Experimental observation of thermal avalanches during the creep responde of a granular material
I will present recent experimental results on the spontaneous deformation of a granular material submitted to a constant stress below its yield stress. We observe non-monotoneous creep deformation of the sample: overall, we observe a logarithmic slowdown of the axial strain, interrupted by bursts of activity, which ultimately leads, in some samples, to spontaneous failure. Those results are surprising as the granular material we use (dry glass beads of typical diameter 100 microns submitted to a low confining pressure) is supposed to be athermal and without any clear damage processes.
Using an interferometric method of measurement allowing us to map the spatial distribution of the strain in the sample during the test, we show that the bursts we observe correspond to avalanches of correlated plastic rearrangements in the system.
Fitting power-law domains with loglog-concave distributions
High variability is a fundamental property in complex systems. Important debates have taken place about how to characterize this high variability, and in particular, about if power-law distributions and power-law-tailed distributions are the best fits for particular real-world data sets. The ``critical’’ issue is to find a threshold value of the variable from which a power-law fit works well. Several heuristic recipes have been proposed [1,2], the most popular one being the Clauset-Shalizi-Newman method.
Here we explain a very different approach, introducing loglog-concave distributions. These are given by a probability density with the only restriction (apart from normalization) of being concave in a double logarithmic representation (concave in the mathematical convention, i.e., concave downwards). Loglog-concave distributions constitute a non-parametric family, encompassing many distributions widely used in complex-systems research, such as log-normal, Weibull, gamma, stretched exponential, Pareto, and generalized extreme-value distributions. Remarkably, the power-law distribution is a limiting case of loglog-concavity (in the sense that it separates loglog-concavity from loglog-convexity).
We explore the maximum likelihood procedure applied to loglog-concave distributions, and explain how the resulting estimator takes the form of a piecewise power-law distribution. This estimator can be computed using an active set algorithm [3]. This characterization provides the thresholds separating the power-law different regions, together with their corresponding power-law exponents, which can be used as a criterion for identifying the ranges where the power-law distribution fit is appropriate.
We apply this method both to simulated data and to representative datasets in extreme-value statistics and complex systems.
[1] Clauset, A. et al., Power-Law Distributions in Empirical Data, SIAM Review 51, 661–703, 2009.
[2] Corral, A. & Deluca, A., A practical recipe to fit discrete power-law distributions, Acta Geophysica 61(6) 1351-1394, 2013.
[3] Dümbgen, L., & Rufibach, K., Active set and EM algorithms for log-concave densities based on complete and censored data, arXiv:0707.4643, 2011.
Symmetry breaking and avalanche shapes in modular neural networks
Experimental evidence suggests that the healthy brain operates near a critical regime, characterized by scale-free neuronal avalanches.
Recent research has increasingly focused on the mean temporal profiles of neuronal avalanches, as a more stringent and reliable test for criticality. Scaling arguments predict that, when appropriately rescaled, the mean temporal profiles of avalanches of widely varying durations should collapse onto a single scaling function, often approximated by an inverted parabola. Experimental measurements have revealed clear departures from perfect symmetry, often displaying leftward skewing and extended tails.
We have investigated the stochastic Wilson-Cowan model on a modular network, in which synaptic strengths differ between intra-module and inter-module connections. The system exhibits a rich phase diagram, comprising symmetry and "broken symmetry" phases.
We found that, at the edge of the transition to a symmetric phase, avalanches are right-ward skewed, as observed also in the non-modular Wilson-Cowan model. On the other hand, at the transition to a "broken symmetry" phase, avalanches become left-ward skewed.
We found that in the latter case avalanches proceed in two stages: an initially cooperative regime, where excitatory activity is prevalent, followed by inhibitory competition that selects one dominant module and suppresses the others. This is the relevant mechanism leading to a fast rise of the avalanche, followed by a slower decay, and therefore to leftward asymmetry.
These findings contribute to a better understanding of the relationship between brain network topology and functional brain activity.
Multiple Percolating Clusters
Inspired by the formation of bigels, we developed a bond percolation model that yields multiple percolating clusters in three dimensions not only at the critical point, but also above it. Our simulations suggest that, in the thermodynamic limit, there is no upper limit to the number of percolating clusters. We show that in finite systems the maximum number of percolating clusters that can be obtained grows logarithmically with the lattice size. For equal initial densities in the thermodynamic limit, all clusters percolate at the same threshold and exhibit critical exponents consistent with the critical exponents of standard percolation. The threshold depends linearly on the initial density of species and the maximum and minimum initial densities decay exponentially with the maximal number of spanning clusters. We also study a percolation model in which we occupy bonds randomly and each time a spanning cluster appears we remove it. The maximum number n_max of spanning clusters one can harvest in this way grows with the system size like a power-law with exponent d-d_f. Also, the variance of n_max and the size distribution of the remaining finite clusters grow like power-laws.
