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# Emergent elasticity in the neural code for space

Edited by Dmitri B. Chklovskii, Simons Foundation, New York, NY, and accepted by Editorial Board Member Charles F. Stevens October 18, 2018 (received for review April 6, 2018)

## Significance

We develop a theoretical model, grounded in known properties of neural dynamics and synaptic plasticity, that can fuse information gathered from the past history of velocity and sequence of encountered landmarks during exploratory behavior, to construct a self-consistent internal representation of space. Moreover, through model reduction techniques, we obtain conceptual insights into how consistent internal spatial representations naturally emerge through an elastic relaxation process in an effective spring–particle system. We verify several experimentally testable predictions of our model involving the spatial behavior of grid cells in the medial entorhinal cortex, as well as suggest additional experiments.

## Abstract

Upon encountering a novel environment, an animal must construct a consistent environmental map, as well as an internal estimate of its position within that map, by combining information from two distinct sources: self-motion cues and sensory landmark cues. How do known aspects of neural circuit dynamics and synaptic plasticity conspire to accomplish this feat? Here we show analytically how a neural attractor model that combines path integration of self-motion cues with Hebbian plasticity in synaptic weights from landmark cells can self-organize a consistent map of space as the animal explores an environment. Intriguingly, the emergence of this map can be understood as an elastic relaxation process between landmark cells mediated by the attractor network. Moreover, our model makes several experimentally testable predictions, including (*i*) systematic path-dependent shifts in the firing fields of grid cells toward the most recently encountered landmark, even in a fully learned environment; (*ii*) systematic deformations in the firing fields of grid cells in irregular environments, akin to elastic deformations of solids forced into irregular containers; and (*iii*) the creation of topological defects in grid cell firing patterns through specific environmental manipulations. Taken together, our results conceptually link known aspects of neurons and synapses to an emergent solution of a fundamental computational problem in navigation, while providing a unified account of disparate experimental observations.

How might neural circuits learn to create a long-term map of a novel environment and use this map to infer where one is within the environment? This pair of problems is challenging because of their nested, chicken and egg nature. To localize where one is in an environment, one first needs a map of the environment. However, in a novel environment, no such map is yet available, so localization is not possible. Similarly, building a map of a novel environment from scratch can be difficult when one cannot even determine one’s own location in the environment. Thus, neural circuits must create a map over time, through exploration in a novel environment, without initially having access to any global estimate of position within the environment. This chicken and egg problem is known in the robotics literature as simultaneous localization and mapping (SLAM) (1).

Here we explore how known aspects of neural circuit dynamics and synaptic plasticity can conspire to self-organize, through exploration, a solution to the problem of creating a global, consistent map of a novel environment. In particular, neural circuits receive two fundamentally distinct sources of information about position: (*i*) signals indicating the speed and direction of the animal, which can be path integrated over time to update the animal’s internal estimate of position, and (*ii*) sensory cues from salient, fixed landmarks in the environment. To create a map of the environment, neural circuits must combine these two distinct information sources in a self-consistent fashion so that sensory cues and self-motion cues are always in coregister.

For example, consider the act of walking from landmark A to landmark B. Sensory perception of landmark A triggers a pattern of neural activity, and subsequent walking from A to B evolves this activity pattern, through path integration, to a final pattern. Conversely, sensory perception of landmark B itself triggers a neural activity pattern. Any circuit that maps space must obey a fundamental self-consistency condition: The neural activity pattern generated by perception of A, followed by path integration from A to B, must match the neural activity pattern triggered by perception of B alone. Only in this manner can neural activity patterns be in one to one correspondence with physical positions in space and become independent of the past trajectory used to reach any physical location.

