Rubin Jonathan E
Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Aug;74(2 Pt 1):021917. doi: 10.1103/PhysRevE.74.021917. Epub 2006 Aug 21.
This work explains a mechanism through which the introduction of excitatory synaptic coupling between two model cells, one of which is excitable and the other of which is tonically active when uncoupled, leads to bursting in the resulting two-cell network. This phenomenon can arise when the individual cells are conditional relaxation oscillators, in that they can be tuned to engage in relaxation oscillations, or when they are conditional square-wave bursters. The mechanism is illustrated with a model for conditional pacemaker neurons in the pre-Bötzinger complex as well as with a reduced form of this model. In the relaxation oscillator case, a periodic bursting solution is proved to exist in the singular limit, under a pair of general conditions. These conditions relate the durations of the silent and active phases of the bursting solution to the locations of certain structures in the phase plane, at appropriate synaptic input strengths. Further, additional conditions on the relative flow rates in the silent and active phases are proved to imply the uniqueness and asymptotic stability of the bursting solution.
这项工作解释了一种机制，通过该机制，在两个模型细胞之间引入兴奋性突触耦合，其中一个细胞是可兴奋的，另一个在未耦合时是持续活跃的，会导致由此产生的双细胞网络中出现爆发。当单个细胞是条件性弛豫振荡器时，即它们可以被调节以进行弛豫振荡，或者当它们是条件性方波爆发器时，就会出现这种现象。该机制通过前包钦格复合体中条件性起搏器神经元的模型以及该模型的简化形式进行说明。在弛豫振荡器的情况下，在一对一般条件下，证明了在奇异极限中存在周期性爆发解。这些条件将爆发解的沉默期和活跃期的持续时间与相平面中某些结构的位置联系起来，处于适当的突触输入强度下。此外，证明了关于沉默期和活跃期相对流速的附加条件意味着爆发解的唯一性和渐近稳定性。
