Bonilla-Quintana Mayte, Wedgwood Kyle C A, O'Dea Reuben D, Coombes Stephen
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD, Nottingham, UK.
Centre for Biomedical Modelling and Analysis, University of Exeter, Living Systems Institute, Stocker Road, EX4 4QD, Exeter, UK.
J Math Neurosci. 2017 Aug 25;7(1):9. doi: 10.1186/s13408-017-0051-7.
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 [Formula: see text] 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 [Formula: see text] current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in [Formula: see text] 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 [Formula: see text] current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions.
内侧内嗅皮层（MEC）中的II层星状细胞表达超极化激活的环核苷酸门控（HCN）通道，该通道可通过Ih电流产生反弹放电，以响应超极化突触输入。哈塞尔莫（Hasselmo）的一项计算建模研究（《英国皇家学会会报B辑：生物科学》，369卷：20120523，2013年）表明，此类细胞的抑制性网络可支持周期行波，其周期由Ih电流的动力学控制。哈塞尔莫提出，这些波可能是网格细胞产生的基础，并且沿背腹轴已知的Ih共振频率差异可以解释观察到的网格细胞放电场之间的大小和间距。在此，我们在一个便于进行分析处理的框架内开发了一个生物物理脉冲发放模型。我们将积分发放神经元的简单性与HCN通道门控动力学的分段线性近似相结合，以开发一个MEC的脉冲发放神经场模型。使用主要来自非光滑动力系统领域的技术，我们展示了如何构建周期行波，特别是确定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，强化了反弹放电是产生网格细胞放电模式的候选机制这一观点。重要的是，我们开展了波稳定性分析，以表明最大允许周期是如何由Ih电流的动力学特性控制的。我们的理论工作通过一维和二维脉冲发放模型的数值模拟得到了验证。
