Statistical Physics seminars

We introduce a statistical physics framework for learning in neural architectures composed of single or interconnected asymmetric attractor networks. These systems can exhibit a manifold of global fixed points capable of implementing sophisticated input-output mappings, which we characterize analytically. Learning from extensive datasets is achieved through the stabilization of fixed points via a fully distributed and local learning process, implemented at the single-neuron level. This simple mechanism yields performance comparable to that of conventional feedforward deep neural networks trained using gradient-based methods. The effectiveness of the model stems from the dense and accessible manifolds of stable fixed points, which encode the internal representations of data. Unlike other approaches to deep learning without backpropagation, our method does not attempt to estimate gradients. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-06-18. Recording duration: 00:45:35.

John Hopfield and Geoffrey Hinton explain how ideas from physics helped establish computational models that store patterns and learn from data. Hopfield develops a physical perspective on collective computation, including associative memories in which network dynamics recover stored patterns from incomplete or distorted inputs. Hinton examines Boltzmann machines, connecting probabilistic neural networks and learning to concepts from statistical physics. Together, the lectures link energy landscapes, interacting units and stochastic behaviour with mechanisms for representation and computation. They provide the scientific background to the neural-network contributions recognised by the 2024 physics prize and the conceptual route from models of collective systems to machine learning.

The factorization of a large matrix into the product of two matrices is an important mathematical problem encountered in many tasks, ranging from dictionary learning to machine learning. Statistical physics can provide on the one hand theoretical limits on the possibility of factorizing matrices in the limit of infinite size, and also practical algorithms. While this program has been successful in the case of finite rank matrices, the regime of extensive rank (scaling linearly with the dimension of the matrix) turns out to be much harder. This talk will describe a new approach to matrix factorization that maps it to neural network models of associative memory: each pattern found in the associative memory corresponds to one factor of the matrix decomposition. A detailed theoretical analysis of this new approach shows that matrix factorization in the extensive rank regime is possible when the rank is below a certain threshold. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-01-03. Recording duration: 00:44:31.

Modern electrophysiological recordings simultaneously capture single-unit spiking activities of hundreds of neurons. These neurons exhibit highly complex coordination patterns. Where does this complexity stem from? One candidate is the ubiquitous heterogeneity in connectivity of local neural circuits. Studying neural network dynamics in the linearized regime and using tools from statistical field theory of disordered systems, we derive relations between structure and dynamics that are readily applicable to subsampled recordings of neural circuits: Measuring the statistics of pairwise covariances allows us to infer statistical properties of the underlying connectivity. Applying our results to spontaneous activity of macaque motor cortex, we find that the underlying network operates in a strongly recurrent regime. In this regime, network connectivity is highly heterogeneous, as quantified by a large radius of bulk connectivity eigenvalues. Being close to the point of linear instability, this dynamical regime predicts a rich correlation structure, a large dynamical repertoire, long-range interaction patterns, relatively low dimensionality and a sensitive control of neuronal coordination. These predictions are verified in analyses of spontaneous activity of macaque motor cortex and mouse visual cortex. Finally, we show that even microscopic features of connectivity, such as connection motifs, systematically scale up to determine the global organization of activity in neural circuits.

The affinity between statistical physics and machine learning has a long history. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the optimization algorithms commonly used for learning.

The three lectures examine how physical theory can make sense of complex systems with fluctuations and many interacting components. Syukuro Manabe describes the construction of climate models that connect atmospheric radiation, heat transport and carbon dioxide to changes in surface temperature. Klaus Hasselmann addresses the relationship between short-term weather variability and long-term climate, including methods for distinguishing human influence from natural variation. Giorgio Parisi examines disordered systems and the coexistence of multiple equilibria, with spin glasses providing a central setting for understanding apparently irregular collective behaviour. The programme preserves both parts of the prize: quantitative climate science and the statistical physics of disorder, rather than treating either as a substitute for the other.

The spontaneous collective motion of self-propelled agents is ubiquitous in the natural world, and it often occurs in complex environments, be it bacteria and cells migrating through polymeric extracellular matrix or animal herds and human crowds navigating structured terrains. Much is known about flocking dynamics in pristine backgrounds, but how do spatio-temporal heterogeneities in the environment impact such collective self-organization? I will present two model systems, a colloidal active fluid negotiating disordered obstacles and a confined dense bacterial suspension in a viscoelastic medium, as controllable platforms to explore this question and highlight general mechanisms for active self-organization in complex environments. By combining theory and experiment, I will show how flocks on disordered substrates organize into a novel dynamic vortex glass phase, akin to vortex glasses in dirty superconductors, while the presence of viscoelasticity can calm the otherwise turbulent swarming of bacteria, allowing the emergence of a large scale coherent and even oscillatory vortex when confined on the millimetre scale.

Granular matter can serve as a prototype for exploring the rich physics of many-body systems driven far from equilibrium. This talk will outline a new direction for granular physics with macroscopic particles, where acoustic levitation compensates the forces due to gravity and eliminates frictional interactions with supporting surfaces in order to focus on particle interactions. Levitating small particles by intense ultrasound fields in air makes it possible to manipulate and control their positions and assemble them into larger aggregates. The small air viscosity implies that the regime of underdamped dynamics can be explored, where inertial effects are important, in contrast to typical colloids in a liquid, where inertia can be neglected. Sound scattered off individual, levitated solid particles gives rise to controllable attractive forces with neighboring particles. I will discuss some of the key concepts underlying acoustic levitation, describe how detuning an acoustic cavity can introduce active fluctuations that control the assembly statistics of small levitated particles clusters, and give examples of how interactions between neighboring levitated objects can be controlled by their shape.

