Synchronization,Resonance And Exit Problem For Neuronal Dynamical Systems

Neurons are the basic units of the structure and the function of the nervous system.The large networks formed by their interconnections play an important role in the organism's perception of external information,learning,emotion,and memory.Neurons receive external stimuli and generate action potentials that process and transmit information in the form of electrical or chemical signals.The electrophysiological properties of neurons are closely related to their dynamical characteristics.Therefore,studying the nonlinear dynamical properties of neurons is crucial for understanding the role and function of neurons in organisms.In this thesis,we study the synchronization of neuronal networks,vibration resonance and exit problem for different neuron models.Due to the finite speed of information transmission in neurons,the time delay between coupled neurons is unavoidable,and has an important influence on the dynamical behaviors of neuron clusters.Because of the spatial distribution among neurons,the distances between them may not be the same.Traditional research generally regards time delay in the network as a constant,while the heterogeneity of the spatial distribution of neurons requires us to consider the heterogeneity of the time delay.The distance-dependent delay is introduced into the ring structured network,and the influences of the time delay on the synchronizations of the neuronal networks and their transition are investigated.Our research contents involve three typical neuron models.By calculating the spatiotemporal patterns under different time delays,it is found that the network spatiotemporal pattern changes with the coupling delay between connected neurons.The robustness of these spatiotemporal patterns for parameters such as network size,network rewiring probability,noise intensity,and coupling strength is verified.Through theoretical analysis and numerical simulation,we find that the network synchronization behavior caused by different time delays is essentially related with the locking between the firing period and the connection delay.By bifurcation analysis,it is revealed that the mechanisms of the synchronizations and the holding and failure of the delay-induced lockings are closely related to the bifurcation structure.In order to quantitatively describe the degree of synchronization between neurons,we propose a definition for measuring the synchronization of neurons,which can be applied to both spiking and bursting neurons.Vibrational resonance has been extensively investigated in recent decades,while research on neuronal systems is mainly concentrated on the suprathreshold part.Because of the multiple timescale of neurons,there are significant differences in dynamical behaviors between the neurons which are of suprathreshold and subthreshold.In this thesis,the FitzHugh-Nagumo neuron is taken as an example to investigate the subthreshold and suprathreshold vibrational resonance separately for the first time,and through which,the different mechanisms of subthreshold and suprathreshold vibrational resonance are unveiled.For the subthreshold vibrational resonance,via the fast-slow variable separation method and the numerical simulation,it is found that if the frequency of high frequency signal is far away from the subthreshold oscillation frequency,vibrational resonance is absent.While if the frequency of high frequency signal is close to the subthreshold oscillation frequency,the vibrational resonance presents and the peak of the response curve corresponds to the largest orbit width.For the suprathreshold vibrational resonance,the global response to the frequency and amplitude of the high frequency signal is analyzed.It is verified that the vibrational resonance occurs at the transition boundary of the phase lockings between the frequency of high frequency signal and the frequency of the neuron firing.In particular,it is found that the two global extremes of vibrational resonance correspond not only to the suprathreshold phase-locking but also to the subthreshold phase-locking.The large deviation theory provides a probabilistic description of the long term behavior for the noise-perturbed systems.In this thesis,the large deviation theory is applied to study the exit process of Class I excitable neurons(the process from the rest state to the threshold).It is found that the extremal dispersion position of the Class I excitable neuron system coincided with the extremal position of the momentum during the exit process.We named this position as the crucial locus of the exit.By introducing Gaussian white noise as well as a pulse signal with a special position for Class I excitable neuronal systems,the significance of the crucial locus to the firing behaviors for Class I excitable neurons is emphasized.In addition,in order to calculate the exit location distribution,we propose a probability evolution method,and apply it to two typical examples and compare the results with the theoretical ones.This method shows a good accuracy for weak noise intensity.The research contents of this thesis are benificial to understanding the mechanisms of synchronization of neuronal networks with time delay,and to distinguishing between the neuronal suprathreshold and subthreshold vibrational resonance,and to identifying the characteristics of noiseinduced firing behavior of neurons.These results may have potential implications for understanding neuronal physiological properties and functions.