In the following, we develop an analytic theory for how neuronal dynamics and synaptic plasticity can generate self-consistent neural maps of space upon exploration through a novel environment. Moreover, our analytic theory makes experimentally testable predictions about neural correlates of space. Indeed, many decades of recordings in multiple brain regions have revealed diverse neural correlates of spatial maps in the brain. In particular, the medial entorhinal cortex (MEC) contains neurons encoding for direction, velocity, and landmarks, as well as grid cells exhibiting striking firing patterns reflecting an animal’s spatial location (2⇓⇓⇓–6). Moreover, the geometry of these firing patterns depends on the shape of the environment being explored (7⇓⇓–10). In particular, these grid firing patterns can be deformed in irregular environments (11, 12), in a manner evocative of deformations of solids forced into an irregular container, suggesting a mechanical model for these deformations (13⇓–15). Also, these firing patterns are not simply driven by current sensory cues; there is evidence for path integration (16⇓–18) in that firing patterns appear almost immediately (2), phase differences are preserved across environments (19), and firing patterns become noisier the longer an animal is away from a landmark (20, 21) and can shift, depending on which landmark the animal most recently encountered (22, 23).

Despite this wealth of experimental observations, no mechanistic circuit model currently explains how known aspects of neuronal dynamics and synaptic plasticity can conspire to learn, through exploration, a self-consistent internal map of a novel environment that both behaves like a deformable medium and also retains, at higher order, some knowledge of recently encountered landmarks. Here, we show how an attractor network that combines path integration of velocity with Hebbian learning (22, 24⇓–26) of synaptic weights from landmark cells can self-organize to generate all of these outcomes. Intriguingly, a low-dimensional reduced model of the combined neuronal and synaptic dynamics provides analytical insight into how self-consistent maps of the environment can arise through an emergent, elastic relaxation process involving the synaptic weights of landmark cells.

## Model Reduction of an Attractor Network Coupled to Sensorimotor Inputs

Our theoretical framework (*SI Appendix*, section II) assumes the existence of three interacting neural components: (*i*) an attractor network capable of realizing a manifold of stable neural activity patterns, (*ii*) a population of velocity-tuned cells that carry information about the animal’s motion, and (*iii*) a population of sensory-driven landmark cells that fire if and only if the animal is in a particular region of space. Our goal is to understand how these three populations can interact together and self-organize through synaptic plasticity, sculpted by experience, to create a self-consistent internal map of the environment. Here, we describe the neuronal and synaptic dynamics of each component in turn, as well as describe a model reduction approach to obtain a low-dimensional reduced description of the entire plastic circuit dynamics. Our low-dimensional description provides insight into how self-consistency of the neural map emerges naturally through an elastic relaxation process between landmarks.

### A Manifold of Stable States from Attractor Network Dynamics.

We first consider a 1D attractor network consisting of a large population of neurons whose connectivity is determined by their position on an abstract ring, as in Fig. 1. For analytical simplicity, we take a neural field approach (27), so that position on the ring of neurons is described by a continuous coordinate u, with the firing rate of a neuron at position u given by

### Motions Along the Attractor Manifold Due to External Inputs.

So far, the attractor network described above has a ring of stable bump activity patterns parameterized by the periodic coordinate

### Path Integration Through Velocity-Conjunctive Attractor Cells.

Following refs. 28 and 29, we achieve path integration by coupling the attractor network to velocity-conjunctive attractor cells such that east (west) movement-selective cells form feedforward synapses onto the attractor ring that are shifted in the positive (negative) u direction (Fig. 2 *A* and *B*). When these inputs are weak compared with the recurrent inputs determining the bump pattern (*SI Appendix*, section III), we can show analytically that this choice of connectivity leads to path integration:*A* and *B* ensures that as the animal moves east (west) along a 1D track, the attractor phase moves clockwise (counterclockwise), at a speed proportional to velocity. Solving Eq. **2** allows us to recover path integration (Fig. 2*C*), where the resulting integrated attractor phase is only a function of current position

However, even though these 1D grid cell firing patterns are now a function of physical space, they still are not yet anchored to the environment. There is as yet no mechanism to set the phase of each cell relative to landmarks, and indeed these grid patterns rapidly decohere without anchoring to landmarks, as demonstrated experimentally (20, 30). Coupling the attractor network to landmark-sensitive cells can solve this problem.

