Efficient Numerical Analysis in Complexity Science

In a tribute to the deep influence Peter Grassberger has always had on my work, I will survey 30 years of efforts to understand the simplest models at the core of the statistical physics of active matter. I will recall that in this field also, “clean and honest” numerical investigations have been crucial. I will explain using key examples why very large-scale simulations are still needed to understand not more complex models nor rare events, but some even basic questions in the field that remain open as of now.

We present a synchronization transition study of the locally coupled Kuramoto model on extremely large graphs. We compare regular $40^5$ 40 and $100^4$ lattice results with those of $12,000^2$ lattice substrates with power-law decaying long links (ll). The latter heterogeneous network exhibits $????_???? > 4$ spectral dimensions. We show strong corrections to scaling and mean-field type of criticality at $????=5$, with logarithmic corrections at $????=4$ Euclidean dimensions. Contrarily, the ll model exhibits a non-mean-field smeared transition, with oscillating corrections at similarly high spectral dimensions. This suggests that the network heterogeneity is relevant, causing frustrated synchronization akin to Griffiths effects.

Permutation entropy is a tool customarily used to assess the complexity of time series. The approach is based on the classification of trajectories on the basis of the mutual relative order of the recorded values, without paying attention to their actual values. I will revisit the approach, by introducing a new observable: the dispersion among all (sample) trajectories encoded by the same ordinal pattern. This variable does not only help to establish the reliability of the symbolic representation, but helps also to disentangle the entropy changes due to an increase of the embedding dimension from those due to the increased resolution of the underlying partition of the phase space. As a result, a more accurate proxy of the Kolmogorov-Sinai entropy can be defined. Furthermore, one can define a more flexible encoding procedure, distinguishing between the window length used for the identification of the proper symbols (letters of the alphabet) from the length used for the construction of the words used in the signal encoding. The quality of the representation can be visually tested, by finally inverting the encoding procedure by returning to the actually recorded variable. Most of the talk is devoted to the analysis of deterministic signals, by the role of observational noise will be also briefly discussed.

In a seminal paper "Performance of different synchronization measures in real data: A case study on electroencephalographic signals" (Phys. Rev. E, 2002), P. Grassberger with co-workers addressed a problem of phase extraction from a scalar time series via the Hilbert transform. In this talk, I show how the Hilbert transform method can be extended to achieve an exact phase from a phase-modulated scalar time series.

Understanding the nature of dynamical phase transitions in chaotic systems is a fundamental challenge in complexity science, with significant applications in fields such as neuroscience, climate research, and socio-economic modeling. The detection and characterization of critical transitions are particularly difficult when the governing equations of motion are unknown, necessitating numerical approaches based on observed time series. In this work, we present a computationally efficient methodology to measure dynamical phase transitions in time series based on the derivative of the order-$q$ Rényi entropy at $q=1$. Our approach builds upon previous theoretical results [Szépfalusy1987, Csordás1989], demonstrating that discontinuities in this entropy measure serve as reliable indicators of tipping points in complex dynamical systems. By formulating the problem within the framework of Markov processes, we derive a closed-form expression for the entropy derivative, and propose an efficient numerical algorithm for its estimation. Furthermore, our method provides insights into the statistical properties of dynamical systems by linking entropy-based measures to physical analogs such as diffusion coefficients and specific heat [Beck1993, Grassberger1983]. The proposed approach serves as a broadly applicable framework as it can integrate various symbolic representations of time series, such as ordinal partitions [Zou2019, Sakellariou2021]. We validate our method through extensive numerical experiments on well-known dynamical systems. The results confirm that the proposed entropy-based measure successfully identifies critical transitions, even in cases where traditional indicators such as variance or autocorrelation functions fail to provide a clear signal. The method exhibits robustness to noise, making it suitable for real-world data analysis. This is particularly relevant for applications in life sciences, where early detection of epileptic seizures or cardiac fibrillation relies on accurate identification of critical transitions in EEG and ECG time series [Lehnertz2023, Toker2020]. Overall, our results contribute to a more precise analysis of phase transitions in dynamical systems. The approach also advances numerical tools in complexity science by providing a scalable, theoretically grounded methodology. By bridging the gap between theoretical formalism and practical applications, it offers a versatile framework for analyzing dynamical transitions in time series. [Szépfalusy1987] Szépfalusy, P., Tél, T., Csordás, A., \& Kovács, Z. (1987). Phase transitions associated with dynamical properties of chaotic systems. Physical Review A, 36(8), 3525. [Csordás1989] Csordás, A., \& Szépfalusy, P. (1989). Singularities in Rényi information as phase transitions in chaotic states. Physical Review A, 39(10), 4767. [Zou2019] Zou, Y., Donner, R. V., Marwan, N., Donges, J. F., \& Kurths, J. (2019). Complex network approaches to nonlinear time series analysis. Physics Reports, 787, 1–41. [Sakellariou2021] Sakellariou, K., Stemler, T., \& Small, M. (2021). Estimating topological entropy using ordinal partition networks. Physical Review E, 103, 022214. [Lehnertz2023] Lehnertz, K., Bröhl, T., \& von Wrede, R. (2023). Epileptic-network-based prediction and control of seizures in humans. Neurobiology of Disease, 181, 106098. [Toker2020] Toker, D., Sommer, F. T., \& D’Esposito, M. (2020). A simple method for detecting chaos in nature. Communications Biology, 3, 11. [Beck1993] Beck, C., \& Schögl, F. (1993). Thermodynamics of Chaotic Systems: An Introduction. Cambridge University Press. [Grassberger1983] Grassberger, P., \& Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D: Nonlinear Phenomena, 9(1-2), 189-208.

