entrainment of randomly coupled oscillator networks hiroshi kori fritz haber institute of max planck...
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Entrainment ofrandomly coupled oscillator networks
Hiroshi KORIFritz Haber Institute of Max Planck Society, Berlin
With: A. S. Mikhailov
Outlook1. Introduction
General motivation biological clocks the problems we consider
2. Model & Dynamics Disappearance of Arnold tongue in hierarchical networks
3. Extension of network Rescue of Arnold tongue
4. Discussion (no more jetlag?)
Entrainment of complex oscillator networks 3Hiroshi KORI
General motivation1. INTRODUCTION
Influence of network architecture on dynamics
A population of oscillators
- coupled by random networks
- under partial external forcing
For this aim, I consider the following system:
(important in neural networks, gene regulation networks, production networks, traffic networks, etc.)
Entrainment of complex oscillator networks 4Hiroshi KORI
Biological clock (circadian rhythm) Endogenous clock embedded in organisms
Even in a dark room, we still act rhythmically Its natural period is close to, but, different from 24h
1. INTRODUCTION
In mammals, produced by Suprachiasmatic Nucleus (SCN) Dense assembly of neurons (>10,000) Each neuron is a genetic oscillator with the period of about 24h
(formed by cyclic expressions of a group of genes)
(Movie from Yamaguchi et al. SCIENCE ’04)
Mutual synchronization ofgene expressions occurs
(without help of external stimulus)
Mathematically, sort of Kuramoto transition
Not the topic of this talk
Entrainment of complex oscillator networks 5Hiroshi KORI
Environmental entrainment1. INTRODUCTION
Other neurons (~90%) are influenced through a complex network inside
SCN
Only ~10% of neurons in SCN are directly influenced by photic inputs
(Kuhlman ’03, Abrahamson ’01)
Natural frequency (e.g., 25h) ≠ environmental rhythm (24h) Change of daylight rhythm (season or long-distance trip)
Adaptation to the environment is essential for the normal function
Light shifts the phase of SCN
(Abrahamson ’01)
Entrainment of complex oscillator networks 6Hiroshi KORI
General problems we consider Size of the Arnold tongue
(The parameter region in which the oscillator network is able to be entrained)
Goal: quantify the dependences of these two quantities on network architectureTool: investigate a general model, and get general results
1. INTRODUCTION
Relaxation time (recovery time from jetlag)
Coupling strength
(Natural frequencyof oscillators)
(Frequency of external forcing)
0
1. Introduction General motivation Entrainment of biological clocks Concrete problems we consider
2. Model & Dynamics Disappearance of Arnold tongue in hierarchical networks
3. Extension of network Rescue of Arnold tongue
4. Discussion (no more jetlag?)
Entrainment of complex oscillator networks 8Hiroshi KORI
A population of identical phase oscillators
A is an asymmetric random matrix
2. MODEL
The model
pN: Connectivity
External forcing
Pacemaker, or,environment
Entrainment of complex oscillator networks 9Hiroshi KORI
Parameters Going to a rotating frame: Rescaling :
2. MODEL
Parameters: (Our interest and in analytical calculations; in simulation)
size of network: N (large; 100)
Connectivity: pN (sparse but large 1 << pN << N; ~10)
# of oscillators directly connected to the pacemaker: N1 (small ; 1~20)
coupling strength inside the network: coupling strength from the pacemaker: (sufficiently large)
Entrainment of complex oscillator networks 10Hiroshi KORI
Hierarchical organization of networks We define hierarchical positions of nodes
2. MODEL
3
3
3
1
1
1
22
2 2
We define the good quantity characterizingthe hierarchy of a given network
depth
(the mean forward distance from PM:It typically takes L steps from PM to a node)
Forward connections Backward + intra-shell connections
by shortest distances from PM
PM
1 1
2 2 2
333333
(only forward connections is displayed)
Set of “shell”s
Entrainment of complex oscillator networks 11Hiroshi KORI
Overview of numerical simulations2. DYNAMICS (NUMERICAL)
Long time frequencies of oscillators NOT directly connected to the pacemaker (i.e., below the 2nd shell)
• Strong correlation between phases and their hierarchical positions
• Entrainment threshold (a certain bifurcation occurs!)
varies largely between different realizations of random networks!
Entrainment of complex oscillator networks 12Hiroshi KORI
Entrainment thresholds2. DYNAMICS (NUMERICAL)
Entrainment thresholds obtained from individually generated networkswith a given connectivity
Exponential dependence on the depth
( N=100, pN=10 )
depth
Entrainment thresholds(log scale)
Entrainment of complex oscillator networks 13Hiroshi KORI
Disappearance of Arnold tongue2. DYNAMICS (NUMERICAL)
(recall that our model is rescaled as →
The Arnold tongue disappears in hierarchical networks(i.e. becomes exponentially smaller with the depth L).
coupling strengthinside the network
(Frequency of external forcing)
0
Practically, only shallow networks has the ability to be entrained!
