The Influence Of Inhibitory Signals On The Criticality Of The Excitatory-inhibitory Network

Self-organized critical states are found for many complex systems in nature,from earthquakes to neuronal avalanches.Several lines of evidence point to the existence of such critical states in brain activity.It was shown that neuronal avalanches may play an important role in cortical information processing and storage.In realistic neural systems,the excitatory and ithibitory neurons coexist in networks.In this paper,we considered a model of excitatory-inhibitory(El)network.We investigated the effect of the number of inhibitory nodes,the number of inhibitory links and inhibitory coupling strength on criticality and dynamic range using both numerical simulations and theoretical analysis.This provides a new insight into the effect of inhibition on complex networks.Specific research were described as follows:First,we simulated the effect of creasing the inhibitory coupling strength on criticality of network.In physiologically realistic,since inhibitory synapses are often closer to the cell body of postsynaptic neurons,it is of practical value to study the effect of increasing the inhibitory coupling strength on criticality.We considered a model of El network on the ER graphs using the KC model.We randomly choosed Ne=feN nodes as excitatory elements,fe=0.8.The rest Ni=fiN nodes as inhibitory ones,fi=1.0-fe.To study the effects of the strength of inhibitory couplings in the model,we set the inhibitory coupling strength is m times larger than excitatory coupling strength.We calculated the average activity when m=1,5,10.We obtained that the critical points are not changed as the strength of inhibitory couplings increases.We also investigated the effects of inhibitory strength on the response of El network to stimulus,and obtained the dynamic range versus the branching ratio for different inhibitory strength.We can conclude that the optimal dynamic range is occur at critical state and dynamic range is slightly decreased with the inhibitory strength m.In the mean-filed approximation,we obtained the evolution of the average activity.The analytic results show that the critical point is independent of the strength of the inhibitory signal.We obtained the evolution of dynamic range and find that the dynamic range is taken to the maximum value in the critical state,and the increase of inhibitory strength can decrease the dynamic range.Moreover,we also simulated the influence of the inhibitory coupling strength in small world network and the regular random network,and obtained the same conclusion as in the random network.Therefore,increasing the inhibitory coupling strength has no effect on the criticality point,and has little effect on dynamic range of network.Then we simulated the effect of decrease the excitation and inhibition.The conclusion of reducing excitation and inhibition in experiments is clear,and we studied this phenomenon by simulation,which is helpful to the study of the mechanism of inhibitory signal action.We changed the network in three ways:removing a fraction(r)of excitatory or inhibitory nodes(0<r<1),deleting a fraction(d)of excitatory or inhibitory links(0<d<1),and weakening excitatory or inhibitory coupling strength by a fraction w(0<w<1)on critical network.Each method of reducing the excitation can recover the phenomenon in experiments.None of the three methods of reducing inhibition can change the dynamic range.In analytical results,we showed that reducing the excitation of the network changes the network in the critical state into the subcritical state.So the dynamic range is reduced.However,decreasing inhibition has no influence on critical point on El network.In addition,we assumed that the excitation probability of the inhibitory nodes is twice times that of the excitatory nodes,and the conclusion that the inhibition has no effect on the critical point is also obtained.Reducing the excitation of the network can reduce the dynamic range of the network,however,reducing the inhibition of the network has no effect on the dynamic range.Finally,we simulated the effect of inhibition on criticality of network with dynamic synapses.The simulation of synaptic dynamics is closer to the reality,so further research in this simulation makes sense.We showed examples of avalanche-size distributions for various branching ratio,and drew the mean-squared deviation from an exact power law,verifying the critical dynamic occur at the range of connectivity parameters.We increased the inhibitory coupling strength,the range of parameters which cause the critical dynamic increases.The simulation results of reducing the inhibitory coupling strength in the network showed that it does not change the criticality of the network.Then,we increased the number of inhibitory nodes in the network,and we found that the value of critical point becomes larger,and the range of critical parameter increases.Finally,we reduced the connection probability of the network,and the results showed that the connectivity of the network does not affect the value and range of the critical parameters.Therefore,it can be concluded that adding inhibition to the neural network with synaptic dynamics can increase the range of critical parameter.Based on above results of simulation and analysis,this paper made a conclusion that the critical point is not affected by inhibitory signal strength,and the increase of inhibitory signal strength will decrease the dynamic range slightly in network with random process of synapses.Moreover,we used three different methods of reducing excitation and inhibition,we showed that reducing the excitation can make the critical state of the network into a subcritical state,thus dynamic range is decreased as the same as experiments.However reducing the inhibition cannot change network's criticality and dynamic range.In addition,when the inhibition increases,range of parameters which cause the critical dynamic increase in network with dynamic synapses.These studies provide important insight into the role of inhibitory elements in excitable neural networks.The analysis is also interesting for other many systems which can be studied using excitable networks.