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Spatial coherence resonance on diffusive and small-world networks of Hodgkin-Huxley neurons

Xiaojuan Sun∗

School of Science, Beihang University, Beijing 100083, China and Institute of Physics,University of Potsdam, PF 601553, 14415 Potsdam, Germany

Matjaz Perc†

Department of Physics, Faculty of Natural Sciences and Mathematics, University ofMaribor, Koroska cesta 160, SI-2000 Maribor, Slovenia

Qishao Lu‡

School of Science, Beihang University, Beijing 100083, China

Jurgen Kurths§

Institute of Physics, University of Potsdam, PF 601553, 14415 Potsdam, Germany

Spatial coherence resonance in a spatially extended systemthat is locally modeled by Hodgkin-Huxley (HH)neurons is studied in this paper. We focus on the ability of additive temporally and spatially uncorrelatedGaussian noise to extract a particular spatial frequency ofexcitatory waves in the medium, whereby examiningalso the impact of diffusive and small-world network topology determining the interactions amongst coupledHH neurons. We show that there exists an intermediate noise intensity that is able to extract a characteristicspatial frequency of the system in a resonant manner provided the latter is diffusively coupled, thus indicatingthe existence of spatial coherence resonance. However, as the diffusive topology of the medium is relaxed viathe introduction of shortcut links introducing small-world properties amongst coupled HH neurons, the abilityof additive Gaussian noise to evoke ordered excitatory waves deteriorates rather spectacularly, leading to thedecoherence of the spatial dynamics and with it related absence of spatial coherence resonance. In particular,already a minute fraction of shortcut links suffices to substantially disrupt coherent pattern formation in theexamined system.

PACS numbers: 05.45.-a, 05.40.-a, 87.18.Sn, 89.75.Kd

Nontrivial effects of noise on nonlinear dynamics havebeen a vibrant topic for many years. It is thoroughly doc-umented and established that noise can play a construc-tive role in different types of nonlinear dynamical sys-tems. Stochastic and coherence resonance are just two,perhaps most prominent, examples of this fact. The no-tion of coherence resonance is particularly inspiring as itmessages that an appropriate intensity of noise alone issufficient to evoke ordered temporal responses of a non-linear dynamical system. Nowadays, however, effects ofnoise on spatially extended systems have gradually slippedinto the focus of many scientists working in diverse fieldsof research, consequently spawning the need to investigatewhether phenomena observed previously for isolated dy-namical system can also be observed, at least conceptuallysimilar, if the latter are coupled. Indeed, it has been shownthat the coherence resonance phenomenon originally re-ported for dynamical systems evolving only in time canalso be observed in spatially extended systems that arelocally described by excitable nonlinear dynamics. Im-portantly thereby is the fact that the previously studiedorder in the temporal dynamics has been ”replaced” bythe order of the noise-induced spatial dynamics, whichmostly manifests as propagating waves of excitatory eventsthroughout the spatial grid. Presently, we aim to extendthe scope of spatial coherence resonance by confirming itspossibility also in models of neuronal dynamics, in par-ticular by employing as the constitutive unit of the spa-

tially extended system the renowned HH model. We showthat while the diffusive connectivity of coupled neuronswarrants the observation of noise-induced pattern forma-tion and with it related spatial coherence resonance, thesmall-world topology is not an appropriate medium forsuch observations. More precisely, even a minute fractionof shortcut links amongst distant neurons prohibits noise-induced waves to be ordered, and hence also precludesthe observation of spatial coherence resonance. Since thepresent study is set-up around a comprehensive HH modelof neuronal dynamics, the presented results should provevaluable not just from the purely theoretical point of view,but hopefully also from the experimental point of view, es-pecially by shedding light into the functioning of neuraltissue.

