How synaptic weights determine stability of synchrony in networks of pulse-coupled excitatory and inhibitory oscillatorsBirgit Kriener Citation: Chaos 22, 033143 (2012); doi: 10.1063/1.4749794 View online: http://dx.doi.org/10.1063/1.4749794 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v22/i3 Published by the American Institute of Physics. Related ArticlesOn the formulation and solution of the isochronal synchronization stability problem in delay-coupled complexnetworks Chaos 22, 033152 (2012) Coherence and decoherence in the brain J. Math. Phys. 53, 095222 (2012) Complex network classification using partially self-avoiding deterministic walks Chaos 22, 033139 (2012) Reverse engineering of complex dynamical networks in the presence of time-delayed interactions based on noisytime series Chaos 22, 033131 (2012) The architecture of dynamic reservoir in the echo state network Chaos 22, 033127 (2012) Additional information on ChaosJournal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors

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How synaptic weights determine stability of synchrony in networksof pulse-coupled excitatory and inhibitory oscillators

Birgit Krienera)

Institute of Mathematical Sciences and Technology, Norwegian University of Life Sciences, As, Norway;Network Dynamics Group, Max Planck Institute for Dynamics and Self-Organization, G€ottingen, Germany;and Bernstein Center for Computational Neuroscience G€ottingen, G€ottingen, Germany

(Received 6 June 2012; accepted 16 August 2012; published online 12 September 2012)

Under which conditions can a network of pulse-coupled oscillators sustain stable collective activity

states? Previously, it was shown that stability of the simplest pattern conceivable, i.e., global

synchrony, in networks of symmetrically pulse-coupled oscillators can be decided in a rigorous

mathematical fashion, if interactions either all advance or all retard oscillation phases (“mono-

interaction network”). Yet, many real-world networks—for example neuronal circuits—are

asymmetric and moreover crucially feature both types of interactions. Here, we study complex

networks of excitatory (phase-advancing) and inhibitory (phase-retarding) leaky integrate-and-fire

(LIF) oscillators. We show that for small coupling strength, previous results for mono-interaction

networks also apply here: pulse time perturbations eventually decay if they are smaller than a trans-

mission delay and if all eigenvalues of the linear stability operator have absolute value smaller or

equal to one. In this case, the level of inhibition must typically be significantly stronger than that of

excitation to ensure local stability of synchrony. For stronger coupling, however, network

synchrony eventually becomes unstable to any finite perturbation, even if inhibition is strong and

all eigenvalues of the stability operator are at most unity. This new type of instability occurs when

any oscillator, inspite of receiving inhibitory input from the network on average, can by chance

receive sufficient excitatory input to fire a pulse before all other pulses in the system are delivered,

thus breaking the near-synchronous perturbation pattern. VC 2012 American Institute of Physics.

[http://dx.doi.org/10.1063/1.4749794]

Synchrony of activity in complex interaction networks

is an ubiquitous phenomenon in physics, chemistry,

biology, or social science, but also in many technical

applications such as mobile communication or power

grid dynamics. Prominent examples, such as synchro-

nous flashing of fireflies, pacemaker cells in the heart,

or firing time synchrony in neuronal networks, can be

described in terms of so-called pulse-coupled oscilla-

tors, i.e., units with periodical dynamics that interact

via discrete signals in time. Synchrony can be both

desired and considered beneficial, as in the case of mo-

bile communication or heart beating, but also detrimen-

tal to system function, as e.g., during epileptic seizures

of the brain. Rigorous assessment of synchrony in

pulse-coupled oscillator networks is often limited to sys-

tems in which pulse interactions between oscillators ei-

ther delay the next pulse emission in the receiving

targets or advance pulse emission. In the mammalian

brain, however, it is the interplay of these two counter-

acting forces that is the fundamental working principle,

and it is instantiated by the two basic neuron classes: in-

hibitory neurons that keep other neurons from firing,

and excitatory neurons that drive other cells to emit a

pulse, the so-called action potential. While recurring

patterns of synchronous action potentials might be

employed by the brain to encode information, an overly

synchronous state can lead to epileptic malfunctions. To

understand what enhances or prevents synchrony in

such oscillator networks is thus of general importance.

Here, we show that mixed-interaction pulse-coupled

networks—in contrast to mono-interaction networks—

have the unique property that their synchronizability

can be changed by either modifying the strength of the

overall coupling, or by tuning the relative inhibitory

coupling strength. Below a certain overall coupling

strength, the necessary condition for stability of global

synchrony can be deduced analogously to mono-

interaction networks. In this weak-coupling regime, sta-

bility depends on the relative strength of inhibition over

excitation in the network: the stronger the relative inhi-

bition, the more likely stability of synchrony becomes.

This is in line with the finding that in purely inhibitory

networks, synchrony is asymptotically stable to small

perturbations, irrespective of the coupling strength. If

the overall coupling strength is increased, however, syn-

chrony becomes unstable even in the presence of strong

inhibition. This can be understood in the following

way: during the reception of pulses, the oscillator

potential moves towards or away from pulse emission

threshold in response to the impinging excitatory and

inhibitory inputs, respectively. Even if the inhibition

is strong enough to, on average, keep the oscillator

far from threshold during this period, its potential

strongly fluctuates and can hit threshold before the os-

cillator received all of its inputs. Thus, it can elicit a

pulse much earlier than other oscillators in the net-

work, effectively breaking down synchrony. This is aa)Electronic mail: birgit.kriener@umb.no.

1054-1500/2012/22(3)/033143/11/$30.00 VC 2012 American Institute of Physics22, 033143-1

CHAOS 22, 033143 (2012)

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novel desynchronization phenomenon that can only

appear in mixed-interaction networks and may have

important functional consequences for the dynamics of

cortical neuron networks.

