Continuous Attractor Dynamics Research Articles

Sensory-memory interactions via modular structure explain errors in visual working memory.

Errors in stimulus estimation reveal how stimulus representation changes during cognitive processes. Repulsive bias and minimum variance observed near cardinal axes are well-known error patterns typically associated with visual orientation perception. Recent experiments suggest that these errors continuously evolve during working memory, posing a challenge that neither static sensory models nor traditional memory models can address. Here, we demonstrate that these evolving errors, maintaining characteristic shapes, require network interaction between two distinct modules. Each module fulfills efficient sensory encoding and memory maintenance, which cannot be achieved simultaneously in a single-module network. The sensory module exhibits heterogeneous tuning with strong inhibitory modulation reflecting natural orientation statistics. While the memory module, operating alone, supports homogeneous representation via continuous attractor dynamics, the fully connected network forms discrete attractors with moderate drift speed and nonuniform diffusion processes. Together, our work underscores the significance of sensory-memory interaction in continuously shaping stimulus representation during working memory.

Causal evidence of a line attractor encoding an affective state

Continuous attractors are an emergent property of neural population dynamics that have been hypothesized to encode continuous variables such as head direction and eye position1–4. In mammals, direct evidence of neural implementation of a continuous attractor has been hindered by the challenge of targeting perturbations to specific neurons within contributing ensembles2,3. Dynamical systems modelling has revealed that neurons in the hypothalamus exhibit approximate line-attractor dynamics in male mice during aggressive encounters5. We have previously hypothesized that these dynamics may encode the variable intensity and persistence of an aggressive internal state. Here we report that these neurons also showed line-attractor dynamics in head-fixed mice observing aggression6. This allowed us to identify and manipulate line-attractor-contributing neurons using two-photon calcium imaging and holographic optogenetic perturbations. On-manifold perturbations yielded integration of optogenetic stimulation pulses and persistent activity that drove the system along the line attractor, while transient off-manifold perturbations were followed by rapid relaxation back into the attractor. Furthermore, single-cell stimulation and imaging revealed selective functional connectivity among attractor-contributing neurons. Notably, individual differences among mice in line-attractor stability were correlated with the degree of functional connectivity among attractor-contributing neurons. Mechanistic recurrent neural network modelling indicated that dense subnetwork connectivity and slow neurotransmission7 best recapitulate our empirical findings. Our work bridges circuit and manifold levels3, providing causal evidence of continuous attractor dynamics encoding an affective internal state in the mammalian hypothalamus.

Winning the Lottery With Neural Connectivity Constraints: Faster Learning Across Cognitive Tasks With Spatially Constrained Sparse RNNs.

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 etal., 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 etal., 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.

Flexible integration of continuous sensory evidence in perceptual estimation tasks

Temporal accumulation of evidence is crucial for making accurate judgments based on noisy or ambiguous sensory input. The integration process leading to categorical decisions is thought to rely on competition between neural populations, each encoding a discrete categorical choice. How recurrent neural circuits integrate evidence for continuous perceptual judgments is unknown. Here, we show that a continuous bump attractor network can integrate a circular feature, such as stimulus direction, nearly optimally. As required by optimal integration, the population activity of the network unfolds on a two-dimensional manifold, in which the position of the network's activity bump tracks the stimulus average, and, simultaneously, the bump amplitude tracks stimulus uncertainty. Moreover, the temporal weighting of sensory evidence by the network depends on the relative strength of the stimulus compared to the internally generated bump dynamics, yielding either early (primacy), uniform, or late (recency) weighting. The model can flexibly switch between these regimes by changing a single control parameter, the global excitatory drive. We show that this mechanism can quantitatively explain individual temporal weighting profiles of human observers, and we validate the model prediction that temporal weighting impacts reaction times. Our findings point to continuous attractor dynamics as a plausible neural mechanism underlying stimulus integration in perceptual estimation tasks.

