Determining Active, Nonuniform Dendritic Membrane Properties from Single and Multipoint Potential Readings

William Marsh Rice University, Houston TX

Investigators

Abstract

Cox 0077728 Information processing along and between nerve cells is achieved via electrodiffusion along branches and across cell membranes. The relatively poor axial conductance of the branches is offset by the myriad of ion channels that perforate the cell membrane. A nerve cell's salient physical, as opposed to geometrical, properties are then its axial conductance, its membrane's capacitance and permeability to one or more ionic species and the kinetics (rules that govern its open/closed state) of the underlying channels. The predictive utility of a mathematical model of course hinges on the accuracy to which these physical properties are known. Unfortunately, the experimental determination of each of these quantities is a formidable task that, in light of recent data suggesting that the permeabilities vary with position in the dendritic tree, requires great investment for all but the simplest geometries. The investigator and his colleagues therefore determine the extent to which the neuron's physical properties may be inferred from more readily available indirect measurements. These indirect measurements are recordings of somatic and distal membrane potential following a known current stimulus to the soma. Assuming current seals at the distal ends, these two potential recordings result in lateral overdetermination of the underlying degenerate-parabolic system of Hodgkin-Huxley equations. The investigator and his colleagues deduce from this overdetermined system a number of well posed problems, and associated algorithms (based on moment, fixed-point and output least squares methods), for the recovery of one or more of the neuron's physical properties. They test these algorithms on data recorded from pyramidal neurons drawn from the rat's hippocampus. In order to repair or reproduce the brain one must have a parts list and a blueprint specifying how the parts are to be connected. At the coarsest level there are but two types of parts, nerves (neurons) and nerve glue (glial cells). Though the human brain has more of each than the Milky Way has stars, it is not their sheer number but rather a subtle combination of interconnectedness and variation in electrical properties that render the brain so powerful. The term `variation' is meant to express the realization that a neuron is not simply a switch within a certain brain center or a wire connecting two such centers, but rather is a tree of wires with electrical properties varying along each of its branches. It is this local variation in a neuron's ability to conduct the brain's principal ions that is thought to be responsible for an individual neuron's ability to perform tasks reminiscent of rudimentary computers. Given however the minute size and variegated nature of a single neuron, the direct experimental determination of its electrical properties has yet to be achieved. The investigator and his colleagues therefore pursue the mathematically challenging task of determining these properties from more readily available, though indirect, experimental measurements. This process is akin to determining the size and location of a leak in a transatlantic telephone cable by comparing what the American said to what the Englishman heard. The success of their endeavor, coupled with the increasingly fine resolution of images of neuronal interconnections, will permit the investigator and his colleagues to produce models of sufficient veracity to be of use by the medical community from construction of prosthetic neuronal circuits to the design and testing of drugs and better treatments.