A Model of Grid Cells Based on a Path Integration Mechanism Alexis Guanella1, and Paul F.M.J. Verschure1,2 1 Institute of Neuroinformatics, University and ETH Z��urich, CH-8057 Z�� urich, Switzerland guanella@ini.phys.ethz.ch 2 ICREA and Technology Department, University Pompeu Fabra E-08002 Barcelona, Spain Abstract. The grid cells of the dorsocaudal medial entorhinal cortex (dMEC) in rats show higher firing rates when the position of the animal correlates with the vertices of regular triangular tessellations covering the environment. Strong evidence indicates that these neurons are part of a path integration system. This raises the question, how such a sys- tem could be implemented in the brain. Here, we present a cyclically connected artificial neural network based on a path integration mecha- nism, implementing grid cells on a simulated mobile agent. Our results show that the synaptic connectivity of the network, which can be rep- resented by a twisted torus, allows the generation of regular triangular grids across the environment. These tessellations share same spacing and orientation, as neighboring grid cells in the dMEC. A simple gain and bias mechanism allows to control the spacing and the orientation of the grids, which suggests that these different characteristics can be generated by a unique algorithm in the brain. Keywords: grid cells, entorhinal cortex, path integration, twisted torus. 1 Introduction Found in the dorsocaudal medial entorhinal cortex (dMEC) of rats, grid cells [1,2] show increased firing frequency when the animal visits regularly distributed regions in an environment. It has been shown, using auto-correlative maps, that these regions (so-called subfields) form regular triangular tessellations, or grids [1]. It is possible to describe these tessellations, and, thus, the characteris- tics of a grid cell, with only a few parameters: the orientation and the phase of the grid, and the spacing (minimal inter-subfields distance, i.e. d in this study) and the size of its subfields. Using these parameters, it was shown that grid cells are topographically organized in the dMEC: first, neighboring cells share common orientation and spacing. Second, the spacing of the grid increases isometrically along the dorsoventral axis (in [1], d varies between 39 to 73 cm). Third, the similitude of neighboring grid cells of different layers of the entorhinal cortex, sharing common orientations and spacing, suggests that they are organized in cortical columns [3]. Corresponding author. S. Kollias et al. (Eds.): ICANN 2006, Part I, LNCS 4131, pp. 740���749, 2006. c Springer-Verlag Berlin Heidelberg 2006

A Model of Grid Cells Based on a Path Integration Mechanism 741 In spite of these observations, the role of grid cells is still poorly understood. Briefly, it has been proposed that grid cells may be part of a generalized path integration system [1,4], and could be the basis of a metric of spatial relation- ships [5]. Many arguments verify this hypothesis. First, the grid cell activity and its regular patterns persist after the deprivation of external landmarks (e.g. in the dark [1]). Second, entorhinal lesions disrupt the return path of rats [6]. Third and fourth, suggesting also hard wired mechanisms, the grid structure is expressed instantaneously in novel environments and the spacing parameter seems to be universal (the grid spacing remains constant when increasing the size of the arena) [1]. Fifth, the periodicity of the grid implies a covering of arbitrary big environments. These arguments raise thus the question, how grid cells could be incorporated into a path integration system. The goal of this article is to describe an artificial neural network implementing grid cells based on a path integration mechanism. In our model, the activity of rate coded neurons is shifted by asymmetric synaptic connections. These con- nections are modulated by the velocity of the animal, represented by a simulated mobile agent exploring randomly a square arena. The neurons of the network represent a population of neighboring grid cells of the dMEC, whose grids share thus same orientation and spacing, but have different phases. A simple gain and bias mechanism allows the control of the spacing and the orientation of the grid (suggesting that exactly the same algorithm may be used to generate grid cells along the dorsoventral locations of the dMEC). The synaptic connectivity of the network is organized cyclically, and can be represented by a twisted torus. This topology is shown to exactly generate the same regular triangular tessellations of space as grid cells. Stability and robust activity is ensured by attractor dynamics and normalization mechanisms. 2 Methods 2.1 Neurons We construct a population of N neurons organized in a matrix covering the repetitive rectangular structure of the subfields of grid cells (Fig. 1a). In order to conserve the ratio between the height and the side of an equilateral triangle (which is the core element of a regular triangular tessellation) and in order to have the same density of cells along both x- and y-axes, the number of cells in each row is approximately 2/ ��� 3 times bigger than the number of cells in each column (Fig. 1b). Activity and Stabilization. The neurons of the network initialized with a random activity uniformly distributed between 0 and 1/ ���are N . The activity of a cell i at time t + 1, i.e. Ai(t + 1) is defined using a linear transfer function Bi(t + 1) given by Bi(t + 1) = Ai(t) + N j=1 Aj (t) wji , (1)