A distributed parameter identification problem in neuronal cable theory models

Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.