A Spatio-temporal Investigation of Dynamics of a Two-dimensional Multi-scale Fitz-Hugh Nagumo Neuronal Network

The multi scale architecture by Breakspear and Stam [2] introduces a framework to consider the dynamical processes specific to a nested hierarchy of spatial scales, from neuronal masses to cortical columns and functional brain regions. They hypothesize that the neural dynamics is a function of the structural properties of the neural system at a certain scale as well as the emergent behaviour of the smaller scale activities. In this paper, we adopt the multi scale framework to investigate a generalized version of the stochastic Fitz-Hugh Nagumo (FHN) neuronal system within the small scale process and their emergent large scale synchronization effects leading to the formation of travelling waves in the large scale system. We extend the multi scale framework to incorporate the nonlinear biological synaptic connectivity at the neuronal mass scale. The modified multi scale scheme utilizes the two-dimensional wavelet decomposition in the plane of dynamical interconnected neurons. In addition, we consider the large-scale spatio-temporal system of FHN reaction-diffusion partial differential equations and evaluate the formation of travelling waves in the simplified context of a cellular neural network (CNN) model. Numerical examples are given to illustrate the response and the isolated influence of the strength of neural connectivity on the travelling wave formation modes.