Modular neural networks outperform nonmodular neural networks on tasks ranging from visual question answering to robotics. These performance improvements are thought to be due to modular networks' superior ability to model the compositional and combinatorial structure of real-world problems. However, a theoretical explanation of how modularity improves generalizability, and how to leverage task modularity while training networks remains elusive. Using recent theoretical progress in explaining neural network generalization, we investigate how the amount of training data required to generalize on a task varies with the intrinsic dimensionality of a task's input. We show theoretically that when applied to modularly structured tasks, while nonmodular networks require an exponential number of samples with task dimensionality, modular networks' sample complexity is independent of task dimensionality: modular networks can generalize in high dimensions. We then develop a novel learning rule for modular networks to exploit this advantage and empirically show the improved generalization of the rule, both in- and out-of-distribution, on high-dimensional, modular tasks.

Frontier AI systems are making transformative impacts across society, but such benefits are not without costs: models trained on web-scale datasets containing personal and private data raise profound concerns about data privacy and security. Language models are trained on extensive corpora including potentially sensitive or proprietary information, and the risk of data leakage, where the model response reveals pieces of such information, remains inadequately understood. Prior work has investigated that sequence complexity and the number of repetitions are the primary drivers of memorization. In this work, we examine the most vulnerable class of data: highly complex sequences that are presented only once during training. These sequences often contain the most sensitive information and pose considerable risk if memorized. By analyzing the progression of memorization for these sequences throughout training, we uncover a striking observation: many memorized sequences persist in the model's memory, exhibiting resistance to catastrophic forgetting even after just one encounter. Surprisingly, these sequences may not appear memorized immediately after their first exposure but can later be “uncovered” during training, even in the absence of subsequent exposures - a phenomenon we call "latent memorization." Latent memorization presents a serious challenge for data privacy, as sequences that seem hidden at the final checkpoint of a model may still be easily recoverable. We demonstrate how these hidden sequences can be revealed through random weight perturbations, and we introduce a diagnostic test based on cross-entropy loss to accurately identify latent memorized sequences.

Composition—the ability to generate myriad variations from finite means—is believed to underlie powerful generalization. However, compositional generalization remains a key challenge for deep learning. A widely held assumption is that learning disentangled (factorized) representations naturally supports this kind of extrapolation. Yet, empirical results are mixed, with many generative models failing to recognize and compose factors to generate out-of-distribution (OOD) samples. In this work, we investigate a controlled 2D Gaussian "bump" generation task with fully disentangled $(x,y)$ inputs, demonstrating that standard generative architectures still fail in OOD regions when training with partial data, by re-entangling latent representations in subsequent layers. By examining the model’s learned kernels and manifold geometry, we show that this failure reflects a "memorization" strategy for generation via data superposition rather than via composition of the true factorized features. We show that when models are forced—through architectural modifications with regularization or curated training data—to render the disentangled latents into the full-dimensional representational (pixel) space, they can be highly data-efficient and effective at composing in OOD regions. These findings underscore that disentangled latents in an abstract representation are insufficient and show that if models can represent disentangled factors directly in the output representational space, it can achieve robust compositional generalization.

Brains excel at robust decision-making and data-efficient learning. Understanding the architectures and dynamics underlying these capabilities can inform inductive biases for deep learning. We present a multi-region brain model that explores the normative role of structured memory circuits in a spatially embedded binary decision-making task from neuroscience. We counterfactually compare the learning performance and neural representations of reinforcement learning (RL) agents with brain models of different interaction architectures between grid and place cells in the entorhinal cortex and hippocampus, coupled with an action-selection cortical recurrent neural network. We demonstrate that a specific architecture–where grid cells receive and jointly encode self-movement velocity signals and decision evidence increments–optimizes learning efficiency while best reproducing experimental observations relative to alternative architectures. Our findings thus suggest brain-inspired structured architectures for efficient RL. Importantly, the models make novel, testable predictions about organization and information flow within the entorhinal-hippocampal-neocortical circuit: we predict that grid cells must conjunctively encode position and evidence for effective spatial decision-making, directly motivating new neurophysiological experiments.

