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This series of papers models the dynamics of a large set of interacting neurons within the framework of statistical field theory. The system is described using a two-field model. The first field represents the neuronal activity, while the…

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can…

We investigate the dynamics of a network consisting of an array of identical cortical units with nearest neighbor interactions under periodic arousal. Each unit consists of two interconnected populations of neurons tuned to a state in which…

Model order reduction in high-dimensional, nonlinear dynamical systems if often enabled through fast-slow timescale separation. One such approach involves identifying a low-dimensional slow manifold to which the state rapidly converges and…

This paper describes a new entropy-style of equation that may be useful in a general sense, but can be applied to a cognitive model with related processes. The model is based on the human brain, with automatic and distributed pattern…

Humans and animals exhibit a range of interesting behaviors in dynamic environments, and it is unclear how our brains actively reformat this dense sensory information to enable these behaviors. Experimental neuroscience is undergoing a…

Deep learning has seen remarkable developments over the last years, many of them inspired by neuroscience. However, the main learning mechanism behind these advances - error backpropagation - appears to be at odds with neurobiology. Here,…

High-order extensions of the Hopfield model are known to exhibit dramatically enhanced storage capacity at equilibrium, while their dynamical retrieval properties remain less well understood. In our previous work, we carried out a dynamical…

We introduce and study a new model of interacting neural networks, incorporating the spatial dimension (e.g. position of neurons across the cortex) and some learning processes. The dynamic of each neural network is described via the elapsed…

Recurrent neural networks (RNNs) trained on neuroscience-inspired tasks offer powerful models of brain computation. However, typical training paradigms rely on open-loop, supervised settings, whereas real-world learning unfolds in…

Quasistationarity is ubiquitous in complex dynamical systems. In brain dynamics there is ample evidence that event-related potentials reflect such quasistationary states. In order to detect them from time series, several segmentation…

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers…

Is the brain really operating at a critical point? We study the non-equilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasi-critical dynamics on the Widom line where the correlation…

Symbolic systems are powerful frameworks for modeling cognitive processes as they encapsulate the rules and relationships fundamental to many aspects of human reasoning and behavior. Central to these models are systematicity,…

In many real-world oscillator systems, the phase response curves are highly heterogeneous. However, dynamics of heterogeneous oscillator networks has not been seriously addressed. We propose a theoretical framework to analyze such a system…

We introduce the delta-homology model of memory, a unified framework in which recall, learning, and prediction emerge from cycle closure, the completion of topologically constrained trajectories within the brain's latent manifold. A…

We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and where the spin dynamics is governed by a synchronous execution of the microscopic update rule…

Neural network modules conditioned by known priors can be effectively trained and combined to represent systems with nonlinear dynamics. This work explores a novel formulation for data-efficient learning of deep control-oriented nonlinear…

The ability of nonlinear dynamical systems to process incoming information is a key problem of many fundamental and applied sciences. Information processing by computation with attractors (steady states, limit cycles and strange attractors)…

Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, the use of…