A spin glass model of path integration in rat medial entorhinal cortex

Abstract

Electrophysiological recording studies in the dorsocaudal region of medial entorhinal cortex (dMEC) of the rat reveal cells whose spatial firing fields show a remarkably regular hexagonal grid pattern (Fyhn et al., 2004; Hafting et al., 2005). We describe a symmetric, locally connected neural network, or spin glass model, that spontaneously produces a hexagonal grid of activity bumps on a two-dimensional sheet of units. The spatial firing fields of the simulated cells closely resemble those of dMEC cells. A collection of grids with different scales and/or orientations forms a basis set for encoding position. Simulations show that the animal's location can easily be determined from the population activity pattern. Introducing an asymmetry in the model allows the activity bumps to be shifted in any direction, at a rate proportional to velocity, to achieve path integration. Furthermore, information about the structure of the environment can be superimposed on the spatial position signal by modulation of the bump activity levels without significantly interfering with the hexagonal periodicity of firing fields. Our results support the conjecture of Hafting et al. (2005) that an attractor network in dMEC may be the source of path integration information afferent to hippocampus.

Figure 1.

Recurrent weight matrix W_ij contains symmetric and asymmetric components. Shown here are the weights for the central unit in the sheet. A, Symmetric component contains angular rings of excitation. B, Asymmetric component contains a ring of inhibition, offset slightly from the center, opposite the preferred direction φ_i of the unit. C, The output weights of a unit (a column of W_ij) are the sum of the weights in A and B. D, The input weights for the unit (a row of W_ij) are approximately symmetrical; the “noise” reflects the variation in preferred directions of the afferent cells. All weights have been raised to the 0.5 power in these plots to better reveal the structure in both components, which differ in magnitude by a factor of 3. E, The structure of the symmetric component is determined by the wave function Ψ, which depends solely on the distance between units on the sheet. The magenta points indicated values that were learned in a neural network simulation, and the green line indicates the function based on numerical integration that was used to construct the symmetric weight matrix W_ij^sym. F, G, These diagrams illustrate how the temporal phase of unit j changes as a function of the direction of propagation of the wave packet, θ_wave, for a given phase of unit i (see Appendix).

Figure 2.

Formation and translation of the bump grid. A, Starting from an all-zero state, at T = 30, the bump array is somewhat disordered; there is a heptagon in the top left quadrant. By T = 200, a regular hexagonal pattern has been established. The next three panels show translation of the grid to the left. B, Spatial firing fields of three grid cells in a simulated square arena.

Figure 3.

Multiple grids yield unique place codes. A, Correlations between place cell population vectors decrease as the number of grids increases. Each line shows how the percentage of positions for which r_max exceeds 0.5 varies as the size of the space is increased. The number superimposed on each line indicates the number of grids used as input to the place cells. B, Correlations are further reduced when the grid orientations vary. C, When grid cells project sparsely to the place fields (2 grids per cell), place coding does not deteriorate.

Figure 4.

Resetting only a subset of the grids during recall of an environment results in partial remapping. A, Similarity to the original place code increases with the number of grids that are reset. The joint distribution over 20 runs, each with different grid spacings and orientations, is shown for each value of N_reset. Box plot tails indicate the central 95% of the joint distribution. B, Double-rotation experiments can produce a range of remapping effects depending on the number of grids aligned with each set of cues. Distributions were based on the average over 20 runs, each with different grid spacings and orientations.

Figure 5.

Sensory modulation of grid cell activity. A, One hundred random input patterns produce population activity vectors that are clearly distinguishable from each other. B, Histogram of correlation coefficients between pattern i and the second presentation of the same pattern (top), the second presentation of the next closest matching pattern (middle), and all patterns other than i (bottom). C, D, Same as A and B except results were recalculated using a 20 unit subset of the place cell population.