Bifurcations in Two-Dimensional Hindmarsh–rose Type Model

Abstract

We analyze a two-dimensional Hindmarsh–Rose type model exhibiting properties of both Class 1 and Class 2 neurons. Although the system is two-dimensional and contains only four parameters, the obtained bifurcation diagrams show that the bifurcation structure satisfies conditions for emergence of both features with constant stimuli.

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

“…The Hindmarsh-Rose model is known to reproduce all dynamical behaviors, such as quiescence, spiking, bursting, irregular spiking, and irregular bursting [4,6]. Bifurcation analysis is examined once more in the past, with respect to one or two bifurcation parameters [7][8][9][10][11]. Local bifurcations and global bifurcations are also analysed and these bifurcation phenomena can be used to explain the transitions between the dynamical behaviors.…”

“…For example, the transition between spiking and bursting in the model can be understood by homoclinic bifurcations [12,13]. More information on bifurcation can be found in [1,4,[8][9][10][11][14][15][16][17][18][19][20][21][22].…”

Bifurcation Analysis in a Neuronal Model without/with Delay

“…It is noteworthy that in a plane system, class I excitability is essentially characterized by the quadratic nonlinearity. To date, many two-and three-dimensional HR models [1], [2], [8]- [10] have been studied. However, most of these papers investigated the bifurcations of the HR model though computer simulations.…”

Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

“…These two-dimensional and three-dimensional Hindmarsh-Rose models have been studied by many papers, see e.g. [19][20][21][22][23][24][25] and references therein. These papers discussed the bifurcations of Hindmarsh-Rose models mostly by computer simulations, but the upper bound of the maximal number of limit cycles bifurcated from the equilibrium via Hopf bifurcation has not been obtained.…”