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Communicated by Nancy
Kopell
Synchrony in Excitatory
Neural
Networks
D. Hansel
Centre de Physique Thiorique UPR0 14 C NR S,
Ecole Polytechnique, 91128 Palaiseau Cedex, France
G. Mato
Racah Institute of Physics and Center for Neural Com putation,
Hebrew University, 91 904 Jerusalem, lsrael
C. Meunier
Centre de Physique Th iorique UPRO14 C N R S,
Ecole Polytechnique, 91128 Palaiseau Cedex, France
Synchronization properties of fully connectec. netwoi,s of identical os-
cillatory neurons are studied, assuming purely excitatory interactions.
We analyze their dependence on the time course of the synaptic in-
teraction and on the response of the neurons to small depolarizations.
Two types of responses are distinguished. In the first type, neurons al-
ways respond to small depolarization by advancing the next spike. In
the second type, an excitatory postsynaptic potential
EPSP)
eceived
after the refractory period delays the firing of the next spike, while
an EPSP received at a later time advances the firing. For these
two
types of responses we derive general conditions under which excita-
tion destabilizes in-phase synchrony. We show that excitation is gen-
erally desynchronizing for neurons with a response of type I but can
be synchronizing for responses of type I1 when the synaptic interac-
tions are fast. These results are illustrated on three models of neurons:
the Lapicque integrate-and-fire model, the model of Connor et
al.,
and
the Hodgkin-Huxley model. The latter exhibits a type I1 response, at
variance with the first
t w o
models, that have type I responses. We then
examine the consequences of these results for large networks, focusing
on the states of partial coherence that emerge. Finally, we study the
Lapicque model and the model
of
Connor
e t al .
at large coupling and
show that excitation can be desynchronizing even beyond the weak
coupling regime.
1
Introduction
Synaptic interactions between neurons are usually classified as excita-
tory or inhibitory according to the value of the reversal potential of the
Neural Computation 7, 307-337
(1995) @ 1995
Massachusetts Institute
of
Technology
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D.
Hansel,
G.
Mato, and C. Meunier
synapses. However, as observed in Kopell (19881, there is no obvious
relationship between this classification and the dynamic behavior of a
network of interconnected neurons. If one focuses on synchronization
properties of neural systems, a more fundamental classification of the in-
teractions should be in terms of synchronizing interactions, that favor
a stable in-phase state (where all the neurons fire at the same time) and
desynchronizing interactions that tend to destabilize this state.
This paper examines the conditions under which excitatory interac-
tions synchronize a network of neurons that fire spikes periodically. In
particular, we will relate the synchronization properties
to
the response
of the neurons to perturbations
of
their membrane potential. For this
purpose we focus on a simple case: a homogeneous and fully connected
network of excitatory neurons. Moreover we do not take into account
interaction delays. Some of the results presented in this paper have been
reported in Hansel et a l . (1993~).
In many cases a small excitatory postsynaptic potential
(EPSP)
sys-
tematically advances the next spike of the neuron, except when it occurs
during the period of refractoriness where it has no effect. As shown
below, this form of response is found, for instance, in simple integrate-
and-fire models and in the model of Connor et
a l .
(1977). We call such a
response to EPSPs a response of type
I.
Using the phase reduction method
(Ermentrout and Kopell 1991; Kuramoto 1984; Neu 19791, a powerful
technique that has been applied recently to neural modeling (Ermentrout
and Kopell 1991; Grannan et
al.
1992; Hansel
et al.
1993a,c; Kopell 19881,
we show that in general two weakly coupled neurons with a response
of type I do not lock stably in-phase. We then illustrate this desynchro-
nizing effect of excitation on specific models of neurons and show that
it occurs for synapses with physiologically relevant time constants (for
non-NMDA synapses).
If the in-phase state of a pair of neurons is unstable, a network of
such neurons cannot synchronize fully. Partially coherent states then
emerge in the network. It is even possible that no coherence can be
achieved and that the asynchronous state turns out to be stable. We
give examples
of
such collective states of large networks, focusing on
the model of Connor et
al.,
which exhibits rotating waves (Kuramoto
1991;
Watanabe and Strogatz 1993) and switching states (Hansel
et
al.
1993b). Our study is based on numerical simulations, but it should be
noted that some properties of these states can be studied analytically in
the framework of phase reduction (Kuramoto 1984; Monnet
et
al . 1994;
Watanabe and Strogatz 1993).
Beyond the weak coupling limit, our general arguments on the desyn-
chronizing nature of excitation for neurons of type
I
no longer hold
and our investigation relies on the study
of
specific models: namely an
integrate-and-fire model and the model of Connor et
al.