I. INTRODUCTION

Synchrony of dynamical units coupled by complex inter-

action topologies is of large relevance in many areas as diverse

as physics,1,2 biology,3 chemistry, or mobile communication4

and power grid dynamics. One type of system that has particu-

lar importance in the modeling of neuronal networks is that of

pulse-coupled oscillator networks.3,5,6 The simplest conceiva-

ble type of synchrony in such systems is global synchrony,

i.e., a network state in which all oscillators emit a pulse in per-

fect unison. In the past, a lot of effort was put into solving the

problem of when this synchronous state is stable to perturba-

tions and, given there is local stability, how large such a per-

turbation can be before synchrony breaks down.3,7–9

Moreover, the crucial role of network topology on the quality

and speed of synchronization has been demonstrated.5,10–12

Rigorous mathematical analysis of asymptotic stability is

often limited to networks of oscillators that are coupled by ei-

ther purely inhibitory (phase retarding) or excitatory (phase

advancing, pulse coupling), where it depends on the properties

of the potential function Uð/Þ that describes the dynamics of

the oscillator whether synchrony is stable or not.9,13

Here, we analyze networks of leaky integrate-and-fire (LIF)

oscillators of mixed interaction structure, a setting with special

relevance in computational neuroscience.14,15 In particular, each

oscillator shall either have only an inhibitory effect on all its tar-

get oscillators, or an excitatory effect, but never both. In neuro-

science, this is known as Dale’s principle.14 In such mixed-

interaction networks, the assessment of synchrony is much more

complicated, because the eigenvalues of the linear stability oper-

ator S can be both larger and smaller than unity, while in net-

works with only one type of interactions, all eigenvalues are

either� 1 (the necessary condition for local stability) or� 1.9

In this paper, we show how the resulting constraints on the

eigenvalue spectra of mixed-interaction networks (in the follow-

ing also called EI i.e. excitatory-inhibitory networks) yield condi-

tions on the type of network structure and weight distributions

that allow for locally stable synchrony. The overall coupling

strength J and the relative coupling strength between inhibition

and excitation g turn out to be the decisive factors in this. In par-

ticular, there is a critical Jc above which EI networks are always

unstable to any finite perturbation, while below Jc networks can

be stable to pulse-time perturbations smaller than the pulse trans-

mission delay, given the eigenvalues of S are smaller than or

equal to unity. In the following, we compare random topologies

and ring topologies, analyze resynchronization time constants,

and finally discuss the effect of neglecting Dale’s principle.

II. OSCILLATOR MODEL AND PHASEREPRESENTATION

The subthreshold dynamics of the membrane voltage

ViðtÞ of a LIF oscillator i is governed by

dViðtÞdt¼ �ViðtÞ

sm

þ RI

sm

þXN

j¼1

aijsjðt� dÞ ; (1)

where sm is the membrane time constant, R is the membrane

resistance, I is an external direct current, aij is the coupling

strength in mV of the synapse from oscillator j to i, j 6¼ i; i; j2 f1;…;Ng; sjðtÞ is the pulse train emitted by oscillator jover time, and d is a fixed pulse transmission delay. When-

ever the potential Vi of oscillator i hits a threshold potential

value H at time t, it instantaneously emits a pulse and is reset

to the reset potential ViðtþÞ ¼ Vr < H. Note that here we

only consider cases where individual input pulses have am-

plitude aij < ðH� VrÞ, such that a single pulse cannot cause

the oscillator to emit a pulse. Moreover, without loss of gen-

erality, we set Vr � 0. If the constant external input RI is

superthreshold, the dynamics of the isolated oscillator

(8i;j2f1;…;Ng aij ¼ 0) is solved by

VðtÞ ¼ RI ð1� e�t=smÞ; (2)

and each neuron oscillates with a fixed period given by

T0 ¼ �sm Log 1� HRI

� �: (3)

Equation (2) can be mapped3 onto a simpler representation by

the introduction of the phase-variable /ðtÞ ¼ t=T0 2 ð�1; 1�,such that Uð/Þ :¼ Vð/T0Þ. Then, / 7! 0, if /ðT0Þ ¼ U�1ðHÞ¼ 1. Incoming pulses from j to i at time t cause an instantane-

ous jump of the membrane potential VðtþÞ ¼ VðtÞ þ aij. In

the phase representation, this event is handled by

Haijð/Þ ¼ U�1ðUð/Þ þ aijÞ ¼

sm

T0

Log e�/ T0=sm � aij

RI

h i�1

:

(4)

It can be shown analytically that in strongly connected

networks of pulse-coupled identical LIF oscillators with

potential function U, such that U0 > 0 and U00 < 0 and

homogeneous delay d, global synchrony is asymptotically

stable if 8i;j2f1;…;Ng aij � 0 and 8i ai :¼P

j aij ¼ a, while it is

unstable if 8i;j2f1;…;Ng aij � 0.9 In the globally synchronous

state, all phases are identical and return to the same value af-

ter the global period

T :¼ d þ T0 1� U�1 Ud

T0

� �þ a

� �� �; (5)

i.e., 8i /iðtÞ ¼ /iðtþ TÞ. A uniform perturbation of the

phases (perturbed phases denoted by p)�dð0Þ :¼

�d ¼ /pðt0Þ

�/ðt0Þ 2 ½�c=2; c=2�N of initial size Dð0Þ :¼ jmaxð/pi Þðt0Þ

�minð/pi Þðt0Þj < c and delivered at some /ðt0Þ > d=T0 and

<ð1� cÞ will break synchrony by advancing some oscilla-

tors, while delaying others. The rank order of this perturba-

tion will in general determine the stroboscopic time- T map

that describes the temporal evolution of the phase-

perturbations and thus also the operator S describing its line-

arized dynamics, such that�dðTÞ¼: S

�dð0Þ. Given a particular

imposed rank order, the elements of S are given by9

033143-2 Birgit Kriener Chaos 22, 033143 (2012)

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Sij ¼pi;n � pi;n�1 if j ¼ jn 2 Pre½i�pi;0 if j ¼ i0 j 62 Pre½i� [ fig;

8<: (6)

with Pre½i� denoting the presynaptic input set of oscillator iand

pi;n ¼U0 U�1 UðdÞ þ

Xn

m¼1

aij

! !

