Abstract
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.
Highlights
The ability to remember specific events occurring at specific places and times plays a major role in our everyday life
As an animal approaches the centre of a grid cell firing field, the spiking output of grid cell will increase in frequency
In contrast to continuous attractor models that rely on the spatial scale of connectivity to control grid spacing, a change in rebound response provides a mechanism of local control via changes in the expression levels of hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels
Summary
The ability to remember specific events occurring at specific places and times plays a major role in our everyday life. The inclusion of important intrinsic biophysical properties into a network has been emphasised by several authors, such as Navratilova et al [27] regarding the contribution of after-spike potentials of stellate cells to theta phase precession, and perhaps most notably by Hasselmo and colleagues for the inclusion of HCN channels [28,29,30,31] This has culminated in a spiking network model of MEC that supports patterns whose periodicity is controlled by a neuronal resonance frequency arising from an Ih current [1]. In contrast to continuous attractor models that rely on the spatial scale of connectivity to control grid spacing, a change in rebound response provides a mechanism of local control via changes in the expression levels of HCN channels This fascinating observation warrants a deeper mathematical analysis.
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