Modeling electrocortical activity through improved local approximations of integral neural field equations

Coombes, S.; Venkov, Nikola; Shiau, LieJune; Bojak, Ingo; Liley, David; Laing, Carlo

Abstract

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice- directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs. © 2007 The American Physical Society.

Citation

Coombes, S., Venkov, N., Shiau, L., Bojak, I., Liley, D., & Laing, C. (2007). Modeling electrocortical activity through improved local approximations of integral neural field equations. Physical Review E, 76(5), https://doi.org/10.1103/PhysRevE.76.051901

• Journal Article Type: Article
• Acceptance Date: Jun 27, 2007
• Publication Date: Nov 1, 2007
• Deposit Date: Oct 3, 2007
• Publicly Available Date: Nov 1, 2007
• Journal: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
• Print ISSN: 2470-0045
• Electronic ISSN: 2470-0053
• Publisher: American Physical Society
• Peer Reviewed: Peer Reviewed
• Volume: 76
• Issue: 5
• DOI: https://doi.org/10.1103/PhysRevE.76.051901
• Keywords: neural fields, brain wave equations, patchy connections
• Public URL: https://nottingham-repository.worktribe.com/output/1016481
• Publisher URL: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.76.051901
• Additional Information: Open Access Journal version at:; http://dx.doi.org/10.1103/PhysRevE.76.051901