The Haken-Kelso-Bunz (HKB) model describes bistable rhythmic coordination between biological systems, capturing a wide range of empirical phenomena in brain and behavioral dynamics. The model has recently been generalized to systems of many oscillators, allowing for a quantitative study of how the dynamics of two coupled oscillators is affected by their "social environment," i.e., by a larger system of oscillators in which both are embedded. In previous work, we studied triads of identical oscillators and showed that bistability can be restored to a monostable dyad by coupling it to a third oscillator. Here, we generalize to triads of nonidentical oscillators with different natural frequencies. We show that a pair of such oscillators, whose frequency difference would cause their dynamics in isolation to be only monostable, can exhibit bistable dynamics when both are coupled to a third oscillator having an intermediate natural frequency. We discuss some applications of this work to social and neuroscience, including gerontology and healthy aging, as well as neurostimulation and the treatment of brain-based diseases, where our findings suggest intervention strategies for promoting coordination between heterogeneous components situated in larger social environments.

Abstract Social isolation threatens the health of older adults, particularly those with Alzheimer’s disease and related dementias (ADRD). Our research uses measures of behavioral coordination to quantify the entrainment of an individual’s behavior to the collective dynamics of a group to which they belong, aiming to identify useful strategies to reduce social isolation for persons with ADRD in therapeutic group contexts. The present study is based on data collected during 27, 30-minute group therapy sessions conducted biweekly via WebEx by staff facilitators at a university-affiliated Memory and Wellness Center (MWC) in 2021, when CoViD-19 restrictions limited patients’ access to existing, in-person day programs. We study conversational entrainment among facilitators and participants in the recorded sessions using previously identified measures such as speech duration; voiced and voiceless interval durations; variations in pitch, intensity, and syllabic rate; and turn breaks. We present preliminary results identifying dynamical characteristics of sessions showing greater and lesser degrees of group social engagement before discussing possible strategies facilitators could use to enhance social coordination within therapy groups. The mental health benefits of professionally facilitated supportive group discussions for older adults with ADRD are well-accepted. Measuring conversational entrainment has the potential to provide quantitative evidence of cognitive benefits for persons with ADRD, and to help shape a therapeutic environment tailored to the needs of this vulnerable population.

The Haken-Kelso-Bunz (HKB) system of equations is a well-developed model for dyadic rhythmic coordination in biological systems. It captures ubiquitous empirical observations of bistability - the coexistence of in-phase and antiphase motion - in neural, behavioral, and social coordination. Recent work by Zhang and colleagues has generalized HKB to many oscillators to account for new empirical phenomena observed in multiagent interaction. Utilising this generalization, the present work examines how the coordination dynamics of a pair of oscillators can be augmented by virtue of their coupling to a third oscillator. We show that stable antiphase coordination emerges in pairs of oscillators even when their coupling parameters would have prohibited such coordination in their dyadic relation. We envision two lines of application for this theoretical work. In the social sciences, our model points toward the development of intervention strategies to support coordination behavior in heterogeneous groups (for instance in gerontology, when younger and older individuals interact). In neuroscience, our model will advance our understanding of how the direct functional connection of mesoscale or microscale neural ensembles might be switched by their changing coupling to other neural ensembles. Our findings illuminate a crucial property of complex systems: how the whole is different than the system's parts.

The authors previously introduced a diffeomorphism-invariant definition of a homogeneous and isotropic sector of loop quantum gravity (LQG), along with a program to embed loop quantum cosmology (LQC) into it. The present paper works out that program in detail for the simpler, but still physically non-trivial, case where the target of the embedding is the homogeneous, but not isotropic, Bianchi I model. The diffeomorphism-invariant conditions imposing homogeneity and isotropy in the full theory reduce to conditions imposing isotropy on an already homogeneous Bianchi I spacetime. The reduced conditions are invariant under the residual diffeomorphisms still allowed after gauge fixing the Bianchi I model. We show that there is a unique embedding of the quantum isotropic model into the homogeneous quantum Bianchi I model that (a) is covariant with respect to the actions of such residual diffeomorphisms, and (b) intertwines both the (signed) volume operator and at least one directional Hubble rate. That embedding also intertwines all other operators of interest in the respective loop quantum cosmological models, including their Hamiltonian constraints. It thus establishes a precise equivalence between dynamics in the isotropic sector of the Bianchi I model and the quantized isotropic model, and not just their kinematics. We also discuss the adjoint relationship between the embedding map defined here and a projection map previously defined by Ashtekar and Wilson-Ewing. Finally, we highlight certain features that simplify this reduced embedding problem, but which may not have direct analogues in the embedding of homogeneous and isotropic LQC into full LQG.

