# Fractional Cable Model for Signal Conduction in Spiny Neuronal Dendrites

## TL;DR

This paper derives explicit solutions for a fractional cable model of neuronal dendrites, capturing anomalous subdiffusion effects with potential to better match experimental membrane potential behaviors.

## Contribution

It provides the first explicit solutions for the fractional cable signaling problem using Efros theorem, extending classical models with fractional derivatives.

## Key findings

- Solutions for impulsive and step inputs are expressed in integral form with Wright functions.
- The fractional model offers adaptable solutions that better fit experimental data.
- Explicit solutions are derived for any input function satisfying Efros theorem conditions.

## Abstract

The cable model is widely used in several fields of science to describe the propagation of signals. A relevant medical and biological example is the anomalous subdiffusion in spiny neuronal dendrites observed in several studies of the last decade. Anomalous subdiffusion can be modelled in several ways introducing some fractional component into the classical cable model. The Chauchy problem associated to these kind of models has been investigated by many authors, but up to our knowledge an explicit solution for the signalling problem has not yet been published. Here we propose how this solution can be derived applying the generalized convolution theorem (known as Efros theorem) for Laplace transforms. The fractional cable model considered in this paper is defined by replacing the first order time derivative with a fractional derivative of order $\alpha\in(0,1)$ of Caputo type. The signalling problem is solved for any input function applied to the accessible end of a semi-infinite cable, which satisfies the requirements of the Efros theorem. The solutions corresponding to the simple cases of impulsive and step inputs are explicitly calculated in integral form containing Wright functions. Thanks to the variability of the parameter $\alpha$, the corresponding solutions are expected to adapt to the qualitative behaviour of the membrane potential observed in experiments better than in the standard case $\alpha=1$.