Computing with continuous attractors: stability and online aspects

Non-image-forming photoreceptors improve visual orientation selectivity and image perception

It has long been a decades-old dogma that image perception is mediated solely by rods and cones, while intrinsically photosensitive retinal ganglion cells (ipRGCs) are responsible only for non-image-forming vision, such as circadian photoentrainment and pupillary light reflexes. Surprisingly, we discovered that ipRGC activation enhances the orientation selectivity of layer 2/3 neurons in the primary visual cortex (V1) of mice by both increasing preferred-orientation responses and narrowing tuning bandwidth. Mechanistically, we found that the tuning properties of V1 excitatory and inhibitory neurons are differentially influenced by ipRGC activation, leading to a reshaping of the excitatory/inhibitory balance that enhances visual cortical orientation selectivity. Furthermore, light activation of ipRGCs improves behavioral orientation discrimination in mice. Importantly, we found that specific activation of ipRGCs in human participants through visual spectrum manipulation significantly enhances visual orientation discriminability. Our study reveals a visual channel originating from "non-image-forming photoreceptors" that facilitates visual orientation feature perception.

Unifying network model links recency and central tendency biases in working memory

The central tendency bias, or contraction bias, is a phenomenon where the judgment of the magnitude of items held in working memory appears to be biased toward the average of past observations. It is assumed to be an optimal strategy by the brain and commonly thought of as an expression of the brain's ability to learn the statistical structure of sensory input. On the other hand, recency biases such as serial dependence are also commonly observed and are thought to reflect the content of working memory. Recent results from an auditory delayed comparison task in rats suggest that both biases may be more related than previously thought: when the posterior parietal cortex (PPC) was silenced, both short-term and contraction biases were reduced. By proposing a model of the circuit that may be involved in generating the behavior, we show that a volatile working memory content susceptible to shifting to the past sensory experience - producing short-term sensory history biases - naturally leads to contraction bias. The errors, occurring at the level of individual trials, are sampled from the full distribution of the stimuli and are not due to a gradual shift of the memory toward the sensory distribution's mean. Our results are consistent with a broad set of behavioral findings and provide predictions of performance across different stimulus distributions and timings, delay intervals, as well as neuronal dynamics in putative working memory areas. Finally, we validate our model by performing a set of human psychophysics experiments of an auditory parametric working memory task.

Short-term postsynaptic plasticity facilitates predictive tracking in continuous attractors

Introduction

The N-methyl-D-aspartate receptor (NMDAR) plays a critical role in synaptic transmission and is associated with various neurological and psychiatric disorders. Recently, a novel form of postsynaptic plasticity known as NMDAR-based short-term postsynaptic plasticity (STPP) has been identified. It has been suggested that long-lasting glutamate binding to NMDAR allows for the retention of input information in brain slices up to 500 ms, leading to response facilitation. However, the impact of STPP on the dynamics of neuronal populations remains unexplored.

Methods

In this study, we incorporated STPP into a continuous attractor neural network (CANN) model to investigate its effects on neural information encoding in populations of neurons. Unlike short-term facilitation, a form of presynaptic plasticity, the temporally enhanced synaptic efficacy resulting from STPP destabilizes the network state of the CANN by increasing its mobility.

Results

Our findings demonstrate that the inclusion of STPP in the CANN model enables the network state to predictively respond to a moving stimulus. This nontrivial dynamical effect facilitates the tracking of the anticipated stimulus, as the enhanced synaptic efficacy induced by STPP enhances the system's mobility.

Discussion

The discovered STPP-based mechanism for sensory prediction provides valuable insights into the potential development of brain-inspired computational algorithms for prediction. By elucidating the role of STPP in neural population dynamics, this study expands our understanding of the functional implications of NMDAR-related plasticity in information processing within the brain.

Conclusion

The incorporation of STPP into a CANN model highlights its influence on the mobility and predictive capabilities of neural networks. These findings contribute to our knowledge of STPP-based mechanisms and their potential applications in developing computational algorithms for sensory prediction.

