Ergodic descriptors of non-ergodic stochastic processes

Empirical methods that provide physical descriptions of dynamic cellular processes

We review empirical methods that can be used to provide physical descriptions of dynamic cellular processes during development and disease. Our focus will be nonspatial descriptions and the inference of underlying interaction networks including cell-state lineages, gene regulatory networks, and molecular interactions in living cells. Our overarching questions are: How much can we learn from just observing? To what degree is it possible to infer causal and/or precise mathematical relationships from observations? We restrict ourselves to data sets arising from only observations, or experiments in which minimal perturbations have taken place to facilitate observation of the systems as they naturally occur. We discuss analysis perspectives in order from those offering the least descriptive power but requiring the least assumptions such as statistical associations. We end with those that are most descriptive, but require stricter assumptions and more previous knowledge of the systems such as causal inference and dynamical systems approaches. We hope to provide and encourage the use of a wide array of options for quantitative cell biologists to learn as much as possible from their observations at all stages of understanding of their system of interest. Finally, we provide our own recipe of how to empirically determine quantitative relationships and growth laws from live-cell microscopy data, the resultant predictions of which can then be verified with perturbation experiments. We also include an extended supplement that describes further inference algorithms and theory for the interested reader.

Multifractal nonlinearity as a robust estimator of multiplicative cascade dynamics

Natural and behavioral sciences increasingly model measurement time series as random multiplicative cascades-nonlinear processes that divide, branch, or aggregate structures across generations, creating multiplicative interactions that break ergodicity. Multifractal formalisms provide a framework for studying cascades, where multifractal spectrum width, Δα, fluctuates with the number of estimable power-law relationships. However, multifractality without surrogate comparison can be ambiguous, as the original measurement series' multifractal spectrum width, Δα_{original}, is sensitive to series length, ergodicity-breaking linear correlations (e.g., fractional Gaussian noise, fGn), or additive dynamics. To address these challenges, we constructed random cascades varying in length, noise type (additive white Gaussian noise, awGn, or fGn), mixtures of awGn and fGn across generations (progressively more awGn, progressively more fGn, or random mixing of awGn and fGn across generations), and noise operations (addition versus multiplication). We found that "multifractal nonlinearity," t_{MF} (a t-statistic comparing Δα_{original} to surrogate spectra width, Δα_{surrogates}), reliably identifies random multiplicative cascades regardless of series length or noise type-t_{MF} is more sensitive to interactivity and number of generations than series length. Unlike random additive cascades, random multiplicative cascades exhibit stronger ergodicity breaking due to their heavy histogram tails and show progressively greater ergodicity breaking with more correlated noise (e.g., fGn as opposed to awGn) and additional generations-independent of series length. Consequently, t_{MF} robustly identifies multiplicative cascade processes, quantifies the depth of cross-scale interactivity, and pinpoints sources of ergodicity breaking while remaining independent of ergodicity-breaking and series length. These properties highlight the potential of t_{MF} as a powerful metric for advancing our understanding of the mechanisms and origins of cascading processes in natural and behavioral sciences. The simulations further suggest its utility in extending findings from group-level analyses to the level of individuals.

Ball Don't Lie: Commentary on Chemero (2024) and Wallot et al. (2024)

The interaction-dominant approach to perception and action, originally formulated in the mid-1990s, has matured and gained remarkable momentum as an entailment of the dynamical hypotheses proposed at that time. This framework seeks to explain the fluid and intricate interplay of causality spanning the entire organism by integrating high-dimensional details with low-dimensional constraints across various scales of behavior. Both Chemero (2024) and Wallot et al. (2024) have skillfully explored the theoretical implications and methodological challenges this perspective introduces. We echo Chemero's (2024) and Wallot et al.'s (2024) focus on multifractality, while also underscoring new efforts to model the synergetic relationships and cascading dynamics inherent in this interaction-dominant approach.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Convergence and equilibrium in molecular dynamics simulations

