This section has shown that the basis for nearly all modern computational models for neurons and biological neural networks dates from the discovery in the early 1950s that information propagated along neural membranes is in the form of transient changes in the transmembrane voltage. These membrane voltage changes could be active, as in the case of nerve action potentials on axons, or passive, as in the case of dendrites.

All nerve axons were shown to have a nearly constant capacitance per square centimeter, and to contain two types of voltage-activated, ion-conducting proteins that under certain conditions would pass either sodium or potassium ions in directions that were determined by concentration gradients and the transmembrane voltage. The behavior of these specific ion-channel proteins was shown to be modeled by nonlinear, voltage-dependent conductances.

Dendrites can be modeled by fixed-parameter, lumped-parameter RC transmission lines, such as illustrated in Figure 1.2-1. Such transmission lines are linear circuits; they are most easily modeled by dividing the dendrite tube into sections of finite length, Ax, and representing each section by a parallel RC circuit emulating the transmembrane leakage conductance and capacitance. The parallel RC sections are connected by resistors representing the longitudinal (axial) resistance inside the dendrite and also outside it.

The presence of myelin beads was shown to increase conduction velocity on axons because high-speed electrotonic conduction occurs on the axon under each bead, and the action potential is regenerated at the nodes of Ranvier between the myelin beads. Both myelinated and unmyelinated axons can be modeled by modified transmission line models. A lumped-parameter model for a myelinated axon is shown in Figure 1.2-7. An active Hodgkin-Huxley circuit with its voltage-dependent sodium and potassium conductances marks the nodes at the left end of a myelin bead, and a passive RC ladder emulates the passive propagation of the action potential under the myelin bead. A circuit consisting of a string of seven or eight of the subunits of Figure 1.2-7 was used to obtain the plot of Figure 1.2-6.

FIGURE 1.2-6 A three-dimensional (transmembrane potential vs. time and distance) plot of saltatory conduction on a myelinated axon model. The distance between nodes of Ranvier is 2 mm. A 30-mA, 10-|s pulse was given at t = x = 0. Each heavy line represents transmembrane potential calculated at and between nodes at a particular time. Note that full depolarization only occurs at the nodes; decremental (passive) propagation occurs under the myelin beads with little attenuation. (Derived from a two-dimensional graph in Plonsey, 1969.)

FIGURE 1.2-6 A three-dimensional (transmembrane potential vs. time and distance) plot of saltatory conduction on a myelinated axon model. The distance between nodes of Ranvier is 2 mm. A 30-mA, 10-|s pulse was given at t = x = 0. Each heavy line represents transmembrane potential calculated at and between nodes at a particular time. Note that full depolarization only occurs at the nodes; decremental (passive) propagation occurs under the myelin beads with little attenuation. (Derived from a two-dimensional graph in Plonsey, 1969.)

Modern computational models for spike generation and propagation still use the basic HH architecture, or modifications of it. The details of ODEs generating the specific ionic conductances are often changed, as are their voltage-dependent coefficients. When modeling synapses, ion channels other than for Na+ and K+ are used to reflect current details known about epsp and ipsp generation.

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