Fisher Information as a tool to probe the vicinity of criticality in intermittent busts of dynamical activity
With the introduction of Self-Organised Criticality by Bak, Tang and Wiesenfeld [1, 2, 3], it became popular to associate criticality with the power law behaviour of probability distributions describing the size and duration of bursts of activity, the so-called avalanches [4, 5]. However, the exact relationship between power-law event distributions and the power-law decay of correlations is not simple. The subtleties are evident, for example, from the fact that the exponents for the duration and size of the burst events in the brain (2 and 3/2) correspond to the exponents for the uncorrelated branching process. How the activity of the brain can be strongly correlated [6, 7] and be characterised by the exponents of the uncorrelated branching process is far from trivial [8]. The uncorrelated branching process is only critical when the branching ratio is equal to 1. The observed approximate power law distributions for measured brain activity have been related to measured effective branching ratios, see e.g. [4, 9]. But even for a pure mathematical branching process with branching probabilities pk, it is difficult to determine the true average branching ratio µ = ∑ₖ pₖ·k from simulations of the empirical activity rate given by the effective branching ratio σ(t) = S(t+1)/S(t) in terms of the activity S(t) at consecutive time steps. This measure of activity can be computed even if the burst activity does not have the form of causal branching trees, which is expected to be the case for bursts of brain activity [8]. These subtleties in quantifying the critical behaviour of the brain suggest that we try to avoid defining avalanches, applying instead more direct methods from information theory that do not rely on such modelling assumptions. For this purpose, in this work we consider the Fisher information metric (FIM) [10]. FIM measures sensitivity to change in parameters. In the case of statistical mechanics, Fisher information is directly related to susceptibility [11, 12], which for infinite systems diverges at the critical point. Hence, it is expected that Fisher information in finite systems will exhibit a peak in the vicinity of critical behaviour. Moreover, it has been found that, for dynamics on finite networks, the detailed behaviour of the Fisher information can relate to an interplay between the underlying network structure and the transition in the dynamics on the network [13]. Importantly, computing FIM only requires the knowledge of the distribution of the relevant observables, which can be empirically estimated from data. To improve our understanding of how the Fisher information behaves in the near criticality in models exhibiting burst dynamics, we study three model systems: the uncorrelated 1 branching process, a tuneable network model of integrate-and-fire dynamics considered to be of relevance to neuronal dynamics and a whole brain model.
[1] P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation for 1/f noise. Phys. Rev. Lett, 859:381–4, 1987.
[2] Henrik Jeldtoft Jensen. Self-organized criticality: emergent complex behavior in physical and biological systems, volume 10. Cambridge university press, 1998.
[3] Gunnar Pruessner. Self-organized criticality: Theory, Models and Charaterisation.
[4] John M Beggs and Dietmar Plenz. Neuronal avalanches in neocortical circuits. Journal of neuroscience, 23(35):11167–11177, 2003.
[5] Enzo Tagliazucchi, Pablo Balenzuela, Daniel Fraiman, and Dante R Chialvo. Criticality in large-scale brain fmri dynamics unveiled by a novel point process analysis. Frontiers in physiology, 3:15, 2012.
[6] P. Expert, R. Lambiotte, D. Chialvo, K. Christensen, H.J. Jensen, D.J. Sharp, and F. Turkheimer. Self-similar correlation function in rest-state fmri. J. R. Soc Interface, 8:472–9, 2010.
[7] F. Lombardi, H/J Herrmann, L. Parrino, D. Plenz, S. Scarpetta, A. E. Vaudano, L de Ar cangleis, and O. Shriki. Beyondn pulsed inhibition: Alpha oscillations modulate atten uation and amplification of neural activity in the awake resting state. Cell Reports, 42.
[8] J Pausch. From neuronal spikes to avalanches: Effects and circumvention of time binning. Phys. Rev. Res., 4:023212, 2022.
[9] Simon-Shlomo Poil, Arjen Van Ooyen, and Klaus Linkenkaer-Hansen. Avalanche dy namics of human brain oscillations: relation to critical branching processes and temporal correlations. Human brain mapping, 29(7):770–777, 2008.
[10] Shun-ichi Amari. Information geometry and its applications. Springer, 2016.
[11] G.E. Crooks. Fisher information and ststatistical mechanis. Tech. Note 008v4, accessed 10.3.2026.
[12] Mikhail Prokopenko, Joseph T Lizier, Oliver Obst, and X Rosalind Wang. Relating fisher information to order parameters. Phys. Rev. E, 84(4):041116, 2011.