U0ðU�1ðUðdÞ þ aÞÞ :

Hence, Sij are, in general, a nonlinear function of the

sumsPn

m¼1 aij, however for the LIF

Sij ¼aij

a� RI e�d=smif i 6¼ j and S0 ¼

RI e�d=sm

RI e�d=sm � a

if i ¼ j ; (7)

independent of the particular perturbation rank order.9,16

Thus, the stability operator S is just a rescaled version of the

coupling matrix A.

III. STABILITY OF SYNCHRONY IN RANDOMEI-NETWORKS

We will first consider random networks of N LIF oscilla-

tors with d-current-pulse coupling, as described in Sec. II,

where the networks contain NE ¼ bN; b 2 ð0; 1Þ excitatory

and NI ¼ N � NE inhibitory oscillators.17 For simplicity,

each oscillator receives exactly KE synapses of strength

aexc ¼ J from KE ¼ pNE; p 2 ½0; 1�, randomly chosen excita-

tory oscillators, and KI synapses of strength ainh ¼ �gJ from

KI ¼ pNI randomly chosen inhibitory oscillators, where we

exclude self-couplings. The coupling matrices A are thus con-

strained random matrix realizations such that aij ¼ f�gJ; 0gif j inhibitory and aij 2 f0; Jg if j excitatory, and for all i :P

j2inh aij ¼ �gpNJð1� bÞ andP

j2exc aij ¼ pNJb, where

“exc” and “inh” denote the excitatory and inhibitory subpo-

pulation, respectively. The results, however, also hold for

more general weight distributions, as long as the row normal-

ization condition is fulfilled.16

Note that the constraints of strict columnwise negativity

or positivity (the output of each oscillator is either com-

pletely inhibitory or completely excitatory, a biological fact

also known as Dale’s principle), induce correlations between

matrix elements that lead to an inhomogeneous distribution

of eigenvalues in the complex plane.18 Due to Eq. (7), the

stability matrix S inherits these properties of A. At

g ¼ b=ð1� bÞ, the dynamical system undergoes a saddle-

node bifurcation and is in a fully regular high firing rate

state,17 with rates only limited by, e.g., a refractory period

for g < b=ð1� bÞ.Here, we are interested in collective synchrony in the

inhibition-dominated regime with g > b=ð1� bÞ where the

system is bistable and can exhibit both global synchrony as

well as asynchronous-irregular activity. We assume identical

oscillators and thus all have the same suprathreshold driving

current I, firing threshold H, resistance R, membrane time

constant sm, and all pulses shall have the same transmission

delay d. Moreover, we neglect refractoriness. Then, if all

oscillators are prepared to have the same initial conditions,

they will reach H at the same time and stay globally

synchronized with a period of T ¼ T0 þ d þ sm Log

e�d=sm � aRI

� �, 8i2f1;…;Ng a ¼

PNj¼1 aij ¼ pNJðb� g ð1� bÞÞ,

cf. Eqs. (4) and (5).

A. Random EI-networks: Weak coupling regime

Because we consider random networks A, and S inherits

the properties of A, random matrix theory (RMT) can be

applied to estimate the eigenvalue of S with the second larg-

est eigenvalue jk2j ¼ maxi½ki : jkij < 1�—which determines

a necessary condition for simple stability of the system9—by

a generalized circle law,18 yielding jk2j ¼ S0 þ rRMT . rRMT

denotes the bulk spectral radius that is derived in the follow-

ing, cf. Eq. (9) and Fig. 4.

jk2j is moreover directly related to the resynchronization

constant ssync / �1=Log½jk2j� since for the asymptotic dy-

namics n!1 of the phase spread10

DðnÞ :¼ jmaxð/pi ÞðnÞ �minð/p

i ÞðnÞj / e�n=ssync : (8)

Each iteration n is in terms of the perturbed period �TiðnÞof one arbitrary reference oscillator i.

S is a sparse random matrix with one eigenvalue k1

equal to one (PN

j¼1 Sij ¼ 1 for all i 2 f1; ::;Ng) and all other

eigenvalues distributed across the complex plane cf. Fig. 1.

For large N, the eigenvalues of S are bound to a disc with

center S0 and radius rRMT ,10,18 cf. Fig. 1(a),

rRMT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNpð1� pÞðS2

ebþ S2i ð1� bÞÞ

q; (9)

with Se ¼ Ja�RIe�d=sm

< 0 and Si ¼ �gJa�RIe�d=sm

> 0. The predic-

tion of rRMT however only becomes exact for N !1,18 and

single eigenvalues of small matrices can lie well beyond the

circle law and typically show a far from homogeneous distri-

bution in the complex plane, as can be seen in Fig. 1(b),

while for larger N, the RMT estimate becomes much better,

cf. Fig. 1(a). Fig. 1 moreover demonstrates the stabilizing

effect of network size in mixed random networks, given a

certain g and p.

If N is sufficiently large, knowing rRMT then allows to

discuss the necessary condition for stability of the synchron-

ized state in terms of network parameters. Thus, in order to

be stable, networks need to fulfill

S0 þ rRMT � 1 ()ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNpð1� pÞðbþ g2ð1� bÞÞ

p� Npðb� gð1� bÞÞ : (10)

Note, that this condition does neither depend on the

absolute coupling strength J nor on oscillator or synapse

parameters such as I, sm, or d. One can then, e.g., calculate

the expected critical value for the relative inhibitory strength

parameter gcrit as a function of N; p; b to fulfill the necessary

condition for simple stability

033143-3 Birgit Kriener Chaos 22, 033143 (2012)

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gcrit ¼Npbð1� bÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� pÞðpð1þ NÞ � 1Þbð1� bÞ

pð1� bÞðpð1þ Nð1� bÞÞ � 1Þ :

(11)

For N !1 or p! 1, we have gcrit ! b=ð1� bÞ (cf.

