Sixteen hundred sixty-five: Huygens discovers a phenomenon he poetically names "Sympathy of the clocks". It is the ineluctable falling into synchrony by two pendulum clocks, driven to coherence by the imperceptible vibration of the suspension beam they share. Since Huygens' inauguration of the entire field of weakly coupled oscillators, the phenomenon has found countless empirical grounds, including in the behavior of interacting people or fireflies, in coupling within and between brains, in electronic, biological and informational systems. In Huygens' experiment, the tenuous vector of the coupling, the vibration of the wooden beams, is countered by the remarkable symmetry of the two clocks. Nonetheless, we now know that systems with lesser symmetry manage to overcome their difference and coordinate through synchrony and metastability. And since adaptation leads to divergence, with weak coupling of nonidentical parts, nature preserves useful coordination mechanisms well past the perfect unison that Huygens revealed, and that many mechanical systems retain. In this talk, I introduce an experimental model of pendulum oscillators with varying natural frequencies, and I provide an empirical analysis of their coordination behavior. My goal is to showcase interdisciplinary research, expose the insights that data analysis offers and engage a discussion on models. My conclusion will be that more theoretical insights on weak coupling and broken symmetry will help us better understand Complexity in Nature.

1. Emmanuelle Tognoli
Center for Complex Systems &
Brain Sciences
Colloquium @ Physics
Department, February 12th,
Hans Erni, La conquête du temps, Museum of Horology, La-Chaux-de-Fond, Switzerland

3. Christiaan Huygens, 1665, sympathy of the clocks
Read more: Willms, A.R., Kitanov, P.M. and Langford,
W.F., 2017. Huygens’ clocks revisited. Royal Society
open science, 4(9), p.170777.
Voyages immobiles:
La Chaux-de-Fond

5. Tognoli
,
Zhang,
Fuchs,
Beetle,
Kelso,
Frontie
rs
Hum.
Neuros
c.
2020
O
V
E
R
V
I
E
W
5

6. Basic toolset: the relative phase f
time
x=0
x=1
x=1
x=0 x=0
x=-1
x=-1
x=0
x=0
x=1
x
.
.
.
. .

8. Trajectories
Kelso
&
Tognoli,
2007,
In
Kozma
&
Perlovsky
(eds)

9. 2 key parameters, coupling (weak), symmetry (broken)
Metastability
Weak coupling
Broken
symmetry
Tognoli, Zhang, Kelso, 2018

10. Tognoli & Kelso, 2014; Tognoli et al., 2020
Phase portrait, viewpoint of broken symmetry dw
Metastability: Loss of attractors

13. Teasing apart ubiquity of electric field and
genuine inter-areal brain synchrony…
…revealed brief tangible episode of coordination (phase-locking or metastability)
Tognoli & Kelso,

14. Relative
phase
Tognoli & Kelso,
Broken symmetry
Homologous left and right
sensorimotor cortex
Brain: it’s metastable!
Provides flexible de-
association and re-
association a.k.a.
adaptability

17. 0
p
-p
Incoher
ent
-
cohere
nt

19. Phase Trajectory of Chimera States
KB’s chimera
(incoherent-
coherent) are in
fact synchronized-
metastable

21. Spatiotemporal metastability in the brain

24. Mass Action, but no flow
Transfer Synchronization
time
space
time
space
Flow, but no mass action

25. Challenge 1: Dark Areas of Complexity

26. How do you like your brain: locked or unlocked?

28. Spatiotemporal metastability in human behavior
Metastable
Phase-locked
Reanalysis of Banerjee et al., 2012

29. Spatiotemporal metastability in the
“synchronized*” fireflies
American fireflies
from the Smoky
Mountains,
presumably Photinus
Carolinus
*Smith, Science, 1935; Buck & Buck, Science, 1968
Tognoli et al., 2018

30. Dominant
interpretation of
fireflies self-organized
collective behavior:
synchrony (+ a little
bit of noise)
1Hz pulse frequency
Spatiotemporally metastable fireflies

31. Temporal view:
-fireflies don’t flash all the time
-flashing is sustained (F, I, J) or
sporadic (G, H) or single events (E)
-phase relationship have inphase
(I), antiphase (E, G) and out-of-
phase (H, J), or clearly changing as
in metastability (F)
Spatiotemporally
metastable fireflies

32. “human” fireflies
Zhang et al., Plos One, 2018;
Royal Interface, 2019
8 people engaged in
sensorimotor coordination

33. Spatiotemporally metastable “human” fireflies
Zhang et al., 2018, 2019
Emergence of temporal scales for coordination
Again resembles theoretical model of metastability
Tognoli & Kelso, in Yamaguchi et al., ICCN, 2013
Model (Zhang et al., 2019) outlines limits to experimental validity
of Kuramoto model (second order coupling term)

35. Attractors or not?
Kelso & Tognoli, 2007, in Kozma and Werbos, Springer
Synchronization
-Good for strongly coupled
systems, your car, your
coffee machine
-States
-Predictable
-Lower dimensional
Metastability
-Good for your brain
-Fluidity in temporal
behavior
-Stationary transients
-Rich compositional
complexity
-Flexibility
Why the difference is essential for brain
complexity & neuroscience Tognoli & Kelso, Frontiers in System
Neuroscience, 2015

36. Now a chalk* talk?
Rationale
-a model system for
broken symmetry in
intrinsic frequencies
-initial phase is fixed and
homogeneous
-generalize with different
initial phases

37. Weak coupling (à la Huygens) + broken symmetry

38. Illusion of coincidence or
metastability?
Analysis of wave pendulum
1
2
3
4
7
Phase patterns

39. Analysis of wave pendulum
But…
No obvious phase attraction
(relative phases wrap and do not
dwell).
Mostly coincidence then?
Relative
phases,
oscillators
1-5
(rad)

40. Illusion of coincidence or metastability?
Analysis of wave pendulum

41. Analysis of wave pendulum
Impression of
multistability, and
patterns of phase
coordination

42. Comments?
To be continued…