Entrainment of complex oscillator networks 14Hiroshi KORI
Relaxation time2. DYNAMICS (NUMERICAL)
N=100, pN=10,=300, >>
Relaxation time
(log scale)
(fixed coupling strength,under entrainment)
When, suddenly, the phase of the pacemaker changes (long-distance trip),how long does it take to relax to the normal entrained state again?
Naïve expectation: the typical time to transmit the information of the pacemaker to the whole network should be proportional to the average distance from the
pacemaker (which is the depth L). So, linear dependence on depth L?
Relaxation time also has the exponential dependence on the
depth
Entrainment of complex oscillator networks 15Hiroshi KORI
Analytical derivation
• the solution under entrainment
• its stability (and relaxation time)
2. DYNAMICS (ANALYTICAL)
The model can be solved by using a mimic of random networks
1 << pN << N
( large connectivity, large network size, but sparse)
Entrainment of complex oscillator networks 16Hiroshi KORI
Structure of random networks 2. DYNAMICS (ANALYTICAL)
# of forward connections received by a node
1
1
1
?
1
PM
1 1
(only forward connections are displayed)
2 2 2
333333
H-1 H-1 H-1
H H
H-2 H-2 H-2 H-2 H-2
# of backward and intra-shell connections received by a node
?
~N
<<N
1 << pN << N
~N
Entrainment of complex oscillator networks 17Hiroshi KORI
Tree approx. in forward connections2. DYNAMICS (ANALYTICAL)
PM
1 1
2 2 2
333333
L L LL L
# of forward connections per oscillator
1
1
1
1
backward
pN
pN
pN
(Intra-shell)
pN
Because all oscillator inside a particular shell have identical connection patterns, phase synchronized state inside each shell exists
PM
1
2
3
L
h
phase
1
1
1
1
Entrainment of complex oscillator networks 18Hiroshi KORI
2. DYNAMICS (ANALYTICAL)
PM
1
2
3
1
1
1
The entrained solution
Forward connections
phase Backward
connections
L
h
1
Phase differences grow exponentially from the deepest shell
(consistent with numerical results)
we get
• Because , the existence condition of the solution is
Forward Backward
1
Entrainment of complex oscillator networks 19Hiroshi KORI
Stability and relaxation time2. DYNAMICS (ANALYTICAL)
Relaxation time(N=100, pN=10,=300, >>
PM
1
2
L
fast
fast
slow
the solution is always stable;it disappears by a saddle-node bifurcation
Entrainment of complex oscillator networks 20Hiroshi KORI
PM
1 1
2 2 2
33333
L L LL L
Mechanism of exponential dependence
3
Accumulation of this asymmetry along forward path(its length is L)
makes the exponential growth of phase differences
3. DYNAMICS (ANALYTICAL)
1
1
1
+
-
Strong asymmetry exists
Forward(+) < Backward(-)
Forward connection : 1
Backward connection : pN
1. Introduction General motivation Entrainment of biological clocks Concrete problems we consider
2. Model & Dynamics Disappearance of Arnold tongue in hierarchical networks
3. Extension of network Rescue of Arnold tongue
4. Discussion (no more jetlag?)
Entrainment of complex oscillator networks 22Hiroshi KORI
Introduction of directivity3. EXTENTION OF NETWORK
PM
1 1
2 2 2
333333
Suppose that we randomly eliminate a certain ratio (1-) of
backward connections
Directivity
# of forward connections
1
backward
pN
: normal random network
: feedforward network
The Arnold tongue is rescued!
1. Introduction General motivation Entrainment of biological clocks Concrete problems we consider
2. Model & Dynamics Disappearance of Arnold tongue in hierarchical networks
3. Extension of network Rescue of Arnold tongue
4. Discussion (no more jetlag?)
Entrainment of complex oscillator networks 24Hiroshi KORI
Design of biological clocks 1. Shallow network (small L)
2. Close to feedforward network (small )
Larger numbers of connections from shallow shells: HUBS
shallow shells
deep shells Small numbers of connections to shallow shells
Similar to the structure of SCN!
(Abrahamson ’01)VIPAVP GRP mENK
4. DISCUSSION
Entrainment of complex oscillator networks 25Hiroshi KORI
Crazy experiment Enlarge the Arnold tongue by training (possible when baby?)
give very fast rhythm (e.g., 20h rhythm) to a baby
NO MORE JETLAG !!If you are expecting a baby, we can discuss the details
4. DISCUSSION
Coupling strength
(Natural frequencyof oscillators)
(Frequency of external forcing)
0
A shallower and more uniformly directed network will be formed.We will have larger Arnold tongue & shorter relaxation time
Entrainment of complex oscillator networks 26Hiroshi KORI
Conclusions
Acknowledgement: Support of Alexander von Humboldt Shifting
Arnold tongue vanishes in hierarchical networks.Practically, only shallow networks can be entrained H. Kori and A.S. Mikhailov, PRL 93, 254101 (2004)
Arnold tongue is rescued in more uniformly directed networks
coupling strength
(Frequency of external forcing)
0