For both models
we find that in an intermediate (but wide) range of coupling strength the
predictions of phase reduction remain qualitatively valid. However, for
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Synchrony
in
Excitatory
Neural
Networks
309
stronger coupling, the deviations from this limit become important. For
the integrate-and-fire model we show analytically that the desynchro-
nizing effect of the excitation is amplified at strong coupling, anti-phase
locking being achieved even at finite coupling. For the model of Connor
et aI. our simulations show that if the rise time of the interaction is large
enough the situation is very similar to what is found for the integrate-
and-fire model. On the other hand, for a short rise time, increasing the
coupling strength can make the excitation synchronizing.
Not all the neurons have a response of type I. Another form of re-
sponse is found, for instance, for the standard Hogdkin-Huxley (HH)
model (Hodgkin and Huxley 1952). There is a region of the limit cycle,
just after the refractory period, where a depolarization delays the firing
of the next spike (for reasons that will become clear later, we will say that
in this region the response is negative).
A
response of this kind will be
called type 11. We show that at weak coupling the region of negative re-
sponse tends to stabilize the in-phase state. The Hodgkin-Huxley model
provides an example in which this stabilizing effect
is
strong enough to
make fast excitatory interactions synchronizing. For slower interactions
excitation is once again desynchronizing.
The paper is organized as follows. In Section 2 we present the basic
types of models of neurons considered in this study. After recalling
the phase reduction method our general results at weak coupling are
established and illustrated on specific examples in Section3. In Section
4
the case
of
large coupling is addressed. Finally, the last section is devoted
to a discussion.
2 The Models
2.1 Conductance-Based Neurons. Conductance-based models
account for spiking by incorporating the dynamics of voltage-dependent
membrane currents (see for instance Tuckwell 1988). In this framework,
the dynamics of a neuron is described by the equation for the membrane
potential V:
d V
C - = I
d t ext
(2.1)
where
C
is the membrane capacitance, and g , and
V;
are, respectively, the
voltage-dependent conductance of the ith ionic current and its reversal
potential. The gating variables
of
the ith current have been denoted here
by
Xi
nd the model must also specify their relaxation dynamics. The
synaptic current
Isyn(t)
s modeled as
(2.2)
syn(f) = - (V
-
Vsyn)gsyn(t)
where Vsyn is the reversal potential of the synapse, and
(2.3)
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D.
Hansel,
G.
Mato, and
C.
Meunier
the summation being performed over all the spikes emitted by the presy-
naptic neurons at times tspike. The synaptic interaction is usually classified
according to whether
Vsyn
is larger or smaller than the threshold potential
Vth, at which the postsynaptic neuron generates spikes. ForVsyn
>
Vth the
interaction is called excitatory, while for
Vsyn
0)
as
no description of the spike is incorporated in the model and the driving
forceVsyn-V remains approximately constant in the subthreshold regime.
Note that the membrane capacitance
C
was assumed to equal 1 and
omitted from 2.7.
One can also introduce in this model a refractory period, if necessary,
by imposing that
V(t)
remains equal to
0
for a time T , after the firing
of a spike. If the neurons are not interacting (8 =
0)
they emit spikes
periodically with a period
TO
=
T ,
70
ln(1
8/lext),
for
Iext
>
8.
Without
loss of generality one can assume 70 = 1, measuring then the time in
units of
70.
3 The Case of Weak Interaction
3.1 Reduction
to
Phase Models. In general, the dynamic equations of
conductance-based neurons cannot be solved analytically and the study
of synchronization in networks of such neurons must rely on numerical
computations. However, if the neurons display a periodic behavior (limit
cycle), if their firing rates all lie in a narrow range, and if the coupling
is weak, a reduction to a phase model can be performed that greatly
simplifies the analysis.
Let us briefly recall the principle of such a reduction (Ermentrout and
Kopell 1991; Kopell 1988; Kuramoto 1984). It is based on an averaging
theorem that enables one to describe the state of each neuron i by a
phase variable i i = 1,. . N, where
N
is the number of nonlinear
cycle oscillators in the system) indicating the position of neuron i on
its limit cycle and to replace the original system of equations for the
N
oscillators by a simpler set of
N
differential equations that governs
the time evolution of the coupled phase variables. This differential
systems reads
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D.
Hansel, G . Mato, and C. Meunier
where wi is the natural frequency of neuron i, that is, its frequency at
zero coupling, while r gives the effective interaction between any two
neurons. I7 depends only on the relative phase on the two neurons. The
system is invariant with respect to a global rotation of all the phases,
that are thus defined up to an arbitrary constant. It is conventional to
choose the phases so that firing occurs for ,
= 0
mod 27r. Note that the
dependence on the relative phases stems from the assumption of weak
coupling. For phase models at arbitrary coupling the interaction between
two neurons depends on the values of both phases; the integrate-and-fire
model studied below provides an example of that situation.
