Nonlinear And Stochastic Dynamics Of Coupled FitzHugh-Nagumo Neurons

Nonlinear dynamic has been paid much attention in neuroscience recently. In this thesis we mainly focus on one of the most complex nonlinear phenomenon–canards and stochastic dynamics in coupled FitzHugh-Nagumo model. Canard solutions occur generically in three-dimensional singularly perturbed systems with a folded two-dimensional critical manifold. They are trajectories that unusually flow the unstable sheet of the slow manifold of these systems from the stable sheet of the manifold for a certain amount of time. As parameters locate near the canard regime, neurons can exhibit much complex behaviors and show great sensitivity to external perturbation. This is very important and meaningful for weak signal processing which guarantees low energy consumption in biological systems.In this thesis, we show the existence of canards in the chemical synaptic coupled FitzHugh-Nagumo (FHN) neurons both analytically and numerically. Also, numeric bifurcation analysis is given to illuminate the influences of certain parameters in this system on the existence of canards. Contrary to the general idea that canards in 3 are robust to small parameter changes [1;2], canards in this system are much sensitive to the external applied current and exist in only a small range of it.We also studied stochastic dynamics of three chemical synaptic coupled neurons with the Gaussian white noisy background, with the comparisons of the well known electrical coupled ones. For both of the coupling cases, optimal coherence resonance and stochastic coherence can be achieved with intermediate noise intensity. Through comparisons and analysis, we can make conclusions that chemical synaptic coupling is more efficient than the well known linear electrical coupling for both coherence resonance and stochastic coherence in the case of local and weak input signal. Also, neurons with parameters locate near the bifurcation point (canard regime) can exhibit the best response of these stochastic dynamics. Besides, the effects of inhomogeneity of the neurons and spatial independence of noise are investigated. Similar to the studies in [3], both of them can enhance information transmission. This is due to the diversities and competitions generated by different parameters and different sources of noise, which are well explored in real systems.