Fig. 2). For small N, there might not be a physically meaning-

ful solution at all, however in praxis small networks can be

stable due to the very sparse eigenvalue spectra that do not

sample the circle law predicted by RMT and hence can have

absolute value well below 1. From the 10 000 matrix realiza-

tions whose eigenvalues are depicted in Fig. 1(b), only about

4% have at least one eigenvalue larger than one. For

the synchronous state for N ¼ 1000; b ¼ 0:8, and p¼ 0.1,

Eq. (11) needs at least a g > gcrit � 5:18 to allow for locally

stable synchrony, cf. Fig. 3. Fig. 3(b) shows exemplarily three

eigenvalue spectra corresponding to three different values of gfor one stability matrix realization each, such that g ¼ gcrit

(red), g < gcrit (purple), and g > gcrit (blue), while Fig. 3(a)

shows the dynamics of the maximal phase spreads DðnÞ for

different g values.

Fig. 4(a) shows an example of the RMT predicition of

ssync, which is the inverse of the slope of the logarithm of

DðnÞ, versus simulation results for a network of size

N¼ 1000. The RMT-prediction slightly overestimates both

the ssync estimated from jk2j of actual coupling matrix instan-

tiations, as well as the one estimated from the dynamical sys-

tem, what is due to the discussed finite-size effects. Fig. 4(b)

FIG. 1. Effect of small network size on eigenvalue spectra of random EI networks: Eigenvalue spectra for mixed random networks (a) of size N¼ 2000,

m¼ 40 realizations and (b) N¼ 20 with m¼ 10 000. Other parameters are g¼ 7, b ¼ 0:8, p¼ 0.2, J¼ 0.1 mV, d¼ 2 ms, sm ¼ 20 ms, I¼ 375 pA,

R ¼ 80 MX; and H ¼ 20 mV.

FIG. 2. Critical relative strength of inhibition gcrit as function of network parameters: (a) The critical strength of inhibition gcrit as a function of the network

size N for b ¼ 0:8. From light to dark gray, the density p varies from sparse to dense (p 2 f0:02; 0:05; 0:1; 0:3; 0:5; 0:8g). The constant black line corresponds

to b=ð1� bÞ ¼ 4. The y-axis is clipped at 0 and 10 because of large negative and positive ecursions of the theoretical value of gcrit for small densities p. (b)

The same as in (a) for hybrid EI-networks (cf. Sec. V). Differences to (a) are subtle and become more pronounced for larger densities p. Hybrid networks

always have a slighlty higher value for gcrit. This is demonstrated in (c) where the curves for p¼ 0.1 (light gray) and p¼ 0.3 (dark gray) are shown in direct

comparison for Dale-conform (dashed) and hybrid (solid) networks. (d) The critical g for Dale-conform ring networks (see, Sec. IV) for p¼ 0.1 (light gray

solid) and p¼ 0.3 (dark gray solid) in comparison to the corresponding curves for random Dale-conform networks (dashed).

033143-4 Birgit Kriener Chaos 22, 033143 (2012)

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demonstrates that, just like in purely inhibitory networks, also

in weakly coupled EI system with g > gcrit maximal perturba-

tion size in phase spread D is limited approximately by the

delay of the sytem (cf. Ref. 19 for a more detailed analysis).

B. Random EI networks: Strong coupling regime

In Sec. III A, we found that the linear stability of a ran-

dom EI network does not depend on the coupling strength J.

This, however, is only valid for small J. If J gets larger, at

one point, the network can become unstable even though all

eigenvalues 6¼ 1 of S are smaller than one, cf. Fig. 5. This is

going to be the case, once a perturbation can have the form

such that at least one oscillator receives all its excitatory

inputs first, and this input is strong enough to push that oscil-

lator to threshold, i.e., for a critical weight,

Jc ¼H� RIð1� e�d=smÞ

bpN: (12)

Moreover, if an oscillator receives most of its excitatory

input first and emits a pulse, it will afterwards receive the

massive barrage of the remaining inhibitory inputs, increas-

ing an existing phase difference even more. Hence, the usu-

ally stabilizing effect of inhibition can act destabilizing once

such an event occurs. Note that there is no direct dependence

on the size of the perturbation, but rather on the perturbation

rank order. Since a perturbation can change rank order over

iterations, such a destabilizing configuration may occur at all

times and even the tiniest perturbation can kick oscillators

above threshold.

A slightly more practical estimation of actual stability

can be obtained by making four reasonable assumptions:

First, we assume that the pulse delivery takes place within

the time given by the perturbation window Dð0Þ, meaning

that arriving pulses do not increase the delivery time beyond

Dð0Þ, i.e., j/ðtreceived; lastÞ � /ðtreceived; firstÞj � Dð0Þ. Given

the synchronous mode is stable, this is to be expected. Sec-

ond, we assume that within this delivery time, oscillator

phases stay approximately constant between incoming

pulses, which also is reasonable, since the phase motion

towards threshold due to only the driving current I is

FIG. 3. The relative amount of inhibition determines stability of global

synchrony: (a) shows the phase spread dynamics DðnÞ :¼ jmaxð/pi ðnÞÞ �

minð/pi ðnÞÞj of sparse random networks of excitatory and inhibitory

oscillators for different values of g. If g < gcrit; DðnÞ diverges (red

lines) and for g > gcrit, it converges back to the synchronous state

(blue lines). The purple line shows the dynamics close to gcrit. The cor-

responding eigenvalue spectra of the stability matrices are shown in (b)

for three selected values (g¼ 4.5, g�gcrit, and g¼ 6). (c) and (d) shows

the same for random networks of hybrid oscillators, cf. Sec. V, (e) and

(f) for Dale-conform ring networks (cf. Sec. IV), and (g) and (h) for

hybrid ring networks. Other parameters: N¼ 1000, p¼ 0.1, b ¼ 0:8,

J¼ 0.1 mV, sm ¼ 20 ms; R ¼ 80 MX; H ¼ 20 mV, I¼ 375 pA, and

d¼ 1.5 ms.