I. INTRODUCTION

Randomness is a common feature in the real world. It iswell known that noise can play a constructive role in differ-ent nonlinear dynamical systems, such as optical devices,1

electronic circuits,2 or neural tissue.3 Stochastic and coher-ence resonance are two prominent examples of such construc-tive effects that rely on the influence of noise on nonlinearsystems.4,5 Remarkably, by the coherence resonance6,7,8 anenhanced ordered behavior results solely from the introduc-tion of noise in the absence of additional weak determinis-

2

tic signals that are otherwise a standard ingredient by the ob-servation of stochastic resonance.4 The phenomenon of co-herence resonance has been studied extensively also in non-identical coupled neurons9,10 as well as one-dimensional ar-rays of nonlinear dynamical systems in general.11,12,13Zhouand Kurths14 showed, for example, that in a weakly coupledregion noise could induce spatio-temporal coherence reso-nance but reduced the degree of synchronization by strongercouplings.

Recently, many authors have shifted their interests to the in-fluences of noise on two-dimensional spatially extended sys-tems, and to their spatial rather than temporal dynamics inparticular.15 Spatio-temporal stochastic resonance has beenreported first in,16 while the spatial coherence resonance hasalso been studied in many different types of spatially extendedsystems, e.g. in chlorine dioxide-iodine-malonic reactions,17

Rulkov maps,18 or excitable biochemical media.19 More pre-cisely, Carrillo et al.17 demonstrated that spatial coherenceresonance could be evoked close to pattern-forming instabil-ities, mimicking one of the mechanisms of standard tempo-ral coherence resonance, while Perc et al.18,19 showed that inlocally discrete excitable spatially extended systems andex-citable biochemical media additive or internal noise couldalsolead to the observation of spatial coherence resonance. Im-portantly, however, there is still a lack of comprehensive stud-ies investigating whether the above results concerning spatialcoherence resonance can be obtained also in more complexspatially extended systems, particularly such that are locallymodeled by realistic nonlinear dynamical systems that faith-fully describe a real-life biological process, as is for examplethe neuronal activity.

In this paper, we would like to address the above-describedvoid and extend the scope of spatial coherence resonanceby examining the possibilities of its existence in a spatiallyextended system that is locally modeled by excitable HHneurons.20 More precisely, we analyze spatial frequency spec-tra of the examined medium in dependence on different lev-els of additive noise and topologies of the interaction networkconstituting the couplings amongst the HH neurons. By calcu-lating the average spatial structure function, we present con-clusive evidences for spatial coherence resonance providedthe neurons are diffusively coupled. In particular, we showthat then there exists an optimal level of additive noise forwhich a particular spatial frequency of excitatory events inthe medium is best pronounced. We emphasize that therebyno additional deterministic inputs are introduced to the sys-tem, and the latter is locally initiated from steady state initialconditions. In contrast with results obtained when employ-ing diffusive nearest-neighbor interactions, however, wefindthat small-world interaction networks21,22 fail to sustain co-herent patterns of spatial noise-induced excitations, andthusdo not warrant the observation of spatial coherence resonance.More precisely, even a minute fraction of rewired links, con-necting two nearest neighbors in case of diffusive coupling,heavily disrupts coherent pattern formation in the medium andinduces decoherence of excitatory waves. We thus reveal theimpact of additive noise and small-world topology on an en-semble of HH neurons, hence providing interesting theoretical

insights into to the functioning of neural tissue.The paper is structured as follows. Section II is devoted to

the description of the employed spatially extended system andthe HH mathematical model as its main ingredient. SectionsIII and IV feature evidences for spatial coherence resonanceon diffusive grids and decoherence of pattern formation onsmall-world networks, respectively, while in the last Sectionwe summarize the results and outline biological implicationsof our findings.

II. MATHEMATICAL MODEL

In 1952, Hodgkin and Huxley presented a mathematicalmodel to predict the quantitative behavior of an isolated squidgiant axon.20 Since then, the model has become a paradigmfor mathematically describing neuron functioning, and manyauthors have studied its nonlinear dynamics. In particular, ithas been discovered that with the change of an external cur-rent injected into the cell membrane, different bifurcationscan occur rendering a stable HH neuron oscillatory, bursting,or even chaotic.23,24,25,26,27,28Moreover, the dynamics of cou-pled HH neurons has also been studied quite extensively inthe past.29,30,31In this paper, we employ the HH model as theconstitutive unit of a spatially extended system, and studytheeffects of different intensities of additive noise and topologiesof the interaction network on the spatial dynamics of excita-tory waves. The equations of a single HH model are given asfollows:

CdV/dt = −gNam3h(V − VNa) − gKn4(V − VK)

