The Poincare map for randomly perturbed periodic mo5on

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1 The Poincare map for randomly perturbed periodic mo5on Georgi Medvedev Drexel University SIAM Conference on Applica1ons of Dynamical Systems May 19, 2013

2 Pawel Hitczenko and Georgi Medvedev, The Poincare map for randomly perturbed periodic mo5on, J. Nonlinear Sci., 2013 Work par5ally supported by NSF through grant DMS

3 Burs5ng in a Single Cell Model C v = I ion (v, n,u) n = n (v) n τ (v) u = ε(i Ca (v) kv) x L x 1 E y sn y bp y t

4 Burs5ng and Spiking in the Single Cell Model x L x 1 E y sn y bp y t x L x 1 E y sn y c y bp y t

5 A Stochas5c Model of Burs5ng C v = I ion (v, n,u)+σ 1 w 1 n = n (v) n +σ 2 w 2 τ (v) u = ε(i Ca (v) kv) n 0.4 v(t) 30 Frequency, v t Number of spikes Hitczenko and Medvedev, SIAM J. Appl. Math., 2009.

6 Stochas5c Mixed- Mode Oscilla5ons C v = I ion (v, n, h)+σ 1 w 1 n = n (v) n τ (v) h = h (v) h τ h (v) n v(t) 1 Frequency, % v t Number of small oscillations

7 Burs5ng in a Coupled Network N v (i) =...+ g a ij (v (i 1) v (i) ) + σ 1 j=1 n (i) (i) =...+ σ 2 w 2 u (i) =..., w 1 (i) i =1,2,...,N Density 0.08 n 0.04 v Number of spikes Medvedev and Zhuravytska, Biol. Cyberne5cs, 2012

8 The exit problem x t = f (x t )+σ P(x t ) W t, P(x) R (d+1) (d+1) O = x = u(θ) : θ { S 1 = R 1 / Z}, u(t +1) = u(t) n v

9 The Local Coordinates x 2! " v(θ), z 1 (θ), z 2 (θ),, z d (θ )}, # x 1 v(θ) = u(θ) u(θ), θ S 1 (cf. Hale, Ordinary Differen5al Equa5ons)

10 In new coordinates x 2 x = u(θ)+ Z(θ)ρ, θ S 1 Z(θ) = ( z 1 (θ), z 1 (θ),, z d (θ)) R (d+1) d x 1 we have θ t =1+ a(θ t ) T ρ t +σ h(θ t ) T W t + ρ t = R(θ t )ρ t +σ H(θ t ) T W t +

11 The Poincare map x 2 θ t =1+ a(θ t ) T ρ t +σ h(θ t ) T W t + ρ t = R(θ t )ρ t +σ H(θ t ) T W t + x 1 Σ : x = u(0)+ Z(0)ρ, ρ <δ τ = inf{t > 0.5 : θ t =1} P : ρ 0 ρ τ

12 The Poincare map ρ = A(I +σ Bζ )ρ +ση +O(σ 2, ρ 2 ) τ σ =1+ b(1) T X(1)ρ 0 σζ 1 + o(σ )+O( ρ 0 2 ) A = X(1), B = X(1) 1 X(1), ζ =ξ-b(1)t η, 1 ξ = h(s) T dw s, η = X(t)X(s) 1 H(s) T dw s, 0 b(t) T = a(s) T X(t)X(s) 1 H(s) T ds

13 The Lineariza5on of the Poincare map ρ n = A(I +σ Bζ n )ρ n +ση n A is a stable matrix, i.e., ρ(a)<1-ε {ζ n },{η n } are Gaussian random processes A! 1 ε, Ax! (1 ε) x!, D h = {x R d : x! h} The first exit time: τ (ρ n, D h ) = min{n : ρ n D h }

14 Y is asympto5cally geometric random variable with parameter p if P(Y = n) P(Y n) p as n

15 The Lineariza5on of the Poincare map ρ n = A(I +σ Bζ n )ρ n +ση n A is a stable matrix, i.e., ρ(a)<1-ε The first exit time: τ (ρ n, D h ) = min{n : ρ n D h } Theorem 1. For all 0<σ <σ 0 (ε), τ is an asymptotically geometric RV with parameter p such that C 1 σ exp{-c 2 σ 2 }(1+O(σ 2 )) p C 2 σ exp{-c 3 σ 2 }(1+O(σ 2 ))

16 Sta5s5cs of the first exit 5me for the full Poincare map ρ n = A(I +σ Bζ n )ρ n +ση n +O(σ 2, ρ 2 ) A is a stable matrix, i.e., ρ(a)<1-ε The first exit time: τ (ρ n, D h ) = min{n : ρ n D h } Theorem 2. For all 0<σ <σ 0 (ε), for all n 2 P(τ = n) P(τ n) p, 0 p Cσ 2/3.

17 The randomly perturbed linear map y n = M n y n 1 + z n, M n = A(I +σξ n B), z n = σgη n A is a stable matrix, i.e., ρ(a)<1-ε ξ n, η n are IID standard normal RVs in R and R d B, G are d d matrices, G is nondegenerate Theorem 3. For all 0<σ <σ 0 (ε), C 1 σ exp{-c 2 σ 2 }(1+O(σ 2 )) p C 2 σ exp{-c 3 σ 2 }(1+O(σ 2 ))

18 Two Technical Lemmas y n = M n y n 1 + z n, M n = A(I +σξ n B), z n = σgη n d Lemma 4. y n " " y; y is independent of (ξ, z). The proof is based on the results due to Kesten (1973). Lemma 5. For some 0 < C 1 < C 2 <1, C 1 P(Ms + z D h ) C 2 uniformly in s D h. The proof uses the fact that A is stable and estimates for normal RVs.

19 The Proof of Theorem 3 P(τ = n +1) = P(y n+1 D h, y k D h, k [n]) = P(y n+1 D h y k D h, k [n])p(y k D h, k [n]) = P(y n+1 D h y n D h )P(τ n +1) Need to show that the ratio below converges to a nonzero limit P(τ = n +1) P(τ n +1) = P(y n+1 D h y n D h ) = P(M n+1 y n + z n D h y n D h ) = P(M n+1 y n + z n D h, y n D h ) P(y n D h )

20 The Proof of Theorem 3 P(τ = n +1) P(τ n +1) = P(M n+1y n + z n D h, y n D h ). P(y n D h ) By Lemma 5, C 1 P(y D h ) P(Ms + z D h )df y (s) C 2 P(y D h ) D h Finally, 0 < C 1 p = lim P(τ = n +1) P(τ n +1) C 2 <1.

21 Conclusions We constructed the Poincare map for the randomly perturbed limit cycle Analyzed sta5s5cs of the first exit 5me Applica5ons: burs5ng and mixed- mode oscilla5ons in neuronal models

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