Controlling excitable media with noise

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The focus of this study is on the influence of fluctuations on coupled excitable systems. For that purpose we first examine numerically the stationary probability distribution as well as the probability flux for an individual FitzHugh–Nagumo system with additive noise. Depending on noise intensity and separation of the timescales different combinations of extrema are found which can be used to classify parameter sets. In one of these sets we find reminiscences of coherence resonance in the distribution. For the investigation of coupled ensembles of excitable systems we use a method based on the central moment dynamics of the corresponding probability distribution. We derive a general expression for a system with N variables per ensemble unit and discuss the quality of different approximation techniques. Noise can not only influence existing excitable dynamics but it can also alter dynamics that are formerly not excitable in such a way that they become excitable. We demonstrate this using a generalization of a well known model for noise-induced phase transitions under the influence of multiplicative noise. With the help of the moment dynamics we obtain the system’s phase diagram that shows regions of noise induced oscillations of the ensemble mean and noise-induced excitability of the mean. Between these two regimes there exists a complicated transition regime. When applying uncorrelated additive noise to each unit of a globally coupled ensemble with FitzHugh–Nagumo kinetics a strikingly similar transition of the mean is observed. We study this transition in detail using the moment dynamics method. Besides period-two oscillations, chaos, intermittent spiking and other regimes we find in the course of the transition a quick increase of a chaotic attractor. This phenomenon is known from non-chaotic oscillations as Canard explosion. We then apply additional global fluctuations to the system but leave the sum of the global and local noise intensities constant. With increasing correlations of the fluctuations the mean of the ensemble exhibits a phenomenon resembling coherence resonance. The coefficient of variation shows a minimum not for a finite nonzero value of the overall noise intensity but of the noise intensity of the global component. We demonstrate the possibility of pattern formation with the help of dichotomous fluctuations using an array of excitable units with nearest neighbor coupling locally obeying FitzHugh–Nagumo kinetics. Depending on the spatial and temporal correlation of the dichotomous fluctuations we find different mechanisms and different parameter ranges for the creation of structure patterns.