1. Introduction - Reproducibility of a Neuron. 3. Introduction Phase Response Curve. 2. Introduction - Stochastic synchronization. θ 1. c θ.

Size: px

Start display at page:

Download "1. Introduction - Reproducibility of a Neuron. 3. Introduction Phase Response Curve. 2. Introduction - Stochastic synchronization. θ 1. c θ."

Meagan Jordan
5 years ago
Views:

1 . Introduction - Reproducibility of a euron Science (995 Constant stimuli led to imprecise spike trains, whereas stimuli with fluctuations produced spike trains with timing reproducible to less than millisecond. - Unreproducible - Reproducible (Synchronization Constant stimuli st trial neuron Fluctuating stimuli st trial nd trial neuron nd trial Common noisy signal may induce synchronization!. Introduction - Stochastic synchronization 3. Introduction Phase Response Curve Asymptotic phase shift of the oscillator caused by external stimuli. i introduction of the phase ii measurement of the PRC Random Signals θ θ sync! θ=.5 θ=.5 ω:const. θ= G(θ G(θ, c c θ Uncoupled limit-cycle oscillators synchronize to each other when driven by common random signals such as Gaussian white noise and Poisson random impulses. θ=.75 - Examples Stuart-Landau FitzHugh-agumo Hodgkin-Huxley What PRC shape yields the best stochastic synchronization?

4. Optimal phase sensitivity for stochastic synchrony If the impulse intensity c is sufficiently small, the oscillators respond linearly to c so that their PRCs can be approximated by means of the

2 4. Optimal phase sensitivity for stochastic synchrony If the impulse intensity c is sufficiently small, the oscillators respond linearly to c so that their PRCs can be approximated by means of the phase sensitivity function Z(θ. G(θ,c cz(θ (exponential growth rate of the phase difference Λ = λ c Z (θ dθ Common random signals always induce stochastic synchronization! J. Teramae and D. Tanaka, PRL (4 - Optimal shape of Z Optimal shape of Z for the stochastic synchronization is sinusoidal. A. Abouzeid and B. Ermentrout, PRE (9 When the nonlinear response is considered, what shape of PRC yields the best stochastic synchrony? 5. Model System - Poisson driven oscillators θ (t = ω + G(θ δ(t t n n= θ (t = ω + G(θ δ(t t n n= G(θ,c : PRC for the impulse intensity c Λ = λ { } : arrival times of impulses t n dθ dcp(cp(θln + G (θ λ : Rate of the Poisson impulses P(c : PDF of the impulse intensity P(θ : PDF of the phase right before impulses 6. Calculation of the Lyapunov exponent Λ = λ - Approximation of the phase distribution In Poisson driven case, the phase distribution can be approximated as uniform. (K. Arai and H. akao, PRE, 8 P(θ - Excitatory impulses The oscillators are driven by the constant-intensity impulses. P(c = δ(c a Λ = λ dθ ln + G (θ where G(θ := G(θ, a dθ dcp(cp(θln + G (θ 7. Optimal PRCs - Trivial optimal PRC Λ = λ dθ ln + G (θ If the impulse is sufficiently strong, optimal solution is a sawtooth. G (θ = Λ = Single impulse can synchronize oscillators! - Weak impulse limit? If the impulse is sufficiently weak so that the oscillator respond linearly to the impulse intensity, the optimal shape is a sinusoid.? What PRC shape is optimal between both limits?

3 8. Constrained Minimization Constraints on PRC - Squared amplitude J[G] = G(θ dθ B = - Overall smoothness K[G] = G (θ dθ C = Lagrange multiplier method Λ + µj + νk = L(G, G, G dθ µ,ν : Lagrange multiplier d L dθ G d L dθ G + L G = Control the amp. of PRC! Large-wavenumber PRC ot sync., but clustering! 9. Euler-Lagrange equation L(G, G, G = λ ln + G (θ + µ(g B + ν( G C νg (4 + λ G ( + G + µg = - We fix the smoothness parameter at ν = 5, which is enough large to exclude large-wavenumber solutions. - Varying the amplitude parameter µ, we look for the solutions which satisfy the periodic boundary conditions, G( = G( =, G ( = G (, G ( = G (, - For simplicity, we set the Poisson rate λ =. The optimal PRCs obey this equation!. Result - Optimal PRCs for Stochastic Synchronization νg (4 + G (+ G + µg = umerically solving the Euler-Lagrange equation with µ >, we obtained the optimal PRCs for the stochastic synchronization.. Result - Optimal PRCs for Stochastic Desynchronization νg (4 + G ( + G + µg = umerically solving the Euler-Lagrange equation with µ <, we obtained the optimal PRCs for the stochastic desynchronization. negative positive The optimal PRC mutates its shape from a sinusoidal to a sawtooth! The optimal PRC have a cusp at θ =.5 and approaches to a sawtooth!

