Control of coherence resonance in multiplex neural networks
CControl of coherence resonance in multiplex neuralnetworks
Maria Masoliver Cristina Masoller and Anna Zakharova Departament de Fisica, Universitat Politecnica de Catalunya, Rambla Sant Nebridi 22,08222 Terrassa, Barcelona, Spain Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623Berlin, Germany
Abstract
We study the dynamics of two neuronal populations weakly and mutually coupledin a multiplexed ring configuration. We simulate the neuronal activity withthe stochastic FitzHugh-Nagumo (FHN) model. The two neuronal populationsperceive different levels of noise: one population exhibits spiking activity inducedby supra-threshold noise (layer 1), while the other population is silent in theabsence of inter-layer coupling because its own level of noise is sub-threshold(layer 2). We find that, for appropriate levels of noise in layer 1, weak inter-layercoupling can induce coherence resonance (CR), anti-coherence resonance (ACR)and inverse stochastic resonance (ISR) in layer 2. We also find that a smallnumber of randomly distributed inter-layer links are sufficient to induce thesephenomena in layer 2. Our results hold for small and large neuronal populations.
Keywords: synchronization, multiplex network, coherence resonance,FitzHigh-Nagumo neuron
1. Introduction
A fundamental challenge of complexity science is to understand synchro-nization and emergent phenomena in complex systems represented by sets ofexcitable units coupled with different topologies. Multilayer networks are receiv-ing increasing attention because they represent many real-world systems [1, 2, 3].Multilayer networks are composed of interconnected layers, where each layer is
Preprint submitted to Chaos, Solitons & Fractals September 22, 2020 a r X i v : . [ n li n . AO ] S e p ormed by a set of N units or nodes, whose interactions are represented by links.In the case when the inter-layer interactions are only vertical, e.g., node i inlayer 1 is linked to node i in layer 2, the network is called multiplex. Multiplexnetworks represent, therefore, a special class of multilayer networks where thelayers contain the same number of nodes and the inter-layer links are allowedonly for replica nodes, i.e., there is a one-to-one correspondence between thenodes in different layers. In this work, we focus on a two-layer multiplex networkwhere each layer is formed by N neurons coupled in a ring configuration (seeFig. 1).We investigate the phenomenon of coherence resonance that correspondsto the state of the network characterized by high temporal regularity of noise-induced oscillations achieved for an intermediate optimal noise intensity [4, 5, 6, 7].This phenomenon is an example of the constructive role of noise in excitabledynamical systems [8]. Coherence resonance has been reported not only inexcitable [9, 10, 11, 12, 13], but also in non-excitable systems [14, 15, 16, 17, 18,19]. In complex networks of FitzHugh-Nagumo units, it has been investigatedin one-layer [6] and two-layer [7] networks. Further topologies include local,nonlocal, global coupling, lattice networks as well as more complex structuressuch as random or small-world networks [20, 21, 22, 23, 6, 24].One of the most relevant and at the same time challenging questions is relatedto the control of coherence resonance. A well-studied mechanism of coherenceresonance control is based on time delay. For example, the control of coherenceresonance in a one-layer network of delay-coupled FitzHugh-Nagumo neuronshas been investigated in [6]. Moreover, time-delayed feedback control has beenapplied to a special type of coherence resonance called coherence-resonancechimera occurring in a ring of nonlocally coupled excitable FitzHugh-Nagumosystems [25, 26, 27].Multilayer networks offer new possibilities of control via the interplay betweendynamics and multiplexing. The advantage of this method is that it allowsregulating the dynamics of one layer by adjusting the parameters of the otherlayer [28, 29]. Recently, the so-called weak multiplexing control has been reported2nd applied to coherence resonance [7] and chimera states [30, 31]. The distictivefeature and the advantage of this control scheme is the possibility of achievingthe desired state in a certain layer without manipulating its parameters and inthe presence of weak coupling between the layers (i.e., the coupling between thelayers is smaller than that inside the layers). While the time-delayed feedbackcontrol of coherence resonance has been well-understood, the multiplexing controlhas been much less investigated.The aim of this work is to study coherence resonance in a two-layer network ofFitzHugh-Nagumo neurons [32, 33] with weak inter-layer coupling. In particular,we focus on the case of unequally noisy layers: a noisy layer (layer 1) thatdisplays spiking activity induced by supra-threshold noise, is multiplexed witha “silent” layer (layer 2), which has subthreshold noise, and whose spikingactivity is induced by weak coupling to layer 1. Recently, the possibility ofinducing coherence resonance in the silent layer has been shown [7]. Here weanalyze the role of the system size and the impact of removing inter-layer links.We find that not only coherence resonance, but also, anti-coherence resonance(ACR) and inverse stochastic resonance (ISR) can be induced in layer 2. ACR ischaracterized by high temporal irregularity of noise-induced oscillations [34] andISR is characterized by noise suppression of oscillations (the average spiking rateof a neuron exhibits a minimum with respect to noise) [35, 36]. We also findthat a small number of randomly distributed inter-layer links can be sufficientto induce these phenomena in layer 2.This paper is organized as follows: Sec. 2 presents the model; sec. 3 presentsthe measures used to quantify the regularity of the neuronal spiking activity,sec. 4 presents the results and sec. 5 summarizes our conclusions.