Why do societies collapse? Some famous examples include Easter Island, the Maya, the Roman Empire, and the Chinese dynasties. The presentation will focus on mathematical models for these historical cases and the methods by which the models can be developed. Particular emphasis will be placed on automated equation discovery methods and their role in uncovering social dynamics. Otherwise, the topics will include a historical background on collapse, the role of complexity and feedback mechanisms, the modelling methodology and criteria, how time scales matter, why networks of societies can be more sustainable and some general lessons for modern society.

Recent years have witnessed significant advances in nanopore technology and DNA sequencing. The translocation of DNA in biological processes involves many constraints; among them, the pore through which it translocates is worthy of attention. We construct two different confining walls, cone-shaped and cylindrically shaped, and investigate the denaturation of DNA molecules during various stages of translocation in a thermal ensemble. We study the effect of cone angle and pore diameter on the melting profile of a DNA molecule in a thermal ensemble. Similarly, for the cylindrically shaped confinement, we investigate the effect of pore diameter on the melting profile of a DNA molecule. By varying the fraction of pairs on either side of a pore, we study the melting behaviour of DNA molecules in these two confinement geometries.

Hallmarks of criticality, such as power-laws and self-similarity typical for order-disorder phase transitions, have been found in many complex systems in nature ranging from fault networks to biological networks. In particular, biological neuronal systems are often thought to be organized in such a way that they optimize their information processing and storage capabilities. This is supported by studies, which have indicated that neuronal networks in vivo and in vitro can self-organize to a critical state. In this talk, I will critically review some of the evidence for criticality — potentially emerging not in the form of a critical point but as an extended critical region or Griffith’s phase — both from a data point of view and a theoretical modelling point of view. The focus will be on the failure of rock samples and fluid-induced seismicity, such as encountered in enhanced geothermal systems and hydraulic fracturing, as well as information processing and memory consolidation in neuronal systems and the spreading of zoonotic diseases.

The emergent dynamics of collective cellular movement are typically considered system-specific, depending on the nature of cell-cell interactions and motility mechanism. Despite the diversity of interactions and mechanisms across biological systems, we find a universal feature in active flows of cells. Combining experiments, statistical analysis, and numerical simulations, we show that flows generated by the collective motion of four different cell types exhibit robust scale and conformal invariance, all described by the Schramm-Loewner Evolution SLE6, which corresponds to the random percolation universality class. Thus, the flows of living biological matter exhibit universal translational, rotational, and scale symmetries independent of the microscopic properties, and can be used to test experimentally predictions from the theories for conformally invariant structures.