How precisely does circuit wiring specify function? This fundamental question is particularly relevant for modern neuroscience, as large-scale electron microscopy now enables the reconstruction of neural circuits at single-synapse resolution across many organisms. To interpret circuit function from such datasets, we must understand the extent to which [measured] structure constrains dynamics. We investigate this question in the drosophila head direction (HD) circuit, which maintains an internal heading estimate through attractor dynamics that integrate self-motion velocity cues. This circuit serves as a sensitive assay for functional specification: continuous attractor networks are theoretically known to require finely tuned wiring, whereas connectomes reveal that biological wiring can be variable and omit key cellular parameters such as synaptic gains, neuronal thresholds, and time constants. We introduce a method that combines self-supervised and unsupervised learning objectives to estimate unknown parameters at the level of cell types, rather than individual neurons and synapses. Given the raw connectivity matrix, our approach recovers a network that robustly exhibits continuous attractor dynamics and accurately integrates a range of velocity inputs, despite minimal parameter tuning on a connectome which notably departs from the symmetric regularity of an idealized ring attractor. We characterize how deviations from the original connectome shape the space of viable solutions. We also perform in-silico ablation experiments to probe the distinct functional roles of specific cell types in the circuit, demonstrating how connectome-derived structure, when augmented with minimal, biologically grounded tuning, can replicate known physiology and elucidate circuit function.

Biological function arises through the dynamical interactions of multiple subsystems, including those between brain areas, within gene regulatory networks, and more. A common approach to understanding these systems is to model the dynamics of each subsystem and characterize communication between them. An alternative approach is through the lens of control theory: how the subsystems control one another. This approach involves inferring the directionality, strength, and contextual modulation of control between subsystems. However, methods for understanding subsystem control are typically linear and cannot adequately describe the rich contextual effects enabled by nonlinear complex systems. To bridge this gap, we devise a data-driven nonlinear control-theoretic framework to characterize subsystem interactions via the Jacobian of the dynamics. We address the challenge of learning Jacobians from time-series data by proposing the JacobianODE, a deep learning method that leverages properties of the Jacobian to directly estimate it for arbitrary dynamical systems from data alone. We show that JacobianODE models outperform existing Jacobian estimation methods on challenging systems, including high-dimensional chaos. Applying our approach to a multi-area recurrent neural network (RNN) trained on a working memory selection task, we show that the “sensory” area gains greater control over the “cognitive” area over learning. Furthermore, we leverage the JacobianODE to directly control the trained RNN, enabling precise manipulation of its behavior. Our work lays the foundation for a theoretically grounded and data-driven understanding of interactions among biological subsystems.

The ability to engineer novel proteins with higher fitness for a desired property would be revolutionary for biotechnology and medicine. Modeling the combinatorially large space of sequences is infeasible; prior methods often constrain optimization to a small mutational radius, but this drastically limits the design space. Instead of heuristics, we propose smoothing the fitness landscape to facilitate protein optimization. First, we formulate protein fitness as a graph signal then use Tikunov regularization to smooth the fitness landscape. We find optimizing in this smoothed landscape leads to improved performance across multiple methods in the GFP and AAV benchmarks. Second, we achieve state-of-the-art results utilizing discrete energy-based models and MCMC in the smoothed landscape. Our method, called Gibbs sampling with Graph-based Smoothing (GGS), demonstrates a unique ability to achieve 2.5 fold fitness improvement (with in-silico evaluation) over its training set. GGS demonstrates potential to optimize proteins in the limited data regime. Code: https://github.com/kirjner/GGS