Molecular dynamics is a powerful tool that has been long used for the simulation of biomolecules. It complements experiments, by providing detailed information about individual atomic motions. But there is an essential and often overlooked assumption that, left unchecked, could invalidate any results from it: is the simulated trajectory long enough, so that the system has reached thermodynamic equilibrium, and the measured properties are converged? Previous studies showed mixed results in relation to this assumption. This has profound implications, as the resulting simulated trajectories may not be reliable in predicting equilibrium properties. Yet, this is precisely what most molecular dynamics studies do. So the question arises: are these studies even valid?Here, we present a thorough analysis of up to a hundred microseconds long trajectories, of several system with varying size, to probe the convergence of different structural, dynamical and cumulative properties, and elaborate on the relevance of the concept of equilibrium, and its physical and biological meaning. The results show that properties with the most biological interest tend to converge in multi-microsecond trajectories, although other properties-like transition rates to low probability conformations-may require more time.

Multifractal foundations of biomarker discovery for heart disease and stroke

Any reliable biomarker has to be specific, generalizable, and reproducible across individuals and contexts. The exact values of such a biomarker must represent similar health states in different individuals and at different times within the same individual to result in the minimum possible false-positive and false-negative rates. The application of standard cut-off points and risk scores across populations hinges upon the assumption of such generalizability. Such generalizability, in turn, hinges upon this condition that the phenomenon investigated by current statistical methods is ergodic, i.e., its statistical measures converge over individuals and time within the finite limit of observations. However, emerging evidence indicates that biological processes abound with nonergodicity, threatening this generalizability. Here, we present a solution for how to make generalizable inferences by deriving ergodic descriptions of nonergodic phenomena. For this aim, we proposed capturing the origin of ergodicity-breaking in many biological processes: cascade dynamics. To assess our hypotheses, we embraced the challenge of identifying reliable biomarkers for heart disease and stroke, which, despite being the leading cause of death worldwide and decades of research, lacks reliable biomarkers and risk stratification tools. We showed that raw R-R interval data and its common descriptors based on mean and variance are nonergodic and non-specific. On the other hand, the cascade-dynamical descriptors, the Hurst exponent encoding linear temporal correlations, and multifractal nonlinearity encoding nonlinear interactions across scales described the nonergodic heart rate variability more ergodically and were specific. This study inaugurates applying the critical concept of ergodicity in discovering and applying digital biomarkers of health and disease.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Temporal organization of stride-to-stride variations contradicts predictive models for sensorimotor control of footfalls during walking

Walking exhibits stride-to-stride variations. Given ongoing perturbations, these variations critically support continuous adaptations between the goal-directed organism and its surroundings. Here, we report that stride-to-stride variations during self-paced overground walking show cascade-like intermittency-stride intervals become uneven because stride intervals of different sizes interact and do not simply balance each other. Moreover, even when synchronizing footfalls with visual cues with variable timing of presentation, asynchrony in the timings of the cue and footfall shows cascade-like intermittency. This evidence conflicts with theories about the sensorimotor control of walking, according to which internal predictive models correct asynchrony in the timings of the cue and footfall from one stride to the next on crossing thresholds leading to the risk of falling. Hence, models of the sensorimotor control of walking must account for stride-to-stride variations beyond the constraints of threshold-dependent predictive internal models.

Multifractal test for nonlinearity of interactions across scales in time series

The creativity and emergence of biological and psychological behavior tend to be nonlinear, and correspondingly, biological and psychological measures contain degrees of irregularity. The linear model might fail to reduce these measurements to a sum of independent random factors (yielding a stable mean for the measurement), implying nonlinear changes over time. The present work reviews some of the concepts implicated in nonlinear changes over time and details the mathematical steps involved in their identification. It introduces multifractality as a mathematical framework helpful in determining whether and to what degree the measured series exhibits nonlinear changes over time. These mathematical steps include multifractal analysis and surrogate data production for resolving when multifractality entails nonlinear changes over time. Ultimately, when measurements fail to fit the structures of the traditional linear model, multifractal modeling allows for making those nonlinear excursions explicit, that is, to come up with a quantitative estimate of how strongly events may interact across timescales. This estimate may serve some interests as merely a potentially statistically significant indicator of independence failing to hold, but we suspect that this estimate might serve more generally as a predictor of perceptuomotor or cognitive performance.