An Introduction to Synchronization
in Complex Systems
M. Biey, M. Bonnin, F. Corinto, and M. Righero
Politecnico di Torino, 2014
March 17, 2014
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Outline of the course
PART I : GENERALITIES
Introduction and examples
Complex systems
Nonlinear dynamical systems
3.1 Discrete time systems: The logistic map
3.2 Continuous time systems
3.2.1 Fundamentals
3.2.2 Fixed points and stability, bifurcations
3.2.3 Limit cycles
3.2.4 Lyapunov exponents and chaos
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Outline of the course
PART II
Limit cycles: Poincaré maps and examples.
Limit cycle stability: Variational equation and Floquet’s
theory.
Harmonic balance (HB) and describing function (DF)
techniques. Examples.
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Outline of the course
PART III
The Master Stability Function for synchronized coupled
systems
General synchronization properties for complex
networks
Synchronization in networks of bio-inspired oscillators.
Seminars on specific topics related to complex systems
can be scheduled during the course.
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Introduction
What is synchronization?
In a classical context, synchronization (from Greek:
syn = the same, common and: chronos = time) means
adjustment of rhythms of self-sustained periodic
oscillators due to their weak interaction (coupling);
this adjustment can be described in terms of phase
locking and frequency entrainment (1).
(1) If you have two vibrating objects with the same natural frequency or corresponding harmonic,
they will both have a forced vibration effect on each other. This process, given time, normally
leads to a condition where both objects synchronize.
Of interest, both oscillators do not, necessarily, must have exactly the same natural frequency.
If there is enough "coupling" between the oscillators, they will sometime "lock-in" with one
another at a slightly shifted frequency: the frequencies become equal or entrained.
The onset of a certain relationship between the phases of these oscillators is often termed
phase locking.
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Introduction (cont.)
What is a self-sustained periodic oscillator ?
1. The oscillator is an active system. It contains an internal source of
energy that is transformed into oscillatory behavior. Being isolated, it
continues to generate the same rhythm until the source of energy
expires. It is described as an autonomous dynamical system.
2. The form of the oscillation is determined by the parameters of the
system and does not depend on initial conditions.
3. The oscillation is stable to (small) perturbations
The above properties are characteristic of nonlinear
oscillators
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Introduction (cont.)
Example: Colpitts oscillator
VC1
Time
Period: 10.1 µs
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Introduction (cont.)
Example: Two identical coupled Van der Pol oscillators
w1 = w2 , l1 = l2
 dx1
 dt

 dy1
 dt
= w1 y1  d  x2  x1 

=  w1 x1  l1  2  x12  y12

 dx2
 dt

dy
y1  2
 dt
Uncoupled: d = 0
2
0
0
-2
x2(t) – x1(t)

4
2
-4

=  w2 x2  l2  2  x22  y22 y2
Coupled: d = 0.1
4
x1(t)
x2(t)
= w2 y2  d  x1  x2 
-2
0
5
10
15
20
25
30
35
40
45
50
-4
4
3
2
2
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
1
0
0
-2
-4
-1
-2
0
5
10
15
20
25
30
35
40
phase difference remains
March 17, 2014
45
50
phase difference vanishes
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Introduction (cont.)
Example: Two identical coupled Van der Pol oscillators
The coupled oscillators synchronize: two different interpretations
for the phases
2  t   1  t 
i =
1  yi
tan