The link between behavior, learning and the underlying connectome is a fundamental open problem in neuroscience. In my talk I will show how it is possible to develop a theory that bridges across these three levels (animal behavior, learning and network connectivity) based on the geometrical properties of neural activity. The central tool in my approach is the dimensionality of neural activity. I will link animal complex behavior to the geometry of neural representations, specifically their dimensionality; I will then show how learning shapes changes in such geometrical properties and how local connectivity properties can further regulate them. As a result, I will explain how the complexity of neural representations emerges from both behavioral demands (top-down approach) and learning or connectivity features (bottom-up approach). I will build these results regarding neural dynamics and representations starting from the analysis of neural recordings, by means of theoretical and computational tools that blend dynamical systems, artificial intelligence and statistical physics approaches.

Physics offers countless examples for which theoretical predictions are astonishingly powerful. But it’s hard to imagine a similar precision in complex systems where the number and interdependencies between components simply prohibits a first-principles approach, look no further than the challenge of the billions of neurons and trillions of connections within our own brains. In such settings how do we even identify the important theoretical questions? We describe a systems-scale perspective in which we integrate information theory, dynamical systems and statistical physics to extract understanding directly from measurements. We demonstrate our approach with a reconstructed state space of the behavior of the nematode C. elegans, revealing a chaotic attractor with symmetric Lyapunov spectrum and a novel perspective of motor control. We then outline a maximally predictive coarse-graining in which nonlinear dynamics are subsumed into a linear, ensemble evolution to obtain a simple yet accurate model on multiple scales. With this coarse-graining we identify long timescales and collective states in the Langevin dynamics of a double-well potential, the Lorenz system and in worm behavior. We suggest that such an ``inverse’’ approach offers an emergent, quantitative framework in which to seek rather than impose effective organizing principles of complex systems.

Neural networks can learn and recognize subtle correlations in high dimensional inputs. However, neural networks are simply many-body systems with strong non-linearities and disordered interactions. Hence, many-body physical systems with similar interactions should be able to show neural network-like behavior. Here we show neural network-like behavior in the nucleation dynamics of promiscuously interacting molecules with multiple stable crystalline phases. Using a combination of theory and experiments, we show how the physics of the system dictates relationships between the difficulty of the pattern recognition task solved, time taken and accuracy. This work shows that high dimensional pattern recognition and learning are not special to software algorithms but can be achieved by the collective dynamics of sufficiently disordered molecular systems.

It is sometimes said that there are two reasons why physics is so successful as a science. One is that it deals with very simple problems. The other is that it attempts to account only for universal aspects of systems at a desired level of description, with lower level phenomena subsumed into a small number of adjustable parameters. It is a widespread belief that this approach seems unlikely to be useful in biology, which is intimidatingly complex, where “everything has an exception”, and where there are a huge number of undetermined parameters. I will try to argue, nonetheless, that there are important, experimentally-testable aspects of biology that exhibit universality, and should be amenable to being tackled from a physics perspective. My suggestion is that this can lead to useful new insights into the existence and universal characteristics of living systems. I will try to justify this point of view by contrasting the goals and practices of the field of condensed matter physics with materials science, and then by extension, the goals and practices of the newly emerging field of “Physics of Living Systems” with biology. Specific biological examples that I will discuss include the following: Universal patterns of gene expression in cell biology Universal scaling laws in ecosystems, including the species-area law, Kleiber’s law, Paradox of the Plankton Universality of the genetic code Universality of thermodynamic utilization in microbial communities Universal scaling laws in the tree of life The question of what can be learned from studying universal phenomena in biology will also be discussed. Universal phenomena, by their very nature, shed little light on detailed microscopic levels of description. Yet there is no point in seeking idiosyncratic mechanistic explanations for phenomena whose explanation is found in rather general principles, such as the central limit theorem, that every microscopic mechanism is constrained to obey. Thus, physical perspectives may be better suited to answering certain questions such as universality than traditional biological perspectives. Concomitantly, it must be recognized that the identification and understanding of universal phenomena may not be a good answer to questions that have traditionally occupied biological scientists. Lastly, I plan to talk about what is perhaps the central question of universality in biology: why does the phenomenon of life occur at all? Is it an inevitable consequence of the laws of physics or some special geochemical accident? What methodology could even begin to answer this question? I will try to explain why traditional approaches to biology do not aim to answer this question, by comparing with our understanding of superconductivity as a physical phenomenon, and with the theory of universal computation. References Nigel Goldenfeld, Tommaso Biancalani, Farshid Jafarpour. Universal biology and the statistical mechanics of early life. Phil. Trans. R. Soc. A 375, 20160341 (14 pages) (2017). Nigel Goldenfeld and Carl R. Woese. Life is Physics: evolution as a collective phenomenon far from equilibrium. Ann. Rev. Cond. Matt. Phys. 2, 375-399 (2011).

The affinity between statistical physics and machine learning has long history, this is reflected even in the machine learning terminology that is in part adopted from physics. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the learning algorithm.