Entrainment of randomly coupled oscillator networks Hiroshi KORI Fritz Haber Institute of Max Planck Society, Berlin With: A. S. Mikhailov 

Outlook 1. 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 networks3Hiroshi KORI General motivation 1. 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 networks4Hiroshi 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 of gene expressions occurs (without help of external stimulus) Mathematically, sort of Kuramoto transition Not the topic of this talk

Entrainment of complex oscillator networks5Hiroshi KORI Environmental entrainment 1. 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 networks6Hiroshi 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 architecture Tool: investigate a general model, and get general results 1. INTRODUCTION Relaxation time  (recovery time from jetlag)   Coupling strength (Natural frequency of 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 networks8Hiroshi 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 networks9Hiroshi 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: N 1 (small ; 1~20) coupling strength inside the network:  coupling strength from the pacemaker:  (sufficiently large)

Entrainment of complex oscillator networks10Hiroshi KORI Hierarchical organization of networks We define hierarchical positions of nodes 2. MODEL We define the good quantity characterizing the 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 (only forward connections is displayed) Set of “shell”s

Entrainment of complex oscillator networks11Hiroshi KORI Overview of numerical simulations 2. DYNAMICS (NUMERICAL) Long time frequencies of oscillators NOT directly connected to the pacemaker (i.e., below the 2 nd 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 networks12Hiroshi KORI Entrainment thresholds 2. DYNAMICS (NUMERICAL) Entrainment thresholds obtained from individually generated networks with a given connectivity Exponential dependence on the depth （ N=100, pN=10 ） depth Entrainment thresholds (log scale)

Entrainment of complex oscillator networks13Hiroshi KORI Disappearance of Arnold tongue 2. 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 strength inside the network (Frequency of external forcing) 0 Practically, only shallow networks has the ability to be entrained!

Entrainment of complex oscillator networks14Hiroshi KORI Relaxation time 2. 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 networks15Hiroshi 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 networks16Hiroshi KORI Structure of random networks 2. DYNAMICS (ANALYTICAL) # of forward connections received by a node ? 1 PM 11 (only forward connections are displayed) H-1 HH H-2 # of backward and intra-shell connections received by a node ? ~N <<N 1 << pN << N ~N

Entrainment of complex oscillator networks17Hiroshi KORI Tree approx. in forward connections 2. DYNAMICS (ANALYTICAL) PM LL LLL # of forward connections per oscillator backward pN (Intra-shell) pN Because all oscillator inside a particular shell have identical connection patterns, phase synchronized state inside each shell exists PM L h phase

Entrainment of complex oscillator networks18Hiroshi KORI 2. DYNAMICS (ANALYTICAL) PM 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 networks19Hiroshi KORI Stability and relaxation time 2. DYNAMICS (ANALYTICAL) Relaxation time (N=100, pN=10,  =300,  >>  PM 1 2 L fast slow the solution is always stable; it disappears by a saddle-node bifurcation

Entrainment of complex oscillator networks20Hiroshi KORI PM LL LLL 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) 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 networks22Hiroshi KORI Introduction of directivity 3. EXTENTION OF NETWORK PM 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 networks24Hiroshi 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 networks25Hiroshi 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 frequency of 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 networks26Hiroshi 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, (2004) Arnold tongue is rescued in more uniformly directed networks    coupling strength (Frequency of external forcing) 0