Bimodular continuous attractor neural networks with static and moving stimuli

We investigated the dynamical behaviors of bimodular continuous attractor neural networks, each processing a modality of sensory input and interacting with each other. We found that when bumps coexist in both modules, the position of each bump is shifted towards the other input when the intermodular couplings are excitatory and is shifted away when inhibitory. When one intermodular coupling is excitatory while another is moderately inhibitory, temporally modulated population spikes can be generated. On further increase of the inhibitory coupling, momentary spikes will emerge. In the regime of bump coexistence, bump heights are primarily strengthened by excitatory intermodular couplings, but there is a lesser weakening effect due to a bump being displaced from the direct input. When bimodular networks serve as decoders of multisensory integration, we extend the Bayesian framework to show that excitatory and inhibitory couplings encode attractive and repulsive priors, respectively. At low disparity, the bump positions decode the posterior means in the Bayesian framework, whereas at high disparity, multiple steady states exist. In the regime of multiple steady states, the less stable state can be accessed if the input causing the more stable state arrives after a sufficiently long delay. When one input is moving, the bump in the corresponding module is pinned when the moving stimulus is weak, unpinned at intermediate stimulus strength, and tracks the input at strong stimulus strength, and the stimulus strengths for these transitions increase with the velocity of the moving stimulus. These results are important to understanding multisensory integration of static and dynamic stimuli.

The Neuroanatomical Ultrastructure and Function of a Biological Ring Attractor

Neural representations of head direction (HD) have been discovered in many species. Theoretical work has proposed that the dynamics associated with these representations are generated, maintained, and updated by recurrent network structures called ring attractors. We evaluated this theorized structure-function relationship by performing electron-microscopy-based circuit reconstruction and RNA profiling of identified cell types in the HD system of Drosophila melanogaster. We identified motifs that have been hypothesized to maintain the HD representation in darkness, update it when the animal turns, and tether it to visual cues. Functional studies provided support for the proposed roles of individual excitatory or inhibitory circuit elements in shaping activity. We also discovered recurrent connections between neuronal arbors with mixed pre- and postsynaptic specializations. Our results confirm that the Drosophila HD network contains the core components of a ring attractor while also revealing unpredicted structural features that might enhance the network's computational power.

Precision multidimensional neural population code recovered from single intracellular recordings

Neurons in sensory cortices are more naturally and deeply integrated than any current neural population recording tools (e.g. electrode arrays, fluorescence imaging). Two concepts facilitate efforts to observe population neural code with single-cell recordings. First, even the highest quality single-cell recording studies find a fraction of the stimulus information in high-dimensional population recordings. Finding any of this missing information provides proof of principle. Second, neurons and neural populations are understood as coupled nonlinear differential equations. Therefore, fitted ordinary differential equations provide a basis for single-trial single-cell stimulus decoding. We obtained intracellular recordings of fluctuating transmembrane current and potential in mouse visual cortex during stimulation with drifting gratings. We use mean deflection from baseline when comparing to prior single-cell studies because action potentials are too sparse and the deflection response to drifting grating stimuli (e.g. tuning curves) are well studied. Equation-based decoders allowed more precise single-trial stimulus discrimination than tuning-curve-base decoders. Performance varied across recorded signal types in a manner consistent with population recording studies and both classification bases evinced distinct stimulus-evoked phases of population dynamics, providing further corroboration. Naturally and deeply integrated observations of population dynamics would be invaluable. We offer proof of principle and a versatile framework.

Nonequilibrium Statistical Mechanics of Continuous Attractors

Continuous attractors have been used to understand recent neuroscience experiments where persistent activity patterns encode internal representations of external attributes like head direction or spatial location. However, the conditions under which the emergent bump of neural activity in such networks can be manipulated by space and time-dependent external sensory or motor signals are not understood. Here, we find fundamental limits on how rapidly internal representations encoded along continuous attractors can be updated by an external signal. We apply these results to place cell networks to derive a velocity-dependent nonequilibrium memory capacity in neural networks.

