Effect of boundaries on grid cell patterns

Open Access

Effect of boundaries on grid cell patterns

Mauro M. Monsalve-Mercado and Christian Leibold

Phys. Rev. Research 2, 043137 – Published 27 October 2020

Abstract

Mammalian grid cells represent spatial locations in the brain via triangular firing patterns that tessellate the environment. They are regarded as the biological substrate for path integration and to provide an efficient code for space. However, grid cell patterns are strongly influenced by environmental manipulations, in particular, exhibiting local geometrical deformations and defects tied to the shape of the recording enclosure, challenging the view that grid cells constitute a universal code for space. We show that the observed responses to environmental manipulations arise as a natural result under the general framework of feedforward models with spatially unstructured feedback inhibition, which puts the development of triangular patterns in the context of a Turing pattern formation process over physical space. The model produces coherent neuronal populations with equal grid spacing, field size, and orientation.

Received 23 January 2020
Accepted 7 October 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043137

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Collective dynamics Learning Network formation & growth Neuroplasticity Neuroscience, neural computation & artificial intelligence

Coherent structures Cortical networks Nervous system

Neuronal network models Pattern formation

Physics of Living SystemsNonlinear DynamicsNetworks

Authors & Affiliations

Mauro M. Monsalve-Mercado ^*

Physik-Department, Technische Universität München, 85748 Garching, Germany and Center for Theoretical Neuroscience, Zuckerman Institute, Columbia University, New York, New York 10027, USA

Christian Leibold ^†

Department Biologie II, LMU Munich, 82152 Martinsried, Germany and Bernstein Center for Computational Neuroscience Munich, 82152 Martinsried, Germany

^*Corresponding author: mauro.m.monsalve@tum.de
^†Corresponding author: leibold@bio.lmu.de

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Vol. 2, Iss. 4 — October - December 2020

