The interaction between membrane kinetics and membrane geometry in the transmission of action potentials in non-uniform excitable fibres: a finite element approach

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Abstract

By solving the partial differential equations for an axonal segment using a finite element method, the interaction between membrane kinetics and axonal inhomogeneities, measured by their influence on propagated action potentials and stochastic spike trains, is investigated for Morris–Lecar and Hodgkin–Huxley membrane models. To facilitate comparisons of both kinetic models, parameter values are matched to give approximately the same speed for propagated action potentials. In all cases examined, the Morris–Lecar membrane model is more sensitive to geometric inhomogeneities than the comparable Hodgkin–Huxley membrane model. This difference in sensitivity can, in part, be attributed to significant differences in the membrane current supplied by each kinetic model ahead of the action potential. Also, the Morris–Lecar membrane model did not generate reflected action potentials whereas these were observed over a narrow range of geometric parameters for the comparable Hodgkin–Huxley membrane model. Simulations using stochastic spike train input showed that the presence of a sharp flare could significantly modify the statistical characteristics of the spike train output. The behaviour of action potentials governed by Morris–Lecar kinetics were more sensitive to changes in axonal geometry than those generated by comparable Hodgkin–Huxley kinetics. As a consequence of the fine balance between membrane kinetics and axon geometry, local changes in membrane properties, such as those caused by synaptic activity, can be expected to have a strong influence on the behaviour of stochastic spike trains at regions of changing axonal geometry.

Introduction

Geometric inhomogeneities in axons, such as abrupt or gradual changes in diameter, have been shown to delay (Manor et al., 1991b) or block the propagation of a single action potential (Berkinblit et al., 1970, Goldstein and Rall, 1974, Zhou and Bell, 1994), as well as selectively blocking action potentials in periodic spike trains (Khodorov et al., 1971). Thus, as suggested by a number of authors (see, Khodorov and Timin, 1975, Manor et al., 1991b, Swadlow et al., 1980, and references therein), axonal inhomogeneities may contribute to the signal processing characteristics of axons. Many naturally occurring spike trains, however, are not periodic and so these initial studies can be usefully extended by considering the effect of geometric inhomogeneities on stochastic spike trains. Furthermore, it is well known that membrane kinetics also interact with geometric inhomogeneities to modulate their effect. For example, a particular axonal morphology may allow action potentials to propagate for one kinetics but not another as implied by the work of Parnas and Segev (1979). These results share a common feature—all are inferences drawn from a mathematical model of the axon. This model, based on partial differential equations, is traditionally solved using either finite difference or compartmental methods (e.g. Berkinblit et al., 1970, Cooley and Dodge, 1966, Manor et al., 1991a, Parnas and Segev, 1979). By contrast, finite element methods have been used with great success to solve problems, which are naturally formulated as partial differential equations over a complex geometric region (e.g. Lindsay et al., 2001). The finite element approach offers significant advantages over the traditional methods through the flexible way in which it resolves local inhomogeneities and boundary conditions for models based on partial differential equations. We use a finite element method to solve the partial differential equations describing propagated action potentials along inhomogeneous axons. The accuracy of this method is demonstrated for Hodgkin and Huxley, 1952, Morris and Lecar, 1981 kinetics by comparing the conduction velocity of action potentials in a uniform axon, estimated by two independent methods. For Hodgkin–Huxley kinetics, the ability of the finite element method to predict this speed and reconstruct faultlessly all the known features of the action potential is a severe test of the method.

Earlier work on the influence that local geometric inhomogeneities in axons exert on the propagation of action potentials is enhanced and extended by including a comparative analysis of the interaction between membrane geometry and ‘matched’ Hodgkin–Huxley and Morris–Lecar kinetics. The finite element procedure modified for axonal segments of infinite length, is also used to investigate the impact of local geometric in homogeneities on trains of propagating action potentials, free from the influence of boundary conditions or reflected waves. The statistical properties of stochastic spike trains are shown to be significantly modified by local geometric inhomogeneities, although a Hodgkin–Huxley membrane model is more resistant to these changes than a Morris–Lecar membrane model.

Section snippets

The finite element method

The mathematical model of an axonal segment consists of a cable equation for an excitable axon and two boundary conditions, one for each end. The traditional cable model assumes that the evolution of the transmembrane potential V(x, t) satisfies the second order partial differential equationP(x)cM∂V(x,t)∂t+J(x,t)∂xgAA(x)∂V(x,t)∂x=0,x∈(0,L),t>0,where x measures axial distance along the segment in centimetres, t measures time in milliseconds and A(x), P(x) are, respectively, the segment

Results

The finite element method is applied to three topics concerned with the propagation of action potentials along excitable fibres, namely, velocity profiles of travelling waves approaching changes of axon geometry; the effect of changes of axon geometry1 on the propagation of

Discussion

In this article, two models of membrane kinetics are used to examine how membrane properties and axonal geometry interact to shape the properties of propagated action potentials and, in turn, those of stochastic spike trains. The membrane potential in the first model, due to Hodgkin and Huxley (1952), is activated by a sodium current and restored by a potassium current, aided by sodium inactivation. In the second model, due to Morris and Lecar (1981), calcium plays the role of sodium, but the

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