Quantitative methods for predicting neuronal behavior

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The electrical behavior of a nerve cell may be described by a system of ordinary first-order differential equations that approximates the partial differential equation of cable theory. The choice among methods for solving this system depends primarily on the physiological questions that are asked, and secondarily on special properties of the neuron. This paper describes several such methods, some of which are new, that are appropriate for the analysis of different classes of physiological questions. For steady-state current flows, the equations may be treated as linear with constant coefficients, and direct matrix manipulations suffice to predict voltage attenuations and input resistances. A simplifying assumption with respect to synaptic currents furnishes an efficient means for calculating steady-state firing rates in a synaptically coupled neuronal circuit. Voltage transients in such a system can be calculated either through numerical evaluation of the matrix exponential or through a direct eigenfunction expansion. Some ‘active’ or nonlinear properties of the neuron can be included within these formulations. More realistic physiological descriptions of synaptic effects and of time- and voltage-dependent ionic conductances lead to nonlinear equations having time-dependent coefficients. This level of description precludes the use of the more direct solution methods and dictates the use of difference techniques. Adoption of a method such as Runge-Kutta numerical integration in turn allows the incorporation of detailed descriptions of membrane and ionic processes. A discussion is given of the theoretical and practical considerations bearing on the choice of solution method.