Toroidal topology of population activity in grid cells

The medial entorhinal cortex is part of a neural system for mapping the position of an individual within a physical environment1. Grid cells, a key component of this system, fire in a characteristic hexagonal pattern of locations2, and are organized in modules3 that collectively form a population code for the animal’s allocentric position1. The invariance of the correlation structure of this population code across environments4,5 and behavioural states6,7, independent of specific sensory inputs, has pointed to intrinsic, recurrently connected continuous attractor networks (CANs) as a possible substrate of the grid pattern1,8–11. However, whether grid cell networks show continuous attractor dynamics, and how they interface with inputs from the environment, has remained unclear owing to the small samples of cells obtained so far. Here, using simultaneous recordings from many hundreds of grid cells and subsequent topological data analysis, we show that the joint activity of grid cells from an individual module resides on a toroidal manifold, as expected in a two-dimensional CAN. Positions on the torus correspond to positions of the moving animal in the environment. Individual cells are preferentially active at singular positions on the torus. Their positions are maintained between environments and from wakefulness to sleep, as predicted by CAN models for grid cells but not by alternative feedforward models12. This demonstration of network dynamics on a toroidal manifold provides a population-level visualization of CAN dynamics in grid cells.

The intrinsic attractor manifold and population dynamics of a canonical cognitive circuit across waking and sleep.

Neural circuits construct distributed representations of key variables-external stimuli or internal constructs of quantities relevant for survival, such as an estimate of one's location in the world-as vectors of population activity. Although population activity vectors may have thousands of entries (dimensions), we consider that they trace out a low-dimensional manifold whose dimension and topology match the represented variable. This manifold perspective enables blind discovery and decoding of the represented variable using only neural population activity (without knowledge of the input, output, behavior or topography). We characterize and directly visualize manifold structure in the mammalian head direction circuit, revealing that the states form a topologically nontrivial one-dimensional ring. The ring exhibits isometry and is invariant across waking and rapid eye movement sleep. This result directly demonstrates that there are continuous attractor dynamics and enables powerful inference about mechanism. Finally, external rather than internal noise limits memory fidelity, and the manifold approach reveals new dynamical trajectories during sleep.

Fast Object Tracking on a Many-Core Neural Network Chip.

Fast object tracking on embedded devices is of great importance for applications such as autonomous driving, unmanned aerial vehicle, and intelligent monitoring. Whereas, most of previous general solutions failed to reach this goal due to the facts that (i) high computational complexity and heterogeneous operation steps in the tracking models and (ii) parallelism-limited and bloated hardware platforms (e.g., CPU/GPU). Although previously proposed devices leverage neural dynamics and near-data processing for efficient tracking, their flexibility is limited due to the tight integration with vision sensor and the effectiveness on various video datasets is yet to be fully demonstrated. On the other side, recently the many-core architecture with massive parallelism and optimized memory locality is being widely applied to improve the performance for flexibly executing neural networks. This motivates us to adapt and map an object tracking model based on attractor neural networks with continuous and smooth attractor dynamics onto neural network chips for fast tracking. In order to make the model hardware friendly, we add local-connection restriction. We analyze the tracking accuracy and observe that the model achieves comparable results on typical video datasets. Then, we design a many-core neural network architecture with several computation and transformation operations to support the model. Moreover, by discretizing the continuous dynamics to the corresponding discrete counterpart, designing a slicing scheme for efficient topology mapping, and introducing a constant-restricted scaling chain rule for data quantization, we build a complete mapping framework to implement the tracking model on the many-core architecture. We fabricate a many-core neural network chip to evaluate the real execution performance. Results show that a single chip is able to accommodate the whole tracking model, and a fast tracking speed of nearly 800 FPS (frames per second) can be achieved. This work enables high-speed object tracking on embedded devices which normally have limited resources and energy.

Integrated information and dimensionality in continuous attractor dynamics.

