denker 2004 breaking

7/25/2019 Denker 2004 Breaking

1/4

Breaking Synchrony by Heterogeneity in Complex Networks

Michael Denker, Marc Timme, Markus Diesmann, Fred Wolf, and Theo Geisel

Max- Planck- Instit ut fur Stromungsforschung and Fak ultat fur Physik, Universitat Gottingen, 37073 Gottingen, Germany(Received 2 September 2003; published 18 February 2004)

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in thepresence of coupling heterogeneity precisely ti med periodic firing patterns replace the state of global

synchrony that exists in homogenous networks only. With increasing disorder, t hese firing patternspersist until a critical temporal extent is reached that is of the order of the interaction delay. For strongerdisorder the periodic firing patterns cease to exist and only asynchronous, aperiodic states are observed.We derive self-consistency equations to predict the precise temporal structure of a pattern from anetwork of given connectivity and heterogeneity. Moreover, we show how to design heterogeneouscoupling architectures to create an arbitrary prescribed pattern.

DOI: 10.1103/PhysRevLett.92.074103 PACS numbers: 05.45.Xt, 87.10.+e, 89.75.Fb, 89.75.Hc

Understanding how the structure of a complex net-work [1] determines its dynamics is currently in the fo-cus of research in physics, biology, and technology [2].Pulse-coupled oscillators provide a paradigmatic classof models to describe a variety of networks that occur

in nature, such as populations of fireflies, pacemaker cellsof the heart, earthquakes, or neural networks [3,4].Synchronization is one of the most prevalent kinds ofcollective dynamics in such networks [5]. Recent theo-retical studies, which analyze conditions for the existenceand stability of synchronous states, have focused on ho-mogenous networks with simple topologies, e.g., globalcouplings [6] or regular lattices [7]. In nature, however,intricately structured and heterogeneousinteractions areubiquitous. Previously, aspects of heterogeneity havebeen studied mostly in globally coupled networks [810]. However, for networks with structured connectivityonly a few studies exist [11] and it is sti ll an open question

how heterogeneity influences the dynamics, in particularsynchronization, in such networks.

In this Letter we present an exact analysis of thedynamics of complex networks of pulse-coupled oscilla-tors in the presence of heterogeneity in the couplingstrengths. We demonstrate that the synchronous state,that exists in homogenous networks, is replaced by pre-cisely timed periodic firing patterns. The temporal extentof these patterns grows with the degree of disorder.Patterns persist below a critical strength of the disorderbeyond which only asynchronous, aperiodic states a reobserved. We show how this transition is controlled bythe interaction delay. Simple criteria for the stability offiring patterns are derived. Furthermore, we present anapproach to predict the relative timings of firing eventsfrom self-consistency conditions for the phases in a givennetwork. Conversely, a prescribed pattern can be createdby designing a heterogeneous coupling architecture in anetwork of specified connectivity.

Consider a fixed network ofNpulse-coupled oscillatorsthat interact via directed connections. A matrix C deter-mines the connectivity of this network, where Cij 1if a

connection from oscillator j to i exists, and Cij 0otherwise (Cii 0). The number ki:

PjCij of inputs

to every oscillator i is nonzero, ki 1, and no furtherrestriction on the network topology is imposed. In simu-lations, we illustrate our findings using random graphs in

which every directed connection is present with proba-bilityp.

The state of an individual oscillatorj is represented bya phase variable j t hat increases uniformly in time [3],

dj=dt 1: (1)

Upon crossing the th reshold 1 at time tfa n oscillatoris instantaneously reset to zero, jt

f 0, and a pulse is

sent. After a delay time , this pulse is received by alloscillators i connected to j (for which Cij 1) andinduces an instantaneous phase jump

itf U1Uitf "ij: (2)

Here,"ijare the coupling strengths from jto i, which aretaken to be either purely inhibitory (all"ij 0) or purelyexcitatory (all "ij 0). T he interaction function U ismonotonically increasing, U0 >0, and represents thesubthreshold dynamics of individual oscillators. We con-sider functions with a curvature of constant sign, i.e.,U00 >0 or U00


2/4

Heterogeneity of the couplings is introduced through

"ij ""ij "JijCij=ki (5)

such that in general the total input is different for everyoscillator, while t he connectivityCij remains fixed. Thematrix Jspecifies the structure of the heterogeneity. Thedegree of heterogeneity is quantified by the disorderstrength ". In numerical simulations, we take the ho-

mogenous part of the coupling strength to be ""ij ""Cij=ki and independently draw the matrix elements Jijfrom the uniform distribution on 1; 1.

