Analysis of nonlocal neural fields for both general and gamma-distributed connectivities

• Published in: Physica. D 2005-04, Vol.203 (1), p.30-54
• Main Authors: Hutt, Axel, Atay, Fatihcan M.
• Format: Article
• Language: English
• Subjects: Bifurcation analysis; Exact sciences and technology; Neuronal populations; Physics; Synaptic connectivity
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2005.03.002

This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation includes synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation, namely the general reaction–diffusion equation and the Swift–Hohenberg equation. Hence, the discussed integro-differential equation generalizes these PDEs. In the case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition–lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.