Probably no mathematical model in physiology has had more impact on the field of computational neurobiology than the Hodgkin-Huxley (HH) (1952) model for nerve impulse generation. The HH model does not describe the propagation of a nerve impulse down a myelinated or unmyelinated axon, but it can easily be modified to do so. The HH model simulates the what-was-then-known behavior of the two major ionic species, Na+ and K+, and describes how they pass through a "unit" patch of active membrane in terms of a set of nonlinear ODEs controlling their specific ionic conductances. The main HH ODE is basically a node equation, which by Kirchoff's current law sums the radial currents passing through the membrane. There is an external (input) current, a capacitive current, and three ionic currents. One is a voltage-dependent Na+ current, the second is a voltage-dependent K+ current, and the third is a linear "leakage" current. The sodium current depolarizes the membrane patch; the potassium current hyperpolarizes it. Both ion currents are nonlinear functions of the transmembrane voltage; their dynamics are determined by three first-order ODEs with nonlinear, voltage-dependent coefficients.

To adapt the HH model to describe spike propagation on an unmyelinated axon, one need only link HH "modules" with linear resistances representing the per-unit length resistance of the axoplasm and the external axial spreading resistance (Plon-sey, 1969). Such a model is shown in Figure 1.4-1. To emulate saltatory spike propagation, one can modify the HH model to include the intervening electrotonic sections of myelin-wrapped axon. These are shown in Figure 1.4-2.

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FIGURE 1.4-1 Lumped-parameter model of a nerve axon. HH "patches" are linked by longitudinal (axial) resistances.

The following sections describe the mathematical details of the HH model.

FIGURE 1.4-2 Lumped-parameter model of a myelinated axon. The HH patches are at the nodes of Ranvier; the length of axon under the myelin bead is represented as a series of linked, passive electrotonic modules with constant parameters.

FIGURE 1.4-2 Lumped-parameter model of a myelinated axon. The HH patches are at the nodes of Ranvier; the length of axon under the myelin bead is represented as a series of linked, passive electrotonic modules with constant parameters.

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