### Landmark Cells.

We model each landmark cell i as a purely sensory-driven cell with a firing rate that depends on location through

Consider for example a single landmark cell whose synaptic strength *D*). When the landmark inputs are weak relative to the recurrent inputs that determine the shape of the bump pattern, then the external landmark inputs will not change the shape of the attractor bump, but will cause it only to change its position. Intuitively, landmark cell firing should cause the attractor bump to move to the location

We confirm this intuition analytically in *SI Appendix*, section III.2 by deriving an effective dynamics obeyed by the attractor phase in response to landmark cell input:*D*). In *SI Appendix*, section III.2, we show how to analytically compute the force law *SI Appendix*, section VI for attractor dynamics where F is exactly

In summary, we have so far described path integration dynamics that enable the attractor phase to move in response to motion in Eq. **2** and pinning dynamics of a landmark cell that force the attractor phase to move to a particular phase **4**. However, there is as yet no mechanism to enforce consistency between the attractor phases arrived at through path integration and the various attractor phases arrived at through pinning by landmark cells. We next introduce Hebbian plasticity of efferent landmark cell synapses during exploration while both path integration and landmark cells are active. This plasticity will self-organize each landmark cell’s pinning phase (i.e., the position of its peak synaptic strength profile onto the attractor network), to yield a self-consistent spatial map.

### Hebbian Learning of Landmark Cell Synapses.

We assume that each synapse *B2*) through (*SI Appendix*, section III.5)

### Combined Neural and Synaptic Dynamics During Navigation.

Because the learned synaptic weight profiles of each landmark cell tend to become localized, we can approximate the entire weight profile *SI Appendix*, section III.6 for details). Also, because the attractor bump is close to this distribution when the landmark cell fires, we can derive simplified, linearized equations for the combined neural and synaptic dynamics valid after the initial steps of learning (*SI Appendix*, section IV):

In essence Eqs. **6** and **7** constitute a significant model reduction of Eqs. **1** and **5**. In this reduction, the entire pattern of neural activity of the attractor network is summarized by a single number

Intuitively, the reduced Eq. **6** describes both path integration and dynamics in which each landmark cell i attempts to pin the attractor phase **7** aligns the learned pinning phase

As we will see below, as an animal explores its environment, these coupled dynamics between attractor phase

## Learning a Simple Environmental Geometry

We now examine solutions to these equations to understand how neuronal dynamics and synaptic plasticity conspire to yield a consistent map of the environment. To build intuition, we first consider the dynamics of Eqs. **6** and **7** for the simple case of an animal moving back and forth between the walls of a 1D box of length L, at a constant speed *A*). In this environment we assume two landmark cells with firing fields localized at the east (west) wall. Their pinning phases

We build intuition in the limit **6**, the west border cell pins the initial attractor phase so that

Before any learning, there is no guarantee that the east border cell pinning phase *B*). However, plasticity described in Eq. **7** will act to move *B2*) yields the iterative dynamics*C*). In particular, the attractor phase assigned by the composite circuit to any point in the interior of the environment now becomes independent of which direction the animal is traveling, in contrast to the case before learning (compare the assigned interior phases in Fig. 4*B* with those in Fig. 4*C*).

## Learning as an Elastic Relaxation Between Landmarks

To gain further insight into the learning dynamics, it is useful to interpret the periodic attractor phase *D*). This enables us to associate physical positions to landmark cells, or more precisely their pinning phases, although these assigned positions are defined only up to shifts of the grid period. By “unrolling” the learning rule of Eq. **7**, we derive the learning and plasticity dynamics of unrolled phase for the landmark cells over the long timescale T of exploration,*SI Appendix*, section V).

