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.
Highlights
Identification of nonlinear dynamics of neuronal electrical behavior is very important for understanding the functions of nervous system [1,2,3,4,5,6]
A comprehensive viewpoint of dynamics of the bursting is presented by comparisons between dynamics acquired with the three manners of fast-slow variable dissection. These results present deep understanding for the complex dynamics underlying the bursting behavior of pacemaker model, which is helpful to modulate the bursting related to pathological pain, and a novel feasible performing process for fast-slow variable dissection with two slow variables related to calcium concentrations, which can be extensively utilized to analyze the bursting of other models with slow variables Cin and Clum
Identification of nonlinear dynamics of bursting which participates physiological or pathological functions is very important for the nonlinear dynamics as well as the neuroscience [1,2,3,4,5,6, 60,61,62]
Summary
Identification of nonlinear dynamics of neuronal electrical behavior is very important for understanding the functions of nervous system [1,2,3,4,5,6]. The pyramidal neuron of CA1 region of hippocampus related to SD or SZ [55,56,57] belongs to case2 Both potassium and sodium concentrations are slow variables, and two slow variables exhibit nearly approximate time scale. A comprehensive viewpoint of dynamics of the bursting is presented by comparisons between dynamics acquired with the three manners of fast-slow variable dissection These results present deep understanding for the complex dynamics underlying the bursting behavior of pacemaker model, which is helpful to modulate the bursting related to pathological pain, and a novel feasible performing process for fast-slow variable dissection with two slow variables related to calcium concentrations, which can be extensively utilized to analyze the bursting of other models with slow variables Cin and Clum.
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