Polyglot entrainment for higher dimensional neuronal models

The entrainment of biological oscillators is a classic problem in the field of dynamical systems and synchronization. This paper explores a novel type of entrainment mechanism referred to as polyglot entrainment [Khan et al., "The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems," Chaos 32, 063137 (2022)] (multiple disconnected 1:1 regions for a range of forcing amplitude) for higher dimensional nonlinear systems. Polyglot entrainment has been recently explored only in two-dimensional slow-fast models in the vicinity of Hopf bifurcations (HBs). Heading toward generality, in this research, we investigate the phenomenon of polyglot entrainment in higher-dimensional conductance-based models including the four-dimensional Hodgkin-Huxley model and its reduced three-dimensional version. We utilize dynamical systems tools to uncover the mechanism of entrainment and geometric structure of the null surfaces to explore the conditions for the existence of polyglot entrainment in these models. In light of our findings, in the vicinity of HB, when an unforced system acts as a damped oscillator and the fixed point is located near a cubic-like manifold, polyglot entrainment is observed.