Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

Abstract

Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution (I h) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without I h current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

Bifurcations for counterintuitive post-inhibitory rebound spike related to absence epilepsy and Parkinson disease

“…Our study finds that PIR spike does not appear for the SNIC far from the SNHO B , but for the SNIC close to the SNHO B . Such a result extends the codimension-1 bifurcations for PIR spike from Hopf and BHom in the previous studies [1,18] to SNIC. Additionally, it reveals a new border between non-PIR spike and PIR spike, which occurs at a certain SNIC bifurcation close to SNHO B .…”

“…More details about the PIR for BHom please refer to Ref. [18]. The SHom bifurcation here is close to the SNHO S in Fig.…”

“…[1] Recently, PIR spike is built relationship to big homoclinic orbit (BHom) bifurcation with a special geometry (see Appendix A of the present paper or Ref. [18]). Moreover, PIR spike is simulated in Fitzhugh-Nagumo (FHN) model, [46] where the nearly vertical n-nullcline (solid black curve) left to the stable equilibrium point (red circle) is analytically approved to be a sufficient condition for PIR spike, as depicted in Fig.…”

“…To distinguish these two thresholds, the former is called "positive" threshold, whereas the latter is named "negative" threshold in the present paper. The spike induced by negative stimulation is known as post-inhibitory rebound (PIR) spike, [1,[16][17][18][19] which is due to the existence of "negative" threshold. Such PIR spike is a paradoxical nonlinear phenomenon that differs from the suppression effect of inhibitory stimulation on firing activity.…”

“…In theory, PIR phenomenon has been related to threshold and bifurcation in multiple two-dimensional neuronal models composed of a variable of membrane potential (V ) and a recovery variable (n). [1,18] In the well-known Morris-Lecar (ML) model, it has been widely accepted for a long time that PIR spike is not be caused for stable node near saddle-node bifurcation on an invariant circle (SNIC), which corresponds to "positive" threshold. [1] Instead, PIR spike can be induced from stable focus near Hopf bifurcation, which corresponds to the "negative" threshold.…”