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, by using simulations in a comoving frame of reference, we present numerical results which show that subcritical bifurcations are also present within FitzHugh-Nagumo. We show that a hysteresis region is present at the boundary of the rigidly rotating spiral waves and the meandering spiral waves for a particular set of parameters, a feature of FitzHugh-Nagumo that has previously not been reported. 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.
Highlights
Spiral waves occur naturally in many physical, chemical and biological systems [1,2,3,4,5,6,7,8,9,10]
In cardiac tissue, the presence of these rotating spiral waves indicates that there is an abnormality in the hearts natural rhythm [11,12,13,14,15]
In 1961, FitzHugh published a paper suggesting a model of nerve cell excitation, a simplified version of the Hodgkin-Huxley model, influenced by the Van der Pol oscillator equations [7]
Summary
Spiral waves occur naturally in many physical, chemical and biological systems [1,2,3,4,5,6,7,8,9,10]. The cells ability to be stimulated in response to external energy is critical in the life cycle of spiral waves [16] Excitable systems, such as the propagation of electrical energy along nerves, have been studied mathematically since 1940 using parameter dependent mathematical models. We present work on a numerical approach to study the transition from rigid rotation (RW) to meander (MRW), and show that in the FHN system of equations, there are regions within the parameter space in which the Hopf bifurcations are subcritical. We analyse these results to confirm their validity and show that within the hysteresis regions where there are two solutions relating to the same set of parameters, it is possible to convert one of these solutions to the other
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