4 . Summary - We examine the optimization problem of the PRC for stochastic synchronization and desynchronization of limit-cycle oscillators. - Minimizing (or maximizing Lyapunov exponent with constraints on the squared amplitude and the overall smoothness. - As the squared amplitude increases - The optimal PRC for stochastic synchronization mutates its shape from a sinusoidal to a sawtooth. - The optimal PRC for stochastic desynchronization have a cusp, and gradually approaches to a sawtooth. - Derivation of the Lyapunov exponent - Excitatory an Inhibitory impulses - Diffusion approx. / limit - -dependence ν This work was supported by the Grant-in-Aid for the Global COE Program "The ext Generation of Physics, Spun from Universality and Emergence" from the MEXT of Japan. Derivation of the Lyapunov exponent Δ n+ = dθ n+ Δ n dθ n = ( + G (θ n Δ n ( Δ = + G (θ n Δ n= θ(t = ω + G(θ δ(t t n n= θ n+ = θ n + G(θ n + ωτ n where τ n := t n+ t n Λ = lim log Δ Δ = lim n= log + G (θ n dθ dcp(θp(clog + G (θ n Replace the sample average by the statistical average! Excitatory and Inhibitory Impulses - Probability density of c P(c = δ(c a + δ(c + a Λ = λ dθ ln G (θ νg (4 + λ { } Assuming that G(θ, a = G(θ,a G (+ G + µg = ( G where G(θ := G(θ,a L(G, G, G = λ ln G (θ + µ(g B + ν( G C

5 umerical Result νg (4 + λ G (+ G + µg = ( G umerically solving the Euler-Lagrange equation with µ >, we obtained the optimal PRCs for the stochastic synchronization. Diffusion approx. / limit G(θ az(θ for a - Excitatory impulses νaz (4 + λ a Z (+ a Z + µaz = a νz (4 + λ Z + µz = - Excitatory and inhibitory impulses νaz (4 + λ a Z ( + a Z + µaz = ( a Z a νz (4 + λ Z + µz = negative The optimal PRC mutates its shape from a sinusoidal to a double sawtooth! Z(θ sin λ ± λ 6µν θ 4ν Consistent with the previous study! (A. Abouzeid and B. Ermentrout, 9 Dependence of the solutions on ν Euler-Lagrange equation νg (4 + λ G ( + G + µg = Rescale the phase variable θ nθ. d 4 ν d(nθ G(nθ + λ 4 d d(nθ G(nθ ( + d G(nθ d(nθ + µg(nθ = d ν d 4 n dθ G(nθ + λ G(nθ dθ 4 + d ( G(nθ n dθ + µn G(nθ = ν n G n (4 (θ + λ G n (θ ( + G n (θ + µ ng n (θ = Dependence of the solutions on ν If G(θ is a solution of the EL equation with (µ,ν, then G n (θ is a solution with (µ n,ν n. µ 4µ ν ν / 4 G n (θ = n G(nθ, µ n = n µ, ν n = ν n where G n (θ = n G(nθ, µ n = n µ, ν n = ν n

Exercises. Chapter 1. of τ approx that produces the most accurate estimate for this firing pattern.

Exercises. Chapter 1. of τ approx that produces the most accurate estimate for this firing pattern. 1 Exercises Chapter 1 1. Generate spike sequences with a constant firing rate r 0 using a Poisson spike generator. Then, add a refractory period to the model by allowing the firing rate r(t) to depend