Stability and Spatiotemporal Bifurcations in Spatially Distributed Neural Networks with Nonlocal Delay

The stability of equilibria and bifurcations of neural networks in a real line with nonlocal delay are presented. A sufficient condition of stable equilibria is declared by the linear part. Eigenvalue analysis implies the existence of bifurcations, and by exploiting typical excitatory and inhibitory connectivity kernels in a neural network, the possible bifurcations are discussed according to various cases. It is an advantageous tool using a multiple-scale method to study the stability of bifurcated travelling waves or spots. As an illustration of our theory, the dynamics of a seashell continuous-Time circular mask model are investigated. It is shown that both the shape and range of active function and synaptic weights can affect the dynamics of the model. Finally, the bifurcation set and the variety of bifurcated patterns of the seashell model are numerically revealed.