利用双慢变量的快慢变量分离分析新脑皮层神经元Wilson模型的复杂电活动

Analysis to dynamics of complex electrical activities in Wilson model of brain neocortical neuron using fast-slow variable dissection with two slow variables

具有重要、广谱的生理功能的大脑新皮层神经元表现出规则和快速峰放电、内源性和连续型簇放电,受到两个慢变量(T型钙和钙激活的钾离子通道)的调控(Wilson模型).若只选取更慢的慢变量进行单慢变量的快慢变量分离,不能获得Wilson模型的簇放电的动力学机制,因为此时的快子系统还含有另一个慢变量.因此,本文采用双慢变量的快慢变量分离方法进行分析.首先获得快子系统在参数平面上的分岔及分岔曲线与簇放电相轨线的交点;基于两个慢变量都不足够慢导致的只有部分交点对应的分岔与簇放电相关,本文进一步获得三维空间中分岔曲线与相轨迹的位置关系,识别出与簇放电相关的分岔,并排除无关的分岔.结果发现,内源性簇放电与不变环上的鞍结分岔有关、与鞍结分岔无关,连续型簇放电与不变环上的鞍结分岔和超临界霍普夫分岔有关、与鞍结分岔无关.本研究全面深入认识了新脑皮层神经元簇放电的动力学机制,为调控放电模式奠定了基础.

The neocortex of the brain plays a most important role in achieving functions of the brain via the electrical activities of neurons.Understanding the transition regularity of firing patterns and underlying dynamics of firing patterns of neurons can help to identify the brain functions and to treat some brain diseases.Different neocortical neurons exhibit regular spiking(RS),fast spiking(FS),intrinsic bursting(IB),and continuous bursting(CB),which play vital roles and wide range of functions.Fast-slow variable dissection method combined with bifurcation analysis has been an effective method to identify the underlying dynamical mechanism of spiking and bursting modulated by a single slow variable.The spiking is related to the stable limit cycle of the fast subsystem,and the bursting is associated with the transitions or bifurcations between the stable limit cycle and resting state of the fast subsystem.Such underlying dynamics of bursting has been widely used to distinguish different bursting patterns and identify complex dynamics of bursting modulated by various different factors such as synaptic current,autaptic current,and stimulations applied at a suitable phase related to the bifurcations,which play important roles in the real nervous system to regulate neural firing behaviors.Unfortunately,the bursting of neocortical neuronal model(wilson model)is modulated by two slow variables,i.e.the gating variable of calcium-activated potassium channel H and the gating variable of T-type calcium channel T,with H being slower than T.Then,the underlying dynamical mechanism of the IB and CB of the neocortical neurons cannot be acquired by the fast-slow variable dissection method when H is taken as the sole slow variable,due to the fact that the fast-subsystem contains the slow variable T.In the present paper,we use the fast-slow variable dissection method with two slow variables(H and T)to analyze the bursting patterns.The bifurcations of the fast subsystem,and the intersections between the bifurcation curves and the phase trajectory of bursting in the parameter plane(H,T)are acquired.Owing to the fact that neither of the two slow variables of the bursting is sufficiently slow,the bifurcations of only some intersections are related to the bursting behaviors,but others not.Then,the position relationship between the bifurcation curves and bursting trajectory in the three-dimensional space(H,T,V)(V is membrane potential of bursting)is further acquired,from which the bifurcations related to bursting behaviors are acquired and bifurcations unrelated to bursting behaviors are excluded.The start phase and the termination phase of the burst of the IB are related to the saddle-node on invariant circle(SNIC)bifurcation,but not to the saddle-node(SN)bifurcation.The start phase and termination phase of the burst of the CB are related to the SNIC bifurcation and the supercritical Andronov-Hopf(SupHopf)bifurcation,respectively,but not to the SN bifurcation.The results present a comprehensive and in-depth understanding of the underlying dynamics of bursting patterns in the neocortical neurons,thereby laying the foundation for regulating the firing patterns of the neocortical neurons.