There has been increasing interest in the integrated information theory (IIT) of consciousness, which hypothesizes that consciousness is integrated information within neuronal dynamics. However, the current formulation of IIT poses both practical and theoretical problems when empirically testing the theory by computing integrated information from neuronal signals. For example, measuring integrated information requires observing all the elements in a considered system at the same time, but this is practically very difficult. Here, we propose that some aspects of these problems are resolved by considering the topological dimensionality of shared attractor dynamics as an indicator of integrated information in continuous attractor dynamics. In this formulation, the effects of unobserved nodes on the attractor dynamics can be reconstructed using a technique called delay embedding, which allows us to identify the dimensionality of an embedded attractor from partial observations. We propose that the topological dimensionality represents a critical property of integrated information, as it is invariant to general coordinate transformations. We illustrate this new framework with simple examples and discuss how it fits with recent findings based on neural recordings from awake and anesthetized animals. This topological approach extends the existing notions of IIT to continuous dynamical systems and offers a much-needed framework for testing the theory with experimental data by substantially relaxing the conditions required for evaluating integrated information in real neural systems.

A Hybrid Oscillatory Interference/Continuous Attractor Network Model of Grid Cell Firing

Grid cells in the rodent medial entorhinal cortex exhibit remarkably regular spatial firing patterns that tessellate all environments visited by the animal. Two theoretical mechanisms that could generate this spatially periodic activity pattern have been proposed: oscillatory interference and continuous attractor dynamics. Although a variety of evidence has been cited in support of each, some aspects of the two mechanisms are complementary, suggesting that a combined model may best account for experimental data. The oscillatory interference model proposes that the grid pattern is formed from linear interference patterns or "periodic bands" in which velocity-controlled oscillators integrate self-motion to code displacement along preferred directions. However, it also allows the use of symmetric recurrent connectivity between grid cells to provide relative stability and continuous attractor dynamics. Here, we present simulations of this type of hybrid model, demonstrate that it generates intracellular membrane potential profiles that closely match those observed in vivo, addresses several criticisms aimed at pure oscillatory interference and continuous attractor models, and provides testable predictions for future empirical studies.

Specific evidence of low-dimensional continuous attractor dynamics in grid cells

We examine simultaneously recorded spikes from multiple grid cells, to elucidate mechanisms underlying their activity. We demonstrate that grid cell population activity, among cells with similar spatial periods, is confined to lie close to a 2-dimensional manifold: grid cells differ only along two dimensions of their responses and are otherwise nearly identical. The relationships between cell pairs are conserved despite extensive deformations of single-neuron responses. Results from novel environments suggest such structure is not inherited from hippocampal or external sensory inputs. Across conditions, cell-cell relationships are better conserved than the responses of single cells. Finally, the system is continually subject to perturbations that were the 2-d manifold not attractive, would drive the system to inhabit a different region of state-space than observed. Together, these findings have strong implications for theories of grid cell activity, and provide compelling support for the general hypothesis that the brain computes using low-dimensional continuous attractors.

Nonlinear enhancement of noisy speech, using continuous attractor dynamics formed in recurrent neural networks

Here, formation of continuous attractor dynamics in a nonlinear recurrent neural network is used to achieve a nonlinear speech denoising method, in order to implement robust phoneme recognition and information retrieval. Formation of attractor dynamics in recurrent neural network is first carried out by training the clean speech subspace as the continuous attractor. Then, it is used to recognize noisy speech with both stationary and nonstationary noise. In this work, the efficiency of a nonlinear feedforward network is compared to the same one with a recurrent connection in its hidden layer. The structure and training of this recurrent connection, is designed in such a way that the network learns to denoise the signal step by step, using properties of attractors it has formed, along with phone recognition. Using these connections, the recognition accuracy is improved 21% for the stationary signal and 14% for the nonstationary one with 0db SNR, in respect to a reference model which is a feedforward neural network.

A Gaussian Attractor Network for Memory and Recognition with Experience-Dependent Learning

Attractor networks are widely believed to underlie the memory systems of animals across different species. Existing models have succeeded in qualitatively modeling properties of attractor dynamics, but their computational abilities often suffer from poor representations for realistic complex patterns, spurious attractors, low storage capacity, and difficulty in identifying attractive fields of attractors. We propose a simple two-layer architecture, gaussian attractor network, which has no spurious attractors if patterns to be stored are uncorrelated and can store as many patterns as the number of neurons in the output layer. Meanwhile the attractive fields can be precisely quantified and manipulated. Equipped with experience-dependent unsupervised learning strategies, the network can exhibit both discrete and continuous attractor dynamics. A testable prediction based on numerical simulations is that there exist neurons in the brain that can discriminate two similar stimuli at first but cannot after extensive exposure to physically intermediate stimuli. Inspired by this network, we found that adding some local feedbacks to a well-known hierarchical visual recognition model, HMAX, can enable the model to reproduce some recent experimental results related to high-level visual perception.

