Route to chaos in a branching model of neural network dynamics

• Published in: Chaos, solitons and fractals Vol. 165; p. 112739
• Main Authors: Williams-García, Rashid V., Nicolis, Stam
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2022; Elsevier
• Subjects: Boundary conditions; Chaos; Chaotic Dynamics; Cognitive Sciences; Condensed Matter; Disordered Systems and Neural Networks; Life Sciences; Neural networks; Neurons and Cognition; Nonlinear dynamics; Nonlinear Sciences; Physics; Chaos; Nonlinear dynamics; Boundary conditions; Neural networks; ChaosNeural networksNonlinear dynamicsBoundary conditions
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2022.112739

Simplified models are a necessary steppingstone for understanding collective neural network dynamics, in particular the transitions between different kinds of behavior, whose universality can be captured by such models, without prejudice. One such model, the cortical branching model (CBM), has previously been used to characterize part of the universal behavior of neural network dynamics and also led to the discovery of a second, chaotic transition which has not yet been fully characterized. Here, we study the properties of this chaotic transition, that occurs in the mean-field approximation to the kin=1 CBM by focusing on the constraints the model imposes on initial conditions, parameters, and the imprint thereof on the Lyapunov spectrum. Although the model seems similar to the Hénon map, we find that the Hénon map cannot be recovered using orthogonal transformations to decouple the dynamics. Fundamental differences between the two, namely that the CBM is defined on a compact space and features a non-constant Jacobian, indicate that the CBM maps, more generally, represent a class of generalized Hénon maps which has yet to be fully understood. •Neural network branching model exhibits novel type of period-doubling bifurcation.•Stable fixed point bifurcates directly to period-4; period-2 orbits notably absent.•Quadratic nonlinearity and model constraints lead to a distinct universality class.•Intricate interplay between finite geometry, parameters, and initial conditions.