A simple vector implementation of the Laplace-transformed cable equations in passive dendritic trees (Q1202333)

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.