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

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, the stability of the steady state activity of a neural field as a function of its connectivity is investigated. The variation of the connectivity is operationalized through manipulation of a heterogeneous two-point connection embedded into the otherwise homogeneous connectivity matrix and by variation of connectivity strength and transmission speed. A detailed discussion of the example of the real Ginzburg-Landau equation with an embedded two-point heterogeneous connection in addition to the homogeneous coupling due to the diffusion term is performed. The system is reduced to a set of delay differential equations and the stability diagrams as a function of the time delay and the local and global coupling strengths are computed. The major finding is that the stability of the heterogeneously connected elements with a well-defined velocity defines a lower bound for the stability of the entire system. Diffusion and velocity dispersion always result in increased stability. Various other local architectures represented by exponentially decaying connectivity functions are also discussed. The 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 via oscillatory instabilities independent of the local connectivity. Non-oscillatory (Turing) instabilities are shown to be independent of any influences of time delay.