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A Three-Threshold Learning Rule Approaches the Maximal Capacity of Recurrent Neural Networks.

Alemi A, Baldassi C, Brunel N, Zecchina R - PLoS Comput. Biol. (2015)

Bottom Line: Understanding the theoretical foundations of how memories are encoded and retrieved in neural populations is a central challenge in neuroscience.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.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.

View Article: PubMed Central - PubMed

Affiliation: Human Genetics Foundation (HuGeF), Turin, Italy; DISAT, Politecnico di Torino, Turin, Italy.

ABSTRACT
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.

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Related in: MedlinePlus

Capacity as a function of the robustness parameter ϵ at sparseness f = 0.2.The theoretical calculations is compared with the simulations for f = 0.2. Note that the capacity in the sparse regime is higher than in the dense regime.
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pcbi.1004439.g006: Capacity as a function of the robustness parameter ϵ at sparseness f = 0.2.The theoretical calculations is compared with the simulations for f = 0.2. Note that the capacity in the sparse regime is higher than in the dense regime.

Mentions: The 3TLR can also be adapted to work in a sparser regime, at a coding level lower than 0.5. However, the average activity level of the network is determined by H0, and their relationship also involves the variance of the distribution of the synaptic weights when f ≠ 0.5 (see Materials and Methods). During the learning process, the variance of the weights changes, which implies that the parameter H0 must adapt correspondingly. In our simulations, this adaptation was performed after each complete presentation of the whole pattern set. In practice, this additional self-stabilizing mechanism could still be performed in an unsupervised fashion along with (or in alternation with) the learning process. Using this adjustment, we simulated the network at f = 0.2 and compared the results with the theoretical calculations. As shown in Fig 6, we can achieve at least 70% of the critical capacity across different values of the robustness parameter ϵ.


A Three-Threshold Learning Rule Approaches the Maximal Capacity of Recurrent Neural Networks.

Alemi A, Baldassi C, Brunel N, Zecchina R - PLoS Comput. Biol. (2015)

Capacity as a function of the robustness parameter ϵ at sparseness f = 0.2.The theoretical calculations is compared with the simulations for f = 0.2. Note that the capacity in the sparse regime is higher than in the dense regime.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4546407&req=5

pcbi.1004439.g006: Capacity as a function of the robustness parameter ϵ at sparseness f = 0.2.The theoretical calculations is compared with the simulations for f = 0.2. Note that the capacity in the sparse regime is higher than in the dense regime.
Mentions: The 3TLR can also be adapted to work in a sparser regime, at a coding level lower than 0.5. However, the average activity level of the network is determined by H0, and their relationship also involves the variance of the distribution of the synaptic weights when f ≠ 0.5 (see Materials and Methods). During the learning process, the variance of the weights changes, which implies that the parameter H0 must adapt correspondingly. In our simulations, this adaptation was performed after each complete presentation of the whole pattern set. In practice, this additional self-stabilizing mechanism could still be performed in an unsupervised fashion along with (or in alternation with) the learning process. Using this adjustment, we simulated the network at f = 0.2 and compared the results with the theoretical calculations. As shown in Fig 6, we can achieve at least 70% of the critical capacity across different values of the robustness parameter ϵ.

Bottom Line: Understanding the theoretical foundations of how memories are encoded and retrieved in neural populations is a central challenge in neuroscience.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.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.

View Article: PubMed Central - PubMed

Affiliation: Human Genetics Foundation (HuGeF), Turin, Italy; DISAT, Politecnico di Torino, Turin, Italy.

ABSTRACT
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.

No MeSH data available.


Related in: MedlinePlus