Continuation and bifurcation analysis of a periodically forced excitable system

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh–Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different “ M : N ” periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main “tree” of period- 2 P branches, where the 1:1, 1:0, 2:2, 2:1, … rhythms are located, and of several closed loops made up of period- { N × 2 P } branches ( N > 2 ) , isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms ( N > 2 ) on one hand, and N: N - 1 or Wenckebach rhythms ( N > 2 ) on the other hand, are located. Stable M : N and M: N - 1 rhythms ( M ⩾ N ) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.