−gL(V − VL) + Iext, (1a)

dm/dt = αm(1 − m) − βmm, (1b)

dh/dt = αh(1 − h) − βhh, (1c)

dn/dt = αn(1 − n) − βnn. (1d)

In Eq. (1)V is the membrane potential of the neuron,Iext isthe external stimulus current, whereasm, h, andn are the gat-ing variables describing the ionic transport through the mem-brane. Moreover, the constantsgNa, gK , andgL are maximalconductances for ion and leakage channels, whileVNa, VK ,andVL are the corresponding reversal potentials. The func-tionsαm, αh, αn, βm, βh, andβn are defined as:

αm = 0.1(V + 40)/1− exp[−(V + 40)/10], (2a)

αh = 0.07 exp[−(V + 65)/20], (2b)

αn = 0.01(V + 55)/1− exp[−(V + 55)/10], (2c)

βm = 4.0 exp[−(V + 65)/18], (2d)

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βh = 1 + exp[−(V + 35)/10]−1, (2e)

βn = 0.125 exp[−(V + 65)/80]. (2f)

In the literature one may find different forms of the HH equa-tions, mainly depending on the zeros of the potential. Herewe choose the form so that the membrane potential at restis V ≈ −61mV. Then according to the studies of Hodgkinand Huxley,20 the parameter values in Eq. (1) are as follows:C = 1µF/cm2, gNa = 120ms/cm2, gK = 36ms/cm2, gL =0.3ms/cm2, VNa = 50mV, VK = −77mV, and VL =−54.4mV.

First, we outline some properties of a single HH neuronevoked by the above parameters dependent on the externalstimulusIext.23 WhenIext < I0 ≈ 6.2 there exists only aglobally stable fixed point. The birth of stable and unstablelimit cycles occurs atIext = I0 ≈ 6.2 via a saddle-node bi-furcation. ForI0 < Iext < I1 ≈ 9.8 there coexist a stablefixed point, a stable limit cycle, and an unstable limit cycle.The unstable limit cycle constitutes the boundary separatingthe attractive basins corresponding to the fixed point and thelimit cycle, respectively. For a more precise description of thebifurcation structure of the HH model, in dependence also onother system parameters, we refer to.23,24,25,26,27,28

Based on the original HH model given by Eqs. (1) and (2),we introduce the spatially extend system

CdVi,j/dt = −gNam3i,jhi,j(Vi,j − VNa)

−gKn4i,j(Vi,j − VK) + σξi,j(t)

−gL(Vi,j − VL) + Iext

+D∑k,l

εi,j,k,l(Vk,l − Vi,j), (3a)

dmi,j/dt = αmi,j(1 − mi,j) − βmi,j

mi,j , (3b)

dhi,j/dt = αhi,j(1 − hi,j) − βhi,j

hi,j , (3c)

dni,j/dt = αni,j(1 − ni,j) − βni,j

ni,j , (3d)

where subscriptsi, j = 1, . . . , N denote each of theN × Ncoupled HH neurons. The sum in Eq. (3) runs over all lat-tice sites wherebyεi,j,k,l = 1 if the site (k, l) is coupledto (i, j) and εi,j,k,l = 0 otherwise. If the fraction of ran-domly rewired linksq, constituting the small-world topologyif 0 < q ≪ 1, equals zero,εi,j,k,l = 1 only if (k, l) isone of the four nearest neighbors of site(i, j) on a regulartwo-dimensional mesh. Thereby, we obtain a diffusively cou-pled network of HH neurons as depicted in Fig. 1(a), whichwill form the backbone for the study of noise-induced spatialdynamics in Section III. Ifq > 0, however, the correspond-ing fraction of nearest-neighbor links is randomly rewiredviathe analogy with the creation of Watts-Strogatz small-worldnetworks,21,22 whereby we preserve the initial connectivityz = 4 (as by the diffusive nearest-neighbor coupling) ofeach HH neuron as depicted in Fig. 1(b) to focus explicitly

(a) (b)

