類Hindmarsh-Rose模型之分岔與動態行為的研究
Abstract
本研究主要的目的是探討Hindmarsh-Rose Type model的分岔研究。該模型是Hodgkin-Huxley(HH)的神經元模型的簡化型。為了簡化HH模型並保留神經元活動的基本特性，FitzHugh-Nagumo以及Hindmarsh–Rose(HR)等人將該模型簡化成多項式的形態，以利於研究單一神經元的活動。我們主要的工作是透過研究HR模型的saddle-node (SN) bifurcation及Andronov-Hopf (AH) bifurcation以了解類型一神經元及類型二神經元的活性化。我們主要的結果是確定SN及AH發生的條件。本研究成果有助於了解單一神經元動態行為及其基本特性。
In the paper, we aim to study some bifurcations and dynamical behaviors of a two dimensional Hindmarsh-Rose type (HRT) model, which is a simplified version of Hodgkin-Huxley neuron mode. Hodgkin suggested that there exist two classes of neurons: one is Class 1 and the other is Class 2. In dynamical systems, these two classes also called Type 1 and Type 2, respectively. We mathematically confirm the occurrences of both saddle-node and Andronov-Hopf bifurcations of the HRT model. Physiologically, the first bifurcation is concerned with Class 1 excitability and spiking, and the second bifurcation is related to Class 2 excitability and spiking. Therefore, the research could help us to understand the dynamical behaviors of a single neuron and its basic characteristics.
In the paper, we aim to study some bifurcations and dynamical behaviors of a two dimensional Hindmarsh-Rose type (HRT) model, which is a simplified version of Hodgkin-Huxley neuron mode. Hodgkin suggested that there exist two classes of neurons: one is Class 1 and the other is Class 2. In dynamical systems, these two classes also called Type 1 and Type 2, respectively. We mathematically confirm the occurrences of both saddle-node and Andronov-Hopf bifurcations of the HRT model. Physiologically, the first bifurcation is concerned with Class 1 excitability and spiking, and the second bifurcation is related to Class 2 excitability and spiking. Therefore, the research could help us to understand the dynamical behaviors of a single neuron and its basic characteristics.
Description
Keywords
類Hindmarsh-Rose模型, saddle-node分岔, Andronov-Hopf分岔, 動態系統, Hindmarsh-Rose type model, saddle-node bifurcation, Andronov-Hopf bifurcation, dynamical systems