Self-organizing continuous attractor networks and path integration: two-dimensional models of place cells

Single-neuron recording studies have demonstrated the existence of neurons in the hippocampus which appear to encode information about the place where a rat is located, and about the place at which a macaque is looking. We describe 'continuous attractor' neural network models of place cells with Gaussian spatial fields in which the recurrent collateral synaptic connections between the neurons reflect the distance between two places. The networks maintain a localized packet of neuronal activity that represents the place where the animal is located. We show for two related models how the representation of the two-dimensional space in the continuous attractor network of place cells could self-organize by modifying the synaptic connections between the neurons, and also how the place being represented can be updated by idiothetic (self-motion) signals in a neural implementation of path integration.     

Self-organizing path integration using a linked continuous attractor and competitive network: path integration of head direction

A key issue is how networks in the brain learn to perform path integration, that is update a represented position using a velocity signal. Using head direction cells as an example, we show that a competitive network could self-organize to learn to respond to combinations of head direction and angular head rotation velocity. These combination cells can then be used to drive a continuous attractor network to the next head direction based on the incoming rotation signal. An associative synaptic modification rule with a short term memory trace enables preceding combination cell activity during training to be associated with the next position in the continuous attractor network. The network accounts for the presence of neurons found in the brain that respond to combinations of head direction and angular head rotation velocity. Analogous networks in the hippocampal system could self-organize to perform path integration of place and spatial view representations.     

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.     

Effective neuronal learning with ineffective Hebbian learning rules

In this article we revisit the classical neuroscience paradigm of Hebbian learning. We find that it is difficult to achieve effective associative memory storage by Hebbian synaptic learning, since it requires network-level information at the synaptic level or sparse coding level. Effective learning can yet be achieved even with nonsparse patterns by a neuronal process that maintains a zero sum of the incoming synaptic efficacies. This weight correction improves the memory capacity of associative networks from an essentially bounded one to a memory capacity that scales linearly with network size. It also enables the effective storage of patterns with multiple levels of activity within a single network. Such neuronal weight correction can be successfully carried out by activity-dependent homeostasis of the neuron's synaptic efficacies, which was recently observed in cortical tissue. Thus, our findings suggest that associative learning by Hebbian synaptic learning should be accompanied by continuous remodeling of neuronally driven regulatory processes in the brain.     

Spike-timing-dependent plasticity in small-world networks

Biologically plausible excitatory neural networks develop a persistent synchronized pattern of activity depending on spontaneous activity and synaptic refractoriness (short term depression). By fixed synaptic weights synchronous bursts of oscillatory activity are stable and involve the whole network. In our modeling study we investigate the effect of a dynamic Hebbian-like learning mechanism, spike-timing-dependent plasticity (STDP), on the changes of synaptic weights depending on synchronous activity and network connection strategies (small-world topology). We show that STDP modifies the weights of synaptic connections in such a way that synchronization of neuronal activity is considerably weakened. Networks with a higher proportion of long connections can sustain a higher level of synchronization in spite of STDP influence. The resulting distribution of the synaptic weights in single neurons depends both on the global statistics of firing dynamics and on the number of incoming and outgoing connections.     

Design of continuous attractor networks with monotonic tuning using a symmetry principle

Neurons that sustain elevated firing in the absence of stimuli have been found in many neural systems. In graded persistent activity, neurons can sustain firing at many levels, suggesting a widely found type of network dynamics in which networks can relax to any one of a continuum of stationary states. The reproduction of these findings in model networks of nonlinear neurons has turned out to be nontrivial. A particularly insightful model has been the "bump attractor," in which a continuous attractor emerges through an underlying symmetry in the network connectivity matrix. This model, however, cannot account for data in which the persistent firing of neurons is a monotonic -- rather than a bell-shaped -- function of a stored variable. Here, we show that the symmetry used in the bump attractor network can be employed to create a whole family of continuous attractor networks, including those with monotonic tuning. Our design is based on tuning the external inputs to networks that have a connectivity matrix with Toeplitz symmetry. In particular, we provide a complete analytical solution of a line attractor network with monotonic tuning and show that for many other networks, the numerical tuning of synaptic weights reduces to the computation of a single parameter.     

