What Is Coordination Dynamics?
How mathematics, biology, and meaning converge in a universal theory of self-organization.
Sungchul Ji, Ph.D.
Emeritus Professor of Theoretical Cell Biology
Ernest Mario School of Pharmacy, Rutgers University, Piscataway, NJ
October 12, 2025
1. A New Science of How Things Work Together
What do neurons firing in synchrony, dancers moving in rhythm, and quantum systems resonating in coherence have in common?
They all reveal a hidden order — coordination — that emerges when independent entities interact. The study of this order, its laws, and transitions between patterns is known as Coordination Dynamics (CD).
Born in the 1980s at the intersection of neuroscience, physics, and complex systems theory, coordination dynamics has evolved into one of the most profound frameworks for understanding how mind, matter, and meaning align in time.
2. The Discovery: Rhythm and the Law of Transition
In the early 1980s, physicist Hermann Haken and neuroscientist J.A. Scott Kelso collaborated to explore a simple but fascinating problem:
How do two hands move in coordination?
When people rhythmically tap or swing their hands, two stable patterns emerge spontaneously:
In-phase — both hands move together (0° phase difference).
Anti-phase — one hand moves opposite to the other (180° difference).
But as the movement frequency increases, the anti-phase pattern suddenly collapses, and the system shifts to in-phase motion. No command from the brain is needed — the transition occurs spontaneously.
This spontaneous shift is a phase transition — the same kind of behavior seen in magnets, chemical reactions, and lasers.
Kelso, Haken, and Bunz captured this phenomenon in a mathematical model that became the foundation of coordination dynamics.
3. The Mathematical Core: The HKB Equation
At the heart of the theory lies a simple yet powerful equation:
where:
= relative phase between two oscillators (e.g., two hands, two neurons)
= rate of change of phase difference
= control parameters describing coupling strength and stability
This equation describes how patterns of coordination emerge, stabilize, or collapse as the parameters change. It predicts bifurcations — sudden transitions between stable coordination states.
It is, in effect, a law of coordination, analogous to Newton’s law for motion but for relations rather than objects.
4. Coordination as a Universal Principle
Kelso realized that the same dynamics appear everywhere: in motor behavior, neural oscillations, speech, music, animal group motion, and even social interactions.
In every case, individual agents (neurons, limbs, or people) are coupled through information exchange, forming patterns that persist until parameters change. These patterns are known as attractors in the language of dynamical systems.
Thus, coordination dynamics provides a mathematical grammar for cooperation — a way to describe how wholes emerge from interacting parts.
5. The Broader Framework
Table 1. The Universality of Coordination Dynamics
Domain
Coordinating Elements
Mathematical Representation
Physics
Coupled oscillators
Phase equations, bifurcation theory
Biology
Cells, organs, organisms
Nonlinear coupling, self-organization
Neuroscience
Neuronal populations
Network synchronization, metastability
Psychology
Perception–action systems
Information flow, phase transitions
Society
Individuals in communication
Game dynamics, coupled agents
Across these scales, the same mathematical motifs recur: coupling, feedback, bifurcation, and metastability — the tendency of systems to hover between order and chaos, allowing both stability and flexibility. Figure 15.16 below was reproduced from [6].It shows the coordination dynamics at different scales lead to different self-organized systems of Prigogine, driven by their unique driving-forces.
6. Metastability: Order at the Edge
One of Kelso’s most profound contributions is the idea of metastability — a dynamic balance between competing tendencies to synchronize and to act independently.
Metastable systems are neither frozen in order nor lost in chaos. They are alive, continuously forming and dissolving patterns.
The human brain, for example, operates at metastable states: constantly coordinating billions of neurons without ever locking them rigidly together. This “liquid coherence” enables creativity, adaptability, and consciousness.
7. Mathematics Meets Meaning
The beauty of coordination dynamics is that it quantifies interaction, not just objects.
In this sense, it is not merely a model of motion but a model of relationship — a bridge between physics and phenomenology.
Information flow between agents can be represented as phase synchronization, where coordination itself carries meaning. Thus, meaning is no longer an afterthought but an emergent property of dynamics.
This makes coordination dynamics a precursor to biosemiotics, complex systems biology, and even theoretical consciousness studies.
8. Triadic Mapping: Coordination Dynamics and the Geometry of Reality
The structure of coordination dynamics naturally forms a triad, perfectly aligning with the Geometry of Reality (GOR) and Gnergy Principle of Organization (GPO).