Objective:We evaluated components of an integrated, mobile health-based intervention "Activate for Life" (AFL) on health outcomes in lower-income older adults (≥ 60 years). Methods:AFL incorporates balance (Otago; OG), physical strength (Gentle Yoga and yogic Breathing; GYYB), and mental engagement (Behavioral Activation; BA) components. Thirty participants were randomly allocated to one of three study arms (n=10): OG (Arm 1), OG+GYYB (Arm 2), or OG+GYYB+BA (Arm 3; a.k.a. "full AFL"). Participants were evaluated for physical, functional, and physiological endpoints at baseline and post-intervention (12-weeks and/or 3-month follow up). Results:Improvements in pain interference and 1,5- anhydroglucitol biomarker levels over time were noted for all arms. No significant changes were observed for other physical, functional, or physiological measures. Discussion:This study illustrates potential benefits of the AFL intervention on the health of lower-income older adults. Lessons learned from this pilot trial will inform design improvements for a large-scale randomized controlled trial.

Abstract Multiple causes converge for older adults to shed social relationships. Lost opportunities for social engagement are tied to weakened cognitive reserve and under-optimal aging in health and disease. For example, a woman, 75, regularly strolls with younger friends. At 80, her reduced motor fitness makes it hard to keep pace and she withdraws her participation. With same-age peers, she might continue this healthy physical and social activity a few more years by unobtrusively shortening the outing or by slowing her pace. A man, 85, loves to debate politics with family, but his turn at talks diminish: his hearing loss (sensory) prevents quick grasp of the discussion; his slower verbal fluency (cognitive) hamper quick-witted replies. Both examples illustrate that social aging is not only a ¬¬¬property of the aging individual. Social context plays an important role. Our recently formed interdisciplinary group (geropsychiatric nurse, mathematical physicist and complexity scientist) is studying the systemic complexities of social aging with experiments and mathematical models. Our aim is to present our model and aging-focused hypotheses, as well as empirical validation in younger adults. Four key variables are group size and heterogeneity, and the strength and adaptability of social coordination. Our current results show that people coordinate better with others like them in pace, but they lose the ability to coordinate with people whose pace is different. We anticipate that our program of research will deliver evidence-based recommendations on social-engineering of activities that maximize opportunities for sustained interactions among older adults.

Humans’ interactions with each other or with socially competent machines exhibit lawful coordination patterns at multiple levels of description. According to Coordination Dynamics, such laws specify the flow of coordination states produced by functional synergies of elements (e.g., cells, body parts, brain areas, people…) that are temporarily organized as single, coherent units. These coordinative structures or synergies may be mathematically characterized as informationally coupled self-organizing dynamical systems (Coordination Dynamics). In this paper, we start from a simple foundation, an elemental model system for social interactions, whose behavior has been captured in the Haken-Kelso-Bunz (HKB) model. We follow a tried and tested scientific method that tightly interweaves experimental neurobehavioral studies and mathematical models. We use this method to further develop a body of empirical research that advances the theory toward more generalized forms. In concordance with this interdisciplinary spirit, the present paper is written both as an overview of relevant advances and as an introduction to its mathematical underpinnings. We demonstrate HKB’s evolution in the context of social coordination along several directions, with its applicability growing to increasingly complex scenarios. In particular, we show that accommodating for symmetry breaking in intrinsic dynamics and coupling, multiscale generalization and adaptation are principal evolutions. We conclude that a general framework for social coordination dynamics is on the horizon, in which models support experiments with hypothesis generation and mechanistic insights.

Coordination in living systems-from cells to people-must be understood at multiple levels of description. Analyses and modelling of empirically observed patterns of biological coordination often focus either on ensemble-level statistics in large-scale systems with many components, or on detailed dynamics in small-scale systems with few components. The two approaches have proceeded largely independent of each other. To bridge this gap between levels and scales, we have recently conducted a human experiment of mid-scale social coordination specifically designed to reveal coordination at multiple levels (ensemble, subgroups and dyads) simultaneously. Based on this experiment, the present work shows that, surprisingly, a single system of equations captures key observations at all relevant levels. It also connects empirically validated models of large- and small-scale biological coordination-the Kuramoto and extended Haken-Kelso-Bunz (HKB) models-and the hallmark phenomena that each is known to capture. For example, it exhibits both multistability and metastability observed in small-scale empirical research (via the second-order coupling and symmetry breaking in extended HKB) and the growth of biological complexity as a function of scale (via the scalability of the Kuramoto model). Only by incorporating both of these features simultaneously can we reproduce the essential coordination behaviour observed in our experiment.

In this paper we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta–Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented.

