Pierluigi Giuliana

Working with AI models, I've arrived at what I consider the most significant pure-theory breakthrough of my research (in theoretical neuroscience) so far: arxiv.org/abs/2610.12426 Chaotic systems show what is colloquially called the butterfly effect, namely, a tiny nudge to the initial state grows exponentially in time. How fast the nudge grows depends on its direction. Each direction has its own growth rate, called a Lyapunov exponent, and the full set of rates is the Lyapunov spectrum. The Lyapunov spectrum is a rich (and diffeomorphism-invariant) characterization of chaos, telling us how many directions are chaotic, how many dimensions the attractor occupies, and how fast the dynamics generate information. In 1988, Sompolinsky, Crisanti, and Sommers (Physical Review Letters) introduced a nonlinear recurrent-network model of N randomly coupled neurons that becomes chaotic when the coupling strength exceeds a critical value. This network has since become a cornerstone of theoretical neuroscience, both as a model of the spontaneous activity of cortex and as the starting point for training recurrent networks on tasks. I have worked on this model for much of my research. In the same paper, the authors obtained the largest Lyapunov exponent from the ground-state energy of a Schrödinger equation and named the full Lyapunov spectrum as an open problem. In 2023, beautiful work by Engelken et al. (Physical Review Research) computed the full Lyapunov spectrum numerically. However, an analytical calculation of the full Lyapunov spectrum has remained out of reach. Four decades after the problem was posed, this paper derives the full Lyapunov spectrum at large N from a simple self-consistent single-site problem. This calculation has three steps. First, we lower every exponent by a constant s, which turns the question "how many exponents lie below s?" into the question "how many tangent directions decay?" We show that, if we drive the shifted tangent dynamics with a source and take the minimum-norm solution, the one-step response is the projector onto the decaying directions. Thus, the trace of this projector counts the decaying directions. This step is exact at finite N. Second, to work with the minimum-norm solution, we cast it as the vanishing-regularization limit of the solution of a least-squares problem. This regularized solution in turn solves a dynamical system with one field running forward in time and another running backward. Third, we apply the cavity method, that is, we add one neuron to the network, along with its forward and backward fields. The effect of the rest of the network on this neuron then reduces to the classical dynamical mean-field description of the activity of this neuron, plus two kernels that couple the forward and backward fields of this neuron to each other across time. Thus, the full Lyapunov spectrum follows from a single-site problem, which boils down to the self-consistent calculation of these kernels. The resulting theory agrees closely with simulations for the Lyapunov spectrum, the attractor dimension, and the entropy rate, and recovers known results, including the Schrödinger equation of Sompolinsky, Crisanti, and Sommers for the largest Lyapunov exponent. As a byproduct, the theory establishes analytically that the chaos is extensive, i.e., the attractor dimension and the entropy rate grow in proportion to N. I had assumed that this problem was analytically intractable. After working independently for 100 minutes with Max thinking, GPT-6 Astra produced the initial derivation, yielding a self-consistent problem whose solution matched simulations of large networks. I found this derivation intriguing, albeit complicated and somewhat mysterious (see AI Methodology in the paper). After investigating the structure of the proposed solution more deeply, together with AI models, I arrived at a physically transparent derivation. I am excited to apply this approach to other network models, trained networks, and, more broadly, to high-dimensional systems to which Lyapunov analysis is relevant, including AI models themselves. Of course, as with any paper, I took care to write up the paper clearly. A few broader thoughts... Neural circuits are high-dimensional nonlinear dynamical systems, and such systems are notoriously hard to understand. I would frame the central problem of theoretical neuroscience as understanding how such systems function, process information, learn, and generate behavior. Many people in the theoretical sciences are anxious about AI. However, in my field at least, we are nowhere close to fully understanding the high-dimensional computations unfolding in our brains. With AI, we could reach a level of understanding that so far we have only gestured toward. So, I am excited.