On the supra-linear storage in dense networks of grid and place cells

TL;DR

This work tackles the limited storage capacity of place-cell networks, which in traditional pairwise Hebbian models scales only linearly with system size. It introduces a minimal two-layer architecture with a grid-cell–like hidden layer and a place-cell–like visible layer, where each place cell couples to pairs of grid cells; marginalizing the place layer yields an effective dense Battaglia–Treves network with four-body interactions, enabling supra-linear storage $K_{ ext{max}} = \alpha_c N^{p-1}$ (with $p=4$ for 3D embeddings). Using Guerra interpolation and replica techniques under replica symmetry, the authors derive the phase diagram and retrieve coherent spatial charts, demonstrating the model’s capacity to support spatial navigation on surfaces embedded in 3D. Numerically, the model shows navigation robustness and reveals that place-field errors accumulate with time in a Gamma-distributed fashion, consistent with experimental observations. Overall, the study links biologically plausible layering of grid and place cells to dense higher-order memory architectures, offering a principled route to scalable spatial cognition in higher dimensions with potential implications for understanding hippocampal–entorhinal circuitry and continuous attractor dynamics.

Abstract

Place-cell networks, typically forced to pairwise synaptic interactions, are widely studied as models of cognitive maps: such models, however, share a severely limited storage capacity, scaling linearly with network size and with a very small critical storage. This limitation is a challenge for navigation in 3-dimensional space because, oversimplifying, if encoding motion along a one-dimensional trajectory embedded in 2-dimensions requires $O(K)$ patterns (interpreted as bins), extending this to a 2-dimensional manifold embedded in a 3-dimensional space -- yet preserving the same resolution -- requires roughly $O(K^2)$ patterns, namely a supra-linear amount of patterns. In these regards, dense Hebbian architectures, where higher-order neural assemblies mediate memory retrieval, display much larger capacities and are increasingly recognized as biologically plausible, but have never linked to place cells so far. Here we propose a minimal two-layer model, with place cells building a layer and leaving the other layer populated by neural units that account for the internal representations (so to qualitatively resemble grid cells in the MEC of mammals): crucially, by assuming that each place cell interacts with pairs of grid cells, we show how such a model is formally equivalent to a dense Battaglia-Treves-like Hebbian network of grid cells only endowed with four-body interactions. By studying its emergent computational properties by means of statistical mechanics of disordered systems, we prove -- analytically -- that such effective higher-order assemblies (constructed under the guise of biological plausibility) can support supra-linear storage of continuous attractors; furthermore, we prove -- numerically -- that the present neural network is capable of recognition and navigation on general surfaces embedded in a 3-dimensional space.

Figures (11)

• Figure 1: A sketch of the model. Left: the place cells $\{\mathbf z\}$ are disposed on the vertices of the grid following the mapping that associates each index $\mu$ to the coordinate $(\mu_x,\mu_y)=(\mu \bmod L, \lfloor \mu/L\rfloor)$. Right: as the animal crosses specific points in the environment, place cells activate accordingly, producing a coherent state in the space of grid cells $\{\mathbf s\}$ and relative map $\mu$.
• Figure 2: Duality of representation. Left: the two-layer neural network where couples of grid cells (green circles in the left layer) are coupled to a place cell (blue circle in the right layer). For the sake of simplicity only one triplet -i.e. one coupling- is shown. Right: the equivalent representation (obtained by marginalizing out the place cells, see eq. \ref{['DualityEq']}) in terms of a dense Battaglia-Treves-like neural network of grid cells only.
• Figure 3: Phase diagram of the dense Battaglia-Treves model for $p=4$. Notice that the free energy eq. \ref{['eq:freeen']} is formally invariant under $d$ (since $d$ always only appears inside the definition of $\alpha$), hence the phase diagram is the same for every value of $d$. As expected, we note the presence of three regions, namely the high noise limit captured by the paramagnetic phase$(PM)$ -where nor computational capabilities neither spin glass features appear- the low noise but too much load regime captured by the spin glass region $(SG)$ (where glassy features are shown but the model is handling too much information and it fails in performing chart recognition) and the ferromagnetic phase$(FM)$ -where retrieval of maps is effectively achieved by the network in this challenging high storage regime where $K \propto N^{3}$.
• Figure 4: Phase diagrams of dense Battaglia-Treves like neural networks of grid cells with (left) $d=2, p=4$, (center) $d=2, p=6$ and (right) $d=2, p=8$ at different values of the inhibition strength $\lambda$ confined to planar motion. We remark two points: The former is that, qualitatively, the existence of a retrieval region is robust with respect to the density of the network. The latter is that, while $\lambda$ actually does not appear in the main text (due to our choice of using $\lambda=1$ in order to simplify the mathematical treatment, as explained in Appendix \ref{['AppendiceZero']}), such working assumption (i.e. $\lambda=1$) is roughly a worse case scenario as proved by the insets of these diagrams, where the maximal storage $\alpha_c$ is shown versus $\lambda$.
• Figure 5: Simplest tests of the dynamical reconstruction of two trajectories, the first one in the shape of a pentagon (left) and the second one a circular trajectory (right) of the 'animal' by the dense network. Black: true velocity of the animal, represented by arrows starting at the prescribed anchor points, for all the involved tiles within the plane where the motion happens. Overall, all these arrows roughly form a circle in the visible space $\mathcal{M}_{visible}$. Red: reconstructed trajectory in the $\mathbf z$ layer of place cells: just by visual inspection, we can appreciate how the reconstructed trajectory resembles the real motion, nevertheless, it also shines that there are errors in the reconstruction (i.e. the two circles do not perfectly overlap as the red arrows sometimes lie inside the black circle some other times outside, preserving zero mean, but not-zero standard deviations).

Theorems & Definitions (27)

• Definition 1: Multi charts
• Remark 1
• Definition 2: Coherent States
• Definition 3: Quenched Average
• Definition 4: pairwise Battaglia-Treves Hamiltonian
• Definition 5: Dense Battaglia-Treves Hamiltonian
• Definition 6: Boltzmann and Quenched Averages
• Definition 7
• Definition 8: Interpolating partition function
• Definition 9: Replica symmetry