Topic: chaos

Giovanni Forni discusses mathematical billiards as a tractable setting for studying dynamical systems. The conversation explores the tension between periodic motion and chaos, why seemingly elementary billiard trajectories lead to difficult questions, and which aspects remain unresolved. The interview was recorded at the European Congress of Mathematics.

Neural circuits exhibit complex activity patterns, both spontaneously and in response to external stimuli. Information encoding and learning in neural circuits depend on the ability of time-varying stimuli to control spontaneous network activity. In particular, variability arising from the sensitivity to initial conditions of recurrent cortical circuits can limit the information conveyed about the sensory input. Spiking and firing rate network models can exhibit such sensitivity to initial conditions that are reflected in their dynamic entropy rate and attractor dimensionality computed from th

Threshold-linear networks (TLNs) display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. Over the past few years, we have developed a detailed mathematical theory relating stable and unstable fixed points of TLNs to graph-theoretic properties of the underlying network. In particular, we have discovered that a special type of unstable fixed points, corresponding to "core motifs," are predictive of dynamic attractors. Recently, we have used these ideas to classify dynamic attractors in a two-parameter family of inhibition-dominate

The remarkable properties of information-processing of biological and of artificial neuronal networks alike arise from the interaction of large numbers of neurons. A central quest is thus to characterize their collective states. The directed coupling between pairs of neurons and their continuous dissipation of energy, moreover, cause dynamics of neuronal networks outside thermodynamic equilibrium. Tools from non-equilibrium statistical mechanics and field theory are thus instrumental to obtain a quantitative understanding. We here present progress with this recent approach [1]. On the experime

Recurrent neural networks (RNNs) are powerful dynamical models, widely used in machine learning (ML) for processing sequential data, and also in neuroscience, to understand the emergent properties of networks of real neurons. Prior theoretical work in understanding the properties of RNNs has focused on models with additive interactions. However, real neurons can have gating i.e. multiplicative interactions, and gating is also a central feature of the best performing RNNs in machine learning. Here, we develop a dynamical mean-field theory (DMFT) to study the consequences of gating in RNNs. We u