Simultaneous embedding of multiple attractor manifolds in a recurrent neural network using constrained gradient optimization

The storage of continuous variables in working memory is hypothesized to be sustained in the brain by the dynamics of recurrent neural networks (RNNs) whose steady states form continuous manifolds. In some cases, it is thought that the synaptic connectivity supports multiple attractor manifolds, each mapped to a different context or task. For example, in hippocampal area CA3, positions in distinct environments are represented by distinct sets of population activity patterns, each forming a continuum. It has been argued that the embedding of multiple continuous attractors in a single RNN inevitably causes detrimental interference: quenched noise in the synaptic connectivity disrupts the continuity of each attractor, replacing it by a discrete set of steady states that can be conceptualized as lying on local minima of an abstract energy landscape. Consequently, population activity patterns exhibit systematic drifts towards one of these discrete minima, thereby degrading the stored memory over time. Here we show that it is possible to dramatically attenuate these detrimental interference effects by adjusting the synaptic weights. Synaptic weight adjustments are derived from a loss function that quantifies the roughness of the energy landscape along each of the embedded attractor manifolds. By minimizing this loss function, the stability of states can be dramatically improved, without compromising the capacity.Comment: To be presetned at the Thirty-seventh Conference on Neural Information Processing Systems (NeurIPS 2023

An Efficient Coding Theory for a Dynamic Trajectory Predicts non-Uniform Allocation of Grid Cells to Modules in the Entorhinal Cortex

Grid cells in the entorhinal cortex encode the position of an animal in its environment using spatially periodic tuning curves of varying periodicity. Recent experiments established that these cells are functionally organized in discrete modules with uniform grid spacing. Here we develop a theory for efficient coding of position, which takes into account the temporal statistics of the animal's motion. The theory predicts a sharp decrease of module population sizes with grid spacing, in agreement with the trends seen in the experimental data. We identify a simple scheme for readout of the grid cell code by neural circuitry, that can match in accuracy the optimal Bayesian decoder of the spikes. This readout scheme requires persistence over varying timescales, ranging from ~1ms to ~1s, depending on the grid cell module. Our results suggest that the brain employs an efficient representation of position which takes advantage of the spatiotemporal statistics of the encoded variable, in similarity to the principles that govern early sensory coding.Comment: 23 pages, 5 figures. Supplemental Information available from the authors on request. A previous version of this work appeared in abstract form (Program No. 727.02. 2015 Neuroscience Meeting Planner. Chicago, IL: Society for Neuroscience, 2015. Online.

An efficient coding theory for a dynamic trajectory predicts non-uniform allocation of entorhinal grid cells to modules

<div><p>Grid cells in the entorhinal cortex encode the position of an animal in its environment with spatially periodic tuning curves with different periodicities. Recent experiments established that these cells are functionally organized in discrete modules with uniform grid spacing. Here we develop a theory for efficient coding of position, which takes into account the temporal statistics of the animal’s motion. The theory predicts a sharp decrease of module population sizes with grid spacing, in agreement with the trend seen in the experimental data. We identify a simple scheme for readout of the grid cell code by neural circuitry, that can match in accuracy the optimal Bayesian decoder. This readout scheme requires persistence over different timescales, depending on the grid cell module. Thus, we propose that the brain may employ an efficient representation of position which takes advantage of the spatiotemporal statistics of the encoded variable, in similarity to the principles that govern early sensory processing.</p></div

Optimization for rat foraging statistics.

<p><b>A.-B.</b> Analysis of trajectories measured from two rats (each panel represents one animal). Both animals randomly foraged for food pellets in a familiar 1.5m square arena with black walls and floor (data courtesy of the Moser lab, NTNU, Trondheim). We evaluated the mean square displacement of an animal over a time interval Δ<i>T</i>. A fit to a power law [see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e015" target="_blank">Eq (8)</a>] was obtained by linear regression of vs. log Δ<i>T</i> in the interval Δ<i>T</i> = [0.32, 0.8] s (see the main text for a discussion on the behavior for large and very small Δ<i>T</i>). For both animals we obtained <i>ϵ</i> ≃ 1.68.<b>C.-F.</b> Optimized parameters for encoding and decoding by grid cells, as in Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g002" target="_blank">2</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g004" target="_blank">4B</a>, but using the optimization for power law statistics of motion, and substituting the parameters <i>ϵ</i> and <i>g</i> [see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e015" target="_blank">Eq (8)</a>] that were computed from animal 1 (<b>A</b>) (blue dots) and from animal 2 (<b>B</b>) (red squares). <b>C.</b> Number of neurons in a module as a function of the module index. <b>D.</b> The grid spacing. <b>E.</b> Ratio between grid spacings in subsequent modules. This ratio approaches ∼ 1.48 in the smallest modules. <b>F.</b> Integration time <i>τ</i> as a function of the module index, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e033" target="_blank">Eq (17)</a>. See main text for parameters. <b>G.</b> Performance of the simple estimator, using the optimization for power law statistics of motion and substituting the results of C-F. Here the average of the posterior probability distribution is over 9350 time points. The MSE and margins of errors were computed based on 100 simulations each lasting ∼ 9.4s.<b>H.</b> Numbers of experimentally identified neurons in different modules as a function of the module index. Data is extracted from Supplementary Figs. 2a and 2e of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.ref017" target="_blank">17</a>] (see also Fig. 1d in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.ref017" target="_blank">17</a>] which includes a subset of the data). The numbers are shown for all the tangential recordings. Different colors correspond to different rats (yellow—14257, red—15444, blue—13473, dark gray—13388, light gray—14760). The light and dark gray traces with square symbols correspond to animals in which the coverage of the dorsoventral axis was highly nonuniform (see middle panels of Supplementary Fig. 2a [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.ref017" target="_blank">17</a>]). In rat 14760 (light gray) the number of recorded cells was also significantly smaller than in the other animals. Black lines: predicted slopes for trajectories with random walk statistics [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e007" target="_blank">Eq (6)</a>,dotted line], and for the power law statistics of measured rat trajectories (∼ 1.59^<i>i</i>, dashed line).</p

Optimal Bayesian decoder.