Below is a diagram showing this alignment:
Diagram description: at the bottom left, a blue circle labeled “Matter (Energy)”; bottom right, a green circle labeled “Mind (Information)”; and at the top center, a gold circle labeled “Coordination (Meaning/Organization)”. Arrows connect all three, forming a dynamic triangle.
9. Triadic Correspondence
In Table 2 below, coordination dynamics becomes the mathematical expression of what the Gnergy and IRVSE (Iterative Reproduction with Variation and Selection by Environment) frameworks describe conceptually:
Iterative reproduction (oscillation),
variation (phase shifts), and
selection by environment (bifurcation) — a universal triad of becoming.
Table 2. The Isomorphism between GOR (Geometry of Reality) and Coordination Dynamics.
GOR Axis
Coordination Dynamics Element
Function
X (Matter / Energy)
Physical oscillators, control parameters
Substrate of dynamics
Y (Mind / Information)
Phase relationships, coupling terms
Communication / information exchange
Z (Spirit / Meaning / Environment)
Metastability, bifurcation thresholds
Context and purpose of coordination
10. Links to Modern Theories
Coordination dynamics resonates with other integrative frameworks:
Synergetics (Haken, 1977) — the study of pattern formation and order parameters.
Free Energy Principle (Friston, 2010) — the brain as a self-organizing inference system minimizing surprise.
Process Philosophy (Whitehead) — reality as relational and dynamic rather than substance-based.
Biosemiotics — life as communication through signs and interpretation.
In each case, coordination, not causation, becomes the central explanatory motif.
11. Toward a Universal Science of Coordination
Kelso has suggested that coordination dynamics could form the “physics of living things” — a universal science describing how nature organizes itself at all levels.
In this view, physics, biology, and consciousness are not separate domains but different faces of the same coordination principle. The equations governing hand movements, neuronal oscillations, and even social or quantum synchrony may be mathematically isomorphic.
This points toward a truly unified science — one that includes energy, information, and meaning within a single framework.
12. Editor’s Note
(by Sungchul Ji)
From the perspective of the Gnergy Principle of Organization (GPO) [5] and IRVSE (Iterative Reproduction with Variation and Selection by Environment) [7], coordination dynamics offers the quantitative backbone of the same universal process S. Ji has described qualitatively.
The oscillatory components correspond to iterative reproduction.
Phase variation maps onto variation.
Bifurcation and metastability correspond to selection by environment.
In essence, coordination dynamics is the mathematical manifestation of IRVSE.
It provides a rigorous model for how energy (E) and information (I) unite as gnergy (G = E + iI) to create organization, adaptation, and ultimately, consciousness.
13. Closing Reflection
Coordination dynamics reveals a simple yet profound truth:
Everything that exists participates in rhythm, coupling, and transformation.
From molecules to minds (see Figure 15.16 above), the universe organizes itself not through mechanical pushes and pulls, but through relations, resonances, and rhythms — the mathematics of togetherness.
And perhaps, as both Kelso and Josephson have suggested in their own ways,
this mathematics of coordination may be the missing bridge between mind and matter, and the language through which the universe knows itself. [8]
References
[1] Haken, H., Kelso, J.A.S., & Bunz, H. (1985). A Theoretical Model of Phase Transitions in Human Hand Movements. Biological Cybernetics, 51, 347–356.
[2] Kelso, J.A.S. (1995). Dynamic Patterns: The Self-Organization of Brain and Behavior. MIT Press.
[3] Tognoli, E. & Kelso, J.A.S. (2014). The Metastable Brain. Neuron, 81(1): 35–48.
[4] Friston, K. (2010). The Free-Energy Principle: A Unified Brain Theory? Nature Reviews Neuroscience, 11(2): 127–138.
[5] Ji, S. (2018). The Cell Language Theory: Connecting Mind and Matter. World Scientific Publishing.
[6] Ji, S. (2012). Micro-Macro Coupling in the Human Body. In:Molecular Theory of the Living Cell: Concepts, Molecular Mechanisms, and Biomedical Applications. Springer, New York. Pp. 534-557.
[7] Ji, S. (2025). The Geometry of Reality (GOR): A Triadic Framework for Matter,
Mind, and Spirit.
https://622622.substack.com/p/geometry-of-reality______________________________________________________________________[8] Ji, S. (2025). Conscio-Genesis, Enzyme Catalysis, and Quantum Processes (Revised by adding the missing Figure 2).