Stochastic resonance in discrete excitable dynamics on graphs

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Abstract

How signals propagate through a network as a function of the network architecture and under the influence of noise is a fundamental question in a broad range of areas dealing with signal processing - from neuroscience to electrical engineering and communication technology. Here we use numerical simulations and a mean-field approach to analyze a minimal dynamic model for signal propagation. By labeling and tracking the excitations propagating from a single input node to remote output nodes in random networks, we show that noise (provided by spontaneous node excitations) can lead to an enhanced signal propagation, with a peak in the signal-to-noise ratio at intermediate noise intensities. This network analog of stochastic resonance is not captured by a mean-field description that incorporates topology only on the level of the average degree, indicating that the detailed network topology plays a significant role in signal propagation.

Highlights

► We explore the propagation of excitations through a network under the influence of noise. ► Special emphasis is on the application to neuroscience. ► A novel labeling technique of signal excitations is introduced and compared to the classical signal-to-noise ratio. ► We show that noise can lead to an enhanced signal propagation, with a peak in the signal-to-noise ratio at intermediate noise intensities.

Introduction

Spatiotemporal patterns formed by excitable elements are a common topic of interest in diverse disciplines, ranging from cell biology (e.g. [1]), neurodynamics (e.g. [2], [3]) to social systems (e.g. rumor spreading [4] or epidemic diseases [5], [6]). The characteristic feature of excitable elements is that they cycle through a well-defined sequence of events: the susceptible element enters an active state as soon as it is reached by a sufficient amount of excitations, then goes through a refractory period, before it returns to the susceptible state. In continuous descriptions, such as Hodgkin–Huxley equations [7], Beeler–Reuter equations [8] or the FitzHugh–Nagumo equations [9], [10], this sequence of events is determined by the shape of the nullclines of the differential equations. At a more general level of abstraction, it is possible to regard the described sequence itself as the dynamical model, operating on discrete time, with the state space of each excitable element consisting only of these three (susceptible, excited, refractory) states. This is the basic model that we explore here, in order to address the fundamental questions of how noise is relayed and processed by an excitable network and how the network architecture can facilitate the functioning of such dynamical systems. This setting can also be considered from the perspective of communication and information theory. From that perspective, we study the transmission of a coherent signal, where the (noisy) channel is the network.

Spatiotemporal patterns arising from coupled excitable elements have been studied for many decades (see, e.g. [11], [12]) and still continue to be of outstanding interest due to, for example, their importance for cardiac dynamics (e.g. [13], [14]) and the general idea of predicting such patterns from the variability in the system’s components [15], [16]. A systematic exploration of excitable dynamics on graphs, however, has been attempted only during the last few years [17], [18]. The key idea is that network patterns (i.e. the “network equivalent” to classical spatiotemporal patterns) reveal themselves as correlations between topology and dynamics [19], [3]. In this context, it is an important challenge to assess the impact of different types of network topology on the observed patterns.

The simple three-state stochastic cellular automaton used for the present minimal model has proved to be a suitable tool for exploring how network topology regulates the patterns formed by excitable dynamics on graphs. In particular, noise (i.e. spontaneous excitations) has been identified as an important parameter regulating such patterns [19], [3]. In [3], two types of correlations between network topology and dynamics were observed with the help of the minimal model: waves propagating from central nodes and module-based synchronization. Remarkably, the dynamical behavior of hierarchical modular networks could switch from one of these modes to the other with a changing rate of spontaneous network activation (see [3] for details). In our subsequent work [20], we could capture the origin of this switching behavior in a mean-field description supplemented with a formalism where excitation waves are regarded as avalanches on the graph.

One of the most surprising effects of noise in the context of spatio-temporal pattern formation in excitable media is the possibility of enhancing wave propagation and spiral wave formation by a suitable amount of noise, while too low noise fails to trigger an excitation wave and too high noise destroys the coherence of the pattern. This phenomenon of spatio-temporal stochastic resonance has been first described by Jung and Mayer-Kress [21] and experimentally verified in a light-sensitive variant of a BZ reaction [22]. It is not a priori clear that the non-trivial path structure between randomly selected nodes in an Erdős–Rényi (ER) random graph still allows for noise-enhanced propagation of a signal, as in the case of spatiotemporal stochastic resonance mentioned above. Indeed, the latter has been observed in an excitable medium, i.e. (qualitatively speaking) when the underlying graph is a regular lattice. A positive answer has been given in [23] for a system consisting of two populations (excitatory and inhibitory) of stochastic binary units (either active or inactive with some probability depending on the neighborhood state) on sparse networks.