[13] A.C. Kalloniatis, M.I. Zupaic, and M. Prokopenko. Fisher information and criticality in the kuramoto model of nonidentical oscillators. Phys. Rev. E, 98:022302, 2018.
Universal Collective Creep and Avalanches of Magnetic Domain Walls: Bridging Theory, Numerics, and Experiment
The thermally activated creep motion of driven elastic interfaces in disordered media is a paradigmatic framework for understanding glassy systems, domain wall dynamics, and out-of-equilibrium transport. While classically described as a sequence of independent, ultra-slow activation events over pinning barriers, recent work reveals that the creep regime itself is inherently intermittent, displaying complex spatiotemporal clustering. In this talk, we present a comprehensive picture of these distinct "creep avalanches" by synthesizing analytical scaling arguments, high-resolution magneto-optical experiments, and state-of-the-art numerical simulations. First, we show that despite microscopic differences across disparate ferromagnetic thin films, macroscopic transport data collapse onto a single, universal pinning energy barrier function. We then focus on the microscopic nature of this activated motion, utilizing time-resolved microscopy and exact numerical optimization techniques—including recent finite-temperature Dijkstra-based algorithms—to isolate the true nature of intermittency deep below the depinning threshold. Crucially, we demonstrate a fundamental decoupling of scales: while a temperature-independent optimal rearrangement scale rules the temporal activation bottleneck, a distinct, temperature-dependent creep avalanche length scale dictates the spatial geometry of the interface. By computing the structure factor and four-point dynamical susceptibilities, we show how the spatial correlations of these collective creep avalanches are uniquely encoded by depinning critical exponents, providing a unified physical picture of intermittent transport in elastic manifolds without confusing the mechanics of activation with zero-temperature depinning.
The average avalanche shape in interface depinning: temporal asymmetries and averaging effects
The average temporal shape of avalanches has emerged as a key quantity characterizing the statistical properties of avalanches in driven systems. It is parametrized by one of the universal scaling exponents defining the universality class, and may exhibit asymmetries that have been interpreted as manifestations of broken time-reversal symmetry in the avalanche dynamics.
Here, I will discuss our recent work on the average avalanche shape in the depinning of 1+1- and 2+1-dimensional interfaces with local elastic interactions. We analyze the velocity signal by decomposing it into the product of the instantaneous number of connected clusters of active interface segments and the average size of such clusters. This reveals the key role of the evolution of the average cluster size during an avalanche in the emergence of temporally asymmetric avalanche profiles.
We also consider two averaging protocols for computing the average shape, in which the amplitude of each avalanche signal is normalized before averaging either by $T^{\gamma-1}$ or by $s/T$, where $s$ and $T$ are the avalanche size and duration, respectively, and $\gamma$ is the scaling exponent relating the average avalanche size to its duration. We show that these two procedures yield slightly different average shapes and trace this difference back to the $s$-dependence of the shape at fixed $T$. Thus, the typical average shape of avalanches with fixed duration emerges as an average over size-dependent shapes.
Avalanches and mechanical noise in soft materials
Random organization is a phenomenon by which a suspension of micrometer-sized hard particles in a fluid, subjected to slow oscillatory shear of a small amplitude, eventually reaches a reversible state where all particles follow a limit cyclic trajectory. This state is very sensitive to perturbations, and shows an avalanching response: when a particle trajectory is destabilized, it further destabilizes other particles trajectories via hydrodynamic couplings that create a long-range "mechanical noise". Similarly, yielding is a phenomenon by which a yield stress fluid subjected to a shear stress close to the yield value is prone to plasticity avalanches: when some part of the system yields plastically, it further destabilizes other parts of the system via elastic stress redistribution that also create a long-range mechanical noise. In this talk, we show numerical evidence that yielding and random organization may actually be two realizations of the same phase transition. In particular, they share the same avalanche behavior: in contrast to systems with short-range interactions, which show depinning-like avalanches that are spatially compact, the long-range mechanical noise in yielding and random organization lead to sparse avalanches. Furthermore the two transitions share the same relations between critical exponents. This suggests the existence of a specific class of transitions with long-range interactions, which show a critical behavior in sharp contrast to other driven disordered systems.