These dynamics for the two landmark cell synapses in unrolled phase are equivalent to those of two particles at physical positions *E*). If the separation

This condition in unrolled phase is equivalent to the fundamental consistency condition for a well-defined spatial map, namely that the phase advance due to path integration equals the phase difference between the pinning phases of landmark cells (Fig. 4*C*). However, the utility of the unrolled phase representation lies in revealing a compelling picture of how a spatially consistent map arises from the combined neuronal and synaptic dynamics, through simple, emergent first-order relaxational dynamics of landmark particles connected by damped springs. As we see below, this simple effective particle–spring description of synaptic plasticity in response to spatial exploration generalizes to arbitrary landmarks in arbitrary 2D environments.

We note that if the environment has not been fully learned or has been recently deformed, the internal representation of landmarks will lag behind the true geometry for a time, leading to “boundary-tethered” firing fields seen in refs. 22, 25, and 31. Additionally (*SI Appendix*, section V), we have solved the dynamics when the firing fields of the border cells have a finite extent **8** and **9**, with a different rest length

## Generalization to 2D Grid Cells

To make contact with experiments, we generalize all of the above to 2D space. Now grid cells live on a periodic 2D neural sheet, where each cell has position **1** is*A*). Attractor dynamics on this sheet can yield a 2D family of steady-state bump activity patterns *SI Appendix*, section VII). The attractor state is now a 2D phase *B*). Velocity-conjunctive attractor cells yield a generalization of Eq. **2**:

We again assume the effect of landmark cells on the attractor network is strong enough to affect the position of the bump pattern, but not strong enough to change its shape. Then learned Hebbian synaptic weights *SI Appendix*, section XI1 for a proof)*C*). Also, each landmark cell i will affect the attractor phase through*C*).

## Combined 2D Neural and Synaptic Dynamics During Exploration

By combining the effects of path integration in Eq. **11** with landmark cells in Eq. **14**, we obtain the full dynamics of the 2D attractor phase driven by both animal velocity and landmark encounters:*D*, we can unroll the attractor phase **15** by summarizing *D* and *SI Appendix*, section VII.4.D). In analogy to Eqs. **6** and **7** this yields dynamics for the 2D internal position and landmark estimates given a 2D animal trajectory

## Spatial Consistency Through Emergent Elasticity

We showed in Eqs. **8** and **9** and Fig. 4 *D* and *E* that the emergence of spatial consistency between path integration and landmarks through Hebbian learning dynamics, during exploration of a simple 1D environment, could be understood as the outcome of an elastic relaxation process between landmark cell synapses, viewed as particles in physical space connected by damped springs. Remarkably, this result generalizes far beyond this simple environment. As long as the exploration dynamics are time reversible [time reversible means that for any *SI Appendix*, section VIII.

Overall, this elastic relaxation process converges toward an internal map where all pairs of landmark cell synapses, viewed as particles in unrolled phase, or physical space, become separated by the physical distance between their firing fields. This convergence ensures a consistent internal environmental map of external space in which velocity-based path integration of attractor phase starting at the pinning phase of landmark i and ending at landmark j will yield an integrated phase consistent with the pinning phase of landmark j itself. These relaxation dynamics explain path-dependent shifts in firing patterns observed in recently deformed environments (22). Also, the experimental observation in ref. 10 that, in multicompartment environments, consistent maps within compartments form before consistent maps between compartments is also explained by these relaxation dynamics. In essence, the longest-lived learning mode of the relaxation dynamics corresponds to differences in maps between compartments.

Furthermore, as we explain in the next three sections, these relaxation dynamics yield several experimental predictions: (*i*) systematic path-dependent shifts in fully learned 2D environments, (*ii*) mechanical deformations in complex environments, and (*iii*) the prediction of creation of topological defects in grid cell firing patterns through specific environmental manipulations.

## Path Dependence in 2D Environments

We saw above that exploration in a simple 1D geometry led to a consistent internal map in which the attractor network phase was mapped onto the current physical position alone, independent of path history (Fig. 4*C*). This consistency arises through the elastic relaxation process in Eqs. **8** and **9**, which makes the distance between the landmark cells in unrolled phase *E*). This situation will generalize to 2D if there are only two landmarks, namely a west and an east border cell (Fig. 6 *A1*). However, it becomes more complex with the addition of a third landmark cell, for example a south border cell (Fig. 6 *A2*).