FIG. 4. Resynchronization speed after perturbations Dð0Þ < d is determined by ssync: (a) The logarithms of the phase spread DðnÞ over the last 200 iterations

before D < 10�8, thus the asymptotic phase of resynchronization. The solid black line shows the predicition from RMT, cf. Eq. (9); and the dashed black line

shows the predicition from the average of the actual second largest eigenvalues. The thick gray line is the mean of the ten example curves denoted by the thin

gray lines. (b) Demonstrates that also in random EI networks, the maximal perturbation is approximately limited by the delay (horizontal dashed line corre-

sponds to d=T0, inset is a zoom-in). Parameters: N¼ 1000, p¼ 0.1, b ¼ 0:8, J¼ 0.1 mV, sm ¼ 20 ms; R ¼ 80 MX; H ¼ 20 mV, I¼ 375 pA, and d¼ 1.5 ms.

033143-5 Birgit Kriener Chaos 22, 033143 (2012)

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negligible for short time intervals, cf. Fig. 6(a). Third, even

though the results hold for more general weight or degree

distributions, as long as the condition 8i

Pjaij ¼ a is ful-

filled, we choose all oscillators to receive exactly K ¼KE þ KI synapses from the network oscillators, and that all

KE excitatory inputs have the same strength J, and all KI in-

hibitory have strength �gJ. Since the system shall be in a

near synchronous orbit, each oscillator thus receives K pulses

during one period. Finally, we assume that upon reception

of an excitatory pulse of weight J or inhibitory pulse of

weight �gJ, the phase of an oscillator is advanced or delayed

by Jef f :¼ HJð5d=4T0Þ, or gef f Jef f :¼ H�gJð5d=4T0Þ, respec-

tively, independent of its current phase (note that here we

assumed a Dð0Þ ¼ d=2T0).

With these assumptions, we can consider pulse reception

on a temporal grid of K slices, and count all possible paths

the phase of an oscillator can take during pulse reception

with corresponding probabilities, see Figs. 7 and 6(b). The

mean lðiÞ and variance r2ðiÞ as a function of pulse reception

events i 2 f1;…;Kg can then easily be computed

FIG. 5. Principal instability does not equal typical instability in EI networks: The phases of a random subset of oscillators after a small perturbation to the syn-

chronous mode for different coupling strengths J, N¼ 1000, p¼ 0.1, and b ¼ 0:8. (a) to (e) shows the phase traces over individual pulse reception events in

the network for 50 randomly picked oscillators. (a) J¼ 0.1 mV and random perturbation rank order. (b) J¼ 0.1 mV and sorted perturbation rank order, such

that all excitatory oscillators are perturbed slightly more towards threshold than the inhibitory oscillators, hence excitation is delivered before inhibition can

come into play. (c) and (d) but with J ¼ Jc � 0:223 mV. Given that the oscillators are perturbed most critically as in (d), the phase perturbation D/ can grow

due to prematurely spiking oscillators. For randomly perturbed phases as in (c), the system may still resynchronize, because inhibition balances excitation. (e)

Same as in (a) but for J¼ 0.8 mV. If weights are just large enough, also random perturbations can cause oscillators to fire prematurely. (f) Phases per iteration

triggered on one randomly picked oscillator with J¼ 0.5 mV. Even though oscillators can fire out of phase, they might or might not cause the system to fall

into an asynchronous state, which in this case, it does at about 70 iterations (blue: inhibitory and red: excitatory oscillator). Other parameters: g¼ 6, d¼ 1.5 ms,

sm ¼ 20 ms, H ¼ 20 mV, I¼ 375 pA, and R ¼ 80 MX.

033143-6 Birgit Kriener Chaos 22, 033143 (2012)

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lðiÞ ¼i Jef f ðKE � gef f KIÞ þ

D/2

� �K

and

r2ðiÞ ¼X

j

pðxðiÞjÞðxðiÞj � lðiÞÞ2 ; (13)

where x(i) is the vector of possible grid coordinates the oscil-

lator phases can assume at event time i and p(x(i)) is the asso-

ciated probability mass function, cf. Fig. 7. For example, after

the first iteration, we expect a fraction KE=K of the oscillator

phases to be increased by an excitatory input Jef f and KI=K to

be decreased by gef f Jef f . Hence, if we assume that the

starting coordinate is x0 ¼ 0, the coordinate vector at i¼ 1 is

xð1Þ ¼ ðJef f ;�gef f Jef f Þ> with probabilites pðxð1Þ1Þ ¼ KE=K;pðxð1Þ2Þ ¼ KI=K. In the next iteration, xð2Þ ¼ ð2Jef f ;Jef f � gef f Jef f ;�2gef f Jef f Þ> with probabilites pðxð2Þ1Þ

¼ KEðKE � 1Þ=KðK � 1Þ; pðxð2Þ2Þ ¼ 2KEKI=KðK � 1Þ;pðxð2Þ3Þ ¼ KIðKI � 1Þ=KðK � 1Þ, and so forth. Due to the

symmetry of the system, the maximum of the variance will be

reached after K/2 steps. For simplicity, we assume that this

point is the most critical in terms of any oscillator reaching

threshold before all pulses are delivered, although this point

might be earlier due to the in general negative slope of lðiÞ(jD/=2j � jJef f ðKE � gef f ÞKIj and gef f > KE=KI).

