Download Lecture 2

Complex Network Dynamics:
Theory & Application
Yuri Maistrenko
E-mail: [email protected]
Lecture 2
Network Science
Dynamical networks
Complex networks versus Oscillatory networks
Ensembles of oscillators
Structure
or
Topology
versus
Dynamics
or
Activity
We want to study
Emergent network dynamics
“In philosophy, system theory, science, and art,
EMERGENCE
is the way how complex systems and patterns arise
out of a multiplicity of relatively simple interactions.”
Wikipedia
Chimera stare in network of 100x100x100 oscillators in 3Dim cube.
Two incoherent rolls in coherent surrounding
What is a network?
Network: a set of
nodes joined by links
oscillators
connections
node, oscillator
link, connection
A network with N=8 nodes and K=10 links (Newman, 2005)
- mathematically, network is a graph
- there is dynamics in each node/oscillator
- our task: to study collective dynamics of the network
We need differential equations
or
iterated maps
Network: Coupled oscillators
Two types of dynamical systems for networks:
• Differential equations:
xi
dxi
 f ( xi )  coupling terms
dt
• Difference equations (iterated maps):
xit 1  f ( xit )  coupling terms
f ( x) : R M  R M
- nonlinear function
xi 1
xi 1
network
Nonlinear Dynamics of Networks
- regular (stationary, periodic ,or quasiperiodic)
- chaotic (strange attractors, chimera states)
THE ESSENCE OF CHAOS
• Process (dynamics) deterministic
fully determined by initial state and system equations
• Long-term behavior unpredictable
butterfly effect
CHAOS
= sensitive dependence on initial conditions
CHAOS = BUTTERFLY EFFECT
Henri Poincaré: (1880)
“ It so happens that small differences
in the initial state of the system
can lead to very large differences
in its final state. A small error in
the former could then produce
an enormous one in the latter.
Prediction becomes impossible,
and the system appears
to behave randomly.”
Ray Bradbury “A Sound of Thunder “ (1952)
EXAMPLES OF CHAOTIC SYSTEMS
•
•
•
•
•
the solar system (Poincare)
the weather (Lorenz)
turbulence in fluids
population growth
lots and lots of other systems…
“HOT” APPLICATIONS
• neuronal networks of the brain (dynamical chaos)
• gene regulatory networks (spatial chaos)
WEATHER UNPREDICTIBILITY
Edward Lorenz: (1963)
Difficulties in the weather forecast
are not related to the complexity
of the Earths’ climate but to
CHAOS
in the global weather dynamics
(given by nonlinear equations).
LORENTZ ATTRACTOR (1963)
butterfly effect
a trajectory in the phase space
The Lorenz attractor is generated by the system of 3 differential equations
dx/dt= -10x
+10y
dy/dt= 28x
-y
dz/dt= -8/3z
+xy.
-xz
Dynamics of oscillators placed in a network:
Coupled oscillators
Dynamical system: a system of one or more variables which evolve in
time according to a given rule
Two types of dynamical systems:
xi
• Differential equations:
dxi
 f ( xi )  coupling terms
dt
• Difference equations (iterated maps):
xit 1  f ( xit )  coupling terms
f ( x) : R M  R M
- nonlinear function
xi 1
xi 1
network
Networks of iterated maps
Networks of chaotic maps: Coherent states
t 1
i
x
 f (x ) 
t
i

iP
t
t
[
f
(
x
)

f
(
x
 j
i )]
2P ji-P
f ( x)  ax(1  x), a  3.8
(chaotic map)
𝑁 = 100 oscillators
Parameter
regions of
coherence
coupling strength
𝝈
Chaotic
synchronization
COHERENCE
coupling radius 𝒓 = 𝑷/𝑵
coupling radius 𝒓 = 𝑷/𝑵
Space-temporal chaos
Network of Lorenz systems: Traveling waves
xi  axi  ayi 

i P
2P
y i  bxi  yi  xzi 
 (x
j i  P

2P
zi  xi yi  czi
j
 xi )
i P
 (y
j i  P
j
Lorenz attractor
 yi )
( i  1,..., N )
𝑎 = 10, 𝑏 = 28, 𝑐 = 8/3
Chaotic
synchronization
𝜎 = 16, 𝑟 = 0.1, 𝑁 = 100
𝜎 = 13.3 , 𝑟 = 0.1, 𝑁 = 100
Space-temporal chaos
𝜎 = 13.8 , 𝑟 = 0.05, 𝑁 = 300
Network of Rössler systems: Chimera states
xi   yi  z i 
y i  xi  ayi
zi  b  czi  xi zi

iP
2P
 (x
j i  P
j
 xi )
(i  1,..., N )
Neuronal Networks
Human brain: ~ 100 000 000 000 neurons
- how complex are neuronal networks in the brain?
- are they locally or globally coupled?
- strength of coupling between individual neurons?
- excitatory and inhibitory neurons, why so?
How to get modelling?
21
22
24
- bidirectional connections are more
common than one can expected,
if the network connections be random
- connection strength distribution differs
significantly from random and
characterized by a “long tail”
- synaptic weights are concentrated among
few strong synaptic connections
Neuronal connectivity represents
a skeleton of stronger connections
in the sea of weaker ones!
25
26
The next UNIT to model
(The Neocortical - column )
(1mm3 )
Size of a pin head
Cortical column and neuronal microcircuits
28
29
Why do we need to build a model?
“I have all these data in the cortex - cell types, their firing
properties, dendritic excitability, connectivity, synaptic
dynamics… But I don’t Understand it. I need to model it.”
Bert Sakmann
Nobel Prize in Physiology and Medicine, 1991
University of Heidelberg
30
How can one model neuronal networks?
Kiss - detailed
Rodin
Kiss – reduced
Brancusi
Alan Lloyd
Hodgkin
Hodgkin-Huxley model (1952)
Andrew Fielding
Huxley
The H&H model; (1) Biophysical, (2) Compact, (3) Predictive
Network of Hodgkin-Huxley neurons
Synaptic current
Coupling function
Network of Hodgkin-Huxley neurons : Chimera state
TIME
Chimera state can “reflect” working memory in neuronal networks
Bell-shaped persisted neural activity
Network architecture
Space-temporal network
activity in a bump state
Renart, Song, Wang "Robust spatial working memory…”(Neuron 2003)