The effective interaction between the phases is given by
(3.2)
This formula can be interpreted as follows. The effective interaction be-
tween the presynaptic neuron j and the postsynaptic neuron i is obtained
by convolving over one period the synaptic current
Isyn(+'l,
),due to the
EPSPs
(or IPSPs) generated by neuron
j,
and the "response function"
Z
of the target neuron i to these perturbations. The function Z is nothing
else than the phase resetting curve of the neuron in the limit of van-
ishingly small perturbations of the membrane potential. If
Z ( )
> 0 a
small and instantaneous depolarization at of the neuron will advance
the next spike; if
Z( )
< 0 the next spike will be delayed. To calculate
r one must implement numerically the rigorous method described in
Ermentrout and Kopell (1984) and Kopell (1988) or the more qualitative
algorithm explained in Hansel et al. (1993a). Note that the 27r-periodic
function r depends only on the single neuron dynamics. Once this effec-
tive phase interaction is determined it can be used to analyze networks
of arbitrary complexity. Note also that the introduction of a delay
A
in
the interaction is immediate in this formalism:
r(+)
s just replaced by
r(4
-
A).
The synaptic current in 3.2 is
I s y n ( 4 ,
)
= -gsyn( ) [V(4) - Vsynl
L y n ( 4 ,
) =
gsyn(l i , ) (3.4)
(3.3)
for an interaction described by equation 2.2 and
for an interaction described by a current independent of the postsynaptic
voltage as in the integrate-and-fire model of Section 2.2. In the two cases
the function
gsyn(+)
must take into account all the spikes emitted by the
presynaptic neuron and has to be computed at the leading order in
g.
It
has period 27r and is defined, for
0
5 u
0.
Therefore:
(3.12)
where ,
= 27rTr/T
is the length of the refractory region expressed in
terms of phase. Let us introduce $* = max($,, $+), where QP s the phase
at which gsyneaches its peak value. We have then
The first contribution to r(0) s negative and tends to stabilize the in-
phase state while the second is positive and tends to destabilize it. There-
fore the stability
of
the in-phase locked state will depend on the balance
between these two terms.
If ,
> $+,
the stabilizing term disappears. This provides a sufficient
condition for the in-phase state to be unstable. This situation will be
encountered, in particular, for interactions with a rise time short with
respect to
T,.
We can estimate in such cases how the unstability rate
depends on 7 1 and 7 2 (note that l lPvaries slowly when
TI
or 7 2 increases).
At fixed
72
the overlap between
Z
and
g
increases with
71
Therefore
r(0)
increases also and the in-phase state becomes more unstable. Similarly,
increasing 7 2 at fixed 71 enhances the instability of the in-phase state.
Another general statement, valid even if
$+
> $+, can be made if
Z
reaches its maximum just before the firing of the spike and then drops
abruptly to
0
(as occurs for the Lapicque model, see below). In that
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Z
0.06
0.04
0.02
0.00
0
V
27r
0
2.x:
V
Figure
1:
(a) The response function
Z
as a function of time along one cycle
of the model of Connor et al. The frequency of the neuron is approximately
57
Hz.
The origin of the time scale is set at the firing of the spike. (b) The
two functions
-Z( )[V( )
- Vsyn] (solid line) and
gsyn($)
dashed line) for the
model of Connor et
a l .
Same frequency as in (a). The scales for both curves are
arbitrary. The interaction is excitatory (Vsyn
=
0). The rise time is
72
= 1 msec
and the decay time is
TI
= 3 msec. The convolution of these two functions
yields the effective interaction
r
of the phase model.
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case the excitation is always desynchronizing. Indeed, since the function
gsyn($) s periodic one has
(3.14)
As
gtyn($)
>
0 in the interval [0,h] he mean value theorem ensures that
for some +; in this interval
J ? 8 ; y " ( m w + -
J P I = Z ( N
o
gsyn($)d@
Similarly there exists some in the interval
[d ,
7r]
such that
(3.15)
(3.16)
Using equation 3.14 and the fact that
Z
is monotonically increasing we
have
~* "g ~y n ( + ) z ( ~ / l ) d ~ j- gLyn(+)Z($)dg
(3.17)
Therefore the desynchronizing contribution to r (0) s predominant.
One can rely on a similar argument to prove that if Z
is
differentiable
everywhere and has only one maximum, an excitatory interaction with
instantaneous rise is desynchronizing whatever its decay time.'
These results can be extended by continuity. It is clear that the two
contributions to
r (0)
can be comparable only if dip and the maximum
of Z are not too far apart. Since &