A theoretical computation of abnormal oscillation propagation in a 2-D excitable neuronal network coupled via gap junction

The propagation of abnormal oscillations in actual neural tissue is often irregular and highly complex. The experiments and theoretical work on it are both very difficult; however, it can be helpful to understand some disorder in the neural system. With the help of microelectrode recording techniques and microdialysis, some experimental results from human beings and animal models have demonstrated that epileptic seizures can occur when either the external cellular environment of neurons is changed drastically from physiological conditions or when the synapses of neurons are extensively induced to release neurotransmitter or other neural signals. Here, we present a theoretical framework to investigate the gap junction's (electrical synapse) effect on the propagation of abnormal oscillations. Although such theoretical work is still very limited in explaining all the mechanistic problems related to the disorder situation, e.g., epilepsy, it is, nevertheless, helpful to our understanding of synaptic effects on the abnormal activity propagation. Now, from ionic channels to neural networks, a two-dimensional (2-D) spatial-temporal partial differential equation (PDE) is built. The implicit scheme of the finite-differential method in the time domain and a multistep algorithm are utilized to solve the PDE and the nonlinear ordinary differential equations, respectively, while the successive overrelaxation method is utilized to compute the large-scale sparse equations. Lyapunov exponent and approximate entropy are further applied, respectively, to the analysis of chaos and complexity in the propagation. Numerical results show that abnormal oscillations can propagate when the coupling strength of the gap junctions is sufficiently large, leading to turbulence in the excitable network, and that the larger the coupling strength is, the greater the nonlinear and the complexity of the propagation are. It is also concluded that the chaos and the complexity of the activity at the periphery point are larger than that at the central point when the abnormal oscillations propagate from the central to the periphery. This theoretical work is helpful to understand the gap junction's effects on the abnormal oscillation propagation in a 2-D excitable neural tissue.