Discrete-Attractor-like Tracking in Continuous Attractor Neural Networks

Continuous attractor neural networks generate a set of smoothly connected attractor states. In memory systems of the brain, these attractor states may represent continuous pieces of information such as spatial locations and head directions of animals. However, during the replay of previous experiences, hippocampal neurons show a discontinuous sequence in which discrete transitions of the neural state are phase locked with the slow-gamma (∼30-50  Hz) oscillation. Here, we explore the underlying mechanisms of the discontinuous sequence generation. We find that a continuous attractor neural network has several phases depending on the interactions between external input and local inhibitory feedback. The discrete-attractor-like behavior naturally emerges in one of these phases without any discreteness assumption. We propose that the dynamics of continuous attractor neural networks is the key to generate discontinuous state changes phase locked to the brain rhythm.

Continuous Attractor Neural Networks: Candidate of a Canonical Model for Neural Information Representation

Owing to its many computationally desirable properties, the model of continuous attractor neural networks (CANNs) has been successfully applied to describe the encoding of simple continuous features in neural systems, such as orientation, moving direction, head direction, and spatial location of objects. Recent experimental and computational studies revealed that complex features of external inputs may also be encoded by low-dimensional CANNs embedded in the high-dimensional space of neural population activity. The new experimental data also confirmed the existence of the M-shaped correlation between neuronal responses, which is a correlation structure associated with the unique dynamics of CANNs. This body of evidence, which is reviewed in this report, suggests that CANNs may serve as a canonical model for neural information representation.

Learning multisensory integration and coordinate transformation via density estimation

Sensory processing in the brain includes three key operations: multisensory integration-the task of combining cues into a single estimate of a common underlying stimulus; coordinate transformations-the change of reference frame for a stimulus (e.g., retinotopic to body-centered) effected through knowledge about an intervening variable (e.g., gaze position); and the incorporation of prior information. Statistically optimal sensory processing requires that each of these operations maintains the correct posterior distribution over the stimulus. Elements of this optimality have been demonstrated in many behavioral contexts in humans and other animals, suggesting that the neural computations are indeed optimal. That the relationships between sensory modalities are complex and plastic further suggests that these computations are learned-but how? We provide a principled answer, by treating the acquisition of these mappings as a case of density estimation, a well-studied problem in machine learning and statistics, in which the distribution of observed data is modeled in terms of a set of fixed parameters and a set of latent variables. In our case, the observed data are unisensory-population activities, the fixed parameters are synaptic connections, and the latent variables are multisensory-population activities. In particular, we train a restricted Boltzmann machine with the biologically plausible contrastive-divergence rule to learn a range of neural computations not previously demonstrated under a single approach: optimal integration; encoding of priors; hierarchical integration of cues; learning when not to integrate; and coordinate transformation. The model makes testable predictions about the nature of multisensory representations.

Dynamics analysis of a population decoding model

Information processing in the nervous system involves the activity of large populations of neurons. It is difficult to extract information from these population codes because of the noise inherent in neuronal responses. We propose a divisive normalization model to read the population codes. The dynamics of the model are analyzed by continuous attractor theory. Under certain conditions, the model possesses continuous attractors. Moreover, the explicit expressions of the continuous attractors are provided. Simulations are employed to illustrate the theory.

