Self-Organized Criticality Model Of Neural Networks In The Brain

It is well known that the brain might be working at the state of self-organized criticality (SOC). The main feature of it is that the size distribution of neural avalanches obeys power-law. Some theoretical and experimental results demonstrated that neural networks exhibit small-world topology. In this paper, we study the self-organized criticality of neuronal network in brain through the approach of complex network.In the first chapter, we introduce the origin of complex network,basic models of networks,the concept of SOC and the characteristics of it in neural networks.In the second chapter of this thesis, inspired by previous models based on two-dimensional square lattice, we take the small-word connection with increased average degree (eight and sixteen, respectively), use the same boundary conditions (periodic on horizontal while fixed on vertical directions, respectively) as the author used and integrate-and-fire mechanism to describe the activity of neural networks. Three most important ingredients for neuronal activity, namely, threshold firing, neuron refractory period and synaptic rewiring are taken into account. With the hint from the experimental results, we consider the diversity of neural firing thresholds, the competition of two time scales for synaptic rebuilding and the neuronal firings in our model. Numerical results show that the avalanche size distribution exhibits power-law for different rewiring probabilities (0.3-0.9) and a larger range of the ratio of two time scales (1,105). Our neural network model reproduces SOC, we find that the diversity of neural firing thresholds is benefit for producing SOC and longer synaptic rewiring period is benefit for producing SOC. The model provides us with a new approach for the further research on the mechanism of neural networks.In the third chapter, inspired by the studies of Karbowski et al. and He et al. , we propose a SOC model for two-dimension(2D) small-world neuronal network by adding new edges with the probability decay with the distance between two nodes. Starting from a regular two-dimensional square lattice of the original model, we add long-range connections between neurons with the probability p ij ~ rij?δto form a small-world network, where ri j is the geometric distance between any pair of sites i and j , andδis a constant decaying exponent under the constraint of certain total cost C for these connections. Neuronal diversity is introduced by considering a random distribution of firing thresholds and different synaptic rewiring periods. Numerical simulations show that three diversities—different probabilities of adding links, heterogeneity of neuronal thresholds and variable synaptic rewiring time scales–give positive contributions in generating SOC characterized by the power-law distribution of avalanche sizes of neural firings in such networks, with the exponentτ= 1.5consistent with medical measurements. Our model can successfully describe the self-organized criticality in neural networks.