This paper summarizes a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, thereby enabling a detailed comparison of results in loop quantum gravity and loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase-space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. Some additional steps, such as constructing a specific embedding of the Hilbert space of loop quantum cosmology into a space of (distributional) states in the full theory, remain incomplete. However, we also describe, as a proof of concept, a complete analysis of an analogous embedding of homogeneous and isotropic loop quantum cosmology into the quantum Bianchi I model of Ashtekar and Wilson-Ewing. Details will appear in a pair of forthcoming papers.

This note describes a local scheme to characterize and normalize an axial Killing field on a general Riemannian geometry. No global assumptions are necessary, such as that the orbits of the Killing field all have period 2 π. Rather, any Killing field that vanishes at at least one point necessarily has the expected global properties.

A formula is derived for the combined motional and gravitational Doppler effect in general stationary axisymmetric metrics for a photon emitted parallel or antiparallel to the assumed circular orbital motion of its source. The same formula is derived by both the eikonal approximation and Killing vector approaches to elucidate connections between observational astronomy and modern relativity. The formula yields expected results in the limits of a moving or stationary source in the exterior Kerr and Schwarzschild metrics and is useful for broad range astrophysical analyses.

There is a growing effort to image single neurons in vivo, and observe their individual contribution to the brain's functional organization. This effort generally relies on two-photon imaging to explore the structure and activity of cortical columns extending beneath the brain's surface. The need to protect living tissue, however, demands the introduction of coverslips and similar objects that can modify the optics of the imaging beam. This paper develops three-dimensional (3D) analytical and numerical models to characterize and correct for the resulting degradation of image quality. We have illustrated the use of these models by describing a simple, practical technique to reduce the effect of spherical aberration for in vivo two-photon fluorescence experiments.

An approximate Killing field may be defined on a compact, Riemannian geometry by solving an eigenvalue problem for a certain elliptic operator. This paper studies the effect of small perturbations in the Riemannian metric on the resulting vector field. It shows that small metric perturbations, as measured using a Sobolev-type supremum norm on the space of Riemannian geometries on a fixed manifold, yield small perturbations in the approximate Killing field, as measured using a Hilbert-type square integral norm. It also discusses applications to the problem of computing the spin of a generic black hole in general relativity.

We review our recent proposal of a method to extend the quantization of spherically symmetric isolated horizons, a seminal result of loop quantum gravity, to a phase space containing horizons of arbitrary geometry. Although the details of the quantization remain formally unchanged, the physical interpretation of the results can be quite different. We highlight several such differences, with particular emphasis on the physical interpretation of black hole entropy in loop quantum gravity.

Isolated horizons model equilibrium states of classical black holes. A detailed quantization, starting from a classical phase space restricted to spherically symmetric horizons, exists in the literature and has since been extended to axisymmetry. This paper extends the quantum theory to horizons of arbitrary shape. Surprisingly, the Hilbert space obtained by quantizing the full phase space of \textit{all} generic horizons with a fixed area is identical to that originally found in spherical symmetry. The entropy of a large horizon remains one quarter its area, with the Barbero-Immirzi parameter retaining its value from symmetric analyses. These results suggest a reinterpretation of the intrinsic quantum geometry of the horizon surface.

Approximate Killing vector fields are expected to help define physically meaningful spins for non-symmetric black holes in general relativity. However, it is not obvious how such fields should be defined geometrically. This paper relates a definition suggested recently by Cook and Whiting to an older proposal by Matzner, which seems to have been overlooked in the recent literature. It also describes how to calculate approximate Killing fields based on these proposals using an efficient scheme that could be of immediate practical use in numerical relativity.

Christopher Beetle, 2 Marco Bruni, Lior M. Burko, ∗ and Andrea Nerozzi 4 Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431 Department of Physics, University of Utah, Salt Lake City, Utah 84112 Institute of Cosmology and Gravitation, Mercantile House, Hampshire Terrace, PO1 2EG, Portsmouth UK Center for Relativity, Department of Physics, University of Texas at Austin, Austin, Texas 78712-1081 (Dated: February 6, 2008)

A set of boundary conditions defining a non-rotating isolated horizon are specified in general relativity. A space-time representing a black hole which is itself in equilibrium but whose exterior contains radiation admits such a horizon. However, the definition is applicable in a more general context, such as cosmological horizons. Physically motivated, (quasi-)local definitions of the mass and surface gravity of an isolated horizon are introduced and their properties analyzed. Although their definitions do not refer to infinity, these quantities assume their standard values in the static black hole solutions. Finally, using these definitions, the zeroth and first laws of black hole mechanics are established for isolated horizons.