A fundamental problem in science is uncovering the effective number of degrees of freedom in a complex system: its dimensionality. A system’s dimensionality depends on its spatiotemporal scale. Here, we introduce a scale-dependent generalization of a classic enumeration of latent variables, the participation ratio. We demonstrate how the scale-dependent participation ratio identifies the appropriate dimension at local, intermediate, and global scales in several systems such as the Lorenz attractor, hidden Markov models, and switching linear dynamical systems. We show analytically how, at different limiting scales, the scale-dependent participation ratio relates to well-established measures of dimensionality. This measure applied in neural population recordings across multiple brain areas and brain states shows fundamental trends in the dimensionality of neural activity—for example, in behaviorally engaged versus spontaneous states. Our novel method unifies widely used measures of dimensionality and applies broadly to multivariate data across several fields of science.
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Scale-dependent halo bias from scale-dependent growth
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The failure of roughness parameters to predict surface properties stems from their inherent scale-dependence; in other words, the measured value depends on how the parameter was measured. Here we take advantage of this scale-dependence to develop a new framework for characterizing rough surfaces: the Scale-Dependent Roughness Parameters (SDRP) analysis, which yields slope, curvature, and higher-order derivatives of surface topography at many scales, even for a single topography measurement. We demonstrate the relationship between SDRP and other common statistical methods for analyzing surfaces: the height-difference autocorrelation function (ACF), variable bandwidth methods (VBMs) and the power spectral density (PSD). We use computer-generated and measured topographies to demonstrate the benefits of SDRP analysis, including: novel metrics for characterizing surfaces across scales, and the detection of measurement artifacts. The SDRP is a generalized framework for scale-dependent analysis of surface topography that yields metrics that are intuitively understandable.more » « less
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Bacteria are efficient colonizers of a wide range of secluded microhabitats, such as soil pores, skin follicles, or intestinal crypts. How the structural diversity of these habitats modulates microbial self-organization remains poorly understood, in part because of the difficulty to precisely manipulate the physical structure of microbial environments. Using a microfluidic device to grow bacteria in crypt-like incubation chambers of systematically varied lengths, we show that small variations in the physical structure of the microhabitat can drastically alter bacterial colonization success and resistance against invaders. Small crypts are uncolonizable; intermediately sized crypts can stably support dilute populations, while beyond a second critical length scale, populations phase separate into a dilute region and a jammed region. The jammed state is characterized by extreme colonization resistance, even if the resident strain is suppressed by an antibiotic. Combined with a flexible biophysical model, we demonstrate that colonization resistance and associated priority effects can be explained by a crowding-induced phase transition, which results from a competition between proliferation and density-dependent cell leakage. The emerging sensitivity to scale underscores the need to control for scale in microbial ecology experiments. Systematic flow-adjustable length-scale variations may serve as a promising strategy to elucidate further scale-sensitive tipping points and to rationally modulate the stability and resilience of microbial colonizers.more » « less