Information processing and integration with intracellular dynamics near critical point

Recent experimental observations suggest that cells can show relatively precise and reliable responses to external signals even though substantial noise is inevitably involved in the signals. An intriguing question is the way how cells can manage to do it. One possible way to realize such response for a cell is to evolutionary develop and optimize its intracellular signaling pathways so as to extract relevant information from the noisy signal. We recently demonstrated that certain intracellular signaling reactions could actually conduct statistically optimal information processing. In this paper, we clarify that such optimal reaction operates near bifurcation point. This result suggests that critical-like phenomena in the single-cell level may be linked to efficient information processing inside a cell. In addition, improving the performance of response in the single-cell level is not the only way for cells to realize reliable response. Another possible strategy is to integrate information of individual cells by cell-to-cell interaction such as quorum sensing. Since cell-to-cell interaction is a common phenomenon, it is equally important to investigate how cells can integrate their information by cell-to-cell interaction to realize efficient information processing in the population level. In this paper, we consider roles and benefits of cell-to-cell interaction by considering integrations of obtained information of individuals with the other cells from the viewpoint of information processing. We also demonstrate that, by introducing cell movement, spatial organizations can spontaneously emerge as a result of efficient responses of the population to external signals.

Neural information processing with feedback modulations

Descending feedback connections, together with ascending feedforward ones, are the indispensable parts of the sensory pathways in the central nervous system. This study investigates the potential roles of feedback interactions in neural information processing. We consider a two-layer continuous attractor neural network (CANN), in which neurons in the first layer receive feedback inputs from those in the second one. By utilizing the intrinsic property of a CANN, we use a projection method to reduce the dimensionality of the network dynamics significantly. The simplified dynamics allows us to elucidate the effects of feedback modulation analytically. We find that positive feedback enhances the stability of the network state, leading to an improved population decoding performance, whereas negative feedback increases the mobility of the network state, inducing spontaneously moving bumps. For strong, negative feedback interaction, the network response to a moving stimulus can lead the actual stimulus position, achieving an anticipative behavior. The biological implications of these findings are discussed. The simulation results agree well with our theoretical analysis.

Dynamical synapses enhance neural information processing: gracefulness, accuracy, and mobility

Experimental data have revealed that neuronal connection efficacy exhibits two forms of short-term plasticity: short-term depression (STD) and short-term facilitation (STF). They have time constants residing between fast neural signaling and rapid learning and may serve as substrates for neural systems manipulating temporal information on relevant timescales. This study investigates the impact of STD and STF on the dynamics of continuous attractor neural networks and their potential roles in neural information processing. We find that STD endows the network with slow-decaying plateau behaviors: the network that is initially being stimulated to an active state decays to a silent state very slowly on the timescale of STD rather than on that of neuralsignaling. This provides a mechanism for neural systems to hold sensory memory easily and shut off persistent activities gracefully. With STF, we find that the network can hold a memory trace of external inputs in the facilitated neuronal interactions, which provides a way to stabilize the network response to noisy inputs, leading to improved accuracy in population decoding. Furthermore, we find that STD increases the mobility of the network states. The increased mobility enhances the tracking performance of the network in response to time-varying stimuli, leading to anticipative neural responses. In general, we find that STD and STP tend to have opposite effects on network dynamics and complementary computational advantages, suggesting that the brain may employ a strategy of weighting them differentially depending on the computational purpose.

How each movement changes the next: an experimental and theoretical study of fast adaptive priors in reaching

Most voluntary actions rely on neural circuits that map sensory cues onto appropriate motor responses. One might expect that for everyday movements, like reaching, this mapping would remain stable over time, at least in the absence of error feedback. Here we describe a simple and novel psychophysical phenomenon in which recent experience shapes the statistical properties of reaching, independent of any movement errors. Specifically, when recent movements are made to targets near a particular location subsequent movements to that location become less variable, but at the cost of increased bias for reaches to other targets. This process exhibits the variance-bias tradeoff that is a hallmark of Bayesian estimation. We provide evidence that this process reflects a fast, trial-by-trial learning of the prior distribution of targets. We also show that these results may reflect an emergent property of associative learning in neural circuits. We demonstrate that adding Hebbian (associative) learning to a model network for reach planning leads to a continuous modification of network connections that biases network dynamics toward activity patterns associated with recent inputs. This learning process quantitatively captures the key results of our experimental data in human subjects, including the effect that recent experience has on the variance-bias tradeoff. This network also provides a good approximation of a normative Bayesian estimator. These observations illustrate how associative learning can incorporate recent experience into ongoing computations in a statistically principled way.