Animals can swiftly adapt to novel tasks, while maintaining proficiency on previously trained tasks. This contrasts starkly with machine learning models, which struggle on these capabilities. We first propose a new task, the sequential Morris Water Maze (sWM), which extends a widely used task in the psychology and neuroscience fields and requires both rapid and continual learning. It has frequently been hypothesized that inductive biases from brains could help build better ML systems, but the addition of constraints typically hurts rather than helping ML performance. We draw inspiration from biology to show that combining 1) a content-addressable heteroassociative memory based on the entorhinal-hippocampal circuit with grid cells that retain shared across-environment structural representations and hippocampal cells that acquire environment-specific information; 2) a spatially invariant convolutional network architecture for rapid adaptation across unfamiliar environments; and 3) the ability to perform remapping, which orthogonalizes internal representations; leads to good generalization, rapid learning, and continual learning without forgetting, respectively. Our model outperforms ANN baselines from continual learning contexts applied to the task. It retains knowledge of past environments while rapidly acquiring the skills to navigate new ones, thereby addressing the seemingly opposing challenges of quick knowledge transfer and sustaining proficiency in previously learned tasks. These biologically motivated results may point the way toward ML algorithms with similar properties.

State estimation is crucial for the performance and safety of numerous robotic applications. Among the suite of estimation techniques, particle filters have been identified as a powerful solution due to their non-parametric nature. Yet, in high-dimensional state spaces, these filters face challenges such as ’particle deprivation’ which hinders accurate representation of the true posterior distribution. This paper introduces a novel resampling-free particle filter designed to mitigate particle deprivation by forgoing the traditional resampling step. This ensures a broader and more diverse particle set, especially vital in high-dimensional scenarios. Theoretically, our proposed filter is shown to offer a near-accurate representation of the desired posterior distribution in high-dimensional contexts. Empirically, the effectiveness of our approach is underscored through a high-dimensional synthetic state estimation task and a 6D pose estimation derived from videos. We posit that as robotic systems evolve with greater degrees of freedom, particle filters tailored for high-dimensional state spaces will be indispensable.

Visual representations become progressively more abstract along the cortical hierarchy. These abstract representations define notions like objects and shapes, but at the cost of spatial specificity. By contrast, low-level regions represent spatially local but simple input features. How do spatially non-specific representations of abstract concepts in high-level areas flexibly modulate the low-level sensory representations in appropriate ways to guide context-driven and goal-directed behaviors across a range of tasks? We build a biologically motivated and trainable neural network model of dynamics in the visual pathway, incorporating local, lateral, and feedforward synaptic connections, excitatory and inhibitory neurons, and long-range top-down inputs conceptualized as low-rank modulations of the input-driven sensory responses by high-level areas. We study this ${\bf D}$ynamical ${\bf C}$ortical ${\bf net}$work ($DCnet$) in a visual cue-delay-search task and show that the model uses its own cue representations to adaptively modulate its perceptual responses to solve the task, outperforming state-of-the-art DNN vision and LLM models. The model's population states over time shed light on the nature of contextual modulatory dynamics, generating predictions for experiments. We fine-tune the same model on classic psychophysics attention tasks, and find that the model closely replicates known reaction time results. This work represents a promising new foundation for understanding and making predictions about perturbations to visual processing in the brain.

Grid cells in the medial entorhinal cortex create remarkable periodic maps of explored space during navigation. Recent studies show that they form similar maps of abstract cognitive spaces. Examples of such abstract environments include auditory tone sequences in which the pitch is continuously varied or images in which abstract features are continuously deformed (e.g., a cartoon bird whose legs stretch and shrink). Here, we hypothesize that the brain generalizes how it maps spatial domains to mapping abstract spaces. To sidestep the computational cost of learning representations for each high-dimensional sensory input, the brain extracts self-consistent, low-dimensional descriptions of displacements across abstract spaces, leveraging the spatial velocity integration of grid cells to efficiently build maps of different domains. Our neural network model for abstract velocity extraction factorizes the content of these abstract domains from displacements within the domains to generate content-independent and self-consistent, low-dimensional velocity estimates. Crucially, it uses a self-supervised geometric consistency constraint that requires displacements along closed loop trajectories to sum to zero, an integration that is itself performed by the downstream grid cell circuit over learning. This process results in high fidelity estimates of velocities and allowed transitions in abstract domains, a crucial prerequisite for efficient map generation in these high-dimensional environments. We also show how our method outperforms traditional dimensionality reduction and deep-learning based motion extraction networks on the same set of tasks. This is the first neural network model to explain how grid cells can flexibly represent different abstract spaces and makes the novel prediction that they should do so while maintaining their population correlation and manifold structure across domains. Fundamentally, our model sheds light on the mechanistic origins of cognitive flexibility and transfer of representations across vastly different domains in brains, providing a potential self-supervised learning (SSL) framework for leveraging similar ideas in transfer learning and data-efficient generalization in machine learning and robotics.

