An equivalent cable model for neuronal trees with active membrane
The dendrites of cortical neurons contain a variety of voltage-gated channels that may be important to neuronal signal processing. Compartmental models are known to cope not only with any complexity of dendritic branching but also with nonlinear membrane properties. The models then contain hundreds of compartments, and thousands of coupled differential equations must be solved at each time step. To find analytical tools to describe the spatio-temporal response behaviour, or at least to speed up simulations, approximations to branching dendritic trees have been used, i.e. models that preserve important properties of the signal flow but are described by a much smaller number of differential or algebraic equations for easier mathematical or numerical analysis. In particular, massive synaptic input typical for most neurons can be studied by means of a coarse classification of different input types with respect to input site, kinetic properties, type of synaptic channels, etc. A specific feature of input organization is layering, e.g., in the cortex. The equivalent cable model has characteristics supporting the analysis of the effects of such layered synaptic input problems, including the effect of different active dendritic ion channels, influence of relative locations and strength (number, electrical parameters) of synaptic inputs, and the impact of different cell morphologies. This paper develops a nonuniform cable model of membrane voltage changes in branching neuronal trees with active ion channels. For passive dendritic trees fulfilling certain symmetry conditions the reduction to an equivalent cable can be justified on theoretical grounds. In several studies the equivalent cable model has been employed for active trees on heuristic grounds. To provide theoretical justification for such reduced models, the authors have developed an equivalent cable model for active dendrites. The model generalizes precursor models by broadening morphological classes of neuronal trees accessible to analytical treatment, and customizing trees endowed with active membranes. The main result of the paper (Theorem 1, Equivalent Cable Theorem) states that a neuronal tree fulfilling some linked geometrical and electrical conditions can be reduced to an equivalent cable. Further, the authors analyze the signal flow in two neuronal trees and their dendritic profiles. One fulfils the paper's reduction criteria whereas the other violates some of them, leading, nevertheless, to appropriate results. Unlike the general case of an equivalent cable, the voltage changes in this paper admit exact solutions of the dynamic voltage distribution. The authors use this to show analytically different response behavior caused by synaptic inputs to dendrites of different geometric type.
scientific article; zbMATH DE number 1164624