Connecting empirical phenomena and theoretical models of biological coordination across scales (1812.00423v1)

Abstract: Coordination is ubiquitous in living systems. Existing theoretical models of coordination -- from bacteria to brains -- focus on either gross statistics in large-scale systems ($N\rightarrow\infty$) or detailed dynamics in small-scale systems (mostly $N=2$). Both approaches have proceeded largely independent of each other. The present work bridges this gap with a theoretical model of biological coordination that captures key experimental observations of mid-scale social coordination at multiple levels of description. It also reconciles in a single formulation two well-studied models of large- and small-scale biological coordination (Kuramoto and extended Haken-Kelso-Bunz). The model adds second-order coupling (from extended Haken-Kelso-Bunz) to the Kuramoto model. We show that second-order coupling is indispensable for reproducing empirically observed phenomena and gives rise to a phase transition from mono- to multi-stable coordination across scales. This mono-to-multistable transition connects the emergence and growth of behavioral complexity in small and large systems.

• The paper presents a novel second-order coupling in the Kuramoto model that bridges small-scale and large-scale biological coordination.
• The model introduces a phase transition from mono-stable to multistable dynamics, validated through human ensemble experiments with varying frequency diversity.
• The findings enable a unified framework for analyzing emergent complexities in biological and neural systems across different scales.

Connecting Empirical Phenomena and Theoretical Models of Biological Coordination Across Scales

Abstract

This research paper investigates the unification of large-scale and small-scale theoretical models of biological coordination, focusing on a model that describes mid-scale systems. Traditional models like Kuramoto for large-scale and extended Haken-Kelso-Bunz (HKB) for small-scale coordination operate independently, each capturing phenomena at their respective scales. This study introduces a second-order coupling to the Kuramoto model, effectively bridging these two approaches, capturing biological coordination phenomena across different scales. The proposed model posits a phase transition from mono- to multi-stable coordination influenced by second-order coupling, suggesting a continuum between small and large-scale coordination frameworks.

Introduction

Biological coordination, fundamental to understanding complexity in living systems, manifests in various forms, from bacterial colonies to human social groups. Existing models address either small-scale systems with a few oscillators or large-scale systems approaching infinite numbers. However, these models operate in isolation. This paper proposes an integrative model directly applicable to mid-scale coordination systems, reconciling both large- and small-scale models. This integration suggests a transition from monostable to multistable coordination and provides insights into the emergent complexity of biological systems.

Model Description

The extended model introduces second-order coupling to the classical Kuramoto model, enhancing its utility for mid-scale coordination systems. Specifically, the dynamics of an oscillator in the system are governed by:

$$\dot{\varphi}_i = \omega_i - \sum_{j = 1}^N a_{ij} \sin (\varphi_i - \varphi_j) - \sum_{j = 1}^N b_{ij} \sin 2 (\varphi_i - \varphi_j)$$

where $a_{ij}$ encapsulates the first-order coupling, while $b_{ij}$ incorporates second-order interactions. This formulation mirrors the extended HKB for small systems and can scale towards large-scale systems by adjusting $b_{ij}$. The second-order term is crucial for modeling antiphase and multistability, observed in experimental data, particularly in data from human ensemble experiments.

Human Experiment and Results

Experiments were conducted with human subjects to compare theoretical predictions against empirical observations. Participants were grouped into ensembles that interacted in rhythmic tasks, with coordination analyzed under varying frequency diversities ($\delta f$). Results indicated that with low frequency diversity, groups showed high integration; increased diversity led to group segregation. The model effectively captured this behavior, demonstrating its applicability to real-world data, specifically the transition to multistable coordination.

Analysis and Implications

The model's introduction of second-order coupling accounts for the multistability observed in human coordination, formally indicating conditions (beyond single oscillator behavior) where small group interactions can reflect complex system dynamics typical of larger scales. This suggests a unified approach to analyzing biological and neural ensemble behaviors across different scales, enhancing the understanding of how micro- and macro-level patterns emerge and stabilize.

Extension and Future Work

The model not only provides a foundation for understanding biological coordination but also poses testable predictions about system behaviors under varying coupling conditions. Future research directions include exploring system dynamical responses to perturbations and verifying theoretical predictions of critical coupling regimes in larger biological systems, such as neural networks.

Conclusion

The study effectively bridges the gap between existing models for small-scale and large-scale biological coordination, demonstrating that phenomena such as multistability emerge naturally from second-order couplings. The implications of these findings extend into practical applications in understanding and predicting system behaviors in biological and artificial intelligence systems operating across scales.