It turns out that the five-fold standard deviation rðK=2Þis a good indicator of where the critical coupling strength

lies that when exceeded will lead to all networks becoming

most certainly unstable in the large-N limit. This is demon-

strated in Fig. 8; Fig. 8(a) shows the predicted critical cou-

pling strengths Jc (cf. 12) and J0c as a function of network

size; while in Fig. 8(b), the fraction of 10 networks that were

still resynchronized after 500 iterations is shown as a func-

tion of J for different network sizes, respectively. The J0cvalues of (a) are shown by vertical lines in (b) for visual

guidance.

IV. STABILITY OF SYNCHRONY IN RING EINETWORKS

We consider ring networks such that each oscillator is

coupled to its j ¼ pN nearest neighbors, j=2 neighbors on

each side, but not to itself, such that the coupling matrix is

given by

aij ¼�gJ if j mod 5 6¼ 0 and ji� jjmod N � j=2

�gJ if j mod 5 ¼ 0 and ji� jjmod N � j=2

�g0 if j ¼ i:

8<:

(14)

Note that we demand that N mod j=2 ¼ 0 and

j mod 5 ¼ 0. Thus, every fifth oscillator on the ring is inhibi-

tory (cf. Fig. 9(a)) and all oscillators receive the same amount

of KE excitatory and KI inhibtory inputs, implying that the

mean input composition a is the same as in the previously dis-

cussed random networks. For this periodic layout, the eigen-

system can be computed analytically by taking into account

that the coupling matrix is invariant to index shifts by five, cf.

Fig. 9(b). If we call the operator that performs that index shift

(c)(a) (b)

FIG. 6. Variance of phase spread during pulse reception events explained by constrained random walk on a grid: (a) The dynamics of the phase trajectories /i

of a subset of 30 of 500 oscillators during the pulse reception period after a small perturbation (gray lines). The black solid line is the trajectory of the ensemble

mean, while the dashed black line is the theoretical expectation l, cf. 13. (b) The same collapsed to mere interaction events only (all oscillators receive K¼ 50

pulses). (c) Shows the standard deviation of the phase trajectories as a function of events as measured (dark gray), as estimated from a constrained random

walk with the constant step sizes Jef f ; gef f Jef f on the grid in Fig. 7 (light gray) and theoretical expectation for the standard deviation of such a walk (dashed

black).

Steps

φ

K0

K E

K

K I

K

K E

K

IEK K 2

K(K−1)

E (K −1)E

K(K−1)

K

I I

K(K−1)

K (K −1)

K I

K

FIG. 7. Transition probabilities of the constrained random walk: The

reduced model of Fig. 6(b) that incorporated the four assumptions outlined

in the text for K¼ 12, KE ¼ 8, and KI ¼ 4. With every event step, the oscil-

lator will receive either an excitatory or inhibitory pulse. The probabilities

for these two events depend on how many of each kind the oscillator already

received. In the first step, it can receive KE excitatory and KI inhibitory. In

the second step, it will either get another excitatory pulse with probability

ðKE � 1Þ=ðK � 1Þ or an inhibitory with probability ðKI � 1Þ=ðK � 1Þ, yield-

ing the expected probability mass densities for the three possibilities for the

oscillator phase to be in as indicated in the figure. In this way, the probabil-

ity mass function as a function of interaction event number can be computed

analytically.

033143-7 Birgit Kriener Chaos 22, 033143 (2012)

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T5, it thus holds ½T5;A� ¼ ½A; T5� ¼ ½T5; S� ¼ ½S; T5� ¼ 0 and

both matrices can be diagonalized in the same eigenbasis.20

Due to the symmetry of the system, the problem is effectively

five-dimensional and each eigenvalue lies in one of five bands,

which define the dispersion relation, i.e., the absolute value of

eigenvalues as a function of wavenumber a, cf. Fig. 10.

A. Ring EI networks: Weak coupling regime

For small J, there is the same bistability between global

synchrony and asynchronous spiking19,21 as found in bal-

anced random networks. Again, for perturbations of the syn-

chronous mode that are smaller than the delay, the network

resynchronizes if the ratio between the strength of inhibition

and excitation g > gcrit. Here, gcrit is once more only de-

pendent on N; b, and p ¼ j=N and determined by the g-value

at which the largest eigenvalue of S switches from a non-

zero wavenumber (i.e., a modulated pattern vc with a > 0

bumbs) to a ¼ 0 (translation invariant flat) that contains the

eigenvalue k1 � 1 with eigenvector ð1;…; 1Þ>, cf. Fig.

10(b). Figure 2(d) shows gringcrit as a function of N for p¼ 0.1

(solid dark gray) and p¼ 0.3 (solid light gray) in comparison

to the same curve for the corresponding random network

(dashed). gringcrit < gcrit and thus ring networks can afford less

inhibition in the network and still be stable to small perturba-

tions in the weak-coupling regime.

B. Ring EI networks: Strong coupling regime

For larger weights J, however there is not only the al-

ready discussed principal instability of the synchronous ac-

tivity mode to basically any perturbation but the rate

distribution can become spatially structured.20 This is the

case when any eigenvalue of the coupling matrix A(J) (cf.

Eq. (14)) has absolute value jkðAðJÞÞcj > H. The spatial pat-

tern that is assumed then is given by the eigenvector

vðAðJÞÞc that corresponds to kðAðJÞÞc. For all other parame-

ters fixed, the weight ~J , such that jkðAð~JÞÞcj > H is usually

larger than Jc as defined in Eq. (12), meaning that the tiniest

pulse reception time perturbation will make the vector of fir-

ing rate become proportional to vðAðJÞÞc, cf. Fig. 11.

