Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an $I_{\text{h}}$ current in response to hyperpolarising synaptic input. A computational modelling study by Hasselmo (Philos. Trans. R. Soc. Lond. B, Biol. Sci. 369:20120523, 2013) showed that an inhibitory network of such cells can support periodic travelling waves with a period that is controlled by the dynamics of the $I_{\text{h}}$ current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in $I_{\text{h}}$ resonance frequency along the dorsal to ventral axis can explain the observed size and spacing between grid cell firing fields. Here we develop a biophysical spiking model within a framework that allows for analytical tractability. We combine the simplicity of integrate-and-fire neurons with a piecewise linear caricature of the gating dynamics for HCN channels to develop a spiking neural field model of MEC. Using techniques primarily drawn from the field of nonsmooth dynamical systems we show how to construct periodic travelling waves, and in particular the dispersion curve that determines how wave speed varies as a function of period. This exhibits a wide range of long wavelength solutions, reinforcing the idea that rebound spiking is a candidate mechanism for generating grid cell firing patterns. Importantly we develop a wave stability analysis to show how the maximum allowed period is controlled by the dynamical properties of the $I_{\text{h}}$ current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions.

中文翻译：

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内侧Enthorinal皮层中的网格细胞发射波的分析。

内肠皮质（MEC）中的第II层星状细胞表达超极化激活的环核苷酸门控（HCN）通道，该通道允许通过I _ {\ text {h}} $电流响应超极化突触输入而发生反弹。Hasselmo（Philos。Trans。R. Soc。Lond。B，Biol。Sci.369：20120523，2013）的计算模型研究显示，此类细胞的抑制网络可以支持周期受行波控制的周期性行波$ I _ {\ text {h}} $当前的动态。Hasselmo建议，这些波可以作为网格单元生成的基础，并且沿背腹轴的$ I _ {\ text {h}} $共振频率的已知差异可以解释观察到的网格单元激发场的大小和间距。在这里，我们在允许分析易处理性的框架内开发了生物物理峰值模型。我们将整合和发射神经元的简单性与HCN通道的门控动力学的分段线性漫画相结合，以开发出尖峰的MEC神经场模型。使用主要来自非光滑动力系统领域的技术，我们展示了如何构造周期性的行波，尤其是决定了波速如何随周期变化的色散曲线。这展示了广泛的长波长解决方案，强化了反弹尖峰是生成栅格单元发射图案的候选机制的想法。重要的是，我们进行了波浪稳定性分析，以显示最大允许时间段如何由$ I _ {\ text {h}} $电流的动力学特性控制。