One of the groundbreaking results in nonlinear dynamics in the last century was the understanding of the emergence of collective behavior and synchronization in coupled oscillators. It helped to understand the dynamics of oscillatory networks in various disciplines and on all scales, from micrometer-scale nanoelectromechanical systems to power grids. Yet, the dynamics of oscillatory networks still raises important, unsolved problems, such as how high-dimensional incoherent states emerge from low-dimensional collective dynamics. This problem touches on one of the big problems in physics, namely how extensive dynamics in space and time emerge from intensive, ordered states. In my talk I will approach this question using a prototypical model of coupled oscillators, namely globally coupled Stuart-Landau oscillators. I will demonstrate two different pathways in which the oscillators may diversify from synchronized states to high-dimensional dynamics, both of which are connected to bifurcation cascades and a hierarchical organization of space.

Stochastic processes with long-range dependent correlations naturally emerge in many systems when degrees of freedom are integrated out, apart from the dynamic of the (tracer) particle of interest. In non-equilibrium situations, the resulting overdamped dynamics often corresponds to fractional Brownian motion (FBM). In disordered systems the observed displacement probability density is often non-Gaussian, and FBM-type processes display scaling exponents varying in time or space. This talk introduces diffusion models with stochastically and deterministically varying diffusion coefficients and scaling exponents. Apart from the more traditional Mandelbrot-van Ness formulation of FBM, Levy's non-equilibrium approach via a fractional integral will also be discussed. Various applications to experimental data will be introduced. Finally, a variance-gamma subordinated FBM will be introduced, leading to fractional Laplace motion with far exponential tails in the probability density function.

Fractional Brownian motion (FBM), a Gaussian, self-similar process characterized by a constant Hurst exponent, is a canonical model of power-law correlated dynamics and has found utilty in diverse systems including biology, hydrology and economics. However, studies of financial time series and recent single particle tracking experiments in biology show that the canonical FBM is insufficient to explain the observations and there is the need to consider generalizations of FBM called multifractional Brownian motion (MBM) characterized by Hurst exponents that change in time along the sample paths. Here we consider two models of MBM and compare with the canonical FBM. We employ Bayesian inference and by analyzing simulated trajectories demonstrate its efficacy in inferring the correct model parameters as well as the model which was used to generate the trajectories. Our results from the analysis of stock price data reveal crucial insights about financial markets and underscore the broad applicability of Bayesian analysis with multifractional processes.

We recently devised a fast, hierarchical, and adaptive algorithm for Metropolis Monte Carlo simulations of systems with long-range interactions that reproduces the single-spin flip dynamics of a standard implementation exactly and hence also allows nonequilibrium studies. The power of our method is demonstrated for algebraically interacting long-range Ising and XY spin models and a Lennard-Jones particle system in two dimensions where we observe speedup factors larger than 10^4. The average complexity is compatible with O(N log N), where N is the number of spins or particles. As physical applications recent nonequilibrium coarsening and aging studies of the long-range Ising model will be discussed.

How does the nonlinear dynamics of networked systems respond to external time-dependent perturbations? As direct numerical simulations alone are unsuitable to comprehensively explore high-dimensional multi-parameter systems such as networks, theory combined with computer algebra and numerical analysis needs to help out. We start by showing how first and second order perturbation theory helps predicting local responses to fluctuations. Tipping points, indicating the transition to non-local responses, however, are not predictable by standard perturbation theory of any finite order. We thus propose a self-consistency condition in terms of harmonic balance equations, yielding algebraic expressions for the tipping point. Yet, the number of coupled nonlinear constraint equations rapidly grows with system dimension, the order of the harmonic approximation and the disparity of the frequency content of the driving signal. We illustrate how a combination of theory, computer algebra and numerical approaches may help making progress in predicting tipping points. Joint work with Moritz Thuemler, Malte Schroeder, Seungjae Lee, Marisa Fischer, Julian Fleck, Gwendolyn Quasebarth, Georg Boerner and others. The work has been partially supported by a DFG Koselleck grant no. TI 629/14-1.

This work investigates ordering and self-organization in an XY model composed of magnetic dipoles, focusing on the spontaneous generation of structures such as vortices and anti-vortices through computational simulations. It explores how simple microscopic interactions lead to large-scale complexities and how spin alignment at low temperatures can be interpreted as the system's effort to minimize its Helmholtz free energy, in accordance with the second law of thermodynamics. We study the dynamics of the system while approaches to the equilibrium, focusing on the study of how good are some quantitative definitions of statistical complexity proposed in literature. This work contributes to a better understanding of the dynamics and thermodynamics of complex systems, offering interdisciplinary perspectives for studying emergent phenomena, phase transitions, and self-organization.

