Neural Fields: Localised States with Piece-Wise Constant Interactions

Abstract

Neural field models are typically cast as continuum integro-differential equations for describing the idealised coarse-grained activity of populations of interacting neurons. For smooth Mexican hat kernels, with short-range excitation and long-range inhibition, these non-local models can support various localised states in the form of spots in two-dimensional media. In recent years, there has been a growing interest in the mathematical neuroscience community in studying such models with a Heaviside firing rate non-linearity, as this often allows substantial insight into the stability of stationary solutions in terms of integrals over the kernels. Here we consider the use of piece-wise constant kernels that allow the explicit evaluation of such integrals. We use this to show that azimuthal instabilities are not possible for simple piece-wise constant Top Hat interactions, whilst they are easily realised for piece-wise constant Mexican hat interactions.

Citation

Gökçe, A., Coombes, S., & Avitabile, D. (2018). Neural Fields: Localised States with Piece-Wise Constant Interactions. In Mathematical and Theoretical Neuroscience: Cell, Network and Data Analysis. Cham, Switzerland: Springer Nature. https://doi.org/10.1007/978-3-319-68297-6_7

• Book Type: Book Chapter
• Acceptance Date: Mar 21, 2018
• Publication Date: Mar 21, 2018
• Deposit Date: Nov 19, 2018
• Publisher: Springer Nature
• Book Title: Mathematical and Theoretical Neuroscience: Cell, Network and Data Analysis
• ISBN: 978-3-319-68297-6
• DOI: https://doi.org/10.1007/978-3-319-68297-6_7
• Public URL: https://nottingham-repository.worktribe.com/output/1217573
• Publisher URL: https://link.springer.com/chapter/10.1007%2F978-3-319-68297-6_7