Network capacity analysis for latent attractor computation

Attractor networks have been one of the most successful paradigms in neural computation, and have been used as models of computation in the nervous system. Recently, we proposed a paradigm called 'latent attractors' where attractors embedded in a recurrent network via Hebbian learning are used to channel network response to external input rather than becoming manifest themselves. This allows the network to generate context-sensitive internal codes in complex situations. Latent attractors are particularly helpful in explaining computations within the hippocampus--a brain region of fundamental significance for memory and spatial learning. Latent attractor networks are a special case of associative memory networks. The model studied here consists of a two-layer recurrent network with attractors stored in the recurrent connections using a clipped Hebbian learning rule. The firing in both layers is competitive--K winners take all firing. The number of neurons allowed to fire, K, is smaller than the size of the active set of the stored attractors. The performance of latent attractor networks depends on the number of such attractors that a network can sustain. In this paper, we use signal-to-noise methods developed for standard associative memory networks to do a theoretical and computational analysis of the capacity and dynamics of latent attractor networks. This is an important first step in making latent attractors a viable tool in the repertoire of neural computation. The method developed here leads to numerical estimates of capacity limits and dynamics of latent attractor networks. The technique represents a general approach to analyse standard associative memory networks with competitive firing. The theoretical analysis is based on estimates of the dendritic sum distributions using Gaussian approximation. Because of the competitive firing property, the capacity results are estimated only numerically by iteratively computing the probability of erroneous firings. The analysis contains two cases: the simple case analysis which accounts for the correlations between weights due to shared patterns and the detailed case analysis which includes also the temporal correlations between the network's present and previous state. The latter case predicts better the dynamics of the network state for non-zero initial spurious firing. The theoretical analysis also shows the influence of the main parameters of the model on the storage capacity.     

Stable Neural Attractors Formation: Learning Rules and Network Dynamics

Different models of attractor networks have been proposed to form cell assemblies. Among them, networks with a fixed synaptic matrix can be distinguished from those including learning dynamics, since the latter adapt the attractor landscape of the lateral connections according to the statistics of the presented stimuli, yielding a more complex behavior. We propose a new learning rule that builds internal representations of input timuli as attractors of neurons in a recurrent network. The dynamics of activation and synaptic adaptation are analyzed in experiments where representations for different input patterns are formed, focusing on the properties of the model as a memory system. The experimental results are exposed along with a survey of different Hebbian rules proposed in the literature for attractors formation. These rules are compared with the help of a new tool, the learning map, where LTP and LTD, as well as homo- and heterosynaptic competition, can be graphically interpreted.     

Analysis of spike statistics in neuronal systems with continuous attractors or multiple, discrete attractor States

Attractor networks are likely to underlie working memory and integrator circuits in the brain. It is unknown whether continuous quantities are stored in an analog manner or discretized and stored in a set of discrete attractors. In order to investigate the important issue of how to differentiate the two systems, here we compare the neuronal spiking activity that arises from a continuous (line) attractor with that from a series of discrete attractors. Stochastic fluctuations cause the position of the system along its continuous attractor to vary as a random walk, whereas in a discrete attractor, noise causes spontaneous transitions to occur between discrete states at random intervals. We calculate the statistics of spike trains of neurons firing as a Poisson process with rates that vary according to the underlying attractor network. Since individual neurons fire spikes probabilistically and since the state of the network as a whole drifts randomly, the spike trains of individual neurons follow a doubly stochastic (Poisson) point process. We compare the series of spike trains from the two systems using the autocorrelation function, Fano factor, and interspike interval (ISI) distribution. Although the variation in rate can be dramatically different, especially for short time intervals, surprisingly both the autocorrelation functions and Fano factors are identical, given appropriate scaling of the noise terms. Since the range of firing rates is limited in neurons, we also investigate systems for which the variation in rate is bounded by either rigid limits or because of leak to a single attractor state, such as the Ornstein-Uhlenbeck process. In these cases, the time dependence of the variance in rate can be different between discrete and continuous systems, so that in principle, these processes can be distinguished using second-order spike statistics.     

Attractor networks for shape recognition

We describe a system of thousands of binary perceptrons with coarse-oriented edges as input that is able to recognize shapes, even in a context with hundreds of classes. The perceptrons have randomized feedforward connections from the input layer and form a recurrent network among themselves. Each class is represented by a prelearned attractor (serving as an associative hook) in the recurrent net corresponding to a randomly selected subpopulation of the perceptrons. In training, first the attractor of the correct class is activated among the perceptrons; then the visual stimulus is presented at the input layer. The feedforward connections are modified using field-dependent Hebbian learning with positive synapses, which we show to be stable with respect to large variations in feature statistics and coding levels and allows the use of the same threshold on all perceptrons. Recognition is based on only the visual stimuli. These activate the recurrent network, which is then driven by the dynamics to a sustained attractor state, concentrated in the correct class subset and providing a form of working memory. We believe this architecture is more transparent than standard feedforward two-layer networks and has stronger biological analogies.     

