arXiv subjects

Samantha J. Fournier

Publications and source records attributed to Samantha J. Fournier.

5 recordsLinked to original sources

arXiv2026

Learning limit cycles via Hebbian synaptic plasticity

We investigate high-dimensional, non-linear dynamical systems when exposed to incoherent periodic inputs and Hebbian-like synaptic plasticity. Our findings reveal a striking phenomenon: depending on the interplay between the strength of the periodic drive and synaptic plasticity, the system's phase diagram can give rise to a region where, once both inputs are removed, the collective dynamics spontaneously settles into a limit cycle. This suggests that periodic drives can imprint lasting rhythmic patterns into the network through plasticity, effectively teaching it to oscillate on its own. Numerical simulations on finite size systems show that the limit cycle phase can be easily detected on single-sample trajectories, while averaged curves are affected by strong finite size effects due to sample-to-sample fluctuations of the period of the limit cycles.

cond-mat.dis-nn↗

arXiv2026

High-dimensional dynamical systems: co-existence of attractors, phase transitions, maximal Lyapunov exponent and response to periodic drive

We study the dynamical properties of a broad class of high-dimensional random dynamical systems exhibiting chaotic as well as fixed point and periodic attractors. We consider cases in which attractors can co-exists in some regions of the phase diagrams and we characterize their nature by computing the maximal Lyapunov exponent. For a specific choice of the dynamical system we show that this quantity can be computed explicitly in the whole chaotic phase due to an underlying integrability of a properly defined Schrödinger problem. Furthermore, we consider the response of this dynamical systems to periodic perturbations. We show that these dynamical systems act as filters in the frequency-amplitude spectrum of the periodic forcing: only in some regions of the frequency-amplitude plane the periodic forcing leads to a synchronization of the dynamics. All in all, the results that we present mirror closely the ones observed in the past forty years in the study of standard models of random recurrent neural networks. However, the dynamical systems that we consider are easier to study and we believe that this may be an advantage if one wants to go beyond random dynamical systems and consider specific training strategies.

cond-mat.dis-nn↗

arXiv2025

Non-reciprocal interactions and high-dimensional chaos: comparing dynamics and statistics of equilibria in a solvable class of models

We investigate a model of high-dimensional dynamical variables with all-to-all interactions that are random and non-reciprocal. We characterize its phase diagram and show that the model can exhibit chaotic dynamics. We show that the equations describing the system's dynamics exhibit a number of equilibria that is exponentially large in the dimensionality of the system, and these equilibria are all linearly unstable in the chaotic phase. Solving the effective equations governing the dynamics in the infinite-dimensional limit, we determine the typical properties (magnetization, overlap) of the configurations belonging to the attractor manifold. We show that these properties cannot be inferred from those of the equilibria, challenging the expectation that chaos can be understood purely in terms of the numerous unstable equilibria of the dynamical equations. We discuss the dependence of this scenario on the strength of non-reciprocity in the interactions. These results are obtained through a combination of analytical methods such as Dynamical Mean-Field Theory and the Kac-Rice formalism.

cond-mat.dis-nn↗

arXiv2025

Generative modeling through internal high-dimensional chaotic activity

Generative modeling aims at producing new datapoints whose statistical properties resemble the ones in a training dataset. In recent years, there has been a burst of machine learning techniques and settings that can achieve this goal with remarkable performances. In most of these settings, one uses the training dataset in conjunction with noise, which is added as a source of statistical variability and is essential for the generative task. Here, we explore the idea of using internal chaotic dynamics in high-dimensional chaotic systems as a way to generate new datapoints from a training dataset. We show that simple learning rules can achieve this goal within a set of vanilla architectures and characterize the quality of the generated datapoints through standard accuracy measures.

cs.LG↗

arXiv2023

Statistical physics of learning in high-dimensional chaotic systems

In many complex systems, elementary units live in a chaotic environment and need to adapt their strategies to perform a task, by extracting information from the environment and controlling the feedback loop on it. One of the main example of systems of this kind is provided by recurrent neural networks. In this case, recurrent connections between neurons drive chaotic behavior and when learning takes place, the response of the system to a perturbation should take into account also its feedback on the dynamics of the network itself. In this work, we consider an abstract model of a high-dimensional chaotic system as a paradigmatic model and study its dynamics. We study the model under two particular settings: Hebbian driving and FORCE training. In the first case, we show that Hebbian driving can be used to tune the level of chaos in the dynamics and this reproduces some results recently obtained in the study of more biologically realistic models of recurrent neural networks. In the latter case, we show that the dynamical system can be trained to reproduce simple periodic functions. To do this, we consider the FORCE algorithm -- originally developed to train recurrent neural networks -- and adapt it to our high-dimensional chaotic system. We show that this algorithm drives the dynamics close to an asymptotic attractor the larger the training time. All our results are valid in the thermodynamic limit thanks to an exact analysis of the dynamics through dynamical mean field theory.

cond-mat.dis-nn↗