FIG. 1: Examples of considered network topologies. For clarity onlya6× 6 excerpt of the whole network is presented in each panel. (a)Diffusively coupled network characterized byq = 0. Each vertex isdirectly connected only to its four nearest neighbors, hence havingconnectivityz = 4. (b) Realization of small-world topology viarandom rewiring of a certain fractionq of links, constrained only bythe requirement that the initial connectivityz = 4 of each unit mustbe preserved.

on the effect of network randomness rather than possible ef-fects originating from different numbers of inputs per neuron.The precise procedure for generating such small-world net-works is given in,32 while results obtained when using themas interaction networks for the noise driven HH spatially ex-tended system will be presented in Section IV. Importantly,each interaction network was generated at the beginning ofa particular simulation and was held fixed, and if necessary,results presented below were averaged over30 different real-izations of the interaction network by eachq. Turning backto Eq. (3),σ is the main control parameter to be varied inthis study, denoting the standard deviation of additive uncor-related Gaussian noiseξi,j that satisfies< ξi,j(t) >= 0 and< ξi,j(t)ξm,n(t

′

) >= δ(t− t′

)δi,mδj,n. Moreover,D = 0.35is the presently employed coupling strength, while all otherparameters are taken the same as mentioned by the descrip-tion of a single HH neuron. In the present study the periodicboundary condition is used, namely:V0,j = VN,j , VN+1,j =V1,j , Vi,0 = Vi,N , andVi,N+1 = Vi,1, and the system size isN = 128 in each spatial dimension of the two-dimensionalgrid. In order to study explicitly the impact of noise on thespatial dynamics of the system, we take the external currentIext = 6.1 (in accordance with the above outlined propertiesof a single HH neuron assuring a unique stable steady statesolution of each coupled unit) and initiate all units from theexcitable steady state(−61.198, 0.08199, 0.46014, 0.37727).Thus, without the addition of noise the medium would remainquiescent forever, and consequently the dynamics of below-discussed spatial patterns is evoked solely by additive Gaus-sian noise.

III. SPATIAL COHERENCE RESONANCE BY DIFFUSIVECOUPLING

As announced, in this section we takeq = 0 wherebythe coupling termD

∑εi,j,k,l(Vk,l − Vi,j) in Eq. (3) reads

4

(c)(b)(a)

FIG. 2: Characteristic snapshots of the spatial profile ofVi,j obtained for (a)σ = 1.1, (b) σ = 1.3 , and (c)σ = 1.9. For the intermediatevalue ofσ the spatial dynamics is clearly most coherent, exhibiting ordered circular waves propagating through the spatial grid.

asD(Vi−1,j + Vi+1,j + Vi,j−1 + Vi,j+1 − 4Vi,j). We startby visually examining three snapshots of the spatial profileof Vi,j obtained by three different values ofσ, as presentedin Fig. 2. Evidently, there exists an intermediate value ofσ,at which nicely ordered circular excitatory waves propagatethrough the spatial domain [Fig. 2(b)]. On the other hand,smaller or larger values ofσ clearly fail to have the same ef-fect evoking either only small-amplitude deviations from thesteady state [Fig. 2(a)] or rather violent and uncorrelatedex-citations throughout the spatial grid [Fig. 2(c)], respectively.Hence, the visual inspection of the snapshots already givessome indication for a typical coherence resonance scenariofor the spatial dynamics of the studied HH medium.

To make the above observations quantitative, we calculatethe spatial structure function of theVi,j field according to

P (kx, ky) = 〈H2(kx, ky)〉, (4)

whereH(kx, ky) is the spatial Fourier transform of theVi,j

field at a particular timet, and〈...〉 is the ensemble averageover noise realizations. The results from numerical simula-tions for differentσ are presented in Fig. 3. The three de-picted panels correspond to the same values ofσ as used al-ready in Fig. 2. Indeed, the results in Fig. 3 fully supportour visual assessments, as it can be observed nicely that forsmall and large noise levels [Fig. 3(a) and (c), respectively] thepresented spectra show no particularly expressed spatial fre-quency, although a close examination ofP (kx, ky) atσ = 1.9still reveals some remnants of structure formation in the sys-tem. Only for intermediate levels of noise the spatial structurefunction develops several well-expressed circularly symmet-ric rings, indicating the existence of a preferred spatial fre-quency induced by additive Gaussian noise. As the noise levelis increased, random fluctuations start to dominate the spatialdynamics and thus, similar as by small noise levels, the char-acteristic waterfall-like outlay ofP (kx, ky) vanishes and nopreferred spatial frequency can be inferred.

