The comprehension of the mechanisms at the basis of the functioning of complexly interconnected networks represents one of the main goals of neuroscience. In this work, we investigate how the structure of recurrent connectivity influences the ability of a network to have storable patterns and in particular limit cycles, by modeling a recurrent neural network with McCulloch–Pitts neurons as a content-addressable memory system. A key role in such models is played by the connectivity matrix, which, for neural networks, corresponds to a schematic representation of the 'connectome': the set of chemical synapses and electrical junctions among neurons. The shape of the recurrent connectivity matrix plays a crucial role in the process of storing memories. This relation has already been exposed by the work of Tanaka and Edwards, which presents a theoretical approach to evaluate the mean number of fixed points in a fully connected model at thermodynamic limit. Interestingly, further studies on the same kind of model but with a finite number of nodes have shown how the symmetry parameter influences the types of attractors featured in the system. Our study extends the work of Tanaka and Edwards by providing a theoretical evaluation of the mean number of attractors of any given length L for different degrees of symmetry in the connectivity matrices.

On the number of limit cycles in asymmetric neural networks / Hwang, Sungmin; Folli, Viola; Lanza, Enrico; Parisi, Giorgio; Ruocco, Giancarlo; Zamponi, Francesco. - In: JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT. - ISSN 1742-5468. - 2019:5(2019). [10.1088/1742-5468/ab11e3]

On the number of limit cycles in asymmetric neural networks

Folli, Viola;Lanza, Enrico;Parisi, Giorgio;Ruocco, Giancarlo;Zamponi, Francesco
2019

Abstract

The comprehension of the mechanisms at the basis of the functioning of complexly interconnected networks represents one of the main goals of neuroscience. In this work, we investigate how the structure of recurrent connectivity influences the ability of a network to have storable patterns and in particular limit cycles, by modeling a recurrent neural network with McCulloch–Pitts neurons as a content-addressable memory system. A key role in such models is played by the connectivity matrix, which, for neural networks, corresponds to a schematic representation of the 'connectome': the set of chemical synapses and electrical junctions among neurons. The shape of the recurrent connectivity matrix plays a crucial role in the process of storing memories. This relation has already been exposed by the work of Tanaka and Edwards, which presents a theoretical approach to evaluate the mean number of fixed points in a fully connected model at thermodynamic limit. Interestingly, further studies on the same kind of model but with a finite number of nodes have shown how the symmetry parameter influences the types of attractors featured in the system. Our study extends the work of Tanaka and Edwards by providing a theoretical evaluation of the mean number of attractors of any given length L for different degrees of symmetry in the connectivity matrices.
2019
limit cycles; neural networks; Hopfield
01 Pubblicazione su rivista::01a Articolo in rivista
On the number of limit cycles in asymmetric neural networks / Hwang, Sungmin; Folli, Viola; Lanza, Enrico; Parisi, Giorgio; Ruocco, Giancarlo; Zamponi, Francesco. - In: JOURNAL OF STATISTICAL MECHANICS: THEORY AND EXPERIMENT. - ISSN 1742-5468. - 2019:5(2019). [10.1088/1742-5468/ab11e3]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1340594
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