A new method for extracting cable parameters from input impedance data

A computational approach for the inverse problem of neuronal conductances determination

The derivation by Alan Hodgkin and Andrew Huxley of their famous neuronal conductance model relied on experimental data gathered using the squid giant axon. However, the experimental determination of conductances of neurons is difficult, in particular under the presence of spatial and temporal heterogeneities, and it is also reasonable to expect variations between species or even between different types of neurons of the same species.We tackle the inverse problem of determining, given voltage data, conductances with non-uniform distribution in the simpler setting of a passive cable equation, both in a single or branched neurons. To do so, we consider the minimal error iteration, a computational technique used to solve inverse problems. We provide several numerical results showing that the method is able to provide reasonable approximations for the conductances, given enough information on the voltages, even for noisy data.

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

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.

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.

Recovering the passive properties of tapered dendrites from single and dual potential recordings

We demonstrate that measurement of the membrane potential at one or more sites on a branched and tapered neuron following a known transient injection of subthreshold somatic current uniquely determines the cell's passive electrical properties. That is, knowledge of the potentials allows recovery of the cell's axial resistance, membrane capacitance, membrane conductance and soma conductance. The argument underlying uniqueness leads immediately to a constructive, robust algorithm that we successfully test on synthetic data. The robustness stems from the fact that the algorithm requires only a few weighted integrals, or moments, of the measured potentials.

Recovering quasi-active properties of dendritic neurons from dual potential recordings

We develop the theory and accompanying algorithm for the recovery of a dendritic neuron's cytoplasmic resistivity, membrane capacitance, leakage conductance, and two maximal channel conductances from weighted averages of simultaneous recordings of somatic and dendritic potential following a somatic current stimulus. We test our results on two model systems with distinct, though prescribed, channel kinetics and branching patterns.