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# An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex

Springer Berlin Heidelberg

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### 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document} current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document} current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions.

##### Electronic Supplementary Material

The online version of this article (doi:10.1186/s13408-017-0051-7) contains supplementary material.

### Most cited references44

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### Microstructure of a spatial map in the entorhinal cortex.

(2005)
The ability to find one's way depends on neural algorithms that integrate information about place, distance and direction, but the implementation of these operations in cortical microcircuits is poorly understood. Here we show that the dorsocaudal medial entorhinal cortex (dMEC) contains a directionally oriented, topographically organized neural map of the spatial environment. Its key unit is the 'grid cell', which is activated whenever the animal's position coincides with any vertex of a regular grid of equilateral triangles spanning the surface of the environment. Grids of neighbouring cells share a common orientation and spacing, but their vertex locations (their phases) differ. The spacing and size of individual fields increase from dorsal to ventral dMEC. The map is anchored to external landmarks, but persists in their absence, suggesting that grid cells may be part of a generalized, path-integration-based map of the spatial environment.
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### Path integration and the neural basis of the 'cognitive map'.

(2006)
The hippocampal formation can encode relative spatial location, without reference to external cues, by the integration of linear and angular self-motion (path integration). Theoretical studies, in conjunction with recent empirical discoveries, suggest that the medial entorhinal cortex (MEC) might perform some of the essential underlying computations by means of a unique, periodic synaptic matrix that could be self-organized in early development through a simple, symmetry-breaking operation. The scale at which space is represented increases systematically along the dorsoventral axis in both the hippocampus and the MEC, apparently because of systematic variation in the gain of a movement-speed signal. Convergence of spatially periodic input at multiple scales, from so-called grid cells in the entorhinal cortex, might result in non-periodic spatial firing patterns (place fields) in the hippocampus.
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• Abstract: found

### Spatial representation in the entorhinal cortex.

(2004)
As the interface between hippocampus and neocortex, the entorhinal cortex is likely to play a pivotal role in memory. To determine how information is represented in this area, we measured spatial modulation of neural activity in layers of medial entorhinal cortex projecting to the hippocampus. Close to the postrhinal-entorhinal border, entorhinal neurons had stable and discrete multipeaked place fields, predicting the rat's location as accurately as place cells in the hippocampus. Precise positional modulation was not observed more ventromedially in the entorhinal cortex or upstream in the postrhinal cortex, suggesting that sensory input is transformed into durable allocentric spatial representations internally in the dorsocaudal medial entorhinal cortex.
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### Author and article information

###### Contributors
Mayte.BonillaQuintana@nottingham.ac.uk
K.C.A.Wedgwood@exeter.ac.uk
reuben.o'dea@nottingham.ac.uk
stephen.coombes@nottingham.ac.uk
###### Journal
J Math Neurosci
J Math Neurosci
Journal of Mathematical Neuroscience
Springer Berlin Heidelberg (Berlin/Heidelberg )
2190-8567
25 August 2017
25 August 2017
2017
: 7
###### Affiliations
[1 ]ISNI 0000 0004 1936 8868, GRID grid.4563.4, Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, , University of Nottingham, ; University Park, NG7 2RD Nottingham, UK
[2 ]ISNI 0000 0004 1936 8024, GRID grid.8391.3, Centre for Biomedical Modelling and Analysis, , University of Exeter, Living Systems Institute, ; Stocker Road, EX4 4QD Exeter, UK
###### Article
51
10.1186/s13408-017-0051-7
5572897
28842863
© The Author(s) 2017

###### Funding
Funded by: FundRef http://dx.doi.org/10.13039/501100004963, Seventh Framework Programme;
Award ID: 289146
Award Recipient :
Funded by: FundRef http://dx.doi.org/10.13039/100004440, Wellcome Trust;
Award ID: WT105618MA
Award Recipient :
###### Categories
Research