LoFaro T, Kopell N
Department of Pure and Applied Mathematics, Washington State University, Pullman 99163-3113, USA.
J Math Biol. 1999 Jun;38(6):479-533. doi: 10.1007/s002850050157.
We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and "singular Poincaré maps" that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold.
我们讨论一种方法，通过该方法，由相互抑制耦合的神经元网络动力学可简化为一维映射。该网络由一对神经元组成，其中一个是内源性爆发神经元，另一个是可兴奋神经元，但在无相位输入时不爆发。后一个细胞有不止一个慢过程。这种简化使用了慢/快过程的标准分离方法；它还利用了关于慢流形上的动力学在有限量的慢时间后如何演化的信息。通过这种简化，我们得到了一个依赖于原始生物物理方程参数的一维映射。在某些参数区域，可以推断出原始方程具有这样的解，即最初可兴奋细胞的活跃期在每次爆发之间是恒定的，而在其他参数区域则不是。这种调节的存在与否对应于一维映射中定性不同的动力学。与简化和动力学分析相关的计算包括使用沿轨迹按时间参数化的坐标，以及“奇异庞加莱映射”，它将沿慢流形的流信息与慢流形分支之间的跳跃信息结合起来。