The synchronous state (4) of homogenous networks(" 0) is replaced by periodic firing patterns in thepresence of heterogeneity (" >0), cf. Fig. 1. After adiscontinuous transition at a certain critical disorder "cthese patterns disappear and aperiodic, asynchronousstates are observed [14].

The firing patterns emerging for " < "c are con-fined to a subinterval of the phase axis. The extent of apattern

maxi i

mini i (6)

(taking into account wraparound effects at the threshold)increases as the disorder strength " is increased,whereas the firing order is determined by the structureJof the heterogeneity. If" "c the periodic pattern

is replaced either directly by an asynchronous state(Fig. 1), or it reaches this state through a sequence ofstates consisting of several separated clusters of firingpatterns (not shown).

To understand the origin of this transition, we note thatthe critical disorder"c is directly related to the criticalpattern extent c before breakdown [cf. Fig. 1(b)]. Aperiodic pattern is always observed for . For

globally connected networks, the transition to an asyn-chronous state occurs at

globc (7)

(see Fig. 2). If the system is initialized with a pattern ofextent >

globc , some oscillators, which we term

critical, send pulses that affect oscillators that have notfired within the period considered, causing divergencefrom the periodic state. In networks that are not globallyconnected, the firing patterns may persist up to a critical

extent c> globc that depends on the specific net-

work structure. For large random networks, we estimate

an upper bound

max

c for c by assuming that thefiring times (i.e., the phases) within the pattern are uni-formly and independently distributed in an interval oflength [15]. We first calculate the probability

P0 1

1

k 1

1

k1

(8)

that a given oscillatoriis critical, i.e., that of the approxi-mately k pN oscillators j receiving input from i, atleast one satisfiesi j > . In a second step, we cal-culate the probabilityP 1 1 P0N that at least oneof the Noscillators is critical. Setting P 1 guarantees

0 0.1 0.20

0.1

0 0.1 0.20

1

0 0.1 0.20.4

0.5

0.6

Phasesi

0 1 2 30

0.5

1

Time

NetworkRate

0 1 2 30

0.1

0.2

Time

NetworkRate

a

c d

b

=

FIG. 1. Heterogeneities in a random network (N 100, p 0:4, b 2, "" 0:1, 0:1) induce desynchronization fol-

lowed by a transition to an asynchronous, aperiodic state.(a) Relative phases (ten selected phases shown in black) for dif-ferent disorder strengths"(fixedJ). (b) Increase of the extentof the pattern shown in (a), until the pattern disappears atc (dashed line at ). Inset: Onset of asynchro-nous state ( 1) for large disorder. (c),(d) Instantaneousnetwork rate of (c) a periodic (" 0:10) and (d) an aperiodic(" 0:15) state in real time computed using a triangularsliding time window. Gray histograms on the right of (c) and(d) indicate rate distribution [excluding peak at zero networkrate in (c)].

0.05 0.1 0.15 0.2

0.05

0.1

0.15

0.2

c

p=1.0p=0.2fit (p=0.2)prediction

0 0.05 0.1 0.150

1

Prob.

Pattern

FIG. 2. The critical extent of a pattern c as a function ofthe interaction delay (N 100,b 2, "" 0:1). Each datapoint is obtained for a different heterogeneity J in fullyconnected networks () and random networks () of differentconnectivityCwithp 0:2. For the random network, data areaveraged over 20 trials (dashed line: fit). Gray area indicatesthe theoretical second-order prediction of the transition zonebetween

globc and maxc . Inset: Probability that a periodic

pattern emerges from synchronous initial conditions for a givendisorder " (histogram of 100 random heterogeneities a ndconnectivities,p 0:2, 0:1).

P H Y S I C A L R E V I E W L E T T E R S week ending20 FEBRUARY 2004VOLUME92, NUMBER7

074103-2 074103-2


3/4

the existence of a critical oscillator and results in animplicit formula for maxc in terms ofN; p and . Tosecond order in we obtain

maxc 12=k

p (9)

as an approximate upper bound on the critical patternextent c. Numerical data are in good agreement withour theoretical predictions

globc and maxc (cf. Fig. 2).

