Direct activation (or field stimulation) occurs if all or essentially all of the tissue is activated simultaneously without the aid of a propagated wave front. According to the model (14.54), it should be possible to stimulate cardiac tissue directly with brief stimuli of sufficiently large amplitude.

A numerical simulation demonstrating how direct stimulation can be accomplished for the bistable equation is shown in Fig. 14.32. In this simulation a one-dimensional array of 200 cells was discretized with five grid points per cell, and a brief, large current was injected at the left end and removed at the right end of the cable. In Fig. 14.29 is shown the response to the stimulus when the cable is uniform. Shown here is the membrane potential, beginning at time t = 0.1, and at later times with equal time steps At = 0.2. The stimulus duration was t = 0.2, so its effects are seen as a depolarization on the left and hyperpolarization on the right in the first trace. As noted above, a wave is initiated from the left from superthreshold depolarization, and a wave from the right is initiated by anode break excitation.

The same stimulus protocol produces a substantially different result if the cable has nonuniform resistance. In Fig. 14.32 is shown the response of the discretized cable with high resistance at every fifth node, at times t = 0.15,0.25, and 0.35, with a stimulus duration of 0.2. The first curve, at time t = 0.15, is blurred because the details of the membrane potential cannot be resolved on this scale. However, the overall effect of the rapid spatial oscillation is to stimulate the cable directly, as seen from the subsequent traces.

To analyze this situation, note that since direct activation occurs without the benefit of propagation, it is sufficient to ignore diffusion and the boundary conditions and simply examine the behavior of the averaged ordinary differential equation

o CL

o CL

100 Cell number

Figure 14.32 Response of a nonuniform cable with regularly spaced high-resistance nodes to a stimulus of duration t = 0.2 applied at the ends of the cable. The traces show the response at time t = 0.15, during the stimulus, and at times t = 0.25, 0.35 after the stimulus has terminated.

For any resting excitable system, it is reasonable to assume that f (V) < 0 for 0 <V <0, where 0 is the threshold that must be exceeded to stimulate an action potential. To directly stimulate a medium that is initially at rest with a constant stimulus, one must apply the stimulus until V > 0. The minimal time to accomplish this is given by the strength-duration relationship,

Clearly, this expression is meaningful only if 0 is sufficiently large that F(V, 0) > 0 on the interval 0 < V < 0. In other words, there is a minimal stimulus level (a threshold) below which the medium cannot be directly stimulated.

While its threshold cannot be calculated in the same way as for direct stimulus, the mechanism of defibrillation can be understood from simple phase-plane arguments. To study defibrillation, we must include the dynamics of recovery in our model equations. Thus, for simplicity and to be specific we take FitzHugh-Nagumo dynamics with f (v) = v(v — 1)(a — v), and assume that the parameters are chosen such that reentrant waves are persistent. This could mean that there is a stable spiral solution, or it could mean that the spiral solution is unstable but some nonperiodic reentrant motion is persistent. Either way, we want to show that there is a threshold for the stimulating current above which reentrant waves are terminated.

The mechanism of defibrillation is easiest to understand for a periodic wave on a one-dimensional ring, but the idea is similar for higher-dimensional reentrant patterns. For a one-dimensional ring, the phase-portrait projection of a rotating periodic wave is a closed loop. From singular perturbation theory, we know that this loop clings to the leftmost and rightmost branches of the nullcline w = f (v) and has two rapid transitions connecting these branches, and these correspond to wave fronts and wave backs (recall

According to our model, the effect of a stimulus is to temporarily change the v nullclines and thereby to change the shape of the closed loop. After the stimulus has ended, the distorted closed loop will either go back to a closed loop, or it will collapse to a single point on the phase portrait and return to the rest point. If the latter occurs, the medium has been "defibrillated."

Clearly, if ¡ is small and the periodic oscillation is robust, then the slight perturbation is insufficient to destroy it. On the other hand, if 0 is large enough, then the change is substantial and collapse may result.

There are two ways that this collapse can occur. First, and easiest to understand, if the nullcline for nonzero 0 is a monotone curve (as in Fig. 14.31 with 0 = 1.2), then

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