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

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 fourdimensional neuronal model to describe firing associated with pathological pain. The lumenal (C-lum) and intracellular (C-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-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-in, chosen as slow variable, C-lum value of the initiation point of burst is far from the saddle-node bifurcation point, due to C-lum not contained in the equation of the membrane potential. To overcome this problem, both Cm and C-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-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.

Automated summary

This paper proposes a novel fast-slow variable dissection method with two slow variables to analyze complex bursting behavior, addressing issues encountered with traditional methods and providing insights for modulating pathological pain.