Authors: Friederike Schmid, Jude Vishnu, Xinxiang Chen ---------------------------------------------------------- Swollen polymer networks such as hydrogels are interesting for a variety of applications due to their high sensitivity to small stimuli. In the talk, we will first consider reversible gels that form via reversible crosslinking of long chains by shorter linker chain. Our system is designed to mimic Protein-RNA mixtures that interact by specific binding processes. Depending on the fraction of binding sites per chain, the specific interactions can induce both phase separation, and gelation without phase separation. We investigate the two possibilities and compare our simulation results with the prediction of a recent mean-field theory due to Danielson, Semenov, and Rubinstein (Danielsen et al, Macromolecules 56, 5661 (2023). Interestingly, nonuniformity of sticker distributions has only little influence on the global phase separation, although the sticker sequence influences the internal structure of the denser phase. In the second part of the talk, we will discuss irreversibly crosslinked regular polymer networks. Specifically, we consider a special class of polymer gels that include as light-driven molecular motors as crosslinking units. These gels contract under irradiation with light caused by irregular twisting of the chains, which makes them interesting for potential applications as artificial muscles. We use molecular simulations to investigate the molecular origin of the contraction and find unexpected universal behavior, which may explain some intriguing experimental observations. References: [1] Sol-gel transition in heteroassociative RNA-protein solutions: A quantitative comparison of coarse-grained simulations and the Semenov- Rubinstein theory X. Chen, J. A. Vishnu, P. Besenius, J. König, F. Schmid, Macromolecules, accepted (2025). doi: 10.1021/acs.macromol.4c03065 [1b] X. Chen et al, in preparation. [2] Scalable approach to molecular motor-polymer conjugates for light-driven artificial muscles X. Yao, J. A. Vishnu, C. Lupfer, D. Hoenders, O. Skarsetz, W. Chen, D. Dattler, A. Perrot, W-Z. Wang, C. Gao, N. Giuseppone, F. Schmid, A. Walther, Advanced Materials 36, 2403514 (2024). doi: 10.1002/adma.202403514 [2b] J. Vishnu et al, in preparation.

Epilepsy is one of the most common serious neurological disorders, affecting approximately 65 million people worldwide. Epilepsy is nowadays conceptualized as a network disease with functionally and/or structurally aberrant connections on virtually all spatial scales. All constituents of large-scale epileptic networks can contribute to the generation, maintenance, spread, and termination of even focal seizures as well as to the many pathophysiologic phenomena seen during the seizure-free interval. An improved characterization of the spatial-temporal brain dynamics in epilepsy could be achieved with tools from nonlinear dynamics, statistical physics, synchronization and network theory. I will summarize these research findings that already have opened promising directions for the development of new therapeutic possibilities and will discuss the impact on computational models of epileptic phenomena.

This talk will be centered around the concept of transfer entropy, as introduced some 25 years ago. An insider's recollection will be presented that outlines the irritation that eventually led to its definition. An outsider's view will then be given of what happened in the years after, including some theoretical advances and applications from fields as diverse as brain science, finance, social sciences and music. Some remaining issues and limitations will be discussed as well. A few more or less promising ideas for further research will conclude the journey.

We tackle the quantification of synchrony in globally coupled populations. Furthermore, we treat the problem of incomplete observations when the population mean field is unavailable, but only a small subset of units is observed. We introduce a new order parameter based on the integral of the squared autocorrelation function and demonstrate its efficiency for quantifying synchrony via monitoring general observables, regardless of whether the oscillations can be characterized in terms of the phases. Under the condition of a significant irregularity in the dynamics of the coupled units, this order parameter provides a unified description of synchrony in populations of units of various complexities. The main examples include noise-induced oscillations, coupled strongly chaotic systems, and noisy periodic oscillations. The most significant advantage of our approach is its ability to infer and quantify synchrony from the observation of a small percentage of the units and even from a single unit, provided the observations are sufficiently long. Next, we extend our approach to quantify synchrony in a large ensemble of interacting neurons from the observation of spiking events. In a simulation study, we efficiently infer the synchrony level in a neuronal population from a point process reflecting the spiking of a small number of units and even from a single neuron. We introduce a synchrony measure (order parameter) based on the Bartlett covariance density; this quantity can be easily computed from the recorded point process. This measure is robust concerning missed spikes and, if computed from observing several neurons, does not require spike sorting. We illustrate the approach by modeling populations of spiking or bursting neurons, including the case of sparse synchrony.

