Dendrites and conformal symmetry

TL;DR

This paper explores the mathematical symmetries of the cable equation in neural dendrites with various geometries, revealing invariance under the Schrödinger and conformal groups, which could enhance understanding of neural signal propagation.

Contribution

It demonstrates that the cable equation for dendrites with cylindrical and parabolic geometries exhibits invariance under Schrödinger and conformal groups, providing new analytic tools for neural modeling.

Findings

Cable equation is invariant under Schrödinger group for cylindrical geometry.

Parabolic dendrite geometry makes the cable equation equivalent to free particle Schrödinger equation.

A family of geometries makes the cable equation equivalent to conformal quantum mechanics.

Abstract

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.