Information capacity of the Hopfield model

The information capacity of general forms of memory is formalized. The number of bits of information that can be stored in the Hopfield model of associative memory is estimated. It is found that the asymptotic information capacity of a Hopfield network of N neurons is of the order N^3b. The number of arbitrary state vectors that can be made stable in a Hopfield network of N neurons is proved to be bounded above by N

Capacity for patterns and sequences in Kanerva's SDM as compared to other associative memory models

The information capacity of Kanerva's Sparse Distributed Memory (SDM) and Hopfield-type neural networks is investigated. Under the approximations used, it is shown that the total information stored in these systems is proportional to the number connections in the network. The proportionality constant is the same for the SDM and Hopfield-type models independent of the particular model, or the order of the model. The approximations are checked numerically. This same analysis can be used to show that the SDM can store sequences of spatiotemporal patterns, and the addition of time-delayed connections allows the retrieval of context dependent temporal patterns. A minor modification of the SDM can be used to store correlated patterns

Input-driven unsupervised learning in recurrent neural networks

Understanding the theoretical foundations of how memories are encoded and retrieved in neural populations is a central challenge in neuroscience. A popular theoretical scenario for modeling memory function is an attractor neural network with Hebbian learning (e.g. the Hopfield model). The model simplicity and the locality of the synaptic update rules come at the cost of a limited storage capacity, compared with the capacity achieved with supervised learning algorithms, whose biological plausibility is questionable. Here, we present an on-line learning rule for a recurrent neural network that achieves near-optimal performance without an explicit supervisory error signal and using only locally accessible information, and which is therefore biologically plausible. The fully connected network consists of excitatory units with plastic recurrent connections and non-plastic inhibitory feedback stabilizing the network dynamics; the patterns to be memorized are presented on-line as strong afferent currents, producing a bimodal distribution for the neuron synaptic inputs ('local fields'). Synapses corresponding to active inputs are modified as a function of the position of the local field with respect to three thresholds. Above the highest threshold, and below the lowest threshold, no plasticity occurs. In between these two thresholds, potentiation/depression occurs when the local field is above/below an intermediate threshold. An additional parameter of the model allows to trade storage capacity for robustness, i.e. increased size of the basins of attraction. We simulated a network of 1001 excitatory neurons implementing this rule and measured its storage capacity for different sizes of the basins of attraction: our results show that, for any given basin size, our network more than doubles the storage capacity, compared with a standard Hopfield network. Our learning rule is consistent with available experimental data documenting how plasticity depends on firing rate. It predicts that at high enough firing rates, no potentiation should occu

Dreaming neural networks: forgetting spurious memories and reinforcing pure ones

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is $\alpha \sim 0.14$, far from the theoretical bound for symmetric networks, i.e. $\alpha =1$. Inspired by sleeping and dreaming mechanisms in mammal brains, we propose an extension of this model displaying the standard on-line (awake) learning mechanism (that allows the storage of external information in terms of patterns) and an off-line (sleep) unlearning$\&$consolidating mechanism (that allows spurious-pattern removal and pure-pattern reinforcement): this obtained daily prescription is able to saturate the theoretical bound $\alpha=1$, remaining also extremely robust against thermal noise. Both neural and synaptic features are analyzed both analytically and numerically. In particular, beyond obtaining a phase diagram for neural dynamics, we focus on synaptic plasticity and we give explicit prescriptions on the temporal evolution of the synaptic matrix. We analytically prove that our algorithm makes the Hebbian kernel converge with high probability to the projection matrix built over the pure stored patterns. Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in order to ensure such a convergence. Finally, we run extensive numerical simulations (mainly Monte Carlo sampling) to check the approximations underlying the analytical investigations (e.g., we developed the whole theory at the so called replica-symmetric level, as standard in the Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size effects, finding overall full agreement with the theory.Comment: 31 pages, 12 figure

Neural Distributed Autoassociative Memories: A Survey

Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors.Comment: 31 page

Correlating matched-filter model for analysis and optimisation of neural networks

A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described