Next, we exploit the circular symmetry ofP (kx, ky) as pro-posed by Carrillo et al.17 to obtain an estimate for the signal-to-noise ratio (SNR) of the spatial dynamics of excitatoryevents for differentσ. In particular, we calculate the circu-

lar average of the structure function according to the equation

p(k) =

∫Ωk

P (~k)dΩk, (5)

where~k = (kx, ky), and Ωk is a circular shell of radiusk = |~k|. Figure 4(a) showsp(k) for the three differentσcorresponding to the values used already in Figs. 2 and 3.The presented results reveal that there indeed exists a partic-ular spatial frequency, marked with the vertical dashed lineat k = kmax, that is resonantly enhanced for some interme-diate level of additive Gaussian noise. Note that subsequentlocal maxima are just higher harmonics ofkmax. To quan-tify the ability of eachσ to extract the characteristic spatialfrequency of the medium more precisely, we calculate theSNR as the peak height atkmax normalized with the back-ground fluctuations in the system; namelyp(kmax)/p, wherep = [p(kmax − ∆ka) + p(kmax + ∆kb)]/2 is an approxi-mation for the level of background fluctuations in the system,whereby∆ka and∆kb mark the estimated width of the peakaroundkmax at the optimalσ. This is the spatial counterpartof the measure frequently used for quantifying constructive ef-fects of noise in the temporal domain of dynamical systems,4

whereas a similar measure for quantifying effects of noise onthe spatial dynamics of spatially extended systems was alsoused in.17 Figure 4(b) shows how the SNR varies withσ. It isevident that there exists an optimal level of additive Gaussiannoise for which the peak of the circularly averaged structurefunction is best resolved, thus clearly indicating the existenceof spatial coherence resonance in the diffusively coupled HHmedium.

We argue that the existence of spatial coherence resonancein the studied HH medium, and with it related remarkable or-der of the spatial dynamics of noise-induced excitatory eventsobserved by intermediateσ, must be primarily attributed tothe characteristic noise-robust excursion time inherent to allexcitable systems.8 The noise-robust excursion time, togetherwith the rate of diffusive spread that is proportional to

√D,

introduces an eigenfrequency of spatial waves that can be res-onantly enhanced by an appropriate intensity of noise, as re-cently shown analytically for a simple toy model of excitable

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FIG. 3: The spatial structure function ofVi,j obtained for (a)σ = 1.1, (b) σ = 1.3, and (c)σ = 1.9. For the intermediate value ofσ thecharacteristic waterfall-like outlay ofP (kx, ky) is evident, indicating the existence of a preferred spatialfrequency of noise-induced excitatoryevents in the medium. Note that only an excerpt of the wholeP (kx, ky) plane is shown in all panels.

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FIG. 4: Evidences for spatial coherence resonance in the studied HH medium. (a) The circular averages of structure functions presented inFig. 3. The dashed vertical line atk = kmax marks the spatial frequency of excitatory events that is resonantly enhanced by an intermediatelevel of additive Gaussian noise. (b) The signal-to-noise (SNR) ratio in dependence onσ (the curve is just a guide to the eye).

dynamics.33 Our results thus extend the scope of spatial co-herence resonance by showing that real-life motivated morecomplex models of excitable dynamics, such as the HH math-ematical model,20 fully conform to findings obtained on sim-pler models, and that indeed this phenomenon may have im-portant implications in several biological systems.