Change-Based Inference in Attractor Nets: Linear Analysis

One standard interpretation of networks of cortical neurons is that they form dynamical attractors. Computations such as stimulus estimation are performed by mapping inputs to points on the networks' attractive manifolds. These points represent population codes for the stimulus values. However, this standard interpretation is hard to reconcile with the observation that the firing rates of such neurons constantly change following presentation of stimuli. We have recently suggested an alternative interpretation according to which computations are realized by systematic changes in the states of such networks over time. This way of performing computations is fast, accurate, readily learnable, and robust to various forms of noise. Here we analyze the computation of stimulus discrimination in this change-based setting, relating it directly to the computation of stimulus estimation in the conventional attractor-based view. We use a common linear approximation to compare the two methods and show that perfect performance at estimation implies chance performance at discrimination.

Continuous attractors of Lotka-Volterra recurrent neural networks with infinite neurons

Continuous attractors of Lotka-Volterra recurrent neural networks (LV RNNs) with infinite neurons are studied in this brief. A continuous attractor is a collection of connected equilibria, and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks. The existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network. A continuous attractor can be stable or unstable. It is shown in this brief that a LV RNN can possess multiple continuous attractors if the synaptic connections and the external inputs are Gussian-like in shape. Moreover, both stable and unstable continuous attractors can coexist in a network. Explicit expressions of the continuous attractors are calculated. Simulations are employed to illustrate the theory.

A Moving Bump in a Continuous Manifold: A Comprehensive Study of the Tracking Dynamics of Continuous Attractor Neural Networks

Understanding how the dynamics of a neural network is shaped by the network structure and, consequently, how the network structure facilitates the functions implemented by the neural system is at the core of using mathematical models to elucidate brain functions. This study investigates the tracking dynamics of continuous attractor neural networks (CANNs). Due to the translational invariance of neuronal recurrent interactions, CANNs can hold a continuous family of stationary states. They form a continuous manifold in which the neural system is neutrally stable. We systematically explore how this property facilitates the tracking performance of a CANN, which is believed to have clear correspondence with brain functions. By using the wave functions of the quantum harmonic oscillator as the basis, we demonstrate how the dynamics of a CANN is decomposed into different motion modes, corresponding to distortions in the amplitude, position, width, or skewness of the network state. We then develop a perturbation approach that utilizes the dominating movement of the network's stationary states in the state space. This method allows us to approximate the network dynamics up to an arbitrary accuracy depending on the order of perturbation used. We quantify the distortions of a gaussian bump during tracking and study their effects on tracking performance. Results are obtained on the maximum speed for a moving stimulus to be trackable and the reaction time for the network to catch up with an abrupt change in the stimulus.

Single-cell and population coding of expected reward probability in the orbitofrontal cortex of the rat

The orbitofrontal cortex (OFC) has been implicated in decision-making under uncertainty, but it is unknown how information about the probability or uncertainty of future reward is coded by single orbitofrontal neurons and ensembles. We recorded neuronal ensembles in rat OFC during an olfactory discrimination task in which different odor stimuli predicted different reward probabilities. Single-unit firing patterns correlated to the expected reward probability primarily within an immobile waiting period before reward delivery but also when the rat executed movements toward the reward site. During these pre-reward periods, a subset of OFC neurons was sensitive to differences in probability but only very rarely discriminated on the basis of reward uncertainty. In the reward period, neurons responded during presentation or omission of reward or during both types of outcome. At the population level, neurons were characterized by a wide divergence in firing-rate variability attributable to expected probability. A population analysis using template matching as reconstruction method indicated that OFC generates a distributed representation of reward probability with a weak dependence on neuronal group size. The analysis furthermore confirmed that predictive information coded by OFC populations was quantitatively related to reward probability, but not to uncertainty.