2. Model
We study a two-layer multiplex network schematically represented in Fig. 1.Each layer is a ring of N FitzHugh-Nagumo (FHN) neurons [32, 33] in theexcitable regime with Gaussian white noise. In each layer each neuron has3wo neighbors, one in each direction of the ring. All the links (intra-layer andinter-layer) are diffusive and bidirectional. The model equations are: (cid:15) du i dt = u i − ( u i ) / − v i + σ i +1 (cid:88) j = i − ( u j − u i ) + µ i σ ( u i − u i ) + (cid:112) D ζ i ( t ) ,dv i dt = u i + a,(cid:15) du i dt = u i − ( u i ) / − v i + σ i +1 (cid:88) j = i − ( u j − u i ) + µ i σ ( u i − u i ) + (cid:112) D ζ i ( t ) ,dv i dt = u i + a. Here u ki and v ki are the activator variable (i.e., voltage-like variable) and theinhibitor variable respectively; index i ( i = 1 . . . N ) denotes the i -th neuron ineach of the two layers while index k ( k = 1 ,
2) denotes the layer in which theneuron is located.The parameter σ denotes the coupling strength between neurons in the samelayer that we refer to as intra-layer coupling. The strength of the couplingbetween the layers (that we refer to as inter-layer coupling) is characterizedby the parameter σ . Here we focus on the “weak multiplexing” situation, inwhich the inter-layer coupling is much weaker than the intra-layer coupling (i.e., σ << σ ).The deterministic bifurcation parameter is a . The uncoupled ( σ = σ = 0)and deterministic ( D = D = 0) neurons undergo a Hopf bifurcation at a = 1:for | a | < | a | > a = 1 .
05 and (cid:15) = 0 .
01 constant. The small parameter (cid:15) is responsible for thetime scale separation of fast activator and slow inhibitor. ζ i ( t ) and ζ i ( t ) represent uncorrelated Gaussian white noise sources whereas D and D represent the noise intensities, respectively. As we are interestedin understanding how the activity of the neurons in layer 1 excite the neuronsin layer 2, D and D are chosen such that D is supra-threshold (i.e., noiseinduces spiking of the neurons in layer 1) while D is sub-threshold (i.e., neuronsin layer 2 are excited through the multiplex coupling σ : if σ = 0, neurons in4 igure 1: Schematic representation of the network under study: a multiplex neural networkconsisting of two layers coupled through the inter-layer coupling σ . Nodes within each layerare coupled through the intra-layer couplings σ . Gaussian white noise is applied to both layers,only the noise applied to layer 1 is supra-threshold. layer 2 do not fire). We vary D as a control parameter and keep D = 2 . · − constant.
3. Methods
To quantify coherence resonance (i.e., noise-induced regularity of the spikingactivity) we use the coefficient of variation, R , of the distribution of inter-spike-intervals [5]. It is computed for neurons in layer k = 1 or in layer k = 2as R k = σ ISIk / (cid:104) ISI (cid:105) k (1)where the mean, (cid:104) ISI (cid:105) k , and the standard deviation, σ ISIk , of the inter-spikeintervals (ISIs) are calculated by averaging over time and over space (i.e., byaveraging the inter-spike intervals in all the spike sequences of all the neuronsin layer k ). If layer k shows coherence resonance, there will be a pronouncedminimum of R k with respect to the noise strength D ( D is kept fixed below the5ring threshold). On the other hand, if a layer shows anti-coherence resonance,there will be a maximum of R k with respect to D [34].
