Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model. (English) Zbl 1323.92043
Summary: We analyze the bifurcations occurring in the 3D Hindmarsh-Rose neuronal model with and without random signal. When under a sufficient stimulus, the neuron activity takes place; we observe various types of bifurcations that lead to chaotic transitions. Beside the equilibrium solutions and their stability, we also investigate the deterministic bifurcation. It appears that the neuronal activity consists of chaotic transitions between two periodic phases called bursting and spiking solutions. The stochastic bifurcation, defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value, or under certain condition as the collision of a stochastic attractor with a stochastic saddle, occurs when a random Gaussian signal is added. Our study reveals two kinds of stochastic bifurcation: the phenomenological bifurcation (P-bifurcations) and the dynamical bifurcation (D-bifurcations). The asymptotical method is used to analyze phenomenological bifurcation. We find that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.{
©2013 American Institute of Physics}
©2013 American Institute of Physics}
MSC:
| 92C20 | Neural biology |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
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