【作者】 张莉；

【导师】 安新磊；

【作者基本信息】 兰州交通大学 ， 数学， 2023， 硕士

【摘要】 在神经系统中,神经元之间的信息传递和转化主要通过突触来实现,突触功能取决于电突触和化学突触之间的协同作用。但由于神经元脉冲沿轴突的传播速度有限,以及突触加工造成的时间延迟,因此神经元系统具有固有的时滞。另外,与电突触耦合相比,由电感线圈嵌入到耦合通道来等效描述的化学突触耦合可能是实现神经元之间同步信息编码的最合适的方式,因为离子的连续泵送可以在细胞内诱导磁场,所有的神经元完全由有效的场能泵送控制。因此我们认为场耦合可能提供了另一种实现信号传播和信息编码的有效途径。本文的主要研究内容如下:1.用电感线圈连接两个HR神经元激发磁场耦合,并且耦合过程中,信息的传递必然存在时滞,因此在耦合模型中引入时滞。首先利用单参和双参的分岔图研究了两个完全相同的HR神经元耦合后的放电特性和分岔模式,发现增大时滞和耦合强度,混沌状态消失,高周期放电减少,分岔模式变得简单。然后通过误差图和时间序列图分析了施加不同刺激的两个HR神经元可能发生的同步模式,证实了同步放电模式不仅与神经元的初始放电状态有关,还与时滞和耦合强度有关,具有一定的规律性。2.在第一部分的基础上,首先分别模拟了双神经元网络中依赖于外部刺激电流、耦合强度、时滞和初始条件的共存放电活动和同步行为,在此基础上揭示了初始值诱导的无限多个放电模式,包括混沌放电、周期放电和静息态。还展示了与初始值相关的同步类型,有完全同步、延迟同步和不同步,并且当时滞不为0时,完全同步时的状态只有静息态和周期1放电。此外,提出利用线性增广方法来控制极端多稳定性,当增大线性系统和非线性系统的耦合强度时,可以将具有不同位置和拓扑结构的共存吸引子都控制到形状相同但位置不同的点吸引子,即异构多稳态可以成功控制到同构多稳态,且经过控制,耦合神经元模型是完全同步的。3.分别用电阻器和电感器连接两个HR神经元来描述实际神经元网络中的电突触耦合和化学突触耦合,并且考虑到神经元信号的传播是非瞬时且非均匀的,在耦合项中引入分布式时滞。在不使用控制方法、使用Lyapunov控制法和反步控制法的情况下,对包含弱核函数和强核函数的电阻或电感耦合模型比较突触耦合权重对同步行为和分岔结构的影响。在此基础上揭示了完全同步和近似同步行为,并且证实了耦合神经元同步的实现和分岔的模式取决于耦合通道和核函数的选择。最后从控制器设计、同步误差和同步范围三个方面评估得出最优的控制方法是反步控制法。更多还原

【Abstract】 In nervous system,information transmission and transformation between neurons are achieved through synapses mainly,and synaptic function depends on the synergistic interaction between electrical and chemical synapses.However,the nervous system has inherent time delays due to the limited propagation speed of neuronal impulses along axons and the lag caused by synaptic processing.In addition,chemical synaptic coupling,which is described by embedding an induction coil into the coupling channel equivalently,may be the most suitable way to achieve synchronous information encoding between neurons,compared with electrical synaptic coupling.Because continuous pumping of ions can induce magnetic fields in cells,and all neurons are controlled by effective field energy pumping completely.Therefore,we believe that field coupling may provide another effective way to realize signal transmission and information encoding.The main research contents of this thesis are as follows:1.Two HR neurons are coupled via an induction coil to excite a magnetic field.During the coupling process,there must be a time delay in the transmission of information,so the time delay is introduced into the coupled model.First,the firing characteristics and bifurcation modes of two identical HR neurons are studied by using the bifurcations of one-parameter and two-parameter.It is found that chaotic and high-period states disappear,and the bifurcation modes become simple with increasing the time delay and coupling strength.Then,the possible synchronization patterns of two HR neurons with different stimulation are analyzed by error diagrams and time series diagrams.It is confirmed that the synchronous firing patterns are not only associated with neurons with high stimulation current,but also with time delay and coupling strength,and with some regularity.2.Based on the first part,the coexisting firing activities and synchronization behaviors in the dual neuron network which depend on the external stimulation current,coupling strength,time delay and initial conditions are simulated,respectively.Then the infinite number of firing modes including chaotic firing,periodical firing,and quiescent state is induced by initial conditions.It also shows the types of synchronization related to the initial value,including complete synchronization,delayed synchronization and asynchronization,and the state of complete synchronization exhibits only quiescent state and period-1 firing when the time delay is not equal to 0.Furthermore,the linear augmentation method is conceived to control extreme multistability.It can be found that the attractors with different positions and topological structures can be controlled to the point attractors with the same shape but with different positions when the coupling strength of linear system and nonlinear system is increased.That is,the heterogeneous multistability can be successfully controlled to the homogeneous multistability,and the coupled neurons can also be achieved complete synchronization after control.3.Resistor and inductor are employed to connect two HR neurons in order to describe the coupling effects of electrical synaptic and chemical synaptic in the actual neuronal network,respectively.Moreover,distributed time delays are introduced into the coupling terms when taking into account that the propagation of neuronal signal is non-instantaneous and non-uniform.For the resistor-coupled and inductor-coupled neuron models containing weak and strong kernel functions,the effects of synaptic coupling enhance on synchronization behavior and the bifurcation structures are compared among the three control modes(without control,using Lyapunov control and using back-stepping control).Then complete synchronization and approximate synchronization behaviors are revealed,and it is confirmed that the implementation of synchronization and bifurcation patterns of coupled neurons depend on the selections of coupling channels and kernel functions.Finally,the optimal control method is back-stepping control method by comparing from three aspects of controller design,synchronization error and synchronization range.更多还原