

Preprint 54/2003
Stability and bifurcations in neural fields with axonal delay and general connectivity
Fatihcan M. Atay and Axel Hutt
Contact the author: Please use for correspondence this email.
Submission date: 30. Jun. 2003
published in: SIAM journal on applied mathematics, 65 (2005) 2, p. 644-666
DOI number (of the published article): 10.1137/S0036139903430884
Bibtex
with the following different title: Stability and bifurcations in neural fields with finite propagation speed and general connectivity
MSC-Numbers: 92C20
Keywords and phrases: neural fields, delay, turing instability, traveling waves, spatio-temporal pattern formation
Abstract:
A stability analysis is presented for neural field equations in the presence
of axonal delays and for a general class of connectivity kernels and synaptic
properties. Sufficient conditions are given for the stability of equilibrium
solutions. It is shown that the delays play a significant role in non-stationary
bifurcations of equilibria, whereas the stationary bifurcations depend only on
the connectivity kernel. In the case of non-stationary bifurcations,
bounds are determined on the frequencies of the resulting oscillatory solutions.
A perturbative scheme is used to calculate the types of
bifurcations leading to spatial patterns, oscillations, and
traveling waves. For high transmission speeds a simple method is derived
that allows the determination of the bifurcation type by visual inspection
of the Fourier transforms of the kernel and its first moment.
Results are numerically illustrated on a class of neurologically plausible
systems with combinations of Gaussian excitatory and inhibitory connections.