Supersymmetric methods in the traveling variable: inside neurons and at the brain scale

TL;DR

This paper explores supersymmetric mathematical techniques applied to neural and brain-scale models, analyzing microtubule excitations and corticothalamic diffusion processes with potential implications for understanding brain dynamics.

Contribution

It introduces new supersymmetric methods to analyze neural microtubule excitations and corticothalamic diffusion equations, providing analytic solutions and interpretations.

Findings

Supersymmetry applied to microtubule kinks and sine-Gordon models.

Analytic solutions for Bessel-type diffusion equations in brain models.

Potential interpretation of diffusion processes in neural contexts.

Abstract

We apply the mathematical technique of factorization of differential operators to two different problems. First we review our results related to the supersymmetry of the Montroll kinks moving onto the microtubule walls as well as mentioning the sine-Gordon model for the microtubule nonlinear excitations. Second, we find analytic expressions for a class of one-parameter solutions of a sort of diffusion equation of Bessel type that is obtained by supersymmetry from the homogeneous form of a simple damped wave equations derived in the works of P.A. Robinson and collaborators for the corticothalamic system. We also present a possible interpretation of the diffusion equation in the brain context