title: Topological and dynamical complexity of random neural networks.
abstract: Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown and, similar to spin glasses, shall be fundamentally related to the behavior of the system. In this Letter, we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler disordered system, suggesting a deep and underexplored link between topological and dynamical complexity.
authors.affiliation: LAGA, Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99 avenue J.B. Clément, F-93430 Villetaneuse, France.
authors.forename: Gilles
authors.initials: G
authors.lastname: Wainrib
authors.affiliation: The Mathematical Neuroscience Laboratory, CIRB/Collège de France (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL*), 11, place Marcelin Berthelot, 75005 Paris, France and BANG Laboratory, INRIA Paris-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France.
authors.forename: Jonathan
authors.initials: J
authors.lastname: Touboul
mesh_terms: D032701:Biomimetics; D008959:Models, Neurological; D016571:Neural Networks, Computer; D009474:Neurons / Q000502:physiology