The relative evolution of two initially close trajectories $\mathbf{x}_1$ and $\mathbf{x}_2$ encodes information about the underlying dynamical system. In this context the inter-orbit cross correlation, $ C_{12}(t) = \left\langle (\mathbf{x}_1(t)-\boldsymbol{\mu})\cdot (\mathbf{x}_2(t)-\boldsymbol{\mu}) \right\rangle/s^2$, can be used for efficiently testing for chaos, together with pairwise distance $D_{12}=|\mathbf{x}_1(t)-\mathbf{x}_2(t)|$. Here $\boldsymbol{\mu}$ and $s^2$ denote the long-term mean and variance of bounded orbits, characterizing the respective attractor, such as fixpoints, limit cycles and strange attactors. For long times, $L_{12}$ scales linearly with the initial distance for limit cycles, but not for chaotic attractors. For the latter, the cross correlation can be used to distinguish between full and particially preditive chaos. An overview of recent and historical developments is given.

Even-offspring directed percolation (also called "even-offspring branching-annihilating random walks" or "directed Ising model") is one of the few counter examples to the Janssen-Grassberger conjecture, since it has two degenerate absorbing states. Previous simulations suggested that the exponent $\eta$ controlling the growth of the number of active sites in the "even" sector is exactly zero, but the dynamical exponent $z$ and the order parameter exponent $\beta$ are not rationals (the latter by about 4 standard deviations). We show that $\beta = 1.000 \pm 0.002$, which leaves only one exponent in this model which is not a simple rational.

The Manhattan lattice is a square directed lattice with directions alternating east-west and north-south, as in the streets of Manhattan. Single clusters grown on sites of this lattice follow statistics of the universality class of ordinary percolation, but the Strongly Connected Component (SCC), comprised of pairs of points connected to each other along different paths and in opposite directions, fall into a different universality class, with fractal dimension $d_f = 1.803$ and $\tau = 2.105$. We also carried out precise studies on the diode model studied by de Noronha et al. [Phys. Rev. E {\bf 98}, 062116 (2018) , the L lattice, and the ice lattice, and find good evidence that all these models fall within the ordinary percolation universality class for single-cluster growth, and that the SCC clusters are all in the same universality class as that of the Manhattan lattice above. We also discuss the comparison to regular, two-direction directed percolation on the square lattice. Work with Peter Grassberger.

Thin polymer films have attracted much attention due to their scienstific and technological relevance as e.g. supporting media in tissue engineering or membranes in separation processes. Thus a simulation guided insight in properties is highly desirable. However, equilibrating free-standing films of highly entangled polymer melts is a challenge for computer simulations. We approach this problem by first studying polymer melts based on a very soft-sphere coarse-grained model confined between two repulsive walls. Then we successively insert more fine grained polymer representations until the underlying microscopic details of the employed bead-spring model are reached [1]. We use a version of the model, which allows for free surfaces and a systematic investigation of the glass transition [2]. For confined films, perfect scaling of the thickness-dependent linear dimension of chain conformations is obtained. The relation between the chain length and the glass transition temperature Tg for bulk polymer melts follows the Fox–Flory equation. For films, no further confinement induced chain length effect is observed. Tg decreases and is well described by Keddie's formula, where the reduction is more pronounced for free-standing films. The deviation from bulk Tg at the characteristic film thickness is related to the intrinsic properties of polymer melts such as the entanglement density and bond orientation [3]. Starting from the fully equilibrated free-standing thick films of highly entangled polymer melts, we study void formation in biaxially expanded films. The void growth is inhibited by entanglement constraints and the resulting thin porous films are easily stabilized by cooling. Motivated by these simulations, a similar process is successfully applied experimentally on films of highly entangled polystyrene. Our results show excellent qualitative and even semi-quantitative agreement [4,5]. [1] H.-P. Hsu and K. Kremer, J. Chem. Phys. 153, 144902 (2020). [2] H.-P. Hsu and K. Kremer, J. Chem. Phys. 150, 091101 (2019). [3] H.-P. Hsu and K. Kremer, J. Chem. Phys. 159, 071104 (2023). [4] H.-P. Hsu, M. K. Singh, Y. Cang, H. Therien-Aubin, M. Mezger R. Berger, I. Lieberwirth, G. Fytas, and K. Kremer, Adv. Sci. 2207472 (2023). [5] H.-P. Hsu and K. Kremer, Macromolecules 57, 2998-3012 (2024).

