entrainment of randomly coupled oscillator networks hiroshi kori fritz haber institute of max planck...

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