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Figure 1
Feedback inhibition allows the development of triangular patterns inside enclosures of arbitrary shape. (a) A layer of cells with spatially selective activity $H$ projects with effective weights $W$ to cells $E$ presumably in the entorhinal cortex. In this paper, entorhinal cells are connected via a random inhibitory coupling $M$ with no spatial structure or dependence on the boundaries. (b) Hebbian learning is implemented by an update rule for the weights that depends on their spatial distance. The effective learning kernel $Γ$ is always of the center-surround type, here captured by a general step function of distance. (c) The learning rule implements a Turing pattern formation process on the development of the weights $W (x)$ (greenish dots). A weight in a certain location (black dot) increases in proportion to weights in a neighboring area (yellow), while it decreases in proportion to weights in a further surrounding region (blue). (d) Simulations of the growth process for weights (green) constrained to enclosures of different shapes reveal that when no feedback inhibition is present (top row), the weights tend to cluster along the boundaries with a preference for corners and regions of high curvature. On the other hand, the presence of feedback inhibition (bottom) produces triangular patterns in all shapes without special treatment of the boundaries, i.e., without the use of periodic or fading boundary conditions. (e) Pure Hebbian learning is inherently unstable with weights growing without bounds. An additional stabilizing nonlinearity only used in this panel after 600 time steps of learning reveals that, far from the boundaries, weights may still form local triangular patterns, even when no feedback inhibition is present (top). With feedback inhibition ( $M \neq 0$ ), the additional nonlinearity removes the apparent fading close to the boundaries observed in (d).
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Figure 2
Physical boundaries introduce a bias into the growth process. (a) Weights close to the boundary are potentiated more compared to far away weights inside the enclosure. A weight (black dot) closer than the spatial range of action of the learning kernel loses a region of depression that would come from activity of cells located outside the enclosure. (b) The result is a bias $B$ (see Appendixes) in the rate of growth of weights depending on their distance d away from the boundary. It is proportional to the overall effect (the integral) of the learning kernel centered at d and constrained to the inside of the enclosure. (c) The growth bias computed numerically for all shapes predicts the locations where weights are predisposed to grow the most. (d) Simulations of the growth process for the case when no inhibition is present reveals that at early times the individual patterns (green) reflect the predicted growth bias. Since cells are uncoupled, they represent different random instances of the growth process. Their average (red) approximates the predicted growth bias, even for later stages of the process.
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Figure 3
Feedback inhibition produces coherent population responses. (a) Simulations of the growth process for the coupled case (strong feedback inhibition) reveal that the average of patterns over space (bar) and over grid cells (angular brackets) grows exponentially in time (blue curve, semi-logarithmic scale). In contrast, the coefficient of variations both over the space $C V (\bar{W})$ (i.e., standard deviation divided by the mean (green curves) where averaging is done over the grid cell dimension) and of the population average $C V (〈 W 〉)$ (i.e., averaging is done over space) saturate to a constant value early on in the growth process (see inset). (b) Definitions of the local geometric measures used to characterize the grid cell pattern. (c) Population distribution of the cell-averaged measures of field size and spacing. The ratio of field size to spacing is consistently approximately $1 / 3$ for all parameters explored. (d) For a population with homogeneous connectivity, the orientations (top) and spatial phases (bottom) are uniformly represented. Bottom shows the population average (left, red) and a histogram of the field centers choosing only the fields closest to the midpoint of the square. (e) The distribution of orientations for noisy connectivity shows that all patterns share a preferred orientation. (f) For noisy connectivity, patterns self-organize into three evenly distributed phase groups (contour plot). The skeleton represents the nearest-neighboring fields (vertices, black dots) of a pattern belonging to one of the phase groups. The left column shows example patterns for each of the groups (as marked by the green and blue dots); the middle pattern is from a reference group (black). The white crosses mark the center of the enclosure. (g) Population averages (red) suggest that patterns self-organize into phase groups independent of the geometry of the enclosure. (h) The time-series evolution of a single pattern (green) and the population average (red) illustrates the self-organization into phase groups and the development of a perfect triangular pattern. (i) Three patterns from different phase groups (green) on top of the population average (red) for weak connectivity strength ( $μ = 0.1$ , only in this panel) illustrate the approximate nonoverlapping nature of the phase groups. Slices of activity of the three patterns (green, blue, and orange lines) show that the population average at any position is mostly governed by one phase group. (j) The patterns' mean activity (vertical axis) is approximately normally distributed. For each pattern, we computed the scaling factor by which the average activity needs to be multiplied to fit the nonzero regions of the pattern (in a minimal mean square sense; all fitting errors are below $1 %$ of each pattern's peak). (k) For packed tangent circles mimicking perfectly nonoverlapping phase groups, the ratio of field size to spacing corresponds to $1 / \sqrt{12} \approx 0.3$ .
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Figure 4
The effects of unfinished learning on the grid cell pattern. (a) Typical defects found during a Turing pattern formation process are splitting fields (top left) and lattice dislocations (penta-hepta defect; bottom left). Examples are firing rate maps from grid cells taken from Ref. [50] with permission. We simulated the developments of patterns on a square enclosure from a random initial state. Stopping learning before spatial stability is achieved results in similar defects on some of the patterns (right column). (b) Extending the recording enclosure reveals the appearance of a consistent pattern [14, 17]. We simulated the pattern in an initial trapezoid until it was spatially stable. After expanding the trapezoid, we continue learning the whole pattern, with the novel region starting from an initial random state. The new fields become a coherent part of the pattern learned previously. (c) Grid cells develop independent patterns for independent environments. After they are revealed to be part of a single joined environment, the pattern transforms into a coherent global representation [12, 18]. We simulated the patterns independently in two halves of a square to produce incompatible representations (top). After the patterns were spatially stable, we continued the learning process of the joined environment (bottom) and both patterns combined into a joined hexagonal grid. (d) For this panel only, we simulate the growth process of a network composed of two subnetworks with a 10% stronger connectivity for cells within each subnetwork (black blocks in the connectivity matrix) compared to the connectivity between subnetworks (yellow blocks). By chance, each subnetwork develops a different solution to the growth process, anchoring to opposite corners of the square enclosure and producing different orientations. Each row shows the average (red) and three individual patterns (green) from each subnetwork. The averages reflect how each subnetwork better represents each corner, with individual patterns showcasing the typical distortions due to the interaction between subnetworks. (e) Stereotypical distortions of the square enclosure [13, 15, 16] include a gradient of orientations (top) and different anchoring solutions (bottom). Firing rate maps in the left column from Ref, [15] (with permission) are reproduced by the weight growth process in the right column. Examples are selected by visual inspection from a simulation of 300 grid units stopped at time step 200. (f) The long axis of a grid pattern bends backward in the top example of a grid cell firing map from Ref. [17] (with permission). The simulated example (bottom) was selected by visual inspection from a simulation of 300 grid units at time step 100.
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Figure 5
Time evolution of orientation and phase clustering. See text in Appendix pp3 for details. Shaded regions denote one standard deviation. Dashed line marks $1 / 3$ .
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