A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees
type: Item
instance of: scholarly article
title: A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees
zbMATH Open document ID: 0758.92006
DOI: 10.1007/BF00203133
published in: Biological Cybernetics
publication date: 1993-02-23T00:00:00Z
review text: Transient potentials in dendritic trees can be calculated by approximating the dendrite by a set of connected cylinders. The profiles for the currents and potentials in the whole system can then be obtained by imposing the proper boundary conditions and calculating these profiles along each individual cylinder. An elegant implementation of this method has been described by \textit{W. R. Holmes} [ibid. 55, 115-124 (1986; Zbl 0598.92005)], and is based on the Laplace transform of the cable equation. By calculating the currents and potentials only at the end of the cylinders, the whole system of connected cylinders can be described by a set of \(n\) equations, where \(n\) denotes the number of internal and external nodes (points of connection and endpoints of the cylinders). The present study shows that the set of equations can be formulated by a simple vector equation which is essentially a generalization of Ohm's law for the whole system. The current and potential \(n\)-vectors are coupled by a \(n\times n\) conductance matrix whose structure immediately reflects the connectivity pattern of the connected cylinders. The vector equation accounts for conductances, associated with driving potentials, which may be local or distributed over the membrane. It is shown that the vector equation can easily be adapted for the calculation of transients over a period in which stepwise changes in system parameters have occurred.
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 78A70
Mathematics Subject Classification ID: 92C05
Mathematics Subject Classification ID: 35Q92
zbMATH DE Number: 108857
zbMATH Keywords: dendritic trees
zbMATH Keywords: connected cylinders
zbMATH Keywords: boundary conditions
zbMATH Keywords: Laplace transform of the cable equation
zbMATH Keywords: vector equation
zbMATH Keywords: generalization of Ohm's law
zbMATH Keywords: conductance matrix
zbMATH Keywords: driving potentials
zbMATH Keywords: membrane
author: Jaap van Pelt
describes a project that uses: Algorithm 368
describes a project that uses: NSPIV
MaRDI profile type: Publication
Wikidata QID: Q43670625
cites work: A continuous cable method for determining the transient potential in passive dendritic trees of known geometry
full work available at URL: https://doi.org/10.1007/bf00203133
OpenAlex ID: W1977738872
description: scientific article; zbMATH DE number 108857