One major drawback of a cellular-automaton-like model such as the one explored here is that the patterns can become prone to artifacts due to the model’s discreteness in time, space and the elements’ state space and the analysis of the patterns becomes difficult. In order to adapt the methods for analyzing the simulated data, we introduce an internal labeling technique for specific excitations better suited to discrete signals. We thus show that within this simple and generic model we are capable of observing noise-enhanced signal propagation, when the system receives a periodic input at a randomly selected node.

Section snippets

The model

We study a minimal model of signal propagation on random graphs. The dynamical process is governed by the three-state model of excitable dynamics explored in [19], [3]. This model consists of three discrete states for each node (susceptible S, excited E, refractory R), which are updated synchronously in discrete time steps according to the following rules: (1) A susceptible node becomes an excited node, if more than a fraction κ of the direct neighbors are in the excited state (see details

Results

At low driver frequencies, the subsequent signal excitations are essentially decoupled. The decoupling depends on the network’s capacity to “store” excitations within cycles over the period length of the driver, as this storage capacity is the basis of an interaction between subsequent signal excitations [32]. Indeed, output nodes have a priori a non-vanishing out-degree, so that recurrent connections are present. Recurrent excitation will presumably mix up with new excitations, resulting in

Conclusion

The main result of the present work is the numerical observation that signal propagation through a random network of excitable units under the influence of a periodic driver is enhanced by noise in a resonant fashion, when noise is provided by random spontaneous excitations.

We considered a single input node (pacemaker), a situation differing from coherence resonance (no input) and array-enhanced resonance (input distributed over all nodes). From our perspective, among the various types of

Acknowledgments

MTH and CCH gratefully acknowledge support by Deutsche Forschungsgemeinschaft (joint Grant HU 937/7-1). MTH furthermore acknowledges support from the IHÉS guest program for the purposes of this work; AL has been a visitor of the ICTS program at Jacobs University during the initial phase of this work.

References (46)

  • M. Müller-Linow et al.

    Organization of excitable dynamics in hierarchical biological networks

    PLoS Comput Biol

    (2008)
  • M. Simoes et al.

    Stochastic fluctuations in epidemics on networks

    J R Soc Interface

    (2008)
  • A.L. Hodgkin et al.

    A quantitative description of membrane current and its application to conduction and excitation in nerve

    J Physiol

    (1952)
  • G.W. Beeler et al.

    Reconstruction of the action potential of ventricular myocardial fibres

    J Physiol

    (1977)
  • R. FitzHugh

    Mathematical models of threshold phenomena in the nerve membrane

    Bull Math Biophys

    (1955)
  • J. Nagumo et al.

    An active pulse transmission line simulating nerve axon

    Proc IRE

    (1962)
  • A.T. Winfree

    Electrical turbulence in three-dimensional heart muscle

    Science

    (1994)
  • M.A. Bray et al.

    Experimental and theoretical analysis of phase singularity dynamics in cardiac tissue

    J Cardiovasc Electrophysiol

    (2001)
  • G. Bub et al.

    Spiral wave generation in heterogeneous excitable media

    Phys Rev Lett

    (2002)
  • D. Geberth et al.

    Predicting spiral wave patterns from cell properties in a model of biological self-organization

    Phys Rev E

    (2008)
  • D. Geberth et al.

    Predicting the distribution of spiral waves from cell properties in a developmental-path model of Dictyostelium pattern formation

    PLoS Comput Biol

    (2009)
  • I. Graham et al.

    Investigation of the forest-fire model on a smallworld network

    Phys Rev E

    (2003)
  • M. Müller-Linow et al.

    Topology regulates the distribution pattern of excitations in excitable dynamics on graphs

    Phys Rev E

    (2006)
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