IV. DECOHERENCE OF SPATIAL DYNAMICS BYSMALL-WORLD INTERACTIONS

Next, it is of interest to investigate what are the impactsof small-world topology on the above-described phenomenonof spatial coherence resonance in the diffusively coupled HHmedium. For this purpose we setq > 0 and examine threecharacteristic snapshots ofVi,j obtained byσ = 1.3, whichaccording to the results presented in Section III is the optimalnoise intensity warranting superbly ordered spatial dynamicsof excitatory events. Noteworthy, since we consider regular

small-world networks warranting an equal number of inputs(z = 4) to each coupled neuron, the optimalσ for coherentpattern formation (if at all possible) does not depend onq, aswill be confirmed also in Fig. 6(a) below. The snapshots pre-sented in Fig. 5 clearly evidence that increasing values ofqprogressively hinder coherent pattern formation. In particular,while for q = 0.0001 [Fig. 5(a)] the spatial periodicity of ex-citatory waves is still eligible, the same is much less true forq = 0.0008 [Fig. 5(b)], and is certainly completely false forq = 0.002 [Fig. 5(c)]. Thus, if as little as0.08 % of nearest-neighbor links are rewired the pattern formation is impaired,while by≈ 0.2 % noisy perturbations are completely unableto induce coherent spatial dynamics of excitatory events. In-deed, the snapshot presented in Fig. 5(c) lacks any orderedstructure, although the same intensity of noise applied to adiffusively coupled HH medium evokes superbly ordered cir-cular waves [see Fig. 2(b)] evidencing fascinating examplesof spatial pattern formation out of noise.

To give a quantitative analysis of the observed phenomenon

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(c)(b)(a)

FIG. 5: Characteristic snapshots of the spatial profile ofVi,j obtained at the optimal value ofσ = 1.3 for (a)q = 0.0001, (b) q = 0.0008, and(c) q = 0.002.

outlined in Fig. 5, we calculate the circularly average struc-ture functionp(k) of the spatial dynamics in dependence onσ andq. The inset in Fig. 6(b) shows threep(k) obtained bythe same parameter values yielding snapshots of spatial pro-files shown in Fig. 5. In accordance with the interpretation ofresults presented in Section III, it can be inferred that increas-ing values ofq indeed impair the ability of noise to evoke or-dered patterns with a predominant spatial frequency, becausethe peak height ofp(k) atk = kmax becomes smaller in com-parison to the level of background noise as the fraction ofrewired links increases. While byq = 0.0001 andq = 0.0008the predominant spatial periodicity is still present, it vanishescompletely byq = 0.002 as the outlay ofp(k) becomes essen-tially flat. These observations can be captured succinctly bycalculating the SNR in dependence onσ by differentq. Theresults presented in Fig. 6(a) evidence clearly that the phe-nomenon of spatial coherence resonance fades continuouslyasq increases, disappearing almost completely byq = 0.002.As already noted, however, the optimalσ for coherent patternformation (as much as allowed by the topology of the under-lying interaction network) remains virtually identical byall q,equalingσ = 1.3, as denoted by the dashed vertical line inFig. 6(a). We exploit this fact in Fig. 6(b), where the SNRis plotted in dependence onq for the optimalσ. Clearly, theSNR decreases continuously asq increases, thus evidencingthe decoherence of noise-induced spatial dynamics due to theintroduction of small-world topology in the studied HH spa-tially extended system.

While the existence of spatial coherence resonance in thediffusively coupled HH medium was attributed to the char-acteristic noise-robust excursion time of the local excitabledynamics,8 which together with the rate of diffusive spreadingproportional to

√D introduces an eigenfrequency, or charac-

teristic spatial scale, of waves that can be resonantly enhancedby an appropriate intensity of noise,33 the decoherence of thespatial dynamics due to the introduction of small-world topol-ogy is simply a consequence of the disruption of this inherentspatial scale.34 In particular, while the excursion time of in-dividual space units remains unaltered by the introductionoflong-range couplings, the spread rate of excitations is indi-rectly very much affected by increasing values ofq. Namely,

the introduction of shortcut links decreases the typical pathlength between two arbitrary sites in comparison to a diffu-sively coupled network. Thus, while the spread rate of exci-tations is still proportional to

√D, the typical path length be-

tween two arbitrary grid units decreases dramatically, whichin turn has the same effect as ifD would increase. Therefore,in a small-world network, a locally induced excitation can in-stantly reach much more distant sites than in case of strictdiffusive coupling, in turn facilitating noise-induced synchro-nization of distinct network units,35 or assuring fast responseabilities of the system.36 Importantly, however, the typicalpath length between two arbitrary sites decreases only on av-erage, meaning that there does not exist an exact path lengthdefining the distance between all possible pairs of sites in asmall-world network. Thus, a given local excitation can, dur-ing the excursion time, propagate to the most distant site orjust to its nearest neighbor, whereby clearly the well-definedinherent spatial scale existing for the diffusive couplingislacking, and this leads to spatial decoherence of noise-inducedspatial patterns already for very small values ofq, as empha-sized throughout this Section.