Space weather is driven by the interaction of the sun’s expanding atmosphere (the solar wind) with earth’s own magnetic and plasma environment. Its impacts include power loss, aviation disruption, communication loss, and disturbance to satellite systems and are becoming increasingly important as our society relies increasingly on being globally interconnected. We are now moving into a data-rich era of solar terrestrial physics with multiple, inhomogeneous, satellite and ground based observations and there are common approaches to characterization and visualization with other fields such as earth climate. A key impact is ground perturbation to magnetic fields and induced currents that can adversely affect power grids. Spatially irregular observations such as the SuperMAG 100+ ground based global magnetometer observations can be tested for spatio-temporal patterns of correlation using dynamical networks. Whilst networks are in widespread use in the data analytics of societal and commercial data, there are additional challenges in their application to physical time-series. We are able to construct dynamical networks direct from SuperMAG. The transient dynamics of the auroral current system (its space weather) is captured by the spatio-temporal patterns of correlation between the magnetometer time-series and can be quantified by (time dependent) network parameters. Power grid vulnerabilities can be quantified from modelled and observed ground induced currents. We can then characterize detailed spatio-temporal pattern with a few parameters, so that many events can be compared with each other and with differing theoretical predictions for the system evolution.

Dynamic simulations with many degrees of freedom play a decisive role in understanding complex systems. However, the computing power requirements are often very large due to the high dimensionality and complex interactions that have to be modelled. As a result, actual simluations are often performed based on approximations, which naturally leads to inaccuracies. To solve this problem, we introduce a data-driven approach to approximate interactions between degrees of freedom that are not of direct interest, thus significantly reducing computational costs. Using a semiconductor laser as an example, we demonstrate the superiority of this method over conventional analytical approximations, both in terms of accuracy and efficiency. Our approach reduces the effort for simulations and shows promising accuracy for the chosen system of a semiconductor laser with complex carrier dynamics.

Brain studies mainly focus on the anatomical wiring of the brain and its topological qualities to ascertain the exact relationship between anatomical structure and function. In this study, we examine the weighted degree distributions of representative large connectomes. We find that the behavior of the node strength (weighted degree) distribution varies according to the scale under consideration. The distributions exhibit power-law behavior on a global scale, with an approximately universal exponent near 3. However, this tendency deviates at the local level as the node strength distributions follow the log-normal distribution suggestive of underlying random multiplicative processes which support non-locality of learning in a brain near the critical state. Furthermore, to gain insight on the dynamics that led to this observed scale-free behavior of the global weight distributions of connectomes, we impose an avalanche-based model with Hebbian learning over Hierarchical modular network (HMN) architecture to simulate the dynamic process of reinforcement [weakening] of preferred [unused] pathways. HMNs recover key features of neuronal activity, making them good representative baseline structures for modeling cortical activity. Via targeted link creation and/or reinforcement for dynamically activated sites in a sandpile-like avalanche, and a random weakening and/or pruning during stasis times, we recover edge weight distributions that closely resemble the empirical connectome edge weight distributions.

Complex systems often exhibit fluctuating noisy behavior with randomly widespread responses that critically challenge real-time inference and control. We present a unified framework based on dynamics-informed neural networks (DINNs) that integrates domain-specific dynamical models into data-driven architectures to enhance predictive accuracy and control. For inference, we apply this approach to the deconvolution of epidemiological time series, enabling super-resolution detection of regime shifts by identifying the precise timing of nonpharmaceutical interventions from COVID-19 death records at single-day resolution [1]. For control, we show that biological systems can improve real-time environmental tracking by adapting not directly to an optimal state but to an “actionable target”—a strategy that integrates current conditions with their rate of change. This strategy enables robust adaptation without requiring explicit future predictions, even under noisy and delayed sensing conditions [2]. By embedding domain-specific dynamics and regularization strategies into neural networks, our methodology bridges the gap between inference and actionability, offering promising applications in adaptive biomolecular circuit design and real-time public health decision-making. References: [1] Vilar, J. M. G., & Saiz, L. (2023). Dynamics-informed deconvolutional neural networks for super-resolution identification of regime changes in epidemiological time series. Science Advances, 9(28), eadf0673. https://www.science.org/doi/10.1126/sciadv.adf0673 [2] Vilar, J. M. G., & Saiz, L. (2025). Actionable forecasting as a determinant of biological adaptation. Advanced Science, in press. Available at https://arxiv.org/abs/2404.07895.

