# Mathematics of nerve signals

## TL;DR

This paper develops a comprehensive mathematical framework using coupled partial differential equations to model nerve signals, capturing electrical, mechanical, and thermal phenomena within nerve fibers.

## Contribution

It introduces a novel coupled model that integrates action potential, mechanical waves, and temperature effects in a unified mathematical system.

## Key findings

- Model describes primary electrical and mechanical components of nerve signals.
- Secondary components like transverse displacement and temperature are derived from primary signals.
- The model provides a basis for understanding complex nerve signal interactions.

## Abstract

Mathematical models describing the signals propagating in nerve fibres are described. The emphasis is on the mathematical structures of governing equations while the extremely rich physiological aspects are here not analysed. Based on models of single waves, a joint coupled model is presented which is able to describe the action potential and the accompanying mechanical effects togehter with temperature changes within one system of partial differential equations. The whole signal is an ensemble which includes primary and secondary components. The primary components of a signal are the action potential itself and longitudinal mechanical waves in axoplasm and surrounding biomembrane. These components are characterized by corresponding velocities. The secondary components of a signal are derived from primary components and include transverse displacement of a biomembrane and the temperature change. These secondary components have no independent velocities in the presented model.