Representations of continuous attractors of recurrent neural networks

A continuous attractor of a recurrent neural network (RNN) is a set of connected stable equilibrium points. Continuous attractors have been used to describe the encoding of continuous stimuli in neural networks. Dynamic behaviors of continuous attractors of RNNs exhibit interesting properties. This brief desires to derive explicit representations of continuous attractors of RNNs. Representations of continuous attractors of linear RNNs as well as linear-threshold (LT) RNNs are obtained under some conditions. These representations could be looked at as solutions of continuous attractors of the networks. Such results provide clear and complete descriptions to the continuous attractors.

Population coding of reward magnitude in the orbitofrontal cortex of the rat

Although single-cell coding of reward-related information in the orbitofrontal cortex (OFC) has been characterized to some extent, much less is known about the coding properties of orbitofrontal ensembles. We examined population coding of reward magnitude by performing ensemble recordings in rat OFC while animals learned an olfactory discrimination task in which various reinforcers were associated with predictive odor stimuli. Ensemble activity was found to represent information about reward magnitude during several trial phases, namely when animals moved to the reward site, anticipated reward during an immobile period, and received it. During the anticipation phase, Bayesian and template-matching reconstruction algorithms decoded reward size correctly from the population activity significantly above chance level (highest value of 43 and 48%, respectively; chance level, 33.3%), whereas decoding performance for the reward delivery phase was 76 and 79%, respectively. In the anticipation phase, the decoding score was only weakly dependent on the size of the neuronal group participating in reconstruction, consistent with a redundant, distributed representation of reward information. In contrast, decoding was specific for temporal segments within the structure of a trial. Decoding performance steeply increased across the first few trials for every rewarded odor, an effect that could not be explained by a nonspecific drift in response strength across trials. Finally, when population responses to a negative reinforcer (quinine) were compared with sucrose reinforcement, coding in the delivery phase appeared to be related to reward quality, and thus was not based on ingested liquid volume.

Dynamics and computation of continuous attractors

Continuous attractor is a promising model for describing the encoding of continuous stimuli in neural systems. In a continuous attractor, the stationary states of the neural system form a continuous parameter space, on which the system is neutrally stable. This property enables the neutral system to track time-varying stimuli smoothly, but it also degrades the accuracy of information retrieval, since these stationary states are easily disturbed by external noise. In this work, based on a simple model, we systematically investigate the dynamics and the computational properties of continuous attractors. In order to analyze the dynamics of a large-size network, which is otherwise extremely complicated, we develop a strategy to reduce its dimensionality by utilizing the fact that a continuous attractor can eliminate the noise components perpendicular to the attractor space very quickly. We therefore project the network dynamics onto the tangent of the attractor space and simplify it successfully as a one-dimensional Ornstein-Uhlenbeck process. Based on this simplified model, we investigate (1) the decoding error of a continuous attractor under the driving of external noisy inputs, (2) the tracking speed of a continuous attractor when external stimulus experiences abrupt changes, (3) the neural correlation structure associated with the specific dynamics of a continuous attractor, and (4) the consequence of asymmetric neural correlation on statistical population decoding. The potential implications of these results on our understanding of neural information processing are also discussed.

Change-based inference for invariant discrimination

Under a conventional view of information processing in recurrently connected populations of neurons, computations consist in mapping inputs onto terminal attractor states of the dynamical interactions. However, there is evidence that substantial information representation and processing can occur over the course of the initial evolution of the dynamical states of such populations, a possibility that has attractive computational properties. Here, we suggest a model that explores one such property, namely, the invariance to an irrelevant feature dimension that arises from monitoring not the state of the population, but rather (a statistic of) the change in this state over time. We illustrate our proposal in the context of the bisection task, a paradigmatic example of perceptual learning for which an attractor-state recurrent model has previously been suggested. We show a change-based inference scheme that achieves near optimal performance in the task (with invariance to translation), is robust to high levels of dynamical noise and variations of the synaptic weight matrix, and indeed admits a computationally straightforward learning rule.