A Three-Threshold Learning Rule Approaches the Maximal Capacity of 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 the attractor neural network scenario, whose prototype is the Hopfield model. The model simplicity and the locality of the synaptic update rules come at the cost of a poor storage capacity, compared with the capacity achieved with perceptron learning algorithms. Here, by transforming the perceptron learning rule, we present an online learning rule for a recurrent neural network that achieves near-maximal storage capacity without an explicit supervisory error signal, relying only upon locally accessible information. The fully-connected network consists of excitatory binary neurons with plastic recurrent connections and non-plastic inhibitory feedback stabilizing the network dynamics; the memory patterns to be memorized are presented online as strong afferent currents, producing a bimodal distribution for the neuron synaptic inputs. Synapses corresponding to active inputs are modified as a function of the value of the local fields 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. We simulated and analyzed a network of binary neurons implementing this rule and measured its storage capacity for different sizes of the basins of attraction. The storage capacity obtained through numerical simulations is shown to be close to the value predicted by analytical calculations. We also measured the dependence of capacity on the strength of external inputs. Finally, we quantified the statistics of the resulting synaptic connectivity matrix, and found that both the fraction of zero weight synapses and the degree of symmetry of the weight matrix increase with the number of stored patterns.Author Summary: Recurrent neural networks have been shown to be able to store memory patterns as fixed point attractors of the dynamics of the network. The prototypical learning rule for storing memories in attractor neural networks is Hebbian learning, which can store up to 0.138N uncorrelated patterns in a recurrent network of N neurons. This is very far from the maximal capacity 2N, which can be achieved by supervised rules, e.g. by the perceptron learning rule. However, these rules are problematic for neurons in the neocortex or the hippocampus, since they rely on the computation of a supervisory error signal for each neuron of the network. We show here that the total synaptic input received by a neuron during the presentation of a sufficiently strong stimulus contains implicit information about the error, which can be extracted by setting three thresholds on the total input, defining depression and potentiation regions. The resulting learning rule implements basic biological constraints, and our simulations show that a network implementing it gets very close to the maximal capacity, both in the dense and sparse regimes, across all values of storage robustness. The rule predicts that when the total synaptic inputs goes beyond a threshold, no potentiation should occur.