Neural field dynamics under variation of local and global connectivity and finite transmission speed

Spatially continuous networks with heterogeneous connections are ubiquitous in biological systems, in particular neural systems. To understand the mutual effects of locally homogeneous and globally heterogeneous connectivity, we investigate the stability of the steady state activity of a neural field as a function of its connectivity. The variation of the connectivity is implemented through manipulation of a heterogeneous two-point connection embedded into the otherwise homogeneous connectivity matrix and by variation of the connectivity strength and transmission speed. Detailed examples including the Ginzburg–Landau equation and various other local architectures are discussed. Our analysis shows that developmental changes such as the myelination of the cortical large-scale fiber system generally result in the stabilization of steady state activity independent of the local connectivity. Non-oscillatory instabilities are shown to be independent of any influences of time delay.