Multiple and complex codimension-2 bifurcations underlying post-inhibitory rebound spike and excitability transition

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR is evoked by inhibitory stimulation instead of excitatory stimulation, presents a challenge the threshold concept. In this paper, 7 codimension-2 degenerate bifurcations 10 codimension-1 acquired in neuronal model, underlying spike. Type I corresponds saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, type II (SN) sub-critical Hopf (SubH) sup-critical (SupH) bifurcation. The from Saddle-Node Homoclinic orbit (SNHO) via SNIC terminates meanwhile big homoclinic (BHom) SN emerge. A BHom limit point (LPC) emerges responsible for spiking II, roles other (Cusp, Bogdanov-Takens, Bautin) discussed. addition, different widely accepted viewpoint that mainly near rather than identified be induced LPC curves resemble complex present comprehensive deep understandings spike, helpful modulation neural firing activities physiological functions.