V. SUMMARY

We have shown that temporally and spatially uncorrelatedadditive Gaussian noise is able to extract a characteristicspa-tial frequency of an excitable HH medium in a resonant man-ner if the latter is coupled diffusively. In particular, it thenexists an optimal level of additive noise for which the spatialdynamics of the system is highly coherent and superbly or-dered. Thereby, no additional deterministic inputs were intro-duced to the system and all units were initiated from steady-state excitable conditions. Thus, the results presented inSec-tion III offer convincing evidences for the existence of spatialcoherence resonance in the diffusively coupled HH medium.On the other hand, when the interactions amongst coupledneurons are governed by a small-world network, the abilityof noise to induce ordered spatial dynamics vanishes quicklyas q increases. Our results suggest that as little as0.2 %

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FIG. 6: Evidences for decoherence of the spatial dynamics inthe studied HH medium with small-world topology. (a) The signal-to-noiseratio (SNR) in dependence onσ for differentq. (b) SNR in dependence onq by the optimalσ denoted by a dashed vertical line in panel (a).Inset features the circular average of the structure function p(k) for the optimalσ and three different values ofq, corresponding to the threesnapshots presented in Fig. 5. The values depicted in the main panels resulted from averaging the results over 30 different realizations of theinteraction network by eachq, whereas curves are just guides to the eye.

of rewired nearest-neighbor links suffice to completely hin-der coherent pattern formation out of noise within the cur-rently employed setup. Noteworthy, we have performed addi-tional simulations using the FitzHugh-Nagumo system37 andthe Goldbeter model for calcium oscillations38 as governingunits of the medium, whereby the former excitable dynam-ics is governed by a steady node while the latter is charac-terized by a stable focus situated prior to a Hopf bifurcation.Irrespective of these particularities, we were able to observecoherent patterns emerging out of noise provided the unitswere diffusively coupled. Conversely, if small-world connec-tivity was introduced the ordered waves vanished quickly al-ready byq ≪ 1, albeit stable foci could sustain some orderin the spatial dynamics up to0.5 % of rewired links due tolarger re-settlement times associated with their excitable dy-namics. Larger re-settlement times following an excitationyield waves with larger wavelengths, and hence then the prob-ability of shortcuts to link two quiescent units increases,thusleaving the spatial dynamics temporarily unaffected. Asidefrom these rather minute differences, however, we concludethat the present findings appear to be robust with respect tovariations in models governing the local excitable dynamicsof diffusive and small-world networks.

Since the present results rely on the use of the HH model forthe description of the local dynamics of the excitable medium,

the present study might also have some biological implica-tions. Specifically for neural systems, it has been argued thatexcitable media guarantee robust signal propagation throughthe tissue in a substantially noisy environment.39 It would thusbe fascinating to elucidate if spatial coherence resonanceinthe neural system can be confirmed also experimentally. Theabove theoretical results indicate that such findings mightin-deed be attainable at least if the units are diffusively coupled,and that it thus seems reasonable to pursue this problem inthe future. Moreover, our results obtained when using small-world networks may help by revealing structural functionalityof complex topologies within the neural apparatus, althoughadditional studies regarding the concept of function-follow-form40,41,42,43in the presence of noise and other uncertaintiesare necessary to clarify the importance of different structuralphysiological properties of such networks.

Acknowledgments

X. Sun and Q. Lu are grateful for the support of the Na-tional Science Foundation of China (Grants 10432010 and10702023). M. Perc acknowledges support from the Slove-nian Research Agency (Grant Z1-9629).

∗ Electronic address: sxiaojuan@ss.buaa.edu.cn† Electronic address (corr.): matjaz.perc@uni-mb.si‡ Electronic address: qishaolu@hotmail.com§ Electronic address: jkurths@agnld.uni-potsdam.de1 A. Fioretti, L. Guidoni, R. Mannella, and E. Arimondo, J. Stat.

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