Criticality And Synchronization In Neuronal Networks

Nowadays,with the help of mathematical analysis,statistical physics(mean field theory,master equation,rate equation,generating function,stochastic process,etc.),complex network and other basic theories,the study of brain is in the stage of vigorous development.Studies of the nonlinear phenomena such as criticality,synchronization,desynchronization and chaos in neuroscience experimental research have always been a hot topic in theoretical research.This thesis focuses on explaining the neuronal mechanism of the nonlinear phenomena of neuronal population activities displaying different macroscopic or microscopic properties by means of statistical physics.In the second chapter,we use the approximate-master-equation approach to study the dynamics of the Kinouchi-Copelli neural model on various networks.By categorizing each neuron in terms of its state and also states of its neighbors,we are able to uncover how the coupled system evolves with respective to time by directly solving a series of ordinary differential equations.Then,we can easily calculate the statistical properties of the network instantaneous response,the network response curve,the dynamic range,and the critical point in the framework of the approximate-master-equation approach,and the results are agree well with the stochastic simulations.In particular,three different ways can be conveniently adopted to determine the location of the critical point in terms of the power-law decay behavior of the network instantaneous response,the exponential growth behavior of the network response curve as a function of weak external stimulus,and the minimum response as a function of the branching probability.In the third chapter,we investigate the occurrence of synchronous population activities in a neuronal network composed of both excitatory and inhibitory neurons and equipped with short-term synaptic plasticity.The collective firing patterns with different macroscopic properties emerge visually with the change of system parameters(synaptic efficacy,relative inhibitory efficacy and external input current).We systematically discuss the pattern-formation dynamics on a microscopic level and find a lot of hidden features of the population activities.The bursty phase with power-law distributed avalanches of the neuronal cascades is observed in which the population activity can be either entire or local periodic-like.In the purely spike-to-spike synchronous regime,the periodic-like phase emerges from the synchronous chaos after the backward period-doubling transition.We also show that the inhibitory neurons can promote the generation of various collective firing activity by depressing the activities of postsynaptic neurons partially or wholly,which significantly increases the diversity of neuronal population activity.In the fourth chapter,we analyze the population activities of neurons between the synchronous and asynchronous phases,and expect to determine the phase transition types of two different desynchronization transitions.Through the statistics of the order parameters,the corresponding standard deviations and the fourth-order Binder cumulants,we prove that the strong population activity of neurons near the desynchronization phase transition fluctuates greatly,and the fourth-order Binder cumulants of order parameters have(positive)minimum values at the phase transition points of the desynchronization transitions,but these results are not enough to judge the phase transition types.In the excitatory-inhibitory networks,when inhibition dominates excitation,the desynchronization transitions of population activities with the change of synaptic efficacy or relative inhibitory efficacy are discontinuous.At the same time,the hysteresis between the synchronous population activities and asynchronous ones is observed.In the fifth chapter,we study the influence of noise on the collective firing population activities with different macroscopic or microscopic properties.The input of noise in the neural network can promote the switch process of the population activity displaying different firing patterns.Under the constant noise,the population activity of neurons are stable when they are periodic or periodic-like.In the excitatory-inhibitory networks,when inhibition dominates excitation,the input of noise can cause the transition of neuronal population from asynchronous to synchronous state(noise-induced synchronization).With the increase of noise intensity,the asynchronous population activity occurs in the case of strong population activity with high average firing rate(noise-induced desynchronization).Through a large number of numerical simulations,we confirm that the noise-induced synchronization phase transition in excitatory-inhibitory networks is discontinuous.In excitatory networks and excitatory-inhibitory networks,there are large fluctuations near the phase transition points of the noise-induced desynchronization transitions,but the statistics of order parameters can not confirm their phase transition types.Further research is necessary to resolve these problems.We hope that the research works in this thesis can improve the understanding of the neuronal mechanism of various nonlinear phenomena discovered from the population activities of neurons in the cortex,which may advance our understanding of the brain cognition and promote the innovation of diagnosis and treatment of the brain diseases.