Abstract
A mathematical model of vibrissa motoneurons (vMN), which has been developed by Harish and Golomb, can show repetitive spiking in response to a transient external stimulation. The vMN model is described by a system of nonlinear ordinary differential equations based on the Hodgkin-Huxley scheme. The vMN model is regulated by various types of ionic conductances, such as persistent sodium, transient sodium, delayed-rectifier potassium, and slow ionic conductances (e.g., slowly activating potassium afterhyperpolarization (AHP) conductance and h conductance). In the present study, a numerical simulation analysis of the vMN model was performed to investigate the effect of variations in the transient sodium and the slow ionic conductance values on the response of the vMN model to a transient external stimulation. Numerical simulations revealed that when both the transient sodium and the AHP conductances are eliminated, the vMN model shows a bistable behavior (i.e., a stimulation-triggered transition between dynamic states). In contrast, none of the following induce the transition alone: 1) elimination of the transient sodium conductance; 2) elimination of the AHP conductance; 3) elimination of the h conductance; or 4) elimination of both the transient sodium and the h conductances.
Highlights
Neurons are examples of nonlinear dynamical systems, and their dynamics is investigated by various types of mathematical models based on the Hodgkin-Huxley equations [1]
As an example of such mathematical models, a vibrissa motoneuron model has been proposed [2]. This model is described by a system of nonlinear ordinary differential equations (ODEs) and comprises several voltage-dependent ionic conductances: the transient sodium conductance, the persistent sodium conductance, the delayed-rectifier potassium conductance, and three types of slow ionic conductances (the slowly activating potassium afterhyperpolarization (AHP) conductance, the hyperpolarization-activated h conductance, and the potassium M conductance)
It is important to understand the relationship between the ionic conductance and various dynamical behaviors of the vMN model, and several results that reveal such relationships have been reported so far, for examples, the transient sodium conductance is essential for generating repetitive spiking [4]; the frequency of repetitive spiking is modulated by the AHP conductance and the h conductance [2], the h conductance is involved in mixed-mode oscillations [3], and the h conductance and the M conductance are involved in bistability between the resting and spiking states [3]
Summary
Neurons are examples of nonlinear dynamical systems, and their dynamics is investigated by various types of mathematical models based on the Hodgkin-Huxley equations [1]. It is important to understand the relationship between the ionic conductance and various dynamical behaviors of the vMN model, and several results that reveal such relationships have been reported so far, for examples, the transient sodium conductance (but not the persistent sodium conductance) is essential for generating repetitive spiking [4]; the frequency of repetitive spiking is modulated by the AHP conductance and the h conductance [2], the h conductance is involved in mixed-mode oscillations [3], and the h conductance and the M conductance are involved in bistability between the resting and spiking states [3]. In the present study, a computer simulation analysis of the vMN model was performed to investigate the effect of variations in the transient sodium and the two types of slow ionic conductance (the AHP conductance and the h conductance) on the dynamical behavior. To make the vMN model simpler, the present study does not consider the M conductance)
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