Embed Size (px)
Citation preview
Watch S. Strogatz’s lecture (2004):
The science of sync at page http://www.ted.com/talks/steven_strogatz_on_sync#t-12468
SYNC: How Order Emerges From Chaos In the Universe, Nature, and Daily Life by S.Strogatz (2004)
===================================================================
A book for reading (no formulas! ) :
Nonlinear Dynamics of Networks
- regular (stationary, periodic ,or quasiperiodic)
- chaotic (strange attractors, chimera states)
Neuronal Networks
Human brain: ~ 100 000 000 000 neurons
5
- 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?
Parkinson’s disease is characterized by pathological
synchronization of neuronal activity in subthalamic
nucleus (STN) and external segment of globus pallidus
(GPe):
Namely, some part of the STN-GPe neurons
starts to fires synchronously
at some frequency in the beta-band 10-30Hz
By contrast, under healthy conditions
these neurons fire in an uncorrelated and
desynchronized manner.
Pathological synchronization in Parkinson’s disease
HOW TO DESYNCHRONIZE?
7
Deep Brain Stimulation (Benabid, 1986)
electrode
target point
generator
permanent high-frequency stimulation >90 Hz
Tremor amplitude as
a function of DBS frequency
From :
Experimental data: beta-band synchronization (~10-30 Hz)
9
Normal monkey
Parkinsonian monkey
Recording individual GPE cells
10 “Parkinsonian” spectrum “Healthy” spectrum
How to desynchronize
pathologic neuronal synchrony in the brain?
Co-workers:
R.Levchenko (Kyiv)
B.Lysyansky (Juelich-Kyiv)
Yu.Maistrenko (Juelich-Kyiv)
J.Rubin (Pittsburg)
O.Sudakov (Kyiv)
P.Tass (Juelich)
National Academy of Sciences of Ukraine Institute of Neuroscience and Medicine Research Center Jülich, Germany Pittsburg University, US
How to switch from pathologic synchronouse
to healthy desynchronized state?
Intuitively: phenomenon of multistability could help for desirable transitions between the characteristic states If so: one should apply deep brain stimulation (DBS) with a hope to turn from synchronization to desynchronization by resetting initial conditions But first: we need to study the phenomenon, to build a model for individual neurons and for connectivity in the network .
13
Network of excitatory (STN) and inhibitory(GPe)
neurons
STN GPe
15
Parameters of STN and GPe neurons
17
Strong connectivity Weak connectivity
Network topology
STN GPe
18
2000 coupled neurons
“Parkinsonian” bursting: space-time dynamics
Gpe
cells
STN
cells
time
20
2000 coupled neurons
“Healthy” irregular spiking: space-time dynamics
Four different system parameters to vary
Coupling coefficient SG
Coupling coefficient GG
Coupling coefficient GS
External current
1.5 – 4.0
0.06 – 0.14
2.0 – 10.0
-0.3 – -1.0
What can we conclude from the modelling study?
• Network connectivity plays a crucial role for the STN-GPe network dynamics: namely, sparser connectivity provokes synchronous bursting
• Two characteristic regimes, chaotic spiking and regular bursting are confirmed by massive parallel supercomputing (~600 trajectories of ~10000 ms for 2000 neurons)
• But, both regimes can exist only at different coupling configurations
How one can help to parkinsonian patients?
• Unfortunately, there is no another way to switch from regular bursting
(synchronization, “pathologic””) to chaotic spiking (desynchronization, “healthy”) except as modifying neuronal connectivity in the network.
Spontaneous synchronization
of coupled limit-cycle oscillators.
We are interested in the spontaneous synchronization
of populations of biological oscillators. For example:
- the pacemaker region of the heart consists of thousands of cells
that produce a regular, collective rhythm of electrical signals.
- fireflies, males congregate in a tree and flash in synchrony to the
Sending signals to their lady friends.
Winfree model (1967)
NiZdt
dj
N
j
iii ,...,1 ),( )(
1
Book 2000
A population of interacting limit-cycle oscillators: - weak coupling - nearly identical oscillators
The oscillators relax to their limit cycles and so, can be characterized solely by their phases
Each oscillator is coupled to the collective rhythm generated by the whole population
i
i - phases of individual oscillators
- frequencies of individual oscillators
“Biological rhythms and the behavior of populations of coupled oscillators” J.Theor. Biol. 1967
Network of N coupled phase oscillators:
ii
- coupling function )( ijijij
- phases of individual oscillators
- frequencies of individual oscillators are distributed according to some
probabilistic density 𝑔(𝜔), like Gaussian distribution.
Nidt
dij
N
j
ijii ,...,1 ),(
1
ii
Kuramoto model (1975)
Book 1984
One-dimensional complex Ginzburg-Landau equation, periodic boundary conditions
Introduce non-local coupling
Integral coupling term
How one can obtain Kuramoto model?
)sin(1
1
ij
N
j
iji GN
Kuramoto model Parameter 𝛼
28
“Standard” Kuramoto model
cKK
cKK : synchronization (phase-locking)
: desynchronization (clustering and chaos)
There exists critical bifurcation value such that: 0cK
.,...,1 ),(sin1
NiN
Kij
N
j
ii
All-to-all sinusoidal coupling function sin)(N
Kij
When the coupling is small compared to the spread of natural frequencies, the system behaves incoherently, with each oscillator running at its natural frequency. As the coupling is increased, the incoherence persists until a certain threshold is crossed — then a small cluster of oscillators suddenly ‘freezes’ into synchrony. For still greater coupling, all the oscillators become locked in phase and amplitude.
cKK
global coupling
Synchronization transition in Kuramoto model
𝑵 = 𝟑 𝑵 = 𝟕
i - average frequency of individual oscillator (Poincare rotation number)
symmetric frequency distribution asymmetric frequency distribution
T
Ti dttT
0
)(1
lim
Complex order parameter
.,...,1 ),(sin1
NiN
Kij
N
j
ii
N
j
ii jeN
re1
1 Order parameter:
𝑟(𝑡) measures the coherence of the oscillator population
𝜓(𝑡) is the average phase
.,...,1 ),sin( NiKr iii
Each oscillator is coupled to the common average
phase with coupling strength given by 𝐾𝑟 i
)(t
31
Cauchy-Lorentz frequency distribution
Books
Y. Kuramoto (1984) Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin). A. Winfree (2000) The Geometry of Biological Times (Springer, New-York). Review papers S.Strogatz (2000). " From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1 -20. S.Strogatz (2001). "Exploring Complex Networks". Nature 410 (6825): 268–276. Yu. Maistrenko et al (2005). “Desynchronization and chaos in the Kuramoto model“. Lect. Notes Phys. 671, 285-306.
Recommended literature