Calibration of the head direction network: a role for symmetric angular head velocity cells

Continuous attractor networks require calibration. Computational models of the head direction (HD) system of the rat usually assume that the connections that maintain HD neuron activity are pre-wired and static. Ongoing activity in these models relies on precise continuous attractor dynamics. It is currently unknown how such connections could be so precisely wired, and how accurate calibration is maintained in the face of ongoing noise and perturbation. Our adaptive attractor model of the HD system that uses symmetric angular head velocity (AHV) cells as a training signal shows that the HD system can learn to support stable firing patterns from poorly-performing, unstable starting conditions. The proposed calibration mechanism suggests a requirement for symmetric AHV cells, the existence of which has previously been unexplained, and predicts that symmetric and asymmetric AHV cells should be distinctly different (in morphology, synaptic targets and/or methods of action on postsynaptic HD cells) due to their distinctly different functions.

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

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 ∼10–100 meters and ∼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.

Grid cells: The position code, neural network models of activity, and the problem of learning

We review progress on the modeling and theoretical fronts in the quest to unravel the computational properties of the grid cell code and to explain the mechanisms underlying grid cell dynamics. The goals of the review are to outline a coherent framework for understanding the dynamics of grid cells and their representation of space; to critically present and draw contrasts between recurrent network models of grid cells based on continuous attractor dynamics and independent-neuron models based on temporal interference; and to suggest open questions for experiment and theory.

Continuous attractors of a class of recurrent neural networks

Recurrent neural networks (RNNs) may possess continuous attractors, a property that many brain theories have implicated in learning and memory. There is good evidence for continuous stimuli, such as orientation, moving direction, and the spatial location of objects could be encoded as continuous attractors in neural networks. The dynamical behaviors of continuous attractors are interesting properties of RNNs. This paper proposes studying the continuous attractors for a class of RNNs. In this network, the inhibition among neurons is realized through a kind of subtractive mechanism. It shows that if the synaptic connections are in Gaussian shape and other parameters are appropriately selected, the network can exactly realize continuous attractor dynamics. Conditions are derived to guarantee the validity of the selected parameters. Simulations are employed for illustration.

Context-dependent retrieval of information by neural-network dynamics with continuous attractors

Memory retrieval in neural networks has traditionally been described by dynamic systems with discrete attractors. However, recent neurophysiological findings of graded persistent activity suggest that memory retrieval in the brain is more likely to be described by dynamic systems with continuous attractors. To explore what sort of information processing is achieved by continuous-attractor dynamics, keyword extraction from documents by a network of bistable neurons, which gives robust continuous attractors, is examined. Given an associative network of terms, a continuous attractor led by propagation of neuronal activation in this network appears to represent keywords that express underlying meaning of a document encoded in the initial state of the network-activation pattern. A dominant hypothesis in cognitive psychology is that long-term memory is archived in the network structure, which resembles associative networks of terms. Our results suggest that keyword extraction by the neural-network dynamics with continuous attractors might symbolically represent context-dependent retrieval of short-term memory from long-term memory in the brain.

漸次的持続活性の神経生理学的および計算論的知見が開く連想記憶の新しい地平

Associative memory in neural-neural network systems has long been described by dynamical systems with discrete attractors. Recent neurophysiological findings of graded persistent activity, however, suggest that memory retrieval in the brain is more likely to be described by dynamical systems with continuous attractors. This paper briefly reviews experimental studies of graded persistent activity and computational studies for modelling its neural mechanisms by continuous attractor dynamics. Furthermore, their implications for associative memory are discussed.