Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit

Abstract

Abstract. The FitzHugh-Nagumo equation has been investigated with a wide array of different methods in the last three decades. Recently a version of the equations with an applied current was analyzed by Champneys, Kirk, Knobloch, Oldeman and Sneyd [5] using numerical continuation methods. They obtained a complicated bifurcation diagram in parameter space featuring a C-shaped curve of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations. We use techniques from multiple time-scale dynamics to understand the structures of this bifurcation diagram based on geometric singular perturbation analysis of the FitzHugh-Nagumo equation. Numerical and analytical techniques show that if the ratio of the time-scales in the FitzHugh-Nagumo equation tends to zero, then our singular limit analysis correctly represents the observed CU-structure. Geometric insight from the analysis can even be used to compute bifurcation curves which are inaccessible via continuation methods. The results of our analysis are summarized in a singular bifurcation diagram.

Mixed-Mode Oscillations with Multiple Time Scales

“…Nagumo's equation [167], which models the evolution of an activator v(x, t) and a slow inhibitor u(x, t), is an example that has been studied extensively as an idealized model for propagation of action potentials. Traveling-wave profiles are found via the ansatz v(x, t) = v(x + σt) = v(τ ) and w(x, t) = w(x + σt) = w(τ ) as homoclinic solutions of a three-dimensional ODE with two fast variables and one slow variable [92]; here, σ is the wave speed. It has been shown that MMOs exist as solutions of this reduced ODE [93].…”

Predicting critical ignition in slow-fast excitable models

“…We repeat these calculations for varied γ, from critical γ = γ c down to very small value to test the singular limit. Note that the singular limit of traveling waves in the classical FitzHugh-Nagumo system is well-studied [10][11][12][13]. The scaling results of the previous section apply directly to this model, and thus serve as a method of veri-fying the numerical results under variations in the modal expansion length M , domain length L, and the application of projection boundary conditions, across several decades in γ.…”

Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

“…More precisely, C 0,l and C 0,r are of saddle-type, since the matrix D x f (p, 0) along them always has two real eigenvalues of opposite sign. We remark that saddle-type critical manifolds have played an important role in the history of fast-slow systems in the context of the travelling wave problem for the FitzHugh-Nagumo equation, see for example [21,27,33].…”

“…Indexing the level set value of H f as θ, the solutions of (20) are level curves {H f (w, z) = θ}. The equilibria of (20) are {z = 0, w = w l , w m , w r }; here w l , w m , w r are the three solutions of 2v − w 3 + w = 0 (21) which depend uponv. We only have to consider the case where there are at least two real equilibria w l and w r which occurs forv…”

“…The three-dimensional fast-slow FHN system has been studied in great detail using various fast-slow systems techniques (see e.g. [7,21,27,33]). One particular approach to investigate the FHN parameter space efficiently is to employ numerical continuation methods [8,22].…”