Snakes and ladders in an inhomogeneous neural field model

Avitabile, Daniele; Schmidt, Helmut

Abstract

Continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity. We analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes-and-ladders bifurcation structure. In the case of Heaviside firing rates, we construct analytically symmetric and asymmetric states and hence derive closed-form expressions for the corresponding bifurcation diagrams. We show that the ideas proposed by Beck and co-workers to analyze snaking solutions to the Swift--Hohenberg equation remain valid for the neural field model, even though the corresponding spatial-dynamical formulation is non-autonomous. We investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the Heaviside limit.

Citation

Avitabile, D., & Schmidt, H. (2015). Snakes and ladders in an inhomogeneous neural field model. Physica D: Nonlinear Phenomena, 294, 24-36. https://doi.org/10.1016/j.physd.2014.11.007

• Journal Article Type: Article
• Acceptance Date: Nov 27, 2014
• Online Publication Date: Dec 5, 2014
• Publication Date: Feb 15, 2015
• Deposit Date: Mar 17, 2014
• Publicly Available Date: Dec 5, 2014
• Journal: Physica D: Nonlinear Phenomena
• Print ISSN: 0167-2789
• Electronic ISSN: 0167-2789
• Publisher: Elsevier
• Peer Reviewed: Not Peer Reviewed
• Volume: 294
• Pages: 24-36
• DOI: https://doi.org/10.1016/j.physd.2014.11.007
• Public URL: https://nottingham-repository.worktribe.com/output/997960
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0167278914002280