Quantitative models of the action potential

[5] Both types of models may be used to understand the behavior of small biological neural networks, such as the central pattern generators responsible for some automatic reflex actions.

[6] Such networks can generate a complex temporal pattern of action potentials that is used to coordinate muscular contractions, such as those involved in breathing or fast swimming to escape a predator.

[7] In 1952 Alan Lloyd Hodgkin and Andrew Huxley developed a set of equations to fit their experimental voltage-clamp data on the axonal membrane.

The initial term Iext represents the current arriving from external sources, such as excitatory postsynaptic potentials from the dendrites or a scientist's electrode.

The probabilities for each gate are assumed to obey first-order kinetics where both the equilibrium value meq and the relaxation time constant τm depend on the instantaneous voltage V across the membrane.

By fitting their voltage-clamp data, Hodgkin and Huxley were able to model how these equilibrium values and time constants varied with temperature and transmembrane voltage.

A less ambitious but generally applicable method for studying such non-linear dynamical systems is to consider their behavior in the vicinity of a fixed point.

[14][15] Based on the tunnel diode, the FHN model has only two independent variables, but exhibits a similar stability behavior to the full Hodgkin–Huxley equations.

Op-amp circuits that realize the FHN and van der Pol models of the action potential have been developed by Keener.

Maxwell's equations can be reduced to a relatively simple problem of electrostatics, since the ionic concentrations change too slowly (compared to the speed of light) for magnetic effects to be important.

Equivalent electrical circuit for the Hodgkin–Huxley model of the action potential. I m and V m represent the current through, and the voltage across, a small patch of membrane, respectively. The C m represents the capacitance of the membrane patch, whereas the four g' s represent the conductances of four types of ions. The two conductances on the left, for potassium (K) and sodium (Na), are shown with arrows to indicate that they can vary with the applied voltage, corresponding to the voltage-sensitive ion channels .
Figure FHN: To mimick the action potential, the FitzHugh–Nagumo model and its relatives use a function g ( V ) with negative differential resistance (a negative slope on the I vs. V plot). For comparison, a normal resistor would have a positive slope, by Ohm's law I = GV , where the conductance G is the inverse of resistance G =1/ R .