The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline. Published under an exclusive license by AIP Publishing.

Automated summary

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo model under external periodic forcing. Polyglot entrainment, characterized by multiple disconnected 1:1 entrainment segments, is observed when the unforced system is in the vicinity of a Hopf bifurcation. The emergence of polyglot entrainment is explained using phase-plane analysis and other dynamical system tools. Similar entrainment effects are also observed in other systems.