Diffusion models are capable of generating photo-realistic images that combine elements which do not appear together in natural images, demonstrating their ability to compositionally generalize. Nonetheless, the precise mechanism of compositionality and how it is acquired through training remains elusive. Here, we consider a highly reduced setting to examine whether diffusion models learn semantically meaningful and fully factorized representations of composable features. We performed extensive controlled experiments on conditional DDPMs trained to generate various forms of 2D Gaussian data. We demonstrate that the models learn factorized, semi-continuous manifold representations that are orthogonal in underlying continuous latent features of independent variations but are not aligned for different values of the same feature. With such representations, models demonstrate superior compositionality but have limited ability to interpolate over unseen values of a given feature. Our experimental results further demonstrate that diffusion models can attain compositionality with a small amount of compositional examples, suggesting a novel way to train DDPMs. Finally, we connect manifold formation in diffusion models to percolation theory in physics, thereby offering insights into the sudden onset of factorized representation learning. Our thorough toy experiments thus contribute a deeper understanding of how diffusion models capture compositional structure in data, paving the way for future research aimed at enhancing factorization and compositional generalization in generative models for real-world applications.

The measure of a machine learning algorithm is the difficulty of the tasks it can perform, and sufficiently difficult tasks are critical drivers of strong machine learning models. However, quantifying the generalization difficulty of machine learning benchmarks has remained challenging. We propose what is to our knowledge the first model-agnostic measure of the inherent generalization difficulty of tasks. Our inductive bias complexity measure quantifies the total information required to generalize well on a task minus the information provided by the data. It does so by measuring the fractional volume occupied by hypotheses that generalize on a task given that they fit the training data. It scales exponentially with the intrinsic dimensionality of the space over which the model must generalize but only polynomially in resolution per dimension, showing that tasks which require generalizing over many dimensions are drastically more difficult than tasks involving more detail in fewer dimensions. Our measure can be applied to compute and compare supervised learning, reinforcement learning and meta-learning generalization difficulties against each other. We show that applied empirically, it formally quantifies intuitively expected trends, e.g. that in terms of required inductive bias, MNIST $<$ CIFAR10 $<$ Imagenet and fully observable Markov decision processes (MDPs) $<$ partially observable MDPs. Further, we show that classification of complex images $<$ few-shot meta-learning with simple images. Our measure provides a quantitative metric to guide the construction of more complex tasks requiring greater inductive bias, and thereby encourages the development of more sophisticated architectures and learning algorithms with more powerful generalization capabilities.

How can we tell whether two neural networks utilize the same internal processes for a particular computation? This question is pertinent for multiple subfields of neuroscience and machine learning, including neuroAI, mechanistic interpretability, and brain-machine interfaces. Standard approaches for comparing neural networks focus on the spatial geometry of latent states. Yet in recurrent networks, computations are implemented at the level of dynamics, and two networks performing the same computation with equivalent dynamics need not exhibit the same geometry. To bridge this gap, we introduce a novel similarity metric that compares two systems at the level of their dynamics, called Dynamical Similarity Analysis (DSA). Our method incorporates two components: Using recent advances in data-driven dynamical systems theory, we learn a high-dimensional linear system that accurately captures core features of the original nonlinear dynamics. Next, we compare different systems passed through this embedding using a novel extension of Procrustes Analysis that accounts for how vector fields change under orthogonal transformation. In four case studies, we demonstrate that our method disentangles conjugate and non-conjugate recurrent neural networks (RNNs), while geometric methods fall short. We additionally show that our method can distinguish learning rules in an unsupervised manner. Our method opens the door to comparative analyses of the essential temporal structure of computation in neural circuits.