We present the dynamic crossover of Ising spins coupled to nonlinear growing interfaces in terms of the detailed statistical properties of Ising degrees of freedom coupled to Kardar-Parisi-Zhang (KPZ) growing interfaces. In particular, we addresse the questions of whether interfering directed paths in 1+1 dimensions can develop sign order, the so-called {\em sign phase transition}, and whether the reconstruction order can exist in the steady states of KPZ growing interfaces. Based on extensive Monte Carlo simulations and heuristic arguments, we rule out the existence of such orders in the thermodynamic limit, and identify the finite-size scaling exponent governing the dynamic crossover between quasi-critically fluctuating Ising domains and mesoscopic coarsening ones, where surface and domain reconstructions also occur due to the rescaling of lattice constant by the effective domain size. Finally, we clarify why the dynamic crossover seems to appear a phase transition for finite experimental systems that was often claimed.

The superstatistics concept, introduced some 22 years ago in [1], is a useful general method borrowed from statistical physics to describe driven nonequilibrium systems in spatio-temporally inhomogeneous environments that exhibit fluctuations of one or several intensive parameters. The method can be quite generally applied to heterogeneous complex systems if there is time scale separation of the underlying dynamics. After a brief introduction to the basic ideas [1,2], I will concentrate onto some recent applications for power grids with a higher fraction of sustainable energy generation and with correlated behaviour of the electricity consumers [3-5]. Computational methods in this context often employ machine learning methods. [1] C. Beck and E.G.D Cohen. Superstatistics. Physica A, 322, 267 (2003) [2] C. Beck. Statistics of 3-dimensional Lagrangian turbulence. Phys. Rev. Lett., 98, 064502 (2007) [3] B. Schaefer, C. Beck, K. Aihara, D. Witthaut and M. Timme. Non-Gaussian power grid frequency fluctuations characterized by Levy-stable laws and superstatistics. Nature Energy, 3, 119 (2018) [4] L. Rydin Gorjão, R. Jumar, H. Maass, V. Hagenmeyer, G.C. Yalcin, J. Kruse, M. Timme, C. Beck, D. Witthaut, B. Schaefer, Nature Communications 11, 6362 (2020) [5] M. Anvari, E. Proedrou, B. Schaefer, C. Beck, H. Kantz, M. Timme, Nature Communications 13, 4593 (2022)

The growing collection of paleoclimate records is a rich source of time-series data about the past history of the Earth system, but analysis of these records poses some serious challenges. The timelines for the data are reconstructed, often imperfectly, and the samples are not evenly spaced in time. The records undergo complex post-deposition processes, ranging from diffusion to mechanical deformation, laboratory error, and mathematically suspect post-processing, that deform and alter the data. Corroborating traces are almost never available because of the difficulty and expense involved in collecting and analyzing a core from a polar ice sheet or the deep ocean. Not long after the advent of nonlinear time-series analysis, there were some efforts [1,2] to perform delay reconstruction on paleorecords and establish the existence of a "climate attractor." This work produced conflicting results---and some controversy about their validity because of the inadequacy of the data. Forty years later, the length and resolution of paleorecords have improved vastly; even so, as I will show, these data do not support conclusive calculations of dynamical invariants. While the jury is still out regarding the existence of a climate attractor, there are other ways to study the dynamics that are captured in paleorecords. One can, for example, use Shannon entropy-based measures to study how information propagates forward in time through these traces. Although this approach does not get at the state-space structure in the classic nonlinear dynamics sense---let alone provide mechanistic explanations for the underlying drivers---it is still quite useful. Estimates of the Shannon entropy rate of water-isotope data from the West Antarctica Ice Sheet (WAIS) Divide ice core, for instance, reveal a variety of meaningful patterns in these records, ranging from data-processing issues [3] to interesting climate signals [4]. Among other things, this information measure correlates with accumulation at the drilling site and may record the influence of geothermal heating effects in the deepest parts of the core. Abrupt warming events during the last glacial period, however, do not appear to leave strong signatures in the information record, suggesting that these events may actually be predictable features of the climate’s dynamics. [1] Nature 311:529 (1984) [2] Nature 323:609 (1986) [3] Entropy 12:931 (2018) [4] Chaos 10:101105 (2019)

