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The brain is in a state of perpetual reverberant neural activity, even in the absence of specific tasks or stimuli. Shedding light on the origin and functional significance of such a dynamical state is essential to understanding how the…
Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training, such as initialization-dependent solutions and incremental learning. These phenomena are typically studied in…
The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms…
Nonlinear network dynamics are notoriously difficult to understand. Here we study a class of recurrent neural networks called combinatorial threshold-linear networks (CTLNs) whose dynamics are determined by the structure of a directed…
Functional networks provide a topological description of activity patterns in the brain, as they stem from the propagation of neural activity on the underlying anatomical or structural network of synaptic connections. This latter is well…
Evidence suggests that brain network dynamics is a key determinant of brain function and dysfunction. Here we propose a new framework to assess the dynamics of brain networks based on recurrence analysis. Our framework uses recurrence plots…
Understanding the evolution of brain functional networks over time is of great significance for the analysis of cognitive mechanisms and the diagnosis of neurological diseases. Existing methods often have difficulty in capturing the…
In this article, we review the literature on statistical theories of neural networks from three perspectives: approximation, training dynamics and generative models. In the first part, results on excess risks for neural networks are…
Modeling neural population dynamics underlying noisy single-trial spiking activities is essential for relating neural observation and behavior. A recent non-recurrent method - Neural Data Transformers (NDT) - has shown great success in…
The interest in dynamic processes on networks is steadily rising in recent years. In this paper, we consider the $(\alpha,\beta)$-Thresholded Network Dynamics ($(\alpha,\beta)$-Dynamics), where $\alpha\leq \beta$, in which only structural…
Neural spike trains, which are sequences of very brief jumps in voltage across the cell membrane, were one of the motivating applications for the development of point process methodology. Early work required the assumption of stationarity,…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
Human cognition is theorized to operate in two modes: fast, intuitive System 1 thinking and slow, deliberate System 2 thinking. While current Large Reasoning Models (LRMs) excel at System 2 thinking, their inability to perform fast thinking…
Large language models (LLMs) often exhibit abrupt emergent behavior, whereby new abilities arise at certain points during their training. This phenomenon, commonly referred to as a ''phase transition'', remains poorly understood. In this…
A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning of such hyperparameters may be difficult and, typically, based on a trial-and-error…
The brain's functional connectivity fluctuates over time instead of remaining steady in a stationary mode even during the resting state. This fluctuation establishes the dynamical functional connectivity that transitions in a non-random…
The understanding of neural activity patterns is fundamentally linked to an understanding of how the brain's network architecture shapes dynamical processes. Established approaches rely mostly on deviations of a given network from certain…
Several approaches to cognition and intelligence research rely on statistics-based models testing, namely factor analysis. In the present work we exploit the emerging dynamical systems perspective putting the focus on the role of the…
The brain is immensely complex, with diverse components and dynamic interactions building upon one another to orchestrate a wide range of functions and behaviors. Understanding patterns of these complex interactions and how they are…
Mathematical approaches from dynamical systems theory are used in a range of fields. This includes biology where they are used to describe processes such as protein-protein interaction and gene regulatory networks. As such networks increase…