To solve the spatial problems of mapping, localization and navigation, the mammalian lineage has developed striking spatial representations. One important spatial representation is the Nobel-prize winning grid cells: neurons that represent self-location, a local and aperiodic quantity, with seemingly bizarre non-local and spatially periodic activity patterns of a few discrete periods. Why has the mammalian lineage learnt this peculiar grid representation? Mathematical analysis suggests that this multi-periodic representation has excellent properties as an algebraic code with high capacity and intrinsic error-correction, but to date, synthesis of multi-modular grid cells in deep recurrent neural networks remains absent. In this work, we begin by identifying key insights from four families of approaches to answering the grid cell question: dynamical systems, coding theory, function optimization and supervised deep learning. We then leverage our insights to propose a new approach that elegantly combines the strengths of all four approaches. Our approach is a self-supervised learning (SSL) framework - including data, data augmentations, loss functions and a network architecture - motivated from a normative perspective, with no access to supervised position information. Without making assumptions about internal or readout representations, we show that multiple grid cell modules can emerge in networks trained on our SSL framework and that the networks generalize significantly beyond their training distribution. This work contains insights for neuroscientists interested in the origins of grid cells as well as machine learning researchers interested in novel SSL frameworks.

Recurrent neural networks (RNNs) are often used to model circuits in the brain and can solve a variety of difficult computational problems requiring memory, error correction, or selection (Hopfield, 1982; Maass et al., 2002; Maass, 2011). However, fully connected RNNs contrast structurally with their biological counterparts, which are extremely sparse (about 0.1%). Motivated by the neocortex, where neural connectivity is constrained by physical distance along cortical sheets and other synaptic wiring costs, we introduce locality masked RNNs (LM-RNNs) that use task-agnostic predetermined graphs with sparsity as low as 4%. We study LM-RNNs in a multitask learning setting relevant to cognitive systems neuroscience with a commonly used set of tasks, 20-Cog-tasks (Yang et al., 2019). We show through reductio ad absurdum that 20-Cog-tasks can be solved by a small pool of separated autapses that we can mechanistically analyze and understand. Thus, these tasks fall short of the goal of inducing complex recurrent dynamics and modular structure in RNNs. We next contribute a new cognitive multitask battery, Mod-Cog, consisting of up to 132 tasks that expands by about seven-fold the number of tasks and task complexity of 20-Cog-tasks. Importantly, while autapses can solve the simple 20-Cog-tasks, the expanded task set requires richer neural architectures and continuous attractor dynamics. On these tasks, we show that LM-RNNs with an optimal sparsity result in faster training and better data efficiency than fully connected networks.

Recent works have examined theoretical and empirical properties of wide neural networks trained in the Neural Tangent Kernel (NTK) regime. Given that biological neural networks are much wider than their artificial counterparts, we consider NTK regime wide neural networks as a possible model of biological neural networks. Leveraging NTK theory, we show theoretically that gradient descent drives layerwise weight updates that are aligned with their input activity correlations weighted by error, and demonstrate empirically that the result also holds in finite-width wide networks. The alignment result allows us to formulate a family of biologically-motivated, backpropagation-free learning rules that are theoretically equivalent to backpropagation in infinite-width networks. We test these learning rules on benchmark problems in feedforward and recurrent neural networks and demonstrate, in wide networks, comparable performance to backpropagation. The proposed rules are particularly effective in low data regimes, which are common in biological learning settings.

