Determining a distributed parameter in a neural cable model via a boundary control method

Abstract

Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.

On the inverse problem of the two‐velocity tree‐like graph

“…. This result can be extended to equations on graphs . More specifically, we can claim that the reduced response matrix function known on the controllability time interval of a system determines the reduced TW matrix function (see, e.g.…”

“…More specifically, we can claim that the reduced response matrix function known on the controllability time interval of a system determines the reduced TW matrix function (see, e.g. ). Therefore, Theorem follows from Theorem .…”

Determining distributed parameters in a neuronal cable model on a tree graph

“…Specifically, this problem is concerned with , with y satisfying the Kirchhoff–Neumann conditions on V ∖ ∂ Ω, and with zero initial conditions. Although this inverse problem is discussed in Avdonin and Bell , for completeness we highlight the solution approach briefly in Section 3.…”

“…We first discuss some relevant background in the next section, along with briefly discussing boundary control constructions needed to solve inverse problem 1, namely (7). Discussion here is a summary of work from Avdonin and Bell [6,9], and references therein. Section 4 considers the way to solve inverse problem 2.…”

Solução Numérica de um Problema Inverso em Neurociência via o Método de Landweber Não Linear.

“…Cox [3] utilizou o método de multiplicadores de Lagrange num modelo não-linear para recuperar a distribuição não uniforme da condutância. Avdonin [1] utilizou o método de controle de fronteira para resolver o problema inverso de recuperar a condutância.…”