Emergence of Function
Emergence and Synchrony
Emergence and Synchrony describe collective behavior
in dynamical systems
Emergence: Collective phenomena or behaviors that are not present in their individual parts.
Synchrony: Coordinated timing of events or actions within a system. Alignment of rhythms, oscillations, …..
Virus Assembly
Examples of Emergence
Ant Bridges
Muscle Formation
Prion Disease
EMERGENCE
Bird flocking, Organization of cities, Protein Folding, Consciousness, Digital Twin,.......
Pacemaker cells in your heart: Keep you alive
Sinoatrial node: 10,000 cells all have to fire in unison
Not always good:
Epilepsy : there is an instance billion of brain cell discharging pathological concert. Pokémon episode upon its initial airing on 16 December 1997, over 700 Japanese viewers, the majority of which were children, found themselves in hospitals due to epileptic seizures.
You don’t have to be alive….�Laser: Different atoms give off light waves that all oscillate in unison��Superconductors: Pair of electrons oscillate in synchrony, allowing electricity to flow with almost no resistance�
Examples of Synchrony
John Bardeen
Digital Twin
The Immune System Simplified
Adaptive Immune System
Finds the Solution
Innate Immune System
Recognizes the Problem
Foreign invader enters body
Invader is broken up into small pieces
Non-self antigens presented
to T cells and recruit B cells
B cells divide, mutate, and
checked by T cells for affinity
Low affinity cells die, while some medium affinity stored as memory
T cells
Naive
B cells
Optimization
Non-self piece!
14-16 days
4-5 days
Plasma
B cells
High affinity cells found the
solution, produce antibodies
Problem
Solution!
Flocking
Expansion
Consensus
Sensing
Memory
B cells
Memory
Adversary
Learning
Alan Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society, 237:27–72 (1953)
Morphogenesis: Lapse of Emergence
Alan Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society, 237:27–72 (1953)
Chua, Leon O. "Local activity is the origin of complexity." International journal of bifurcation and chaos 15.11 (2005): 3435-3456.
Simple example
Rajapakse I, and Smale S. Emergence of Function from Coordinated Cells in a Tissue. Proceedings of the National Academy of Sciences 114.7 (2017): 1462-1467.
Gradient Dynamics: Nodes
Diffusion Dynamics: Edges
The Model
Emergence:
Emergence of Function
Emergence of Function
Following the spirit of Turing’s paper, we may combine two dynamics (genome dynamics within the cell and diffusion dynamics between cells) into a system
Rajapakse I, and Smale S. Emergence of Function from Coordinated Cells in a Tissue. Proceedings of the National Academy of Sciences 114.7 (2017): 1462-1467.
Brockett, Roger W. Finite dimensional linear systems. Society for Industrial and Applied Mathematics, 2015.
Emergence of Function
Kuramoto Model (1975)
Yoshiki Kuramoto
Kuramoto Model: Two Oscillators
Kuramoto Model: Three Oscillators
Morse - Smale System
The primary concern of Morse Theory is the relation between spaces and functions. The center of interest lies in how the critical points of a function defined on space affect the topological shape of the space, and conversely, how the shape of a space controls the distribution of the critical points of a function.
More-Smale system: The simplest dynamical systems, structurally stable and have intimate connections to the topology of manifolds
Kuramoto Potential
Equilibria of the Kuramoto System
Equilibria of the Kuramoto System
Saddle Circles
Three Potential Regions of the Kuramoto Flow
Global Dynamics: Stable and Unstable manifold structure
Dynamics of the Kuramoto System
Kuramoto System: - Time
m = 5, Index 2 saddle = (pi,pi,0,0,0)