Modeling And Dynamical Properties Of Complex Biological Neural Networks

Hodgkin and Huxley who are the biologist of Britain put forward the notability nonlinear dynamical equation of giant axon of Loligo, namely Hodgkin-Huxley(H-H)equation, and described rightly the electrochemistry mechanism of nerve discharging in 1952. This model can be used to describing nonlinear phenomenon appearing in neurilemma, such as the problem of self-oscillation, chaos, and multi-stability etc. It also provides the basic frame to study the excitement of neuron. How do thousands upon thousands neurons couple with each other to fire is an important issue both theoretically and experimentally. Recently, the development of complex networks theory provide another measure to research that problem. In this dissertation, we applied statistical method, nonlinear system theory to the complex biological neural networks, constructed three artificial neural networks models, and then studied their dynamic properties. The studies show that these neural models exhibit some characteristic, which have some extent comparability with the real biological neural networks. Our study might shed some light on studying the excitement, the discharge rhythm, synchronization of neural networks.The main contents and originalities in this dissertation can be summarized as follows:(1) The study on coherence resonance and optimal system size in biological neural networks. The network of collective dynamical behaviors have beening the hot topics for the study. In this dissertation, we studied the reciprocal relationship between long-range connection and local connection in the biological neural network. We construct a globally coupled biological neural network model with side-inhibit mechanism, and numerically investigate its coherence resonance. We have considered side-inhibitory mechanism in a globally coupled H-H neurons model, and then studied the spike coherence behavior. We find that the collective behavior of the system is the most regular when the connection strength A and ratioμhave an optimal value, but the optimal island does not exist if the noise strength is small. The CR and size resonance have also been investigated at the different connection strength A and different ratioμ. The ratioμhas little effect on R when the coupling strength A is small. As the coupling strength increases, the influence ofμbecome more distinguishable.(2) The study on excitement, optimality properties and synchronization in small-world biological neural networks with time-varying weights Recently, the biological neural network models are unweighted models or the connection weights are just a constant independent time. But, the connection weights of real-world biological neural network are updated dependent on time. In this dissertation, according to biological neural networks have small-world property and updating connections weights with time, we propose a new model of small-world biological neural networks with time-varying weights. Then we numerically study excitement statistical properties of this model under the external stimulus, and get some results which is consistent with real-world biological neural network properties. The significance results show that there is optimal learning rat value b* with the same condition of structure of networks, parameters employed and external stimulating, which make the excitement strength of biological neural networks reach the biggest. On the other hand, we study the effect of updated weights on synchronization in this model. We find there exist an optimal synchronization state when the coupling connection weights of neurons are adjusted reasonably by changing the learning rate. The larger external stimulus will make it easier to tame the synchronization and the connection-rewiring probability which possessed the small-world networks structure just has little effect on the synchronization.(3) The study on synchronization induced by external stimulus in biological neural networkThe synchronization has very importance both in theory and in practical applications. So, the synchronization of biological neural system has attracted increasing attentions. In this dissertation, the results shown that different frequency of external stimulus can induce synchronous activities in neural network. We also found that the coupling strength of network have effect on the synchronization. But the reason that induced this phenomenon is different.