2012/10/31 by Gilles Wainrib, Jonathan Touboul · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Artificial intelligence #Combinatorics #Complex system #Computer science #Divergence (linguistics) #Dynamical systems theory #Exponent #Lyapunov exponent #Mathematics #Measure (data warehouse) #Neural Networks and Applications #Neural dynamics and brain function #Phase transition #Physics #Quantum mechanics #Scaling #Statistical physics #Topology (electrical circuits) #cond-mat.dis-nn #math-ph #math.MP #q-bio.NC #stochastic dynamics and bifurcation
paper · pdf · doi:10.1103/physrevlett.110.118101
published as Physical Review Letters 110, 118101 (2013)
openalex publication_date 2013/03/11 · arxiv created 2013/03/15 · arxiv updated 2013/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.