Fast–slow variable dissection with two slow variables related to calcium concentrations: a case study to bursting in a neural pacemaker model

Abstract

Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear behavior alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast–slow variable dissection method with two slow variables is proposed to analyze the complex bursting simulated in a four-dimensional neuronal model to describe firing associated with pathological pain. The lumenal (\(C_\mathrm{lum}\)) and intracellular (\(C_\mathrm{in}\)) calcium concentrations are the slowest variables, respectively, in the quiescent state and burst duration. Questions encountered when the traditional method with one slow variable is used. When \(C_\mathrm{lum}\) is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With \(C_\mathrm{in}\) chosen as slow variable, \(C_\mathrm{lum}\) value of the initiation point of burst is far from the saddle-node bifurcation point, due to \(C_\mathrm{lum}\) not contained in the equation of the membrane potential. To overcome this problem, both \(C_\mathrm{in}\) and \(C_\mathrm{lum}\) are regarded as slow variables; the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing the \(C_\mathrm{lum}\) dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast–slow variable dissection and a deep understanding of the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.