 
 xi 

 0
t 
phase synchronization
for the states
 x2 
 x1 
t

  
  t 
 y2 
 y1 

 0
t 
complete (or identical) synchronization
generalization: to non identical systems generalization: to systems with any behavior
limitation: to systems where a phase
can be defined  rhythmic behavior
March 17, 2014
limitation: to identical or approximately
identical systems
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Introduction (cont.)
Example: Phase synchronization in Two Van der Pol
oscillators with different parameters:
l1 = 0.2, l2 = 2, w1 = 1, w2 = 1.1, d = 0.1
2
1
0
-1
-2
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
4
2
0
-2
-4
Each trajectory follows approximately the periodic trajectory
of each (isolated) component system. The phases are locked (i.e.
constant difference) but the corresponding states are different.
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Introduction (cont.)
Examples
In-phase synchronization
Anti-phase synchronization
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Introduction (cont.)
Synchronization with an arbitrary phase shift
No synchrony
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Introduction (cont.)
Modern concept covers also chaotic systems; in this
case one distinguishes between different forms of
synchronization (complete, lag, generalized, phase,
imperfect), the most notable being complete (or
identical) and phase synchronization (but in this case
chaos must be rhythmic) [*].
Example:
Phase synchronization of two coupled chaotic oscillators
[*] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, The Synchronization
of Chaotic Systems, Physics Reports 366, pp. 1-101, 2002
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Introduction (cont.)
Synchronization properties are also influenced
by the general properties of the oscillator
network: complex systems can be more or less
prone to synchronize due to their specific
features.
Synchronization requires knowledge of both
nonlinear dynamics and of complex systems.
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Introduction (cont.)
Sync: How order emerges from chaos (*)
(*) S.H. Strogatz, Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life,
Hyperion, 2004
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Introduction (cont.)
Synchronization: a historical perspective
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Introduction (cont.)
Synchronization: a historical perspective
1665: Huygens observation of pendula
α1
α2
α1
α2
ω1
ω2
ω1
ω2
When pendula are on a common support,
they move in synchrony, if not, they slowly
drift apart
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Introduction (cont.)
Synchronization: a historical perspective
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Introduction (cont.)
Synchronization: a historical perspective
Sleep-Wake rhythms: biological systems can adjust their
rhythms to external signals. Under natural conditions,
biological clocks tune their rhythms (i.e. synchronize) in
accordance with the 24-hour period of the Earth’s daily
cycle
(First observed by J.J. Dortous de Mairan, 1729)
Synchronization of triode oscillators
(Appleton, van der Pol, van der Mark,1922-1928)
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Introduction (cont.)
Synchronization: a historical perspective
Mutual synchronization of cardiac pacemaker cells
(E.E. Verheijek et al., “Pacemaker synchronization of electrically coupled rabbit sinoatrial node
cells,” J. Gen. Physiol., vol. 11 I , pp. 95-112, January 1998)
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Introduction (cont.)
Synchronization: why ?
Synchronization is a parameter of a (possibly complex) network which
gives important insight in observable phenomena in real networks.
Synchronization phenomena are pervasive in biology and are related
to several central issues of neuroscience [1].
Synchronization may allow distant sites in the brain to communicate
and cooperate with each other. For example, synchronization between
areas of the visual cortex and parietal cortex, and between areas of the
parietal and motor cortex was observed during the visual-motor
integration task in awake cats [2].
_____________________
[1] W. Singer and C. M. Gray, “Visual features integration and the temporal correlation hypothesis,” Annual Rev. Neurosci.,
vol. 18, pp. 555–586, 1995.
[2] P. R. Roelfsema, A. K. Engel, P. Knig, and W. Singer, “Visuomotor integration is associated with zero time-lag
synchronization among cortical areas,” Nature, vol. 385, pp. 157–161, 1997
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Introduction (cont.)
Synchronization: why ?
Direct participation of synchrony in a cognitive task was
experimentally demonstrated in humans [3].
Synchronization may help protect interconnected neurons
from the influence of random perturbations (intrinsic
neuronal noise) which affect all neurons in the nervous
system [4].
_____________________
[3] E. Rodriguez, N. George, J.-P. Lachaux, J. Martinerie, B. Renault, and F. J. Varela, “Perception’s shadow: Long
distance synchronization of human brain activity,” Nature, vol. 397, pp. 430–433, 1999.
[4] N.Tabareau, J-J. Slotine, Q. Pham, “How synchronization protects from noise”, PLoS Computational Biology, pp. 1-9,
Vol. 6, N. 1, 2010
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Introduction (cont.)
Synchronization: why ?
Spiking neurons, like any other physical, chemical, or biological oscillators,
can synchronize and exhibit collective behavior that is not intrinsic to any
individual neuron.
Partial synchrony in cortical networks is believed to generate various brain
oscillations, such as the alpha and gamma EEG (electroencephalography)
rhythms.
However, increased synchrony may result in pathological types of activity,
such as epilepsy.
Coordinated synchrony is needed for locomotion and swim pattern
generation in fish.
Depending on the circumstances, synchrony can be good or bad, and it is
important to know what factors contribute to synchrony and how to control it.
______________________
(Extracted from: E. M. Izhikevich , Dynamical Systems in Neuroscience: The
Geometry of Excitability and Bursting, Ch. 10, MIT Press, Cambridge, MA,
USA, 2007)
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Introduction (cont.)
References
1.
2.
A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization, A universal
concept in nonlinear sciences, Cambridge University Press, Cambridge,
UK, 2001.
S. H. Strogatz, Nonlinear dynamics and chaos: With applications to
physics, biology, chemistry, and engineering, Perseus Books,
Cambridge, MA, USA, 1994.
3.
H. K. Khalil, Nonlinear systems, McMillan Publishing Co., New York,
1992.
4.
E. M. Izhikevich, Dynamical Systems in Neuroscience, CH. 10, The MIT
Press, Cambridge, MA, USA, 2007
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