V. EFFECT OF WEIGHT DISTRIBUTION: DALE’SPRINCIPLE LEADS TO FASTERRESYNCHRONIZATION

In cortex, mature neurons group into (at least) two func-

tionally distinct groups, i.e., excitatory and inhibitory neu-

rons, meaning that excitatory neurons can only shift the

phase of all its postsynaptic targets towards pulse emission

threshold, while inhibitory ones can only shift it away from

threshold (Dale’s principle). If only the mean input composi-

tion a of an oscillator would matter, one could conceive just

as well networks of oscillators that can be both excitatory

and inhibitory at the same time. We will call such oscillators

“hybrid” oscillators. If we have identical network structures

as before, however allow for a fraction b of outgoing synap-

ses of each oscillator to be excitatory, while ð1� bÞ shall be

inhibitory, the resulting expected RMT predicition for the

second largest eigenvalue of such a hybrid oscillator network

SH becomes

rHRMT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNpðbS2

e þ ð1� bÞS2i Þ � ða=Nða� RIe�d=smÞÞ2

q:

For random EI networks, we find that (where the expres-

sions are defined, g > gHcrit and J < Jc) rH

RMT � rRMT and thus

Dale-conform random networks synchronize faster than their

hybrid counterparts, whereas if we keep the mean input aconstant, but use purely inhibitory oscillators (i., e. weights

are 8i;j Wij ¼ fa=pN; 0g), synchronization becomes even

faster, see Fig. 12.

In analogy to condition 10, we obtain

FIG. 8. Principal versus typical instability of synchrony in EI networks: (a) The critical coupling strength Jc from Eq. (12) and J0c from the assumption that syn-

chrony becomes unstable 5r � 1 (dots) as a function of network size. (b) Shows the fraction of 10 networks that became unstable over 500 iterations as a function

of coupling strength. From light to dark gray K¼f50;100;150;200;250;300;350;400g with N¼10K and KE ¼ 0:8K. Other parameters as in Fig. 5.

FIG. 9. Layout of the ring network model: (a) The ring network consists of

NE excitatory (white circles) and NI ¼ NE=4 inihibitory neurons (gray

circles). The neurons are periodically distributed across the ring, such that

every fifth neuron is inhibitory. Each neuron is connected to its j nearest

neighbors (here j ¼ 4). (b) Due to the periodic structure, the corresponding

coupling matrix is invariant to index shifts by ‘ ¼ 5. We call the operator

that executes this shift T‘.

033143-8 Birgit Kriener Chaos 22, 033143 (2012)

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gHcrit ¼

pbð1� bÞð1þ N2Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibð1� bÞNðp� Nð1� pNÞÞ

pð1� bÞðpð1� bÞð1þ N2Þ � NÞ

(15)

and see that gHcrit > gRMT

crit and does again not depend on J.

For hybrid ring EI-networks, the critical g value is also

given by gHcrit > gring

crit (cf. Figs. 2(c) and 2(d)) since the eigen-

value spectra are very similar to those of hybrid random

networks, cf. Fig. 3(h). Only for g > gHcrit some singular eigen-

values � 1 become visible to the right of the bulk spectrum

cf. Fig. 3(h). The dynamics of phase differences DðnÞ does

thus also not look notably different for hybrid random and

ring networks, cf. Figs. 3(c) and 3(d), unlike those of Dale-

conform random and ring networks, cf. Figs. 3(a) and 3(e).

The strong coupling regime J > Jc does mainly depend

on the input composition, which is identical in all considered

cases and thus if the network is Dale-conform or not has no

impact on the respective value of Jc.

VI. DISCUSSION

Here, we analyzed and discussed globally synchronous

spiking in networks of pulse-coupled excitatory and inhibitory

leaky integrate-and-fire (LIF) oscillators. It was previously

shown that synchrony is asymptotically stable in purely inhibi-

tory strongly connected LIF oscillator networks, while it is

unstable in networks of excitatory LIF oscillators.9 The mixed

case considered here is inherently different in that no rigorous

mathematical answer to the question of stability can be given:

EI-networks are non-normal and even if all eigenvalues of Sare smaller than or equal to unity, perturbations can transiently

grow,9,22 as can also be seen e. g. in Fig. 3(e). Still, we found

that for small absolute coupling strength J and large network

size N, the stability prediction via inspection of the eigenvalues

of S generally leads to the right assessment of stability.

A. Stability of the synchronous mode for weakcoupling

As long as J < Jc as defined by Eq. (12), the considered

networks appear to be stable to small pulse emission time

perturbations <d, and the criterion whether synchrony is sta-

ble depends solely on network parameters, i.e., network size

N, connection density p, and the fraction of excitatory oscil-

lators b. Functionally this means that changing the relative

gain g of excitation vs. inhibition is a means to make a net-

work either resynchronizable or not. If oscillators do not

comply with Dale’s principle and can be both excitatory and

inhibitory at the same time, the critical g-value gHcrit is always

larger than gcrit. That means that Dale-conform networks can

afford less local inhibition and still sustain stable synchrony.

FIG. 10. Eigenvalues of the stability matrix of the EI ring network as a function of wavenumber: The dispersion relations, i.e., the eigenvalues as a function of

wavenumber a of the stability matrix S derived for ring networks defined by Eq. (14). For smaller g�4 but g < gcrit, cf. (a), the largest eigenvalue has wave-

number a ¼ ac >6¼ 0; while for g > gcrit, the largest eigenvalue is unity and corresponds to wavenumber a ¼ 0, cf. (c). When g ¼ gcrit, the eigenvalue unity

has multiplicity three, cf. (b).

