Noise-induced bursting. Georgi Medvedev

Transcription

1 Noise-induced bursting Georgi Medvedev Department of Mathematics, Dreel University September 16, 27

2 Two problems Noise-induced bursting in single cell models (with Pawel Hitczenko, Dreel U.) Bursting in arrays of electrically coupled cells in the presence of noise (with Sveta Zhuravitska, Dreel U.)

3 citable systems under stochastic forcing { ɛ v = f(v) u + σẇt u = v u b ongtin; Collins, Chow, & Imhoff, 1995; Baltanos & Casado, 1998; rmentrout & Gutkin; Rubin, Terman, & