Elsevier

Neurocomputing

Volume 70, Issues 10–12, June 2007, Pages 1723-1727
Neurocomputing

Quantified symmetry for entorhinal spatial maps

https://doi.org/10.1016/j.neucom.2006.10.050Get rights and content

Abstract

General navigation requires a spatial map that is not anchored to one environment. The firing fields of the “grid cells” found in the rat dorsolateral medial entorhinal cortex (dMEC) could be such a map. dMEC firing fields are also thought to be modeled well by a regular triangular grid (a grid with equilateral triangles as units). We use computational means to analyze and validate the regularity of the firing fields both quantitatively (using summary statistics for geometric and photometric regularity) and qualitatively (using symmetry group analysis). Upon quantifying the regularity of real dMEC firing fields, we find that there are two types of grid cells. We show rigorously that both are nearest to triangular grids using symmetry analysis. However, type III grid cells are far from regular, both in firing rate (highly non-uniform) and grid geometry. Type III grid cells are also more numerous. We investigate the implications of this for the role of grid cells in path integration.

Introduction

Place cell representation is context [24], [13] and task-specific [12], while general navigation requires a more abstract map that is not anchored to one environment. It has been hypothesized that there exists a representation upstream of the rat hippocampus that is context-independent [20], [18], [23]. This general spatial map is just a component of a distributed network [18] which is the basis for navigation [25], [15]. One potential possibility for such a map is the multi-peaked firing field found in dorsolateral medial entorhinal cortex (dMEC), which accurately represent the rat's position [5] in a near-regular grid [7]. This is in sharp contrast to place cells’ one-peak firing fields [15], [16]. The regularity of these firing fields and their context-independence have been used in recent models of path integration. Using symmetry analysis [10], [9], we investigate to what degree dMEC firing fields are regular grids. (see also [2]). We show that there are in fact two types of grid cells, one that is strikingly regular, and another a greater departure from regularity. Both types of firing fields are shown to be nearest to a p6m symmetry group (a triangular grid). However, the departure from regularity of the majority of grid cells analyzed means that the role of dMEC in path integration must be rethought.

Section snippets

Methods

A symmetry g of a geometric set S is a distance-preserving transformation that keeps S setwise invariant (i.e. g(S)=S). All symmetries of S form a group G [3] and is called the symmetry group of S. Thus the symmetry group of a lattice is a collection of all transformations which leave the representation invariant. The lattice can be generated by a single tile. This tile can be translated in 2D such that it produces a covering (no gaps) and a packing (no overlaps) of the plane [6]. There are

Analysis and results

Hafting et al.'s data on grid cells suggests that dMEC firing fields are near-regular [7]. Accordingly, we applied Liu et al.'s [10] algorithm on Hafting et al.'s [7] data (from the 2 m diameter circular enclosure) to quantify such regularity qualitatively (symmetry group type) and quantitatively (nearest regular lattice). The results of using Liu et al.'s lattice regularization algorithm on the firing fields are illustrated in Fig. 2. To the left (in Fig. 2) are the firing fields, whose

Summary and discussion

Using Liu et al.'s symmetry group classification [10], [9] algorithm, we are able to identify the symmetry group of the firing fields as p6m, and quantify grid cells into two types of regularity. Because the statistical regularity of the grids found in dMEC is of two kinds: one highly regular (type I), the other much further from regularity (type III), we can speculate what this means for path integration. Our findings indicate that not all grid cell firing fields are well-described by a

Acknowledgments

This research is funded in part by an NSF “Research Experiences for Undergraduates” (REU) grant associated with the NSF Grant IIS-0099597.

Erick Chastain is currently a graduate student in the Neurobiology and Behavior Ph.D. program at the University of Washington. Recently he graduated (with school of computer science honors) from Carnegie Mellon University with a B.S. in Computer Science and a minor in Theoretical Neuroscience. The subject of his honors thesis was a hierarchical bayesian model of object recognition based on the eccentricity bias fMRI studies.

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  • Cited by (0)

    Erick Chastain is currently a graduate student in the Neurobiology and Behavior Ph.D. program at the University of Washington. Recently he graduated (with school of computer science honors) from Carnegie Mellon University with a B.S. in Computer Science and a minor in Theoretical Neuroscience. The subject of his honors thesis was a hierarchical bayesian model of object recognition based on the eccentricity bias fMRI studies.

    Yanxi Liu received her B.S. degree in physics/electrical engineering in Beijing and her Ph.D. degree in computer science for group theory applications in robotics from University of Massachusetts. Her postdoctoral training was performed at LIFIA/IMAG, Grenoble, France. She also spent one year at DIMACS (NSF center for Discrete Mathematics and Theoretical Computer Science) with an NSF research-education fellowship award. Dr. Liu joins the Computer Science Engineering and Electrical Engineering departments of Penn State University Fall of 2006. Dr. Liu has been a faculty member in the Robotics Institute (RI) of Carnegie Mellon University and affiliated with the Machine Learning department of CMU. She is an adjunct associate professor in the Radiology Department of University of Pittsburgh, and a guest professor of Computer Science Department, Huazhong University of Science and Technology in China. Dr. Liu's research interests span a wide range of applications in computer vision, computer graphics, robotics and computer aided diagnosis in medicine, with two central themes: computational symmetry and discriminative subspace learning. With her colleagues, Dr. Liu won the first place in the clinical science category and the best paper overall at the Annual Conference of Plastic and Reconstructive Surgeons for the paper “Measurement of Asymmetry in Persons with Facial Paralysis.” Dr. Liu chaired the First International Workshop on Computer Vision for Biomedical Image Applications (CVBIA) in conjunction with ICCV 2005 and co-edited the book: “CVBIA: Current Techniques and Future Trends” (Springer-Verlag LNCS). Dr. Liu serves as a reviewer/committee member/panelist for all major journals, conferences as well as NIH/NSF panels on computer vision, pattern recognition, biomedical image analysis, and machine learning. She had been a chartered study section member for Biomedical Computing and Health Informatics at NIH. She is a senior member of IEEE and the IEEE Computer Society.

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