FIG. 11. Pattern formation in Dale-conform ring networks: Pattern formation in the strong coupling regime, such that J > ~J ; Jc. (a) Shows the pulse emission

times of individual oscillators over time. At t¼ 20 ms, a pulse time jitter perturbation <d was added such that the fully synchronous mode breaks up and an in-

homogeneous distribution of pulse emission rates across oscillators develops (b) that is proportional to the critical eigenvector of the underlying coupling ma-

trix (c) (amplitude inhomogeneity due to finite simulation time and stochasticity of the spiking dynamics). Parameters: N¼ 2000, p ¼ j=N ¼ 0:1; b ¼ 0:8,

J¼ 1 mV, g¼ 6, I¼ 750 pA, and d¼ 1.5 ms. All other parameters as before.

033143-9 Birgit Kriener Chaos 22, 033143 (2012)

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As previously shown for inhibitory networks,10,16 it can

be shown that also in EI networks, it holds that ssync �!N!1

0,

if the number of connections K between oscillators scales

with network size N, while it goes to a finite value, if K is

kept fixed (see Refs. 10 and 16 for the argument). If we com-

pare these two EI-network setups to the case of purely inhibi-

tory couplings, such that 8ij Wij 2 fa=pN; 0g and the mean

input is the same as in the EI-networks, we find that these

network synchronize fastest, however lack the tuning capa-

bility of synchronizability via g. On the opposite end of the

random network topology are regular ring networks. We

compared random networks with a fraction of b ¼ 0:8 exci-

tatory oscillators to rings where each fifth oscillator was in-

hibitory and each oscillator is coupled to its j ¼ pN nearest

neighbors. The eigenvalues can then be computed analyti-

cally. We find that the critical g value gringcrit is always smaller

than for random networks, implying that the system can

afford less inhibition and still enable resynchronization. sringsync

for the same parameters can be quite larger or smaller in ring

topology networks as was demonstrated in Fig. 12.

B. Stability of the synchronous mode for strongcoupling

If J is increased at one point, EI networks will become

inherently unstable to perturbations of any finite size, because

then the situation can arise that all excitatory input oscillators

of any given oscillator fire first and can thus push the oscillator

over threshold before all pulses in the system are delivered.

This critical weight is given by Jc in Eq. (12). For weights

J�Jc, this scenario rarely happens, and even if networks often

still resynchronize due to the stabilizing effect of the dominant

inhibition. The larger J becomes, however, the more likely it is

that a random input fluctuation will drive the oscillator to

threshold prematurely. For practical stability, it is thus the size

of the fluctuations of the oscillator potential during the period

of pulse reception that matters. As a quantitative criterion for

this practical stability, the mean together with the variance of

phases during pulse reception can be used and we found that

networks appear to be practically stable as long as the peak

mean plus five-fold standard deviation is smaller than the dis-

tance to threshold. We call this second critical weight that cor-

responds to this critical point J0c and presented how to

compute it in Sec. III B. This moreover does not much depend

on the underlying EI-weight distribution or topology, since the

input composition is identical and pulse reception times are

highly correlated anyway. Note however that for purely inhibi-

tory networks, there is no upper Jc, since these networks are

always stable9 to small perturbations and larger J simply leads

to increased speed of resynchronization.10

In theoretical neuroscience, the random network and

neuron parameter setup studied here is often considered a

FIG. 12. Synchronization speed as a function of network parameters and weight distribution: Synchronization time in EI-networks with Dale-conform (black)

vs. ring (darker gray), vs. hybrid (dark gray) vs. purely inhibitory networks with same mean (light gray) weight distribution as a function of (a) N (p¼ 0.3,

g¼ 5.2, and J¼ 0.1 mV), (b) J (p¼ 0.3, N¼ 2000, g¼ 4.5, and J¼ 0.1 mV), (c) p (N¼ 2000, g¼ 8, and J¼ 0.2 mV), and (d) g (N¼ 2000, p¼ 0.4, and

J¼ 0.2 mV).

033143-10 Birgit Kriener Chaos 22, 033143 (2012)

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minimal model for local cortical networks and referred to as

balanced random network.17,23 The main interest there is that

these type of networks can produce asynchronous irregular

dynamics, which is akin to observed spontaneous spiking in

awake behaving animals. In this context, the results pre-

sented here can help to determine parameters for which the

network is particularly unlikely to fall into periods of global

synchrony, which can correspond to an epileptic network

state, for example, by keeping values for the relative inhibi-

tion g below its critical value, i.e., close to exact balance,

with not too strong inhibition.

The presented analysis can be extended to more com-

plicated network structures such as small-world networks,

i.e., structures characterized by dense local connectivity

with sparse longrange projections, a structural correlate that

is observed in many real world networks, including the

brain.24,25 A recent paper12 presented a mean-field approach

to analytically compute the full eigenvalue spectrum of

mono-interaction Watts-and-Strogatz small-world net-

works.26 This can be extended to mixed-interaction small-

world networks by taking into account the constraints due

to Dale’s principle,20 in order to analyze the effect of dense

local connectivity with a certain fraction of longrange con-

nectivity on the stability of sychnrony. Since these net-

works interpolate between ring and random topology, their

stability properties lie between those limit cases.

Another topic of neuroscientific interest from the coding

point of view is the question of which networks can stably

produce certain spike patterns,19,27–31 however in systems

with transmission delays analysis was so far limited to net-

works without mixed interaction structure. The study of such

patterns in mixed-interaction networks with delays presents

thus an important extension of the current analysis.

ACKNOWLEDGMENTS

We thank Marc Timme for helpful comments. We

gratefully acknowlegde funding by the eScience program of

the Research Council of Norway under Grant 178892/V30

(eNeuro) and by the German Federal Ministry for Educa-

tion and Research (BMBF, Grant 01GQ1005B to BCCN

G€ottingen). Part of the simulations was carried out with

NEST (http://www.nest-initiative.org).

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