When response variability increases neural network robustness to synaptic noise

Cortical sensory neurons are known to be highly variable, in the sense that responses evoked by identical stimuli often change dramatically from trial to trial. The origin of this variability is uncertain, but it is usually interpreted as detrimental noise that reduces the computational accuracy of neural circuits. Here we investigate the possibility that such response variability might in fact be beneficial, because it may partially compensate for a decrease in accuracy due to stochastic changes in the synaptic strengths of a network. We study the interplay between two kinds of noise, response (or neuronal) noise and synaptic noise, by analyzing their joint influence on the accuracy of neural networks trained to perform various tasks. We find an interesting, generic interaction: when fluctuations in the synaptic connections are proportional to their strengths (multiplicative noise), a certain amount of response noise in the input neurons can significantly improve network performance, compared to the same network without response noise. Performance is enhanced because response noise and multiplicative synaptic noise are in some ways equivalent. So if the algorithm used to find the optimal synaptic weights can take into account the variability of the model neurons, it can also take into account the variability of the synapses. Thus, the connection patterns generated with response noise are typically more resistant to synaptic degradation than those obtained without response noise. As a consequence of this interplay, if multiplicative synaptic noise is present, it is better to have response noise in the network than not to have it. These results are demonstrated analytically for the most basic network consisting of two input neurons and one output neuron performing a simple classification task, but computer simulations show that the phenomenon persists in a wide range of architectures, including recurrent (attractor) networks and sensorimotor networks that perform coordinate transformations. The results suggest that response variability could play an important dynamic role in networks that continuously learn.     

A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks

We present a mathematical analysis of the effects of Hebbian learning in random recurrent neural networks, with a generic Hebbian learning rule, including passive forgetting and different timescales, for neuronal activity and learning dynamics. Previous numerical work has reported that Hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations. Here, we interpret these results mathematically and show that these effects, involving a complex coupling between neuronal dynamics and synaptic graph structure, can be analyzed using Jacobian matrices, which introduce both a structural and a dynamical point of view on neural network evolution. Furthermore, we show that sensitivity to a learned pattern is maximal when the largest Lyapunov exponent is close to 0. We discuss how neural networks may take advantage of this regime of high functional interest.     

Correlations and population dynamics in cortical networks

The function of cortical networks depends on the collective interplay between neurons and neuronal populations, which is reflected in the correlation of signals that can be recorded at different levels. To correctly interpret these observations it is important to understand the origin of neuronal correlations. Here we study how cells in large recurrent networks of excitatory and inhibitory neurons interact and how the associated correlations affect stationary states of idle network activity. We demonstrate that the structure of the connectivity matrix of such networks induces considerable correlations between synaptic currents as well as between subthreshold membrane potentials, provided Dale's principle is respected. If, in contrast, synaptic weights are randomly distributed, input correlations can vanish, even for densely connected networks. Although correlations are strongly attenuated when proceeding from membrane potentials to action potentials (spikes), the resulting weak correlations in the spike output can cause substantial fluctuations in the population activity, even in highly diluted networks. We show that simple mean-field models that take the structure of the coupling matrix into account can adequately describe the power spectra of the population activity. The consequences of Dale's principle on correlations and rate fluctuations are discussed in the light of recent experimental findings.     

Weakly pulse-coupled oscillators, FM interactions, synchronization,and oscillatory associative memory

We study pulse-coupled neural networks that satisfy only two assumptions: each isolated neuron fires periodically, and the neurons are weakly connected. Each such network can be transformed by a piece-wise continuous change of variables into a phase model, whose synchronization behavior and oscillatory associative properties are easier to analyze and understand. Using the phase model, we can predict whether a given pulse-coupled network has oscillatory associative memory, or what minimal adjustments should be made so that it can acquire memory. In the search for such minimal adjustments we obtain a large class of simple pulse-coupled neural networks that ran memorize and reproduce synchronized temporal patterns the same way a Hopfield network does with static patterns. The learning occurs via modification of synaptic weights and/or synaptic transmission delays.     