Unsupervised learning from a continuous stream of data is arguably one of the most common and most challenging problems facing intelligent agents. One class of unsupervised models, collectively termed feature models, attempts unsupervised discovery of latent features underlying the data and includes common models such as PCA, ICA, and NMF. However, if the data arrives in a continuous stream, determining the number of features is a significant challenge and the number may grow with time. In this work, we make feature models significantly more applicable to streaming data by imbuing them with the ability to create new features, online, in a probabilistic and principled manner. To achieve this, we derive a novel recursive form of the Indian Buffet Process, which we term the Recursive IBP (R-IBP). We demonstrate that R-IBP can be be used as a prior for feature models to efficiently infer a posterior over an unbounded number of latent features, with quasilinear average time complexity and logarithmic average space complexity. We compare R-IBP to existing offline sampling and variational baselines in two feature models (Linear Gaussian and Factor Analysis) and demonstrate on synthetic and real data that R-IBP achieves comparable or better performance in significantly less time.

Content-addressable memory (CAM) networks, so-called because stored items can be recalled by partial or corrupted versions of the items, exhibit near-perfect recall of a small number of information-dense patterns below capacity and a ’memory cliff’ beyond, such that inserting a single additional pattern results in catastrophic loss of all stored patterns. We propose a novel CAM architecture, Memory Scaffold with Heteroassociation (MESH), that factorizes the problems of internal attractor dynamics and association with external content to generate a CAM continuum without a memory cliff: Small numbers of patterns are stored with complete information recovery matching standard CAMs, while inserting more patterns still results in partial recall of every pattern, with a graceful trade-off between pattern number and pattern richness. Motivated by the architecture of the Entorhinal-Hippocampal memory circuit in the brain, MESH is a tripartite architecture with pairwise interactions that uses a predetermined set of internally stabilized states together with heteroassociation between the internal states and arbitrary external patterns. We show analytically and experimentally that for any number of stored patterns, MESH nearly saturates the total information bound (given by the number of synapses) for CAM networks, outperforming all existing CAM models.

Research in Neuroscience, as in many scientific disciplines, is undergoing a renaissance based on deep learning. Unique to Neuroscience, deep learning models can be used not only as a tool but interpreted as models of the brain. The central claims of recent deep learning-based models of brain circuits are that they make novel predictions about neural phenomena or shed light on the fundamental functions being optimized. We show, through the case-study of grid cells in the entorhinal-hippocampal circuit, that one may get neither. We begin by reviewing the principles of grid cell mechanism and function obtained from first-principles modeling efforts, then rigorously examine the claims of deep learning models of grid cells. Using large-scale architectural and hyperparameter sweeps and theory-driven experimentation, we demonstrate that the results of such models may be more strongly driven by particular, non-fundamental, and post-hoc implementation choices than fundamental truths about neural circuits or the loss function(s) they might optimize. We discuss why these models cannot be expected to produce accurate models of the brain without the addition of substantial amounts of inductive bias, an informal No Free Lunch result for Neuroscience. Based on first principles work, we provide hypotheses for what additional loss functions will produce grid cells more robustly. In conclusion, circumspection and transparency, together with biological knowledge, are warranted in building and interpreting deep learning models in Neuroscience.

Natural data are often well-described as belonging to latent clusters. When the number of clusters is unknown, Bayesian nonparametric (BNP) models can provide a flexible and powerful technique to model the data. However, algorithms for inference in nonparametric mixture models fail to meet two critical requirements for practical use: (1) that inference can be performed online, and (2) that inference is efficient in the large time/sample limit. In this work, we propose a novel Bayesian recursion to efficiently infer a posterior distribution over discrete latent variables from a sequence of observations in an online manner, assuming a Chinese Restaurant Process prior on the sequence of latent variables. Our recursive filter, which we call the Recursive Chinese Restaurant Process (R-CRP), has quasilinear average time complexity and logarithmic average space complexity in the total number of observations. We experimentally compare our filtering method against both online and offline inference algorithms including Markov chain Monte Carlo, variational approximations and DP-Means, and demonstrate that our inference algorithm achieves comparable or better performance for a fraction of the runtime.

