Neural Fields: Localised States with Piece-Wise Constant Interactions.


A. Gökçe, S. Coombes, D. Avitabile


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.



  title={Neural Fields: Localised States with Piece-Wise Constant Interactions},
  author={G{\"o}k{\c{c}}e, Ayt{\"u}l and Coombes, Stephen and Avitabile, Daniele},
  booktitle={Mathematical and Theoretical Neuroscience},