hansel 1995 synchrony

7/25/2019 Hansel 1995 Synchrony

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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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310

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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Synchrony in Excitatory Neural Networks

315

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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316

D. Hansel, G. Mato, and C. Meunier

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 &

hansel 1995 synchrony

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