The Influence Of Refractory Period On The Critical State That Can Excite The Network

The excited network is a simplified model described the activity of multiple complex systems.It is used to simulate the behavior dynamic and information transmission in cortical network of the brain.The distribution of the avalanche size for neuronal activity in cortical network of the brain satisfies the power-law distribution.Many experiments revealed that the activity of brain was at a critical condition.The quantities of information transmission,information capacity and dynamic range are maximized in cortical networks at critical condition.In this paper,we study the critical state of an excitable network,which is often used as a simplified neural network model.We mainly study the effect of refractory period on critical point,the dynamic range and the distribution of avalanche for complex networks.The paper is divided into two sections as follow:In first section,we study the effect of refractory period on criticality in ER random networks.We study from the transition of response from zero to nonzero and dynamic range.The research shows the theory that the criticality occurs at ?= 1.0 is valid for networks without refractory states.However,it is invalid for networks with networks with refractory states.Through numerical simulations,we found the criticality is governed by the number of refractory states.The critical largest eigenvalue and the optimal dynamic range increase monotonously with the refractory period.And the dynamic range is maximized at critical condition.On this basis,we also study the effect of network size and the mean degree on criticality.It is shown that there is size-effect when the network size is small.However,the criticality isn't influenced when it is large enough.The critical largest eigenvalue decreases monotonously and tends to a fixed value with the mean degree.Based on the simulation by computer,we give the condition of criticality's occurring and the formula of calculating the maximal dynamic range by theoretical analysis.By comparing theoretical results with numerical simulation results,the analytical analysis is proved to be correct.At last,we study the distribution of avalanche for ER random networks.We found the distribution of avalanche in ER random network obeys the power-law,and it doesn't turn out that criticality occur when the largest eigenvalue is one.In addition,all nodes are just:excited and some are excited repeatedly in network at critical condition.In second section,we study the effect of refractory period on criticality for BA scale-free networks and WS small-world networks.First,we study BA scale-free networks.These conclusions obtained by simulating are the same as ER random networks,but the critical largest eigenvalues are different.Then,we study WS small-world networks.Disagreed with the first two types of networks,the largest eigenvalue increases monotonously and then shifts to a fixed value.And the theory that the criticality occurs at when the largest eigenvalue is one is invalid for WS small-world networks without refractory states.And there exists no size-effect.Furthermore,we study the effect of the probability of rewiring on criticality in WS small-world networks.The result shows that the criticality is governed by the probability of rewiring.In addition,in the critical condition,the distribution of avalanche obeys power-law if the avalanche size is large enough.However,the distribution of avalanche is influenced by the length of refractory states,the mean degree and the probability of rewiring.In conclusion,we obtained the criticality depended on the refractory states in theory and the mechanism of how refractory states influenced.These results provide a new recognition of theory for understanding the criticality of neural networks and other complex networks in nature.