Emergence of Self-Organized Criticality in Stochastic Memristive Nanowire Networks
Memristive nanowire networks (NWNs) have emerged as promising self-organizing neuromorphic systems, exhibiting rich emergent behavior arising from the collective interaction of a large number of memristive junctions [1]. Despite extensive experimental and theoretical investigations, establishing a unified framework that links these complex dynamics to the information-processing capabilities of the network remains a significant challenge. Here, we show that NWNs can be described as stochastic dynamical systems in which signal propagation emerges from the interplay between deterministic evolution and stochastic fluctuations associated with noise and switching events. Through a combined experimental and modelling approach, we demonstrate that the network dynamics can be effectively captured within a stochastic dynamical framework, providing a quantitative description of the evolution of its internal states [2]. Furthermore, we show that under specific operating conditions, the stochastic dynamics exhibit clear signatures of self-organized criticality, revealing the emergence of localized critical states within the network. Using a multi-terminal experimental approach, we identify and characterize these local critical regimes and investigate their relationship with the computational capabilities of the physical substrate within the framework of physical reservoir computing [3]. Beyond their relevance for neuromorphic computing, our results establish NWNs as versatile experimental platforms for the investigating critical phenomena in complex adaptive systems and for exploring how criticality shapes information processing in physical networks.
[1] Milano, G., et al. "In materia reservoir computing with a fully memristive architecture based on self-organizing nanowire networks." Nature materials 21.2 (2022): 195-202.
[2] Milano, G., et al. "Self-organizing neuromorphic nanowire networks as stochastic dynamical systems." Nature Communications 16.1 (2025): 3509.
[3] Michieletti, F., et al. "Self‐organized criticality in neuromorphic nanowire networks with tunable and local dynamics." Advanced Functional Materials 35.30 (2025): 2423903.
Ekhard Salje: avalanches, acoustic emission and jerky processes
Ekhard Salje was a world-leading scientist who made outstanding contributions across an exceptionally broad range of fields, including materials physics, mathematics, mineralogy, and geophysics. In materials physics, he is widely recognized for his work on ferroelastics, ferroelectrics, superconductors, and multiferroics, among many other topics. Over the past twenty years, he was particularly active in the study of avalanches, not only in connection with phase transitions in ferroelastic and ferroelectric materials, but also across a wide variety of systems, including many of geophysical and biomechanical interest. He had a profound influence on the broad community working on diverse types of avalanche phenomena and was the driving force behind the initiation of this series of workshops. In this talk, I will recall how he became deeply engaged with this subject and will review some of his most significant contributions to the field.
Emerging behavior in engineered neuronal cultures: from activity patterns to criticality
Neurons grown in vitro as neuronal cultures provide a versatile model system to investigate a wide range of questions from the perspective of the physics of complex systems. These cultures are intrinsically active, exhibiting spontaneous, network-wide action potentials in the absence of external stimulation. Such activity emerges from the interplay between intrinsic neuronal dynamics, biological noise, and the underlying connectivity. However, standard neurons cultured on flat substrates typically display quasi-synchronous activity, which fails to capture the rich dynamical repertoire observed in the brain. To overcome this limitation, we investigated whether breaking connectivity isotropy through topographical modulation of neuronal spatial organization could enrich network dynamics. We observed that neurons grown on such structured substrates exhibited diverse spatiotemporal activity patterns, ranging from localized events to culture-wide propagation, resulting in increased dynamical complexity compared to standard flat cultures. To characterize these regimes, we analyzed network activity using a phenomenological renormalization group approach, exploring the emergence of critical dynamics. Topographically structured cultures displayed non-trivial critical exponents consistent with scale invariance, supporting the criticality hypothesis in neuronal systems. In parallel, in silico networks grown on equivalent topographies reproduced similar scaling behavior, allowing a detailed investigation of structure-function relationships and highlighting the role of heterogeneous and modular architectures in sustaining critical dynamics.
Modelling earthquakes interactions from thermally activated spring-sliders
Unlike meteorological hazards, tectonic earthquakes remain hardly predictable, reinforcing their deadly character. This relates to an out-of-equilibrium, intermittent dynamic associated with a strong time asymmetry, i.e. few and non-systematic foreshocks sometimes preceding large earthquakes, while aftershocks sequences are ubiquitous and have been known for a long time. However, 130 after Omori, the physical origin of this time asymmetry and of aftershocks remains highly debated. Here, we model earthquake interactions and natural seismicity from a spring-slider model based on a minimal number of fundamental mechanism, namely elastic stress transfer and reaction rate theory applied to the simplest form of static friction. This allows introducing a microscopic timescale as well as temperature in a physically meaningful way, and to strikingly reproduce many aspects of seismicity and earthquake interactions. This includes (a) a power law distribution of seismic moments, (b) an Omori's as well as productivity laws for aftershocks, (c) a clustering of aftershocks nearby the edge of the mainshock rupture zone and (d) a strong time asymmetry of the seismic cycle, which however weakens with increasing temperature, and so depth. Future developments of the model, such as a more realistic representation of elastic interactions, or the introduction of a healing of the frictional strength of the sliders, will be also discussed.