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## The geometry of excitabliity

Let's begin by considering the Morris-Lecar model in the case that there is a unique rest state and a threshold-like behavior for action potential generation. Figure 1A shows the V-responses to brief current pulses of different amplitudes. The peak V is graded, but the variation occurs over a very narrow range of stimuli; in this case, as in the standard HH model, the threshold phenomenon is not discrete, but rather, steeply graded. In Figure 1B these same responses are represented in the V-w plane. The solution path in the space of dependent variables is called a trajectory, and direction of motion along a trajectory is often indicated by an arrowhead. In Figure 1B, the flow is generally counterclockwise. All the trajectories shown here ultimately lead to the rest point: . One says that the rest state is globally attracting. Each trajectory has a unique initial point, a horizontal displacement from the rest point corresponding to instantaneous depolarization by a brief current pulse. The trajectory of an action potential shows the following features: an upstroke with rapid increase in V (trajectory is moving rightward) and then the transient plateau with a slower increase in w corresponding to the opening of more channels. When w is large enough, the downstroke in V occurs - the trajectory moves leftward rapidly, as V tends toward . Finally, as w decreases (the potassium channels close) the state point returns to rest with a slow recovery from hyperpolarization.

In the phase plane, the slope of a trajectory at a given point is which is just the ratio of to , and these quantities are evaluated from the righthand sides of the differential equations (4)-(5). (The program XPP has a command to plot short vectors which indicate the flow pattern generated by the equations. This allows a global view of the flow without having to compute the trajectories. The program also computes nullclines, defined next.) Thus a trajectory must be vertical or horizontal where or , respectively. These conditions:  define curves, the V- and w-nullclines, which are shown dashed in Figure 1B. This provides a geometrical realization for where V and w can reach their maximum and minimum values along a trajectory in the V-w plane; notice how the trajectories cross the nullclines either vertically or horizontally in Figure 1B. The w-nullcline is simply the w-activation curve, . The V-nullcline, from equation (9), corresponds to V and w values at which the instantaneous ionic current plus applied current is zero; below the V-nullcline, V is increasing and above it, V is decreasing. The cubic-like shape seen here reflects the N-shaped instantaneous I-V relation, versus V with w fixed (equation (6)), which is typical of excitable membrane models in which the V-gated channels carrying inward current activate rapidly. From another viewpoint, which is motivated by the slower time scale of w, suppose we fix w, say, at a moderate value. Then the three points on the V-nullcline at this w correspond to three pseudo-steady states; at the low-V state, small outward and inward currents cancel while at the high-V state, both currents are larger but are again in balance. These states are transiently visited during the plateau phase and the return-to-rest phase of an action potential. Notice how the trajectory is near the right and left branches of the V-nullcline during these phases.

If were smaller still, then the phase plane trajectories (except when near the V-nullcline) would be nearly horizontal (since would be small). In this case, the action potential trajectory during the plateau and recovery phases would essentially cling to, and move slowly along, either the right or left branch of the V-nullcline. The downstroke would occur at the knee of the V-nullcline. The time course would be more like that of a cardiac action potential. Also, in the case of smaller , the threshold phenomenon would be extremely steep; the middle branch of the V-nullcline would act as an approximate separatrix between sub- and super-threshold initial conditions. In contrast, for larger , the response amplitude is more graded. This theoretical conclusion led to an experimental demonstration for squid axon  that, at higher temperatures, the action potential does not behave in an ``all-or-none'' manner.

We remark that phase plane methodology applies to systems which are autonomous, i.e. in which there is no explicit time dependence in the equations. This means that the nullclines and flow field do not change with time. This could not be done if, for example, I were periodic in t; the treatment of periodic stimuli will be covered later. However, the phase plane method extends to cases where a step change in a parameter occurs. At the time a parameter's value jumps, the nullclines would change instantaneously, but not the present location of V and w. FitzHugh  uses this trick to interpret anodal break excitation and Somers and Kopell  have used this to analyze the behavior of coupled Morris-Lecar oscillators when is very small.   Next: Oscillations emerging with Up: Understanding Dynamics via Previous: Understanding Dynamics via

Bard Ermentrout
Mon Jul 29 17:47:46 EDT 1996