In this case, east and west landmark particles will be connected by a spring of rest length *B*. However, the reason is completely different. In Fig. 4*B*, the landmark particles are not separated by the rest length of the spring connecting them because the environment is not fully learned and so the particles are out of equilibrium, whereas in Fig. 6 *A2*, the particles are not separated by the rest length, even in a fully learned environment, because additional springs from the south landmark create excess compression (See *SI Appendix*, section IX for details).

This theory makes a striking experimentally testable prediction, namely that even in a fully learned 2D environment, grid cell firing fields, when computed on subsets of mouse trajectories conditioned on leaving a particular border, will be shifted toward that border (Fig. 6*B*). This shift occurs because at any given position, the attractor phase depends on the most recently encountered landmark. In particular, on a west to east (east to west) trajectory, the attractor phase will be advanced (retarded) relative to an east to west (west to east) trajectory. Thus, on a west to east trajectory, the advanced phase will cause grid cells to fire earlier, yielding west-shifted grid cell firing fields as a function of position. Similarly on an east to west trajectory, grid fields will be east shifted. In summary, the theory predicts grid cell firing patterns conditioned on trajectories leaving the west (east) border will be shifted west (east). While we have derived this prediction qualitatively using the conceptual mass–spring picture in Fig. 6 *A2*, we confirm this intuition through direct numerical simulations of the full circuit dynamics in Eqs. **13** and **15** (Fig. 6 *C2*). Under reasonable parameters, our simulations can yield path-dependent shifts of up to ∼2 cm toward the wall the animal last touched (*SI Appendix*, section XII).

We searched for such subtle shifts in a population of 143 grid cells from 14 different mice that had been exploring a familiar, well-learned, 1-m open field (*SI Appendix*, section XII.2), using two separate analyses, based on cross-correlations and spike shifts with respect to field centers.

### Cross-Correlations.

One method for detecting a systematic firing-field shift across many grid fields is to cross-correlate firing-rate maps conditioned on trajectories leaving two different borders (*SI Appendix*, section IX.1.A). For example, for each cell, we can ask how much and in what direction we must shift its west border-conditioned firing field to match, or correlate as much as possible with, the same cell’s east-conditioned firing field. In particular, for each cell, we can compute the correlation coefficient between a spatially shifted west-conditioned field and an unshifted east-conditioned field and plot the average correlation coefficient as a function of this spatial shift. The theory predicts that we will have to shift the west-conditioned firing field eastward to match the east-conditioned firing field (Fig. 7*A*). This prediction is confirmed by a peak in the cross-correlation as a function of spatial shift when the spatial shift is positive or directed east (Fig. 7*B*). A similar logic holds for north and south.

Overall, this analysis shows that grid patterns are shifted toward the most recently encountered wall, both for the north–south (NS) walls (3 cm,

### Firing-Field Centers.

These results can be corroborated by computing shifts in spikes relative to firing-field centers, when conditioning spikes on the path history (*SI Appendix*, section IX.1.B). For each firing-field center, we calculate the average spike position within that firing field conditioned on the animal having last touched a particular wall. For each cell, we calculate the average shift across all firing fields and examine how the shifts depend on which wall the animal last touched. Again, the patterns are shifted toward whichever wall the animal last touched (Fig. 7*C*) for both the NS walls (0.5 cm,

## Mechanical Deformations in Complex Environments

Another experimental observation that can be reproduced by our theory is the distortion (11) of grid cell patterns seen in an irregular environment (Fig. 8*A*). Landmark cells with firing fields distributed across an entire wall will pull the attractor phase to its associated landmark pinning phase regardless of where along the wall the animal is. The presence of a diagonal wall then causes the average attractor phase as a function of position to curve toward the wall, yielding spatial grid cell patterns that curve away from the wall (Fig. 8 *B1*, *B2*, *C**1*, and *C2*). Previous theoretical accounts of this grid cell deformation have relied on purely phenomenological models that treated individual grid cell firing fields as particles with mostly repulsive interactions (15), without a clear mechanistic basis underlying this interaction. Our model instead provides a clear mechanistic basis for such deformations, grounded in the interaction between attractor-based path integration and plastic landmark cells. Such dynamics yield an emergent elasticity where the particles are landmark cell synapses rather than individual firing-field centers.

