1. Study on synchronization of coupled oscillators using the Fokker-Planck equation H.Sakaguchi Kyushu University, Japan Fokker-Planck equation: important equation in statistical physics Synchronization in coupled oscillators

2. Langevin equation Stochastic differential equation Time evolution in a noisy environment Langevin equation X(t): a stochastic variable such as a position or a membrane potential ξ(t): a random force, Gaussian white noise

3. Random forces Gaussian: The probability distribution function is Gaussian with average 0 and variance 2D. White: There is no time correlation. Consider a large number of independent stochastic variables which obey the Langevin eq. The stochastic variables are randomly distributed, since the random forces are different.

4. Fokker-Planck equation Consider the probability density P(x,t) that the stochastic variable takes a value x. P obeys the Fokker-Planck equation Drift termDiffusion term The probability density drifts with velocity F(x) and diffuses owing to the random force

5. Random walk No drift force Langevin equation Fokker-Planck equation

6. Ornstein-Ulenbeck process Linear force ( ｖ : velocity,- k ｖ viscous force) Fokker-Planck equation Maxwell distribution Stationary distribution Brownian motion of a small particle such as a pollen on water surface

7. Thermal equilibrium distribution Potential force Fokker-Planck equation D=k B T; T: Temperature Thermal equilibrium distribution No probability flow: Detailed balance

8. Fokker-Planck equation for coupled Langevin equations Coupled Langevin equations for two variables Fokker-Planck equation for P(x,y)

9. Synchronization of coupled biological oscillators Synchronization of flashing of fireflies. Synchronization of cell activity in suprachiasmatic nucleus which control the circadian rhythms Sleep spindle waves are brain waves which appear in the second stage of sleep. Spindle waves are created by synchronous firing of inhibitory neurons in thalamus. (Steriade et al.) Human EEG

10. Phase oscillators Limit cycle oscillation Phase description: phase variables Two-coupled phase oscillators Mutual Entrainment

11. Globally coupled noisy phase oscillators Uniform state σ ＝０， P=1/2πUnstable for Ｋ＞２Ｄ P(φ): The probability that the phase takes φ Mean field coupling: Each oscillator interacts with all other oscillators by the same coupling. Order parameter ＜ sin(φ) ＞ = 0 by the symmetry

12. Self-Consistent Method Order parameter

13. Order-Disorder transition Weak interaction, Large noise Phases are randomly distributed. Disordered phase. Strong interaction, small noise Phases gather together. ＜ cos(φ ）＞ is nonzero. Ordered state Order-Disorder transition Phase transition from ferromagnetism to paramagentism If a magnet is heated above a critical temperature, the magnetism disappears.

14. Globally coupled oscillators with different frequencies Phase oscillator model Kuramoto model

15. Self-Consistent analysis ω

16. Globally coupled oscillators with different frequencies and external noises g(ω): Distribution of the natural frequency ω

17. Stationary solution of the Fokker-Planck equation for nonzero ω Flow of probability: average circulation of phase Stationary but non-equilibrium distribution

18. Phase transition via synchronization Complete entrainment is impossible owing to noises

19. Integrate-and-fire model Hodgkin-Huxley equation Detailed dynamics of membrane potential and several ion channels IF model simplest model of the neural firing x:membrane potential

20. Synchronization of two IF neurons Instantaneous interaction Response time 0 Complete synchronization for t>80 δ ｘ＝ｘ １ －ｘ ２

21. Noisy integrate-and-fire model and the Fokker-Planck equation reset process

22. Stochastic resonance in the noisy IF model Stochastic resonance Response of excitable systems to periodic force + noises Response is maximum for intermediate strength of noise

23. Direct simulation of the Fokker-Planck equation Firing rate Oscillation of P Oscillation of J 0 D=0.005,0.0015b=1.1,e=0.05

24. Phase transition in a globally coupled IF models Oscillation amplitude vs. D Disorder Order b=0.8,D=0.215,g=0.6,τ=0.01 τ ： response time

25. Phase transition in a nonlocally coupled IF model Nonlocal interaction Maxican-hat type Excitatory in the neighborhood Inhibitory in far regions Synaptic coupling is nonlocal. Synaptic current at y is determined by the firing rate at y’ by the integral.

26. Propagating pulse states Uniform state is unstable Pulse propagation Oscillation amplitude of I(y,t)J 0 (y,t) Order-disorder phase transion from a uniform state to a traveling wave state Inhibitory interaction suppresses global synchronization D=0.01

27. Another IF model and inhibitory network Thalamus(thalamic reticular neurons) Synchronization occurs among inhibitory neurons. Synchronization between two inhibitory IF neurons is possible if the response time τtakes a suitable value. Another IF model including the dynamics of excited state V>V T Excited state VTVT V1V1 V T2 V2V2

28. Two IF neurons with inhibitory coupling Synchronization of two inhibitory IF neurons -I s inhibitory synapse V 1 and V 2 are synchronized K=0.5,V 0 =-18 Synchronization becomes easier owing to finite duration of excited state

29. Phase transition in globally coupled IF models with mutual inhibition Langevin equation Fokker-Planck equation

30. Oscillatory phase transition in inhibitory systems Oscillation amplitude vs. K τ Phase diagram D=0.2,τ=20D=0.2 Finite response time is preferable for global oscillation

31. V i and the average at K=10, D=0.2V i and the averageat K=2, D=0.2 Time evolution of P at K=10,D=0.2Time evolution of P at K=2,D=0.2 Two types of oscillation Oscillation is synchronized. Firing is not synchronized. Fokker-Planck equation Langevin equation of 1000 neurons Synchronized firing The firing of some neurons suppresses the firing of the other neurons

32. Integrate-and-fire-or-burst model Low threshold Ca 2+ current: I T (t) plays important role for thalamic neurons This current flows for a short time after the potential V goes over V h.

33. Phase transition in globally coupled IFB models with mutual inhibition h(t) is a stochastic variable.

34. Bistability of globally coupled IFB model I 0 =1.6,K=40, D=0.2 and τ=20. h≠0, in one mode (rebound mode, burst mode). h=0, and I T does not work, in another mode (tonic mode). Average membrane potential E(t) vs.I 0. Ｖ ｈ =-70 I 0 is external input Two modes are bistable for 0.55<I 0 <2.2

35. Summary 1 Phase transition via mutual synchronization 2 Direct simulation of Fokker-Planck equation 3 Phase oscillator model and IF models 4 Transition to traveling wave states 5 Mutual synchronization in inhibitory systems Intermittent firing in strongly inhibited systems

36. Discussions and Problems Good points of the Fokker-Planck equation: 1. Stationary distribution might be solved. 2. Numerical results are clear, since it is a deterministic equation. Weak points of the Fokker-Planck equation: If the number of stochastic variables is not one or two, numerical simulations are rather hard. Langevin simulations may be efficient for realistic equations such as noisy Hodgkin-Huxley equations.

37. References Y.Kuramoto, “Chemical Oscillations, Waves and Turbulence” Springer (Berlin, 1984). H.Sakaguchi, Prog.Theor.Phys. Vol.79(1988) 39 S.H.Strogatz, Physica D Vol. 143 (2000) 1. H.Sakaguchi, Phys.Rev. E Vol.70(2004) 022901. M.Steriade and R.R.Llinas, Phsiol.Rev. Vol.68 (1988) 649