Dynamics of subthreshold and suprathreshold resonance modulated by hyperpolarization-activated cation current in a bursting neuron

Resonance of both subthreshold and suprathreshold membrane potentials has been found in brain or motor neurons involving in the information processing or locomotion control via frequency response, which is mainly related to the hyperpolarization-activated cation current (Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document}). In the present paper, the modulations of Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document} current on resonance are acquired in a bursting neuron model, which closely match experimental observations. Firstly, the increase of Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document} current induces stable node changed to stable focus related to Hopf bifurcation, and to bursting via complex bifurcations including the big homoclinic orbit related to type I excitability. Secondly, the subthreshold resonance can be evoked by small stimulation from focus instead of node, resembling the experimental observation that subthreshold resonance appears at strong (Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document}) current. Thirdly, the modulation regularity of Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document} current on frequency and amplitude of resonance is simulated, closely matching the experimental observations, which can be well explained with the changing trends of imaginary and real part of eigenvalues of the focus. Last, the suprathreshold bursting is induced by strong stimulation from the stable focus near big homoclinic orbit, resembling that observed in the experiment. The bursting exhibits “suprathreshold resonance” similar to those evoked near Hopf bifurcation corresponding to type II excitability. A novel condition for “suprathreshold resonance” and the underlying codimension-2 bifurcations are acquired. The results present the comprehensive viewpoints on the subthreshold resonance and extend the condition for the “suprathreshold resonance”, which are helpful for deep understanding resonance in the brain or motor neurons with Ih\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{{\text {h}}}$$\end{document} current.