Neural fields with rebound currents: Novel routes to patterning

Modhara, Sunil; Lai, Yi Ming; Thul, Rüdiger; Coombes, Stephen

Abstract

The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. Replicating this success for subcortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies and, most important, the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one- and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed in the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from nonsmooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic traveling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic traveling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution.

Citation

Modhara, S., Lai, Y. M., Thul, R., & Coombes, S. (2021). Neural fields with rebound currents: Novel routes to patterning. SIAM Journal on Applied Dynamical Systems, 20(3), 1596-1620. https://doi.org/10.1137/20M1364710

• Journal Article Type: Article
• Acceptance Date: May 29, 2021
• Online Publication Date: Sep 13, 2021
• Publication Date: Sep 13, 2021
• Deposit Date: Jun 3, 2021
• Publicly Available Date: Sep 13, 2021
• Journal: SIAM Journal on Applied Dynamical Systems
• Print ISSN: 1536-0040
• Electronic ISSN: 1536-0040
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 3
• Pages: 1596-1620
• DOI: https://doi.org/10.1137/20M1364710
• Keywords: Neural field; pattern formation; non-smooth dynamical systems; synchrony; periodic travelling waves; Evans function
• Public URL: https://nottingham-repository.worktribe.com/output/5623716
• Publisher URL: https://epubs.siam.org/doi/abs/10.1137/20M1364710