Dendrites and conformal symmetry

Progress toward characterization of structural and biophysical properties of neural dendrites together with recent findings emphasizing their role in neural computation, has propelled growing interest in refining existing theoretical models of electrical propagation in dendrites while advocating novel analytic tools. In this paper we focus on the cable equation describing electric propagation in dendrites with different geometry. When the geometry is cylindrical we show that the cable equation is invariant under the Schr\"odinger group and by using the dendrite parameters, a representation of the Schr\"odinger algebra is provided. Furthermore, when the geometry profile is parabolic we show that the cable equation is equivalent to the Schr\"odinger equation for the 1-dimensional free particle, which is invariant under the Schr\"odinger group. Moreover, we show that there is a family of dendrite geometries for which the cable equation is equivalent to the Schr\"odinger equation for the 1-dimensional conformal quantum mechanics.Comment: 19 page

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.Comment: 10 figures. arXiv admin note: substantial text overlap with arXiv:1702.0532

SOLVING THE CABLE EQUATION, A SECOND-ORDER TIME DEPENDENT PDE FOR NON-IDEAL CABLES WITH ACTION POTENTIALS IN THE MAMMALIAN BRAIN USING KSS METHODS

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and other than Cable Equation to the study of Neurodegenerative diseases like multiple sclerosis, Alzheimer’s, Parkinsons etc. The ultimate goal would be to be able to study a broad application of numerical methods to understand features of the human brain and its functions without involving medically invasive procedures. i

Dendritic cable with active spines: a modeling study in the spike-diffuse-spike framework

The spike-diffuse-spike (SDS) model describes a passive dendritic tree with active dendritic spines. Spine-head dynamics is modelled with a simple integrate-and-fire process, whilst communication between spines is mediated by the cable equation. Here we develop a computational framework that allows the study of multiple spiking events in a network of such spines embedded in a simple one-dimensional cable. This system is shown to support saltatory waves as a result of the discrete distribution of spines. Moreover, we demonstrate one of the ways to incorporate noise into the spine-head whilst retaining computational tractability of the model. The SDS model sustains a variety of propagating patterns

Emergence of Fractional Kinetics in Spiny Dendrites

Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models.~The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erd\'ely-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a {\it ggBm-like} construction of the random variable.Comment: 8 page