Spatial coherence resonance in a spatially extended system that is locally modeled by Hodgkin–Huxley (HH) neurons is studied in this paper. We focus on the ability of additive temporally and spatially uncorrelated Gaussian noise to extract a particular spatial frequency of excitatory waves in the medium, whereby examining the impact of diffusive and small-world network topology that determines the interactions amongst coupled HH neurons. We show that there exists an intermediate noise intensity that is able to extract a characteristic spatial frequency of the system in a resonant manner provided the latter is diffusively coupled, thus indicating the existence of spatial coherence resonance. However, as the diffusive topology of the medium is relaxed via the introduction of shortcut links introducing small-world properties amongst coupled HH neurons, the ability of additive Gaussian noise to evoke ordered excitatory waves deteriorates rather spectacularly, leading to the decoherence 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 the examined system.
Nontrivial effects of noise on nonlinear dynamics have been a vibrant topic for many years. It is thoroughly documented and established that noise can play a constructive role in different types of nonlinear dynamical systems. Stochastic and coherenceresonance are just two, perhaps most prominent, examples of this fact. The notion of coherenceresonance is particularly inspiring in that an appropriate intensity of noise alone is sufficient to evoke ordered temporal responses of a nonlinear dynamical system. Nowadays, however, effects of noise on spatially extended systems have gradually slipped into the focus of many scientists working in diverse fields of research, consequently spawning the need to investigate whether phenomena observed previously for isolated dynamical system can also be observed, at least conceptually similar, if the latter are coupled. Indeed, it has been shown that the coherenceresonance phenomenon originally reported for dynamical systems evolving only in time can also be observed in spatially extended systems that are locally described by excitable nonlinear dynamics. Importantly there is the fact that the previously studied order in the temporal dynamics has been “replaced” by the order of the noise-induced spatial dynamics, which mostly manifests as propagating waves of excitatory events throughout the spatial grid. Presently, we aim to extend the scope of spatialcoherenceresonance by confirming its possibility also in models of neuronal dynamics, in particular by employing as the constitutive unit of the spatially extended system the renowned HH model. We show that while the diffusive interactions between coupled neurons warrants the observation of noise-induced pattern formation and with it related spatialcoherenceresonance, the small-world topology is not an appropriate medium for such observations. More precisely, even a minute fraction of shortcut links amongst distant neurons prohibits noise-induced waves to be ordered, and hence also precludes the observation of spatialcoherenceresonance. Since the present study is setup around a comprehensive HH model of neuronal dynamics, the presented results should prove valuable not just from the purely theoretical point of view, but hopefully also from the experimental point of view, especially by shedding light into the functioning of neural tissue.
X. Sun and Q. Lu are grateful for the support of the National Science Foundation of China (Grant Nos. 10432010 and 10702023). M. Perc acknowledges support from the Slovenian Research Agency (Grant No. Z1-9629).
II. MATHEMATICAL MODEL
III. SPATIALCOHERENCERESONANCE BY DIFFUSIVE COUPLING
IV. DECOHERENCE OF SPATIAL DYNAMICS BY SMALL-WORLD INTERACTIONS
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