## Topological Defects in Grid Fields: A Prediction

While the dynamics of the linearized Eqs. **16** and **17** will always flow to the same relative landmark representations **13** and **15**, which can learn multiple different stable landmark cell synaptic configurations. One striking example of this is the ability of the learning dynamics to generate “topological defects,” where the number of firing fields traversed is not the same for two different paths (Fig. 9 *A* and *B* and *SI Appendix*, section XII). An environmental geometry capable of supporting these defects will yield a set of firing patterns that depends not only on the final geometry, but also on the history of how this geometry was created (Fig. 9*C*).

## Discussion

Overall, we have provided a theoretical framework for exploring how sensory cues and path integration may work together to create a consistent internal representation of space. Our framework is grounded in biologically plausible mechanisms involving attractor-based path integration of velocity and Hebbian plasticity of landmark cells. Moreover, systematic model reduction of these combined neural and synaptic dynamics yields a simple and intuitive emergent elasticity model in which landmark cell synapses act like particles sitting in physical space connected by damped springs whose rest length is equal to the physical distance between landmark firing fields. This simple emergent elasticity model not only provides a conceptual explanation of how neuronal dynamics and synaptic plasticity can conspire to self-organize a consistent map of space in which sensory cues and path integration are in register, but also provides predictions involving small shifts in firing fields even in fully learned environments, potential topological defects in grid cells, and grid cell mechanical deformation when borders are irregular. Furthermore, a special case of this model can explain sudden grid cell remapping events in altered virtual reality environments as the disagreement between landmarks and path integration is gradually increased; the model remaps due to sudden phase transitions in its dynamics (32).

This work opens up many interesting avenues. For example, further explorations of the nonlinear regime of our model may yield interesting experimental signatures that distinguish different modes of interaction between attractors, path integrators, and landmark cells. Incorporating heterogeneity of neural representations observed in MEC (33) into our framework is another intriguing direction. Also, as the reliability of sensory and velocity cues changes, it is interesting to ask what higher-order mechanisms may differentially regulate the effect of landmarks and velocity on the internal representation of space. More generally, our theory provides a unified framework for understanding how systematic variations in environmental geometry and the statistics of environmental exploration interact to precisely sculpt neural representations of space.

## Acknowledgments

We thank Daniel Fisher for helpful discussions and Kenneth Harris for helpful comments on the manuscript. We thank Malcolm Campbell, Robert Munn, and Caitlin Mallory for assistance in gathering data. We thank the Office of Naval Research, the Simons and James S. McDonnell Foundations (L.M.G. and S.G.), the McKnight Foundation (S.G.), the New York Stem Cell Foundation, Whitehall Foundation, National Institutes of Mental Health MH106475, a Klingenstein–Simons fellowship (L.M.G.), an Urbanek fellowship (S.A.O.), and a Stanford Interdisciplinary Graduate Fellowship (K.H.) for support.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: samocko{at}gmail.com.

Author contributions: S.A.O., K.H., L.M.G., and S.G. designed research; S.A.O., L.M.G., and S.G. performed research; S.A.O. analyzed data; and S.A.O., K.H., and S.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. D.B.C. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805959115/-/DCSupplemental.

Published under the PNAS license.

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- Abstract
- Model Reduction of an Attractor Network Coupled to Sensorimotor Inputs
- Learning a Simple Environmental Geometry
- Learning as an Elastic Relaxation Between Landmarks
- Generalization to 2D Grid Cells
- Combined 2D Neural and Synaptic Dynamics During Exploration
- Spatial Consistency Through Emergent Elasticity
- Path Dependence in 2D Environments
- Mechanical Deformations in Complex Environments
- Topological Defects in Grid Fields: A Prediction
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