Archive ouverte HAL – On the number of limit cycles in asymmetric neural networks

Sungmin Hwang 1 Viola Folli 2 Enrico Lanza 3 Giorgio Parisi 3 Giancarlo Ruocco 2, 3 Francesco Zamponi 4

Sungmin Hwang, Viola Folli, Enrico Lanza, Giorgio Parisi, Giancarlo Ruocco, et al.. On the number of limit cycles in asymmetric neural networks. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2019, 2019 (5), pp.053402. ⟨10.1088/1742-5468/ab11e3⟩. ⟨hal-02147155⟩

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.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. Sapienza@IIT Laboratory - Center for Life Nano Science [Genova]
  • 3. Università degli Studi di Roma "La Sapienza" [Rome]
  • 4. LPTENS - Laboratoire de Physique Théorique de l'ENS