Recent research challenges the traditional view that the brain is composed of densely connected neural networks, suggesting instead that it relies on highly sparse connectivity. This "low connectivity" structure, combined with neurons interacting through discrete synaptic pulses (often referred to as "shot noise"), challenge earlier theories [1, 2] that used diffusion approximations [3] (i.e., continuous white Gaussian noise) to model synaptic inputs to the neurons. In this work, we present a rigorous mean-field theory that encompasses sparsity in the connections and synaptic shot noise. Our theory [4] predicts new dynamical regimes, not captured by classical diffusion approximations. Notably, these regimes feature self-sustained (global) neural oscillations emerging at low connectivity and strong synaptic coupling. These neural oscillations exhibit distinct features when compared to the ones predicted by diffusion-based approximations [2]. For low (high) excitatory drive (inhibitory feedback) global oscillations emerge via continuous or hysteretic transitions, correctly predicted by our approach, but not from the diffusion approximation. At sufficiently low in-degrees the nature of these oscillations changes from drift-driven to cluster activation. Furthermore, the latter ones could correspond to the γ rhythms reported in mammalian cortex and hippocampus and attributed to ensembles of inhibitory neurons sharing few synaptic connections [5]. REFERENCES [1] C. van Vreeswijk and H. Sompolinsky, Chaos in neuronal networks with balanced excitatory and inhibitory activity, Science 274 (1996) 1724 (1996). [2] N. Brunel and V. Hakim, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural computation 11 (1999) 1621 (1999). [3] R. Capocelli and L. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybernetik 8 (1971) 214. [4] D.S. Goldobin, M. Di Volo, and A. Torcini, Discrete synaptic events induce global oscillations in balanced neural networks, Physical Review Letters, 133 (2024) 238401. [5] G. Buzsáki and K. Mizuseki, The log-dynamic brain: how skewed distributions affect network operations, Nature Reviews Neuroscience 15 (2014) 264.

Connecting the different levels of the hierarchy of complexity in which climate models operate, and comparing the assumptions that apply at each level, has led to much progress in climate science and continues to provide a challenge to numerics. A particularly notable success was Klaus Hasselmann’s use of Brownian motion to inspire his linear Markovian stochastic energy balance model (EBM), the history of which was recently summarised by me (Watkins [2024]). Another informative, but lateral, connection and comparison is that between either studying climate through the lens of stochastic physical models and doing so via statistical methods. This presentation showcases how comparing these approaches can sometimes surprise us. It has been asserted that because the Hasselmann stochastic EBM has a mean-reverting term due to feedbacks, this property must also be detected in global mean temperature time series by statistical models such as the well-known Box-Jenkins ARIMA family. Conversely its absence has been taken as an indication of fundamental difficulties with anthropogenic driving. By fitting Hasselmann models, with and without anthropogenic driving, to an ARFIMA model with automatically selected parameters I will show that in this instance the absence of a prominent autoregressive term has quite the opposite meaning, and is instead be a clear indication of strong driving. I will also report preliminary thoughts about the extent to which the presence of long range memory due to the multiple time scales present in the coupled ocean-atmosphere can affect the above conclusions, updating the work reviewed by Watkins et al [2024]. I thank Nick Moloney and Dave Stainforth for collaboration and many insightful suggestions. Watkins, N. W., "Brownian motion as a mathematical superstructure to organise the science of climate and weather", In Foundational Papers in Complexity Science, Volume 3, pp. 1481–1510. Edited by David C. Krakauer. Santa Fe, NM: SFI Press. DOI: 10.37911/9781947864542.51 (2024). Watkins, N. W., R. Calel, S. C. Chapman, A. Chechkin, R. Klages and D. Stainforth, The Challenge of Non-Markovian Energy Balance Models in Climate. Chaos. 34, 072105 . DOI:10.1063/5.0187815 (2024).