Pattern storage and processing in attractor networks with short-time synaptic dynamics

Neurophysiological experiments show that the strength of synaptic connections can undergo substantial changes on a short time scale. These changes depend on the history of the presynaptic input. Using mean-field techniques, we study how short-time dynamics of synaptic connections influence the performance of attractor neural networks in terms of their memory capacity and capability to process external signals. For binary discrete-time as well as for firing rate continuous-time neural networks, the fixed points of the network dynamics are shown to be unaffected by synaptic dynamics. However, the stability of patterns changes considerably. Synaptic depression turns out to reduce the storage capacity. On the other hand, synaptic depression is shown to be advantageous for processing of pattern sequences. The analytical results on stability, size of the basins of attraction and on the switching between patterns are complemented by numerical simulations.     

Synaptic scaling stabilizes persistent activity driven by asynchronous neurotransmitter release

Small networks of cultured hippocampal neurons respond to transient stimulation with rhythmic network activity (reverberation) that persists for several seconds, constituting an in vitro model of synchrony, working memory, and seizure. This mode of activity has been shown theoretically and experimentally to depend on asynchronous neurotransmitter release (an essential feature of the developing hippocampus) and is supported by a variety of developing neuronal networks despite variability in the size of populations (10-200 neurons) and in patterns of synaptic connectivity. It has previously been reported in computational models that "small-world" connection topology is ideal for the propagation of similar modes of network activity, although this has been shown only for neurons utilizing synchronous (phasic) synaptic transmission. We investigated how topological constraints on synaptic connectivity could shape the stability of reverberations in small networks that also use asynchronous synaptic transmission. We found that reverberation duration in such networks was resistant to changes in topology and scaled poorly with network size. However, normalization of synaptic drive, by reducing the variance of synaptic input across neurons, stabilized reverberation in such networks. Our results thus suggest that the stability of both normal and pathological states in developing networks might be shaped by variance-normalizing constraints on synaptic drive. We offer an experimental prediction for the consequences of such regulation on the behavior of small networks.     

Competition between synaptic depression and facilitation in attractor neural networks

We study the effect of competition between short-term synaptic depression and facilitation on the dynamic properties of attractor neural networks, using Monte Carlo simulation and a mean-field analysis. Depending on the balance of depression, facilitation, and the underlying noise, the network displays different behaviors, including associative memory and switching of activity between different attractors. We conclude that synaptic facilitation enhances the attractor instability in a way that (1) intensifies the system adaptability to external stimuli, which is in agreement with experiments, and (2) favors the retrieval of information with less error during short time intervals.     

Speed of feedforward and recurrent processing in multilayer networks of integrate-and-fire neurons.

The speed of processing in the visual cortical areas can be fast, with for example the latency of neuronal responses increasing by only approximately 10 ms per area in the ventral visual system sequence V1 to V2 to V4 to inferior temporal visual cortex. This has led to the suggestion that rapid visual processing can only be based on the feedforward connections between cortical areas. To test this idea, we investigated the dynamics of information retrieval in multiple layer networks using a four-stage feedforward network modelled with continuous dynamics with integrate-and-fire neurons, and associative synaptic connections between stages with a synaptic time constant of 10 ms. Through the implementation of continuous dynamics, we found latency differences in information retrieval of only 5 ms per layer when local excitation was absent and processing was purely feedforward. However, information latency differences increased significantly when non-associative local excitation was included. We also found that local recurrent excitation through associatively modified synapses can contribute significantly to processing in as little as 15 ms per layer, including the feedforward and local feedback processing. Moreover, and in contrast to purely feed-forward processing, the contribution of local recurrent feedback was useful and approximately this rapid even when retrieval was made difficult by noise. These findings suggest that cortical information processing can benefit from recurrent circuits when the allowed processing time per cortical area is at least 15 ms long.     

An evolutionary approach to associative memory in recurrent neural networks

In this paper, we investigate the associative memory in recurrent neural networks, based on the model of evolving neural networks proposed by Nolfi, Miglino and Parisi.Experimentally developed network has highly asymmetric synaptic weights and dilute connections, quite different from those of the Hopfield model.Some results on the effect of learning efficiency on the evolution are also presented.     

Stationary bumps in networks of spiking neurons.

We examine the existence and stability of spatially localized "bumps" of neuronal activity in a network of spiking neurons. Bumps have been proposed in mechanisms of visual orientation tuning, the rat head direction system, and working memory. We show that a bump solution can exist in a spiking network provided the neurons fire asynchronously within the bump. We consider a parameter regime where the bump solution is bistable with an all-off state and can be initiated with a transient excitatory stimulus. We show that the activity profile matches that of a corresponding population rate model. The bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all-off state. This can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network. A bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance.