2019/04/27 by Éva Kaslik, Kaslik, Eva, Emanuel-Attila Kökövics +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Gene Regulatory Network Analysis #Nonlinear Dynamics and Pattern Formation #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1904.12108
openalex publication_date 2019/04/27 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
The traditional Wilson-Cowan model of excitatory and inhibitory mean field\ninteractions in neuronal populations considers a weak Gamma distribution of\ntime delays when processing inputs, and is obtained via a time-coarse graining\ntechnique that averages the population response. Previous analyses of the\nstability of the Wilson-Cowan model focused on more simplified cases, where the\ndelays were either not present, constant or were of a specific type. Since\nthese simplifications may significantly alter the behavior of the model, we\nfocus on understanding the behavior of the system before time-course graining,\nand for a wider range of delay distributions.\n For these generalized delay equations, we perform stability and bifurcation\nanalyses with respect to parameters that capture both the coupling profile, and\nthe time delay. The investigation is done through the examination of the\nsystem's associated characteristic equation. Under mild assumptions, we give\ncomplete mathematical proofs of our theoretical results, for the model with\ngeneral delay distributions and prove the transversality condition for the\npossible Hopf bifurcations, in a generalized context. The stability region in\nthis parameter space is described theoretically for several types of delay\nkernels, and numerical simulations are presented to substantiate the\ntheoretical results.\n We illustrate these theoretical principles in an application to a basal\nganglia circuit, in which \β-band oscillations have been associated with\nParkinson's Disease.\n