straight lines (Figure 48). Also, UP(θ0) is the straight line whose equation is
n + h ≡ n0 + h0.
Thus, no solution in ∏0 starting at DOWN(θ0) ever crosses UP(θ0). There exist, therefore, Ω-periodic and single pulse solutions, but no burst or finite wave train solutions for small ε and δ. Recent work by Evans, Fenichel, and Feroe [ 1980] implies that if ε is sufficiently large, so that two of the eigenvalues at the rest point become complex conjugate, and if certain other hypotheses are satisfied, then there exist finite wave train solutions of the FitzHugh-Nagumo equations. Computations by Feroe [ 1980] for a piecewise-linear model and by Hastings [ 1980] for a model with cubic G-function indicate that the necessary hypotheses may be satisfied in certain parameter ranges. These Evans-Fenichel-Feroe wave trains have qualitative properties different from those discussed previously: the spikes are widely- and nearly evenly-spaced, with each spike an approximate copy of a single spike. They resemble a finite set of low-frequency Ω-periodic spikes.
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