<p>The posterior probability distribution obtained in simulations of the optimal Bayesian decoder, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e022" target="_blank">Eq (12)</a>, shifted by the true position of the animal, and averaged over 1350 time points. Three different allocations of neurons to modules are shown: <b>A.</b> optimal allocation as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g002" target="_blank">Fig 2B</a>, <b>B.</b> equal number of neurons in each module, and <b>C.</b> reversed allocation. The MSE and margins of error noted on the left bottom of each panel were computed as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g001" target="_blank">Fig 1D and 1E</a>, based on 100 simulations each lasting ∼ 1.4s. All the parameters are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g002" target="_blank">Fig 2</a>, with <i>N</i> = 10⁴.</p

Optimized code: Analytical results.

<p><b>A.</b> An illustration of the role of the parameter <i>β</i> [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e006" target="_blank">Eq (5)</a>]. For large <i>β</i> (left), ambiguities in the location readout may arise due overlap of multiple possible locations from the posterior distribution of module <i>i</i> + 1 with the posterior distribution based on module <i>i</i>. A sufficiently small value of <i>β</i> rules out these ambiguities (right). <b>B.</b> Number of neurons in a module as a function of the module index (10 modules, ordered by grid spacing starting from the largest spacing). The total number of neurons is <i>N</i> = 10⁴ (blue) and <i>N</i> = 10⁵ (red). <b>C.</b> Grid spacings in the optimized code. <b>D.</b> Ratios between grid spacings in successive modules. The ratio approaches in the smallest modules. In all three panels, λ₁ = 5m, <i>D</i> = 0.05m² /s, <i>β</i> = 0.1, and the receptive fields of the cells are Gaussians with maximal firing rate <i>r</i>_max = 10Hz.</p

Modular organization and dynamic decoding.

<p><b>A.</b> Experimental evidence for the modular organization of grid cells (adapted from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.ref017" target="_blank">17</a>], Supplementary Figure 2e, Rat 14257): grid spacing in a single rat, where each dot corresponds to a single cell. Note the dramatic decline in the number of cells for larger grid spacings (see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.g005" target="_blank">Fig 5H</a> and additional Figs in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.ref017" target="_blank">17</a>]). <b>B.</b> Schematic illustration of the posterior distribution over position, inferred from spikes generated by all cells in a single module. The posterior has the same periodicity λ as the single neuron tuning curves. The local MSE, denoted by Δ², is proportional to the variance of the local probability distribution around each peak. <b>C.</b> Schematic illustration of a decoder for a dynamic variable, which follows the statistics of a simple random walk (shown for simplicity in one dimension, and for non-periodic receptive field). Black: a random walk trajectory . Red lines represent spikes emitted by a population of neurons with different tuning curves, where the red <i>y</i> -axis represents the neuron index. The decoder estimates the animal’s position at time <i>t</i>₀, based on all the spikes that occurred up to that time. <b>D.-E.</b> Local MSEs of an optimal decoder, estimating position based on spikes from a single module, as a function of the number of neurons, for a static two-dimensional variable (D), and a dynamic random variable, following the statistics of a simple random walk in two dimensions (E). Logarithmic scales are used in both panels. Blue dots: measurements of the local MSE from simulations of an optimal decoder, responding to spikes generated by neural populations of varying size. Each dot represents an average over 300 realizations, where in (D) the averaging is over a single readout time interval from each simulation lasting Δ<i>T</i> = 100 ms, starting with a uniform prior over positions, and in (E) we average the local MSE in each simulation also over time (realizations lasting at least ∼ 200 ms). Error bars: 1.96 standard deviations of the local MSEs obtained from each simulation, divided by square root of the number of simulations (corresponding to a confidence interval of 95%). The receptive fields of the cells consist of a sum over periodically translated Gaussians with maximal firing rate <i>r</i>_max = 10Hz and standard deviation <i>σ</i> = λ/10, and the grid spacing is λ = 1 m. In the dynamic case (E) <i>D</i> = 0.0125 m² /s. Red lines: theoretical predictions from Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e002" target="_blank">1</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e003" target="_blank">2</a>) (D) and Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e003" target="_blank">2</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005597#pcbi.1005597.e004" target="_blank">3</a>) (E).</p

Grid-cell modules remain coordinated when neural activity is dissociated from external sensory cues

The representation of an animal’s position in the medial entorhinal cortex (MEC) is distributed across several modules of grid cells, each characterized by a distinct spatial scale. The population activity within each module is tightly coordinated and preserved across environments and behavioral states. Little is known, however, about the coordination of activity patterns across modules. We analyzed the joint activity patterns of hundreds of grid cells simultaneously recorded in animals that were foraging either in the light, when sensory cues could stabilize the representation, or in darkness, when such stabilization was disrupted. We found that the states of different modules are tightly coordinated, even in darkness, when the internal representation of position within the MEC deviates substantially from the true position of the animal. These findings suggest that internal brain mechanisms dynamically coordinate the representation of position in different modules, ensuring that they jointly encode a coherent and smooth trajectory