| Spiral wave is one kind of common ordered spatiotemporal pattern,which is generated in various systems far from the equilibrium state.It is a type of non-linear wave that can self-maintain without the need of a wave source.The occurrence of spiral waves in neuronal networks is usually related to certain neurological diseases.For example,the short-lived small spiral wave can be observed frequently in the cerebral cortex in vivo during the seizure,and it prefers to occur in discharge ending stage.The spiral waves of spreading depression have also been observed in hypoglycemic chicken retina,which can cause macular degeneration.Therefore,understanding the formation of brain dynamic pattern is very important for clinical treatment of neurological diseases such as epilepsy.Experimental and theoretical investigations show that noise and inhibitory coupling can induce spiral waves in neuronal networks.However,there is still a lack of extensive research on the factors related to the spontaneous generation of spiral wave except noise and inhibition coupling.In order to explore the mechanism of spontaneous generation of spiral waves in neuronal networks,this paper uses the Hindmarsh-Rose neuronal model and the two-dimensional neuronal network to study whether the system evolved from the chaotic initial phase distribution state can spontaneously generate spiral waves.Entropy is introduced to describe the order of neuronal network pattern and to explore the mechanism of spiral wave generation.The content of the article is arranged as follows:The first chapter is the introduction.The background and significance of investigation,the structure of the neuron and its some mathematical models,the synchronization and the spontaneous generation of spiral waves in the neuron network are briefly introduced.The second chapter is our first research work.The Hindmarsh-Rose neuron model is used to study the dynamic behavior of the two-dimensional(2D)neuronal network with double-coupling layer,which is composed of nearest-neighbor excitatory coupling and long-range repulsive coupling layers and evolves from an initial state with a random phase distribution.An improved cluster entropy is proposed to describe the spatiotemporal pattern of the neuronal network.The numerical simulation results show that the repulsive coupling can either promote the formation of ordered patterns or suppress the formation of ordered patterns.When the repulsive coupling strength and excitatory coupling strength are appropriately selected,the chaotic network can spontaneously generate single spiral wave,multiple spiral wave,traveling wave,the coexistence of spiral wave and others wave state,the coexistence of target wave and others wave state,the coexistence of traveling wave and standing wave,etc.The probabilities with which spiral wave and traveling wave occur reach 0.4555 and 0.1667 respectively.The probabilities with which target wave and other states co-occur,and the probability with which the traveling wave and the standing wave co-occur,are 0.0389 and 0.1056,respectively.These ordered wave patterns and chaotic states can be distinguished by using the proposed cluster entropy.When the repulsive coupling strength is large enough,the neuronal network is generally in chaotic state.It is found by calculating cluster entropy that a large cluster can appear in the neuronal network when both the excitatory coupling strength and repulsive coupling strength are weak.These results can conduce to understanding the self-organization phenomena occurring in the experiments and also to treating various neurological diseases.The third chapter is our second research work.The Hindmarsh-Rose neural model is used to study the spontaneous generation of spiral wave and other ordered waves in neuronal networks.In order to explore the mechanism of spontaneous generation of these ordered waves,the long-range unidirectional coupling is added to two-dimensional neuronal networks with nearest-neighbor diffusive coupling form.The network evolves from an initial state with chaotic initial phase.At the same time,the information transmission entropy is introduced to depict the direction of information transmission in neuronal network.The numerical simulation results show that when the intensity of next nearest neighbor long-range unidirectional coupling and nearest-neighbor diffusive coupling are properly selected,the network can spontaneously appear all kinds of ordered waves such as single spiral wave,multi-spiral wave,the coexistence of single spiral wave with other states,the coexistence of multi-spiral wave with other states,small traveling wave,big traveling wave,the coexistence of big traveling wave with other states,the coexistence of plane-wave with other states,the coexistence of target wave with other states,etc.The multiple self-sustaining long plane waves and the transition from spiral wave to target wave are observed in the network.The average probability of appearance of spiral wave,small traveling wave,big traveling wave,plane wave and target wave reaches 0.55,0.0611,0.2027,0.133 and 0.0528,respectively,which indicates that the next nearest-neighbor unidirectional coupling can promote the formation of spiral wave.With the increase of long-range unidirectional coupling distance,the appearance probability of spiral wave decreases sharply,and the appearance probability of plane wave increases firstly and then decreases.Increasing the distance of long-range unidirectional coupling can promote the formation of plane wave and traveling wave.The generation of ordered waves caused by long-range unidirectional coupling is related to the intermittent directional transmission of information in the network.When single spiral wave and multi-spiral waves are generated in the network,entropy resonance occurs.The formation of single spiral wave and multi-spiral wave induced by noise,long-range inhibitory coupling and long-range repulsive coupling is also related to entropy resonance.The existence of self-sustaining long plane-waves is due to the existence of continuous strong information directional transmission in the network.These studies lay a foundation for exploring the spontaneous generation of ordered waves such as spiral wave in complex neuronal networks. |
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