective neural
subpopulation of the cortical network.
Random connection and critical sparseness
One of our main objectives was to assess how dense synaptic
interconnections must be for the maintenance of the synchronized
network oscillations. The degree of network coherence depended on the
average number of synaptic contacts per cell,
Msyn, in a highly nonlinear fashion. A minimal
Msyn value was identified, below which the
network becomes totally asynchronous. For Msyn
above its critical value, the degree of network coherence becomes
nonzero and increases with Msyn. We demonstrated
that this critical connectedness does not increase in proportion with
the number of neurons, hence, is not sensitive to the network size.
However, further analysis is needed to determine whether there is a
very weak (e.g., logarithmic) dependence. Such a dependence is
expected, for instance, if the required connectedness is related to the
minimal number of links for a random network to be topologically
connected at large scales (Erdös and Rényi, 1960). The
existence of a small critical connectedness has also been found in
other neural network models (Barkai et al., 1990; Wang et al., 1995),
suggesting that this may be a general feature of sparsely connected
random neural networks.
Note that the actual value of this minimal connectedness depends on the
details of single-cell and network properties, as well as on the
cooperative dynamics under consideration. Indeed, we found that
synchronization of oscillations at higher frequencies requires tighter
interneuronal connections (Fig. 7A). Further
analyses are needed to provide a theoretical understanding of this
simulation result. For rhythmicities in the gamma frequency range, our
model network require 60 or more synaptic contacts from an interneuron
to other interneurons. This number is comparable with the estimated
divergence/convergence factor of CA1 basket cells in the hippocampus
(Sik et al., 1995).
We also demonstrated that synchronization of oscillatory neurons
usually cannot be attributed simply to the presence of monosynaptic
couplings between the respective cells. The macroscopic network
synchronization is thus a truly emergent phenomenon of large neuronal
aggregates.
Frequency selection for synchronization
An important finding of the present study is that interneuronal
networks can be synchronized by GABAA synapses
preferentially within the gamma frequency range. This happens despite
the wide range (0-400 Hz) of possible single-neuron firing rates. For
instance, when the synaptic time constant (syn),
external drive (Iµ), or coupling strength
(gsyn) is varied gradually, the neuronal firing
frequencies change monotonically and cover a wide frequency range. In
all three cases, however, the degree of network synchronization shows a
relatively narrow peak within the gamma frequency range (20-80
Hz).
This phenomenon of frequency selection may be qualitatively
understood in terms of the following three neural and network
properties: (1) the high network sensitivity to heterogeneities at low
frequencies, attributable to the steep frequency/current curve of
single neurons; (2) the minimal connectedness for the synchrony, which
is larger at higher oscillation frequencies; (3) the ``dynamical
effect,'' namely, network synchronization is impossible or fragile
against heterogeneities if the ratio between syn and the
oscillation period T (syn/T)
becomes too small. Hence, given syn 10 msec, if the
oscillation frequency is lower than 20 Hz (T > 50 msec), the network coherence may be abolished by a high sensitivity to
input heterogeneity (see Fig. 1B) and by a
reduced syn/T ratio. On the other
hand, if the frequency is higher than 80 Hz, the synchrony may no
longer be possible because the interneuronal connectivity is not
sufficiently tight (see Fig. 7A). Consequently, the
network can be highly synchronized only at 20-80 Hz.
We would like to emphasize that this frequency band for coherent
oscillations cannot be determined in an absolute precision, and its
quantitative limits do depend on the details of the model. However, our
results robustly demonstrated that the synaptic time constant delimits
a frequency band of coherent network oscillations, and the
GABAA-type synapse (with syn 10 msec)
seems especially suitable for the gamma rhythmicity. The limited
frequency range of population gamma oscillations has been observed in
the behaving rat (Bragin et al., 1995) and in hippocampal slices
(Whittington et al., 1995; Traub et al., 1996). Our findings suggest
that the synchronization mechanism by an interneuronal network would be
especially effective if the fast oscillations are generated in the
gamma frequency range by pacemaker neurons (Llinás et al., 1991;
McCormick et al., 1993). On the other hand, even in the absence of
pacemaker neurons, coherent field oscillations could still be observed
in the gamma frequency range, not because possible firing rates of
inhibitory interneurons are restricted to a narrow frequency band, but
because outside this frequency range the large-scale coherence is not
possible (by this particular synaptic mechanism).
Physiological implications
Recurrent excitatory connections have long been regarded as a
possible synaptic substrate underlying correlated neural firings in
general and massively synchronous brain rhythms in particular. Although
coherent slow (epileptic) rhythmic bursting may indeed emerge in a
disinhibited pyramidal cell network (Chagnac-Amitai and Connors, 1989;
Traub et al., 1993), modeling studies suggest that mutual excitation
via the AMPA-type synapse often cannot synchronize neural oscillations
in the gamma frequency range, at least for simple repetitive spiking
neurons (Hansel et al., 1995; van Vreeswijk, et al., 1995) (present
work). An alternative mechanism for the fast entrainment of principal
cells has emerged recently. Previous work (Bragin et al., 1995;
Whittington et al., 1995; Traub et al., 1996) and the present model
suggest that networks of interneurons are critically involved in the
generation of coherent gamma oscillations. The advantage of such
``synchronizer'' function of interneuronal networks is the
maintenance of subthreshold and coherent modulation of the large,
sparsely connected pyramidal cell populations and the resulting precise
timing of their action potentials (Buzsáki and Chrobak, 1995;
Hopfield, 1995).
FOOTNOTES
Received May 5, 1996; revised June 25, 1996; accepted July 31, 1996.
This work was supported by the National Institute of Mental Health
(MH53717-01), Office of Naval Research (N00014-95-1-0319), and the
Sloan Foundation to X.J.W.; and HFSP and the National Institute of
Neurological Disease and Stroke (NS34994) to G.B. and X.J.W.
Simulations were partly performed at the Pittsburgh Supercomputing
Center. We thank D. Golomb, D. Hansel, J.-C. Lacaille, and C. McBain
for discussions, A. Sik for preparing Figure 2, and L. Abbott, J. Lisman, and R. Traub for carefully reading this manuscript.
Correspondence should be addressed to Xiao-Jing Wang, Center for
Complex Systems, Brandeis University, Waltham, MA
02254.
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