We study how recurrent neural networks (RNNs) solve a hierarchical inference task involving two latent variables and disparate timescales separated by 1-2 orders of magnitude. The task is of interest to the International Brain Laboratory, a global collaboration of experimental and theoretical neuroscientists studying how the mammalian brain generates behavior. We make four discoveries. First, RNNs learn behavior that is quantitatively similar to ideal Bayesian baselines. Second, RNNs perform inference by learning a two-dimensional subspace defining beliefs about the latent variables. Third, the geometry of RNN dynamics reflects an induced coupling between the two separate inference processes necessary to solve the task. Fourth, we perform model compression through a novel form of knowledge distillation on hidden representations -- Representations and Dynamics Distillation (RADD)-- to reduce the RNN dynamics to a low-dimensional, highly interpretable model. This technique promises a useful tool for interpretability of high dimensional nonlinear dynamical systems. Altogether, this work yields predictions to guide exploration and analysis of mouse neural data and circuity.

We shed light on the potential of entorhinal grid cells to efficiently encode variables of dimension greater than two, while remaining faithful to empirical data on their low-dimensional structure. Our model constructs representations of high-dimensional inputs through a combination of low-dimensional random projections and "classical" low-dimensional hexagonal grid cell responses. Without reconfiguration of the recurrent circuit, the same system can flexibly encode multiple variables of different dimensions while maximizing the coding range (per dimension) by automatically trading-off dimension with an exponentially large coding range. It achieves high efficiency and flexibility by combining two powerful concepts, modularity and mixed selectivity, in what we call "mixed modular coding". In contrast to previously proposed schemes, the model does not require the formation of higher-dimensional grid responses, a cell-inefficient and rigid mechanism. The firing fields observed in flying bats or climbing rats can be generated by neurons that combine activity from multiple grid modules, each representing higher-dimensional spaces according to our model. The idea expands our understanding of grid cells, suggesting that they could implement a general circuit that generates on-demand coding and memory states for variables in high-dimensional vector spaces.

Neural network models of memory and error correction famously include the Hopfield network, which can directly store---and error-correct through its dynamics---arbitrary N-bit patterns, but only for ~N such patterns. On the other end of the spectrum, Shannon's coding theory established that it is possible to represent exponentially many states (~e^N) using N symbols in such a way that an optimal decoder could correct all noise upto a threshold. We prove that it is possible to construct an associative content-addressable network that combines the properties of strong error correcting codes and Hopfield networks: it simultaneously possesses exponentially many stable states, these states are robust enough, with large enough basins of attraction that they can be correctly recovered despite errors in a finite fraction of all nodes, and the errors are intrinsically corrected by the network’s own dynamics. The network is a two-layer Boltzmann machine with simple neural dynamics, low dynamic-range (binary) pairwise synaptic connections, and sparse expander graph connectivity. Thus, quasi-random sparse structures---characteristic of important error-correcting codes---may provide for high-performance computation in artificial neural networks and the brain.

Self-localization during navigation with noisy sensors in an ambiguous world is computationally challenging, yet animals and humans excel at it. In robotics, {\em Simultaneous Location and Mapping} (SLAM) algorithms solve this problem through joint sequential probabilistic inference of their own coordinates and those of external spatial landmarks. We generate the first neural solution to the SLAM problem by training recurrent LSTM networks to perform a set of hard 2D navigation tasks that require generalization to completely novel trajectories and environments. Our goal is to make sense of how the diverse phenomenology in the brain's spatial navigation circuits is related to their function. We show that the hidden unit representations exhibit several key properties of hippocampal place cells, including stable tuning curves that remap between environments. Our result is also a proof of concept for end-to-end-learning of a SLAM algorithm using recurrent networks, and a demonstration of why this approach may have some advantages for robotic SLAM.

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat's velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of approximately 10-100 meters and approximately 1-10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.