# On solutions related to FitzHugh-Rinzel type model

## TL;DR

This paper analyzes a FitzHugh-Rinzel type dynamical system, deriving explicit solutions for the initial value problem and incorporating control terms, with potential applications in neuroscience modeling.

## Contribution

It provides explicit solutions for a nonlinear integro-differential equation derived from a FitzHugh-Rinzel model, including control modifications.

## Key findings

- Explicit fundamental solution $H(x,t)$ determined.
- Initial value problem solved in the whole space.
- Explicit solutions obtained with added control term.

## Abstract

A ternary autonomous dynamical system of FitzHugh-Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The fundamental solution $ H(x,t) $ is explicitly determined and the initial value problem is analyzed in the whole space. The solution is expressed by means of an integral equation involving $ H(x,t) $. Moreover, adding an extra control term, explicit solutions are achieved.