4. Results
We begin by analyzing the most simple configuration: one neuron in eachlayer (i.e., two diffusely and bidirectionally coupled FHN neurons, one withsupra-threshold noise, and the other, with sub-threshold noise).Figures 2(a), (b) display R and R vs. the level of supra-threshold noise, D , for different values of the coupling strength, σ . For neuron 1, R showsthe characteristic minimum of coherence resonance, and we see that R iseither unaffected (for strong noise) or only slightly affected (for weak noise)by the coupling to neuron 2. This is due to the fact that we consider “weakmultiplexing”, i.e., the two neurons are weakly coupled.For neuron 2, R , in addition of showing the characteristic minimum ofcoherence resonance, displays a maximum at a higher noise level that indicatesanti-coherence resonance. We also note that the coupling strength σ affectsthe level of noise for which coherence and anti-coherence resonances occur: forincreasing σ , both the minimum and the maximum shift to the right, i.e.,towards higher noise intensity.In Fig. 2(c) we see that the average ISI of neuron 1 monotonically decreaseswith the level of noise: as expected, the supra-threshold noise induces spikes andthe spiking rate increases (i.e., the average ISI decreases) with D . However, inFig. 2(d) we see that the average ISI of neuron 2 has a non-monotonic variationwith D : for high levels of noise, inverse stochastic resonance (ISR) [35] occurs.ISR is the phenomenon by which noise inhibits neuronal activity: the spike rateis minimum (and therefore, the average ISI is maximum) at a certain level ofnoise.A similar behavior is seen in Fig. 3, where we analyze two rings with N = 3neurons each. In layer 1, the intra-layer coupling only affects the neuronalactivity when the noise is weak; for strong noise, R is unaffected by σ [Fig. 3(a)].6 igure 2: Characterization of the spiking activity of two coupled neurons for various valuesof the coupling strength. (a) R , (b) R , (c) (cid:104) ISI (cid:105) and (d) (cid:104) ISI (cid:105) as a function of thesupra-threshold noise intensity of neuron 1, D . In other words, the complex interplay of coupling and noise defines the dynamics:for weak noise, the coupling dominates the dynamics and for large values of noiseintensity, noise governs the dynamics. In more detail, for small noise, weak intra-layer coupling supports oscillations with higher regularity. For stronger couplingbetween the neurons inside the layer, it becomes harder to bring the nodesacross the threshold by weak noise (i.e., the coupling dominates). For strongnoise, the dynamics is governed predominantly by stochastic input and the inter-layer coupling does not have an impact on the regularity of the noise-inducedoscillations.On the other hand, σ has almost no effect on the activity of layer 1,regardless of the noise level [Fig. 3(c)]. This is again due to the fact thatthe chosen coupling parameters correspond to weak multiplexing. In layer 2[Figs. 3(b), (d)], by tuning σ or σ we can achieve anti-coherence resonance forboth weak and strong noise. For intermediate noise levels, if σ or σ are large7 igure 3: R and R as a function of the noise intensity D for two coupled layers with N = 3neurons each. In (a), (b) σ = 0 .
01 is kept constant and σ is varied; in (c), (d), σ = 0 . σ is varied; other parameters are indicated in the text. enough, we observe coherence resonance.Next, we study the dynamics of two large inter-connected layers. We consider N = 500 neurons in each ring; qualitatively similar results were found for othervalues of N . Figure 4 shows that weak multiplexing induces coherence resonancein layer 2 (as shown in [7]). Interestingly, the second layer presents anti-coherenceresonance ( R displays a maximum).In order to visualize the underlying dynamics we display the activity of theneuronal populations in layer 1 and in layer 2 using space-time plots. The resultsare presented in Fig. 4. We consider the noise levels that produce maximum orminimum regularity in layer 2 [points marked A and B, respectively in Fig. 4(e)].We see that in point A both layers have the same firing rate, and the neuronsfire synchronously; in point B, the firing dynamics in layer 1 is still quite regular,while in layer 2 it is quite irregular, in space and in time.Similar results were obtained with other network sizes. In fact, Fig. 5 shows8 igure 4: Characterization of the spiking activity of two coupled layers with N = 500 neuronseach. Space-time plots of neurons in layer 1 (a, b) and in layer 2 (c, d) when the noise levelproduces maximum (a, c) and minimum (b, d) coherence (see points labeled A and B in panel(e)). R and R (e) and (cid:104) ISI (cid:105) and (cid:104) ISI (cid:105) (f) as a function of the noise intensity D . Thecoupling strengths are σ = 0 . σ = 0 .