To examine the stability of periodic firing patterns inthe presence of heterogeneous couplings "ij, we considera pattern

izT 0 i;0; z 2 Z; (10)

of period T where 0 represents a common referencephase and i;0 defines t he relative phase shif t of oscil-lator i within the pattern. For simplicity, we order thefiring events according to i;1 > i;2 > >i;kiwherei;ndenotes the phase shift of the oscillator fromwhich oscillator i receives its nth signal within a givenperiod. For example, if j0 fires first of all oscillatorsconnected toi, then i;1 j0;0, cf. Eq. (10).

By definition, all phases increase uniformly in timeexcept for two kinds of discrete events: sending andreceiving pulses. We follow the dynamics of a periodicstate, i0 iT for all i, event by event (cf. [12])starting with a reference phase 0 chosen such that alloscillators have fired but not yet received the generatedpulses within a given period. The collective period of theunperturbed firing pattern is given by

T 1 i;0 i;ki i;ki

; (11)

wherei;0: and

i;n: U1U

i;n1 i;n1 i;n "i;n (12)

(here with ) are the recursively defined phasesof oscillatori right after having received the first n of itski inputs ("i;1; . . . ; "i;ki , ordered accordingly) in a givenperiod.

Adding small per turbationsi;00 :i;0 satisfying

maxiji;0j < min

ijji;0 j;0j=2 (13)

to the phases of the oscillators i before firing ensurespreservation of the ordering of the firing events: Thus,denoting initial phases as i0 0 i;0, wherei;0 : i;0 i;0, and sorting these relative phaseshifts, i;1 >i;2 > >i;ki , it follows that

i;n

i;n

i;n for all i and for all n. Followingthe evolution of the perturbed state event by event weobtain an expression for the time Ti

2 1i;ki

i;ki to the next firing of i. This leads to a nonlinearperiod-Tmap

i;0T T

Ti

2i;0

i;ki

i;ki

i;ki

(14)

of the perturbations, where we used definition (12) with and . Approximating

i;ki

i;ki

Xkin1

i;n1 i;npi;n1 (15)

to first order in i;0, where pi;n :Qki1

ln U0i;l

i;l i;l1=U0

i;l1 for n < ki and pi;ki 1,

yields

i;0T

XN

j1

Mijj;0: (16)

Here, the matrix Mij has diagonal elements Mii pi;0,and the nonzero, off-diagonal elements are Mij pi;n pi;n1, where j is the index of the oscillator sending thenth signal to i (i.e., j;0 i;n). Time translationinvariance implies

PjMij 1 for all i. Furthermore, if

eitherU00 >0and "" >0orU00


4/4

describes the relative firing times up to " "c [cf.Fig. 3(b)].

The above method may be reversed to design hetero-geneous coupling architectures in order to create a pre-scribed pattern defined by. Here, we fix the periodT0

via the reference oscillatorl, a nd solve each of the planeequationsiderived by linearizing (18) withT0 Tl for asuitable set of"ij. Examples of two networks of differentconnectivity designed to exhibit the same prescribedpattern are shown in Fig. 3(c).

In summary, we have demonstrated that for smallcoupling disorder the synchronous state is replaced byprecisely timed periodic firing patterns. We have shownhow to predict the precise timing of pulses from a givencoupling architecture, and reversely how to design net-works in order to create prescribed patterns. There is acritical disorder strength, which we relate to the i nterac-tion delay, beyond which only asynchronous, aperiodicstates are observed. Similar behavior is expected for othersources of parameter heterogeneity that likewise induce adistribution of effective frequencies in the individualoscillators, such as distributions of delays or interactionfunctions U. The continuous transition found forsmall disorder is very different from that found previouslyfor threshold-induced synchronization where oscillatorsare split into one synchronous and one asynchronoussubpopulation due to heterogeneity [8]. It has been shownrecently that periodic firing patterns can be obtained and

even learned in networks with global inhibition and nointeraction delays as a perturbation of the asynchronousstate [16]. In contradistinction, the patterns discussed inthis Letter a re not related t o (but coexist with) asynchro-nous states, and emerge from the synchronous state due toheterogeneity. The simplicity of the original synchronousstate aided us in clarifying how heterogeneity of networkswith a complicated structure controls their precise dy-

namics. These results are a first step towards understand-ing and designing the dynamics of real world networksthat exhibit complex structure and heterogeneity.

