Time Fractional Cable Equation And Applications in Neurophysiology

We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analitically via Laplace Transform, and result be written in terms of known special functions. This generalization could be useful to describe anomalous diffusion phenomena with leakage as signal conduction in spiny dendrites. The presented solutions are computed in Matlab and plotted. Chaos, Solitons and Fractals (2017): Special Issue on Future Directions in Fractional Calculus. Guest Editors: Mark M. Meerschaert, Bruce J. West, Yong Zhou. Published on line 29 April 2017; http://dx.doi.org/10.1016/j.chaos.2017.04.043 Introduction The one dimensional cable model is treated in neurophysiology to model the electrical conduction of non-isopotential excitable cells, we remind as example to the textbooks of Johnston and Wu (1994) [8], Weiss (1996) [29] and Tuckwell (1988) [27], and for some mathematical details to Magin (2006) [12]. In particular it describes the spatial and the temporal dependence of transmembrane potential Vm(x, t) along the axial x direction of a cylindrical nerve cell segment. The membrane behaviour is summarized by an electrical circuit with an axial internal resistance ri, and a transmembrane capacitance cm and a transmembrane resistance rm in parallel, connecting the inner part to the outside [8]. External axial resistance could be eventually included. Transmembrane potential is generated by ionic concentration gradient across the membrane, and is maintained non null at rest (no current) by a combination of passive and active cell mechanisms. Equivalent models can in fact be derived from the Nernst-Planck equation for electro-diffusive motion of ions , see Qian and Sejnowski (1989)[22]. Date: May 19, 2017.