Neural fields with distributed transmission speeds and corticothalamic feedback delays
Fatihcan M. Atay and Axel Hutt
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Submission date: 18. Apr. 2005
published in: SIAM journal on applied dynamical systems, 5 (2006) 4, p. 670-698
DOI number (of the published article): 10.1137/050629367
with the following different title: Neural fields with distributed transmission speeds and long-range feedback delays
MSC-Numbers: 92C20, 34K99, 37N25, 37G10
Keywords and phrases: synaptic networks, non-local interaction, distributed delays, spatio-temporal patterns, bifurcation analysis
We introduce distributed axonal transmission speeds and a non-local feedback loop into the standard neural field model. We derive conditions for the stability of spatially homogeneous equilibrium solutions and analyze their bifurcations for general connectivity kernels. By studying reduced models based on the assumption of small delays, we determine the effects of the delays on the stability and bifurcations. We show that delayed excitatory feedback generally facilitates stationary bifurcations and Turing patterns, while suppressing the bifurcation of periodic solutions and traveling waves. The reverse conclusion holds for inhibitory feedback. Furthermore, feedback connections with a sufficiently large spatial variance tend to induce spatially homogeneous bifurcations. In case of oscillatory bifurcations, the variance of the distributed delays affect the frequency of periodic solutions and the phase speed of traveling waves. The theoretical findings are illustrated by numerical simulations.