These are composed into a single vector representing position by a similarity-preserving, conjunctive vector-binding operation.

These are composed into a single vector representing position by a similarity-preserving, conjunctive vector-binding operation.

Rodents navigating in a well-known environment can rapidly learn and revisit observed reward locations, often after a single trial. While the mechanism for rapid path planning is unknown, the CA3 region in the hippocampus plays an important role, and emerging evidence suggests that place cell activity during hippocam-pal "preplay" periods may trace out future goal-directed trajectories. Here, we show how a particular mapping of space allows for the immediate generation of trajectories between arbitrary start and goal locations in an environment, based only on the mapped representation of the goal. We show that this representation can be implemented in a neural attractor network model, resulting in bump-like activity profiles resembling those of the CA3 region of hippocampus. Neurons tend to locally excite neurons with similar place field centers, while inhibiting other neurons with distant place field centers, such that stable bumps of activity can form at arbitrary locations in the environment. The network is initialized to represent a point in the environment, then weakly stimulated with an input corresponding to an arbitrary goal location. We show that the resulting activity can be interpreted as a gradient ascent on the value function induced by a reward at the goal location. Indeed, in networks with large place fields, we show that the network properties cause the bump to move smoothly from its initial location to the goal, around obstacles or walls. Our results illustrate that an attractor network with hippocampal-like attributes may be important for rapid path planning.

After learning the novel classes, the model is then evaluated on the overall classification performance on both base and novel classes. To this end, we propose a meta-learning model, the Attention Attractor Network, which regularizes the learning of novel classes. In each episode, we train a set of new weights to recognize novel classes until they converge, and we show that the technique of recurrent back-propagation can back-propagate through the optimization process and facilitate the learning of these parameters.

Continuous attractor networks (CANs) are widely used to model how the brain temporarily retains continuous behavioural variables via persistent recurrent activity, such as an animal's position in an environment. However, this memory mechanism is very sensitive to even small imperfections, such as noise or heterogeneity, which are both common in biological systems. Previous work has shown that discretising the continuum into a finite set of discrete attractor states provides robustness to these imperfections, but necessarily reduces the resolution of the represented variable, creating a dilemma between stability and resolution. We show that this stability-resolution dilemma is most severe for CANs using unimodal bump-like codes, as in traditional models. To overcome this, we investigate sparse binary distributed codes based on random feature embeddings, in which neurons have spatially-periodic receptive fields. We demonstrate theoretically and with simulations that such grid-cell-like codes enable CANs to achieve both high stability and high resolution simultaneously. The model extends to embedding arbitrary nonlinear manifolds into a CAN, such as spheres or tori, and generalises linear path integration to integration along freely-programmable on-manifold vector fields. Together, this work provides a theory of how the brain could robustly represent continuous variables with high resolution and perform flexible computations over task-relevant manifolds.

Episodic memory enables humans to recall past experiences by associating semantic elements such as objects, locations, and time into coherent event representations. While large pretrained models have shown remarkable progress in modeling semantic memory, the mechanisms for forming associative structures that support episodic memory remain underexplored. Inspired by hippocampal CA3 dynamics and its role in associative memory, we propose the Latent Structured Hopfield Network (LSHN), a biologically inspired framework that integrates continuous Hopfield attractor dynamics into an autoencoder architecture. LSHN mimics the cortical-hippocampal pathway: a semantic encoder extracts compact latent representations, a latent Hopfield network performs associative refinement through attractor convergence, and a decoder reconstructs perceptual input. Unlike traditional Hopfield networks, our model is trained end-to-end with gradient descent, achieving scalable and robust memory retrieval. Experiments on MNIST, CIFAR-10, and a simulated episodic memory task demonstrate superior performance in recalling corrupted inputs under occlusion and noise, outperforming existing associative memory models. Our work provides a computational perspective on how semantic elements can be dynamically bound into episodic memory traces through biologically grounded attractor mechanisms. Code: https://github.com/fudan-birlab/LSHN.

