Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA.
J Math Neurosci. 2012 Mar 12;2(1):4. doi: 10.1186/2190-8567-2-4.
We present a novel extension of fast-slow analysis of clustered solutions to coupled networks of three cells, allowing for heterogeneity in the cells' intrinsic dynamics. In the model on which we focus, each cell is described by a pair of first-order differential equations, which are based on recent reduced neuronal network models for respiratory rhythmogenesis. Within each pair of equations, one dependent variable evolves on a fast time scale and one on a slow scale. The cells are coupled with inhibitory synapses that turn on and off on the fast time scale. In this context, we analyze solutions in which cells take turns activating, allowing any activation order, including multiple activations of two of the cells between successive activations of the third. Our analysis proceeds via the derivation of a set of explicit maps between the pairs of slow variables corresponding to the non-active cells on each cycle. We show how these maps can be used to determine the order in which cells will activate for a given initial condition and how evaluation of these maps on a few key curves in their domains can be used to constrain the possible activation orders that will be observed in network solutions. Moreover, under a small set of additional simplifying assumptions, we collapse the collection of maps into a single 2D map that can be computed explicitly. From this unified map, we analytically obtain boundary curves between all regions of initial conditions producing different activation patterns.
我们提出了一种新的方法，用于对具有内在动力学异质性的三个细胞耦合网络的聚类解进行快速-缓慢分析。在我们关注的模型中，每个细胞由一对一阶微分方程描述，这些方程基于最近的呼吸节律发生的简化神经元网络模型。在每对方程中，一个因变量在快速时间尺度上演变，一个在慢时间尺度上演变。细胞通过在快速时间尺度上开启和关闭的抑制性突触耦合。在这种情况下，我们分析了细胞依次激活的解，允许任何激活顺序，包括在第三个细胞的连续激活之间对两个细胞的多次激活。我们的分析是通过推导出一组对应于每个周期中非活跃细胞的慢变量对之间的显式映射来进行的。我们展示了如何使用这些映射来确定给定初始条件下细胞将激活的顺序，以及如何在其域中的几个关键曲线上评估这些映射来约束将在网络解中观察到的可能激活顺序。此外，在一小组额外的简化假设下，我们将映射集合折叠成一个可以显式计算的单个 2D 映射。从这个统一的映射中，我们通过分析得到了产生不同激活模式的所有初始条件区域之间的边界曲线。
