Subcritical Hopf Bifurcations in the FitzHugh-Nagumo Model

Abstract

It had been shown that the transition from a rigidly rotating spiral wave to a meandering spiral wave is via a Hopf bifurcation. Many studies have shown that these bifurcations are supercritical, but we present numerical studies which show that subcritical bifurcations are also present within FitzHugh-Nagumo. Furthermore, we present a evidence that this bifurcation is highly sensitive to initial conditions, and it is possible to convert one solution in the hysteresis loop to the other.

Figures (9)

• Figure 1: Parametric Portrait for FHN for $\gamma=0.5$Winfree-1991.
• Figure 2: Quotient data for a meandering spiral wave with $\beta=0.751$. Time v/s (top left) $c_x$, (top right) $c_y$, (bottom left) $\omega$. The full limit cycle is shown (bottom right).
• Figure 3: The $\beta$-$Q_s$ plot. Each dot represents a single single simulation. Shown are reconstructed tip trajected from the quotient data. Simulations $A$ and $F$ are RW; $B$ and $C$ are MRW with outward facing "petals"; $D$ is near the 1:1 resonance line and has an extremely large core radius (something which previous authors could not simulate quickly); and $E$ is MRW with inward facing "petals".
• Figure 4: Bifurcation diagram for the model parameter $\beta$ for the reverse run across the $\beta$-range from $0.990$ to $0.570$ with the $\beta$-step of $0.001$.
• Figure 5: Bifurcation diagram: the $\beta$-$Q_s$ plots depicting the hysteresis region. The black curve represents forward run whereas red curve represents reverse run across the chosen $\beta$-range. Both figures are from the same set of data with the full set of data shown (left) and the data around the hysteresis region shown (right). Dots were used for (left) to illustrate the coinciding of the solutions for most of the data, and red lines and black crosses for the smaller data region (right).