01, other parameters are as indicated in the text. that the dynamics becomes insensitive to the number of neurons if the ring islarge enough: the variation of R and R with the noise level is very similar for N = 50 and N = 100.Figure 4(f) demonstrates that weak multiplexing induces ISR in the secondlayer also for larger system size ( N = 500).To gain further insight into the role of the weak multiplexing, and how thespiking activity of layer 1 generates a spiking activity in layer 2, we analyze theeffect of randomly removing a certain percentage of inter-layer links. In Fig. 6we see that coherence and anti-coherence resonances are induced in layer 2 evenwhen up to 80% of the inter-layer links are removed. The minimum amount oflinks that can be removed depends on the size of the rings. For example, for tworings with 50 neurons each, we could remove up to 70% of the inter-layer links,and still be able to observe coherence resonance in layer 2 (not shown).Because we consider weak multiplexing ( σ << σ ) link removal has almost9 igure 5: Effect of the number of neurons, N , in each ring. The coupling strengths are σ = 0 . σ = 0 .
01, other parameters are as indicated in the text. no effect in the spiking activity in layer 1. As it was shown in Fig. 3(c) for N = 3, R is almost unaffected by σ , and this holds also for larger N .However, when there are only few inter-layer links, their position in the ringstrongly affects the spiking activity of layer 2. For instance, three inter-layerlinks in neighboring neurons can be enough to induce a spiking activity in layer2, but the same number of inter-layer links distributed among non-neighboringneurons might not be sufficient to induce a spiking activity in layer 2. A detailedstudy of this effect is left for future work.
5. Conclusions
We have studied the dynamics of two neuronal populations weakly coupledin a multiplexed configuration, and subject to different levels of noise (onepopulation has supra-threshold noise, while the other, sub-threshold noise). Theactivity of the neurons was simulated with the FHN model. We found thatcoherence, anti-coherence and inverse stochastic resonances can be induce inlayer 2 (with subthreshold noise), for appropriate levels of supra-threshold noisein layer 1. The results were found to be robust to the number of neurons in eachneuronal population.While the coupling topology considered here is not biologically realistic, it10 igure 6: R as a function of the noise intensity D , when a given percentage of randomlyselected inter-layer links are removed. The parameters are as in Fig. 4. The variation of R isnot shown because the removal of the weak inter-layer links has almost no effect in the activityof layer 1 [the plot of R vs. D is very similar to that shown in Figs. 3(c), 4(e) or 5(a)]. is a simple toy model to characterize how noise-induced spiking activity in onelayer can propagate and induce spiking activity in another layer. We have foundthat a small percentage of randomly distributed inter-layer links can be sufficientto induce spikes in the “silent” layer. Further work will aim at using moreadvanced data analysis tools, such as symbolic ordinal analysis [37, 38, 39], tofurther characterize the regularity of the neuronal activity induced in layer 2.Our work yields light into the complex nonlinear dynamics of excitablestochastic units coupled in a simple bilayered structure. Further work using morecomplex structures is of course necessary in order to advance the understandingof the role of noise and multiplexing in biologically realistic neuronal models,such as cortical networks [40]. Acknowledgments
C.M. acknowledges partial support from Spanish Ministerio de Ciencia, Inno-vacin y Universidades grant PGC2018-099443-B-I00 and ICREA ACADEMIA,Generalitat de Catalunya. M.M. and A.Z. acknowledge support by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) - Project No.163436311 - SFB 910. 11 eferenceseferences