[1] R. A lbert and A.-L. Barabasi, Rev. Mod. Phys. 74, 47(2002); S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys.51, 1079 (2002); M. E. J. Newman, SIAM Rev. 45, 167(2003).

[2] S. Strogatz, Nature ( London) 410, 268 (2001).[3] R. E. Mirollo and S. H. Strogatz, SIAM J Appl. Math.50,

1645 (1990); U. Ernst, K. Pawelzik, and T. Geisel, Phys.Rev. Lett. 74 , 1570 (1995).

[4] A.T. Winf ree, J. Theor. Biol.16, 15 (1967); J. Buck andE. Buck, Science 159, 1319 (1968); C. S. Peskin,

Mathematical Aspect s of Hear t Physiology (CourantInst. Math. Sci., New York, 1975); A.V. M. Herz and J. J.Hopfield, Phys. Rev. Lett. 75, 1222 (1995).

[5] A. T. Winfree, The Geometry of Biological Time(Springer, New York, 2001); L. Glass, Nature (London)410, 277 (2001).

[6] D. Hansel, G. Mato, and C. Meunier, Neural Comput. 7,307 (1995); W. Gerstner, J. L. van Hemmen, and J. D.Cowan, Neural Comput. 8, 1653 (1996); C. van Vreeswijk,Phys. Rev. E 54 , 5522 (1996).

[7] P. C. Bressloff, S. Coombes, and B. de Souza, Phys. Rev.Lett. 79, 2791 (1997); A. Corral, C. J. Perez, A. Daz-

Guilera, and A. Arenas, Phys. Rev. Lett.75, 3697 (1995).[8] M. Tsodyks, I. M itkov, and H. Sompolinsky, Phys. Rev.

Lett.71, 1280 (1993).[9] D. Golomb and J. Rinzel, Phys. Rev. E 48, 4810 (1993).

[10] L. Neltner, D. Hansel, G. Mato, and C. Meunier, NeuralComput. 12 , 1607 (2000).

[11] D. Golomb a nd D. Hansel, Neural Comput. 12, 1095(2000); C. Borgers and N. Kopell, Neural Comput. 15,509 (2003).

[12] M. Tim me, F. Wolf, and T. Geisel, Phys. Rev. Lett. 89,258701 (2002).

[13] M. Denker, M. Timme, M. Diesmann, and T. Geisel (to bepublished).

[14] Numerical simulations start with phases initialized inthe synchronous state (4) of the homogenous network.The disorder"is increased (at fixed J) from" 0i nincrements of103. For each value of", the phases areevolved for 300 firings of a reference oscillator, plotted,and used as initial conditions for the simulation at thenext higher value of".

[15] Numerically we observe a unimodal phase distribution.The assumption of a uniform distribution, however,yields a good approximation (cf. Fig. 2).

[16] D. Z. Jin, Phys. Rev. Lett.89, 208102 (2002); I. J. MatusBloch and C. Romero Z., Phys. Rev. E 66, 036127 (2002).

0 10 200.5

0.52

0.54

Time (Periods)

Phase

0 0.05

0.5

0.54

0.58

Phase

Asynch.0 0.05

0

5

x 103

0 0.02

c

ba

c

FIG. 3. Predicting firing patterns and designing couplingarchitectures to create a prescribed pattern (b 2, "" 0:1, 0:1, predictions to first order). (a),(b) Comparison of pre-diction (gray lines) and simulation (markers) of selected phases

of a pattern (N 100, p 0:4). (a) Phases shown period byperiod for " 0:25. Heterogeneity is instantaneously intro-duced at time t 0. The attractor is reached asymptotically.(b) Pattern for increasing disorder "(Jfixed). Inset: standarddeviation of prediction from simulation. (c) Coupling strengths

(indicated by shading) in two different heterogeneous networks

(N 11, left: p 0:6, right: p 0:3) designed to create aprescribedfiring pattern (middle row, gray). Patterns in blackare obtained by simulating these designed networks (initialized

to the synchronous state).

P H Y S I C A L R E V I E W L E T T E R S week ending20 FEBRUARY 2004VOLUME92, NUMBER7

074103-4 074103-4

denker 2004 breaking

Documents