Spin Glass Theory and High-Dimensional Landscapes in Biology

The lecture will consist of three parts. First, I will introduce the mathematical framework for modeling high-dimensional fitness landscapes, including in particular the structure of sequences spaces as Hamming graphs; series representations of functions on these graphs; and the quantification of navigability in terms of rank order properties of landscapes. Second, I will discuss a selection of empirical data sets and describe some of the emerging patterns that need to be explained by models. Third, I will present two case studies of specific landscape models with close connections to spin glass theory: The House-of-Cards/Random Energy model, and the genotype-based version of Fisher’s geometric model, which is equivalent to an antiferromagnetic Hopfield model.

This talk will focus on dynamical systems with many positive variables interacting through random and non-reciprocal couplings, which can be used to model the dynamics of species abundances in large ecosystems. These systems are characterized by multistability, with an exponentially large number of fixed points coexisting as possible attractors of the dynamics. A central question is to what extent the dynamics can be understood from the statistical properties of these attractors. I will discuss how to characterize the statistical properties of equilibria in a standard model (generalized Lotka-Volterra model) and show that they undergo transitions as the parameters of the system are varied. I will then discuss what these transitions may imply for the dynamics, and highlight open questions concerning the connection between the geometry of the attractor landscape and the resulting dynamical behavior.

We experimentally measured the effects of all possible single mutations and all pairs of mutations in an essential transcription factor across a phylogenetic series of reconstructed ancestral proteins that span more than 500 million years of evolution.  With these experiments we provide a complete description of the protein’s first- and second-order epistatic architecture and its relationship to protein structure, how that architecture changed over evolutionary time, and how epistatic drift modified the evolutionary accessibility of mutations as substitutions accumulated during long-term protein evolution.

In the first part of this talk, I address the feature learning/overfitting puzzle in modern machine learning. Deep learning commonly employs large, overparameterized neural networks—so expressive that they can fit pure noise. Yet, when trained on structured data, these models extract latent features. I argue that overfitting and feature learning coexist in such models but unfold on distinct timescales during training. This conclusion arises from analyzing gradient descent dynamics in overparameterized two-layer neural networks trained on single-index data, where the separation of timescales stems from the gradual increase in model complexity when starting from low initial complexity.
In the second part, I broaden this analysis to neural ordinary differential equations (neural ODEs), a framework that captures models where training and inference dynamics interact, such as generative models and recurrent neural networks. I present a class of nonlinear, controlled, high-dimensional dynamical systems and explain how their training and inference dynamics can be studied using Dynamical Mean Field Theory.
Based on:
– Montanari, Urbani, Dynamical Decoupling of Generalization and Overfitting in Large Two-Layer Networks. NeurIPS 2025.
– Urbani, Theory of learning of high-dimensional controlled non-linear dynamical systems (I): models and methods. arXiv:2606.07247v2

Neural populations are composed of subpopulations, each associated with a genetically distinct neuron type. Since neurons of the same type can express considerable variance in their structural and response properties, a model of neural diversity should reflect the contributions of both distinct cell types and of heterogeneity across cells within a given type. We have extended an existing mean-field theory approach to study the macroscopic dynamics of networks of Izhikevich spiking neurons that incorporate both within-type and between-type heterogeneity. Bifurcation analysis produces phase diagrams that identify the various macroscopic dynamical regimes that arise at the population level and their relation to the underlying microscopic neural heterogeneity. As an example, our results indicate that the level of heterogeneity within inhibitory populations controls resonance and hysteresis properties of E-I networks composed of both excitatory and inhibitory neurons.

In the first part of the talk I will introduce unsupervised generative models learned on protein sequence data, in particular Restricted Boltzmann Machine.

I will then describe a Monte-Carlo Path sampling algorithm to sample mutational paths in the sequence landscape inferred by our model.

Ithe mutational paths algorithm will be used to sample evolutionary paths for the the WW domain,both between two different natural proteins with the same binding specificity and between two natural proteins with different binding specificities.

I will finally apply the path sampling algorithm and its mean-field derivation to sample evolutionary paths for the Receptor Binding Domain of the Spike protein in SARSCoV2 under the pressure pf antibody escaping.

We examine the dynamics of recurrent neural network models of cortical circuits. Each network unit represents the collective activity of a cortical neural assembly, whose size is encoded in the unit’s self-coupling, while recurrent couplings are random and uncorrelated. As the self-coupling increases, the network first undergoes a phase transition from chaos to a multistable regime with an exponentially large number of stable states, dominated by a marginal manifold. For larger self-coupling values, the dynamics become trapped and aging occurs. Perturbing the network with an external input resets aging and induces rejuvenation. Our results demonstrate that aging can occur in the absence of reciprocal couplings and it relies on a self-coupling interaction, consistent with the assembly structure observed in cortical circuits.

The Sachdev-Ye-Kitaev (SYK) is a quantum mean-field model studied in condensed matter physics, physics of black holes and theoretical computer science. Its structural properties were derived heuristically in physics using a combination of the replica method and path integration techniques. Analyzing it mathematically rigorously, however, turned out to be notoriously difficult. In this paper we rigorously compute the partition function for this model at high enough but constant temperature. Our results are in numerical agreement with the results derived by physics methods. Our method of proof is novel and is different from the physics approach. It is based on the theory of the component structure of sparse random graphs and large deviations techniques. It provides another intriguing example how physics heuristics methods, such as path
integration and replica method, can be validated mathematically, albeit using different techniques.

The talk will be self-contained and require no background in quantum computing.

Joint work with Francisco Pernice (MIT), Alexander Schmidhuber (MIT), Alexander Zlokapa (MIT)

An insertion of four amino acids into the genome of a coronavirus of somewhat unclear origin eventually led to the 2019 pandemic. Unusual genetic events, including insertions and non-homologous recombination, have shaped the evolutionary history of several pathogens. Yet many models of high-dimensional fitness landscapes predict that such landscapes are highly accessible. If one can easily walk anywhere in the landscape through point mutations, why would evolution need to fly through rare genetic events? The role of dimensionality in accessibility, reversibility, and recombination is an active area of research. This talk discusses insights from graphs and polytopes, with a focus on new mechanisms and changes that emerge as the number of loci and alleles increases.