Attractor dynamics are a hallmark of many complex systems, including the brain. Understanding how such self-organizing dynamics emerge from first principles is crucial for advancing our understanding of neuronal computations and the design of artificial intelligence systems. Here we formalize how attractor networks emerge from the free energy principle applied to a universal partitioning of random dynamical systems. Our approach obviates the need for explicitly imposed learning and inference rules and identifies emergent, but efficient and biologically plausible inference and learning dynamics for such self-organizing systems. These result in a collective, multi-level Bayesian active inference process. Attractors on the free energy landscape encode prior beliefs; inference integrates sensory data into posterior beliefs; and learning fine-tunes couplings to minimize long-term surprise. Analytically and via simulations, we establish that the proposed networks favor approximately orthogonalized attractor representations, a consequence of simultaneously optimizing predictive accuracy and model complexity. These attractors efficiently span the input subspace, enhancing generalization and the mutual information between hidden causes and observable effects. Furthermore, while random data presentation leads to symmetric and sparse couplings, sequential data fosters asymmetric couplings and non-equilibrium steady-state dynamics, offering a natural extension to conventional Boltzmann Machines. Our findings offer a unifying theory of self-organizing attractor networks, providing novel insights for AI and neuroscience.

This paper explores the integration of ring attractors, a mathematical model inspired by neural circuit dynamics, into the reinforcement learning (RL) action selection process. Ring attractors, as specialized brain-inspired structures that encode spatial information and uncertainty, offer a biologically plausible mechanism to improve learning speed and predictive performance. They do so by explicitly encoding the action space, facilitating the organization of neural activity, and enabling the distribution of spatial representations across the neural network in the context of deep RL. The application of ring attractors in the RL action selection process involves mapping actions to specific locations on the ring and decoding the selected action based on neural activity. We investigate the application of ring attractors by both building them as exogenous models and integrating them as part of a Deep Learning policy algorithm. Our results show a significant improvement in state-of-the-art models for the Atari 100k benchmark. Notably, our integrated approach improves the performance of state-of-the-art models by half, representing a 53% increase over selected baselines. This paper addresses the challenge of efficient action selection in reinforcement learning (RL), particularly in environments with spatial structures. Our primary contribution is the novel integration of ring attractors (Kim et al., 2017), a neural circuit model from neuroscience, into the RL framework. This approach improves spatial awareness in action selection and provides a mechanism for uncertainty-aware decision making in RL, leading to more accurate and efficient learning in complex environments.

Transformers are one of the most successful architectures of modern neural networks. At their core there is the so-called attention mechanism, which recently interested the physics community as it can be written as the derivative of an energy function in certain cases: while it is possible to write the cross-attention layer as a modern Hopfield network, the same is not possible for the self-attention, which is used in the GPT architectures and other autoregressive models. In this work we show that it is possible to obtain the self-attention layer as the derivative of local energy terms, which resemble a pseudo-likelihood. We leverage the analogy with pseudo-likelihood to design a recurrent model that can be trained without backpropagation: the dynamics shows transient states that are strongly correlated with both train and test examples. Overall we present a novel framework to interpret self-attention as an attractor network, potentially paving the way for new theoretical approaches inspired from physics to understand transformers.

The proper handling of out-of-distribution (OOD) samples in deep classifiers is a critical concern for ensuring the suitability of deep neural networks in safety-critical systems. Existing approaches developed for robust OOD detection in the presence of adversarial attacks lose their performance by increasing the perturbation levels. This study proposes a method for robust classification in the presence of OOD samples and adversarial attacks with high perturbation levels. The proposed approach utilizes a fully connected neural network that is trained to use training samples as its attractors, enhancing its robustness. This network has the ability to classify inputs and identify OOD samples as well. To evaluate this method, the network is trained on the MNIST dataset, and its performance is tested on adversarial examples. The results indicate that the network maintains its performance even when classifying adversarial examples, achieving 87.13% accuracy when dealing with highly perturbed MNIST test data. Furthermore, by using fashion-MNIST and CIFAR-10-bw as OOD samples, the network can distinguish these samples from MNIST samples with an accuracy of 98.84% and 99.28%, respectively. In the presence of severe adversarial attacks, these measures decrease slightly to 98.48% and 98.88%, indicating the robustness of the proposed method.