A fractional-order Wilson-Cowan formulation of cortical disinhibition

• Published in: Journal of computational neuroscience Vol. 52; no. 1; pp. 109 - 123
• Main Author: González-Ramírez, L. R.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-02-2024; Springer Nature B.V
• Subjects: Biomedical and Life Sciences; Biomedicine; Brain; Brain research; Convulsions & seizures; Dynamic structural analysis; Dynamics; Epilepsy; Excitation; Homeostatic plasticity; Human Genetics; Humans; Mathematical models; Memory; Models, Neurological; Neurological diseases; Neurological disorders; Neurology; Neurons - physiology; Neurosciences; Numerical models; Ordinary differential equations; Parameters; Response functions; Seizures; Theory of Computation; Wilson-Cowan model; 92C20; Caputo’s fractional-order derivative; Disinhibited cortex; 37N25; 26A33; Fractional-order limit cycle
• ISSN: 0929-5313; 1573-6873; 1573-6873
• DOI: 10.1007/s10827-023-00862-y

This work presents a fractional-order Wilson-Cowan model derivation under Caputo’s formalism, considering an order of
0
<
α
≤
1
. To that end, we propose memory-dependent response functions and average neuronal excitation functions that permit us to naturally arrive at a fractional-order model that incorporates past dynamics into the description of synaptically coupled neuronal populations’ activity. We then shift our focus on a particular example, aiming to analyze the fractional-order dynamics of the disinhibited cortex. This system mimics cortical activity observed during neurological disorders such as epileptic seizures, where an imbalance between excitation and inhibition is present, which allows brain dynamics to transition to a hyperexcited activity state. In the context of the first-order mathematical model, we recover traditional results showing a transition from a low-level activity state to a potentially
pathological
high-level activity state as an external factor modifies cortical inhibition. On the other hand, under the fractional-order formulation, we establish novel results showing that the system resists such transition as the order is decreased, permitting the possibility of staying in the low-activity state even with increased disinhibition. Furthermore, considering the memory index interpretation of the fractional-order model motivation here developed, our results establish that by increasing the memory index, the system becomes more resistant to transitioning towards the high-level activity state. That is, one possible effect of the memory index is to stabilize neuronal activity. Noticeably, this neuronal stabilizing effect is similar to homeostatic plasticity mechanisms. To summarize our results, we present a two-parameter structural portrait describing the system’s dynamics dependent on a proposed disinhibition parameter and the order. We also explore numerical model simulations to validate our results.