The instability of the spiral wave induced by the deformation of elastic excitable media

There are some similarities between the spiral wave in excitable media and in cardiac tissue. Much evidence shows that the appearance and instability of the spiral wave in cardiac tissue can be linked to one kind of heart disease. There are many models that can be used to investigate the formation and instability of the spiral wave. Cardiac tissue is excitable and elastic, and it is interesting to simulate the transition and instability of the spiral wave induced by media deformation. For simplicity, a class of the modified Fitzhugh–Nagumo (MFHN) model, which can generate a stable rotating spiral wave, meandering spiral wave and turbulence within appropriate parameter regions, will be used to simulate the instability of the spiral wave induced by the periodical deformation of media. In the two-dimensional case, the total acreage of elastic media is supposed to be invariable in the presence of deformation, and the problem is described with Lx × Ly = N × ΔxN × Δy = L'xL'y = N × Δx'N × Δy'. In our studies, elastic media are decentralized into N × N sites and the space of the adjacent sites is changed to simulate the deformation of elastic media. Based on the nonlinear dynamics theory, the deformation effect on media is simplified and simulated by perturbing the diffusion coefficients Dx and Dy with different periodical signals, but the perturbed diffusion coefficients are compensatory. The snapshots of our numerical results find that the spiral wave can coexist with the spiral turbulence, instability of the spiral wave and weak deformation of the spiral wave in different conditions. The ratio parameter e and the frequency of deformation forcing play a deterministic role in inducing instability of the spiral wave. Extensive studies confirm that the instability of the spiral wave can be induced and developed only if an appropriate frequency for deformation is used. We analyze the power spectrum for the time series of the mean activator of four sampled sites which are selected symmetrically in different cases, such as the condition that the spiral wave coexists with the spiral turbulence, spiral wave without evident deformation, complete instability of the spiral wave (turbulence) and weak deformation of the spiral wave. It is found that more new peaks appear in the power spectrum and the distribution of frequency becomes sparser when the spiral wave encounters instability.