Title: Fractional Cable Model for Signal Conduction in Spiny Neuronal Dendrites,
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 $\alpha=1$.