Cerebellar stellate cells form inhibitory synapses with Purkinje cells, the sole output of the cerebellum. Upon stimulation by a pair of varying inhibitory and fixed excitatory presynaptic inputs, these cells do not respond to excitation (i.e., do not generate an action potential) when the magnitude of the inhibition is within a given range, but they do respond outside this range. We previously used a revised Hodgkin–Huxley type of model to study the nonmonotonic first-spike latency of these cells and their temporal increase in excitability in whole cell configuration (termed run-up). Here, we recompute these latency profiles using the same model by adapting an efficient computational technique, the two-point boundary value problem, that is combined with the continuation method. We then extend the study to investigate how switching in responsiveness, upon stimulation with presynaptic inputs, manifests itself in the context of run-up. A three-dimensional reduced model is initially derived from the original six-dimensional model and then analyzed to demonstrate that both models exhibit type 1 excitability possessing a saddle-node on an invariant cycle (SNIC) bifurcation when varying the amplitude of Iapp. Using slow-fast analysis, we show that the original model possesses three equilibria lying at the intersection of the critical manifold of the fast subsystem and the nullcline of the slow variable hA (the inactivation of the A-type K+ channel), the middle equilibrium is of saddle type with two-dimensional stable manifold (computed from the reduced model) acting as a boundary between the responsive and non-responsive regimes, and the (ghost of) SNIC is formed when the hA-nullcline is (nearly) tangential to the critical manifold. We also show that the slow dynamics associated with (the ghost of) the SNIC and the lower stable branch of the critical manifold are responsible for generating the nonmonotonic first-spike latency. These results thus provide important insight into the complex dynamics of stellate cells.

中文翻译：

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小脑星状细胞兴奋性响应一对抑制性/兴奋性突触前输入的转换：动力学系统视角

小脑星状细胞与浦肯野细胞形成抑制性突触，浦肯野细胞是小脑的唯一输出。在受到一对不同的抑制性和固定的兴奋性突触前输入的刺激时，当抑制的幅度在给定范围内时，这些细胞不会对兴奋做出反应（即，不产生动作电位），但它们确实在该范围之外做出反应. 我们之前使用修正的 Hodgkin-Huxley 类型的模型来研究这些细胞的非单调首次尖峰潜伏期以及它们在整个细胞配置中的兴奋性随时间的增加（称为启动）。在这里，我们通过采用一种高效的计算技术，即两点边值问题，并与延拓方法相结合，使用相同的模型重新计算这些延迟配置文件。然后，我们扩展了研究，以研究在突触前输入的刺激下，反应性的转换如何在助跑的情况下表现出来。一个三维简化模型最初是从原始的六维模型派生出来的，然后进行分析以证明当改变 Iapp 的幅度时，这两个模型都表现出 1 型兴奋性，在不变循环 (SNIC) 分岔上具有鞍节点。使用慢-快分析，我们表明原始模型具有三个平衡点，位于快速子系统的临界流形与慢变量 hA（A 型 K+ 通道的失活）的零斜线的交点处，中间平衡是马鞍型的，二维稳定流形（从简化模型计算）作为响应和非响应机制之间的边界，当 hA 零斜线（接近) 与临界流形相切。我们还表明，与 SNIC（的幽灵）相关的缓慢动态和临界流形的较低稳定分支负责产生非单调的第一次峰值延迟。因此，这些结果提供了对星状细胞复杂动力学的重要见解。我们还表明，与 SNIC（的幽灵）相关的缓慢动态和临界流形的较低稳定分支负责产生非单调的第一次峰值延迟。因此，这些结果提供了对星状细胞复杂动力学的重要见解。我们还表明，与 SNIC（的幽灵）相关的缓慢动态和临界流形的较低稳定分支负责产生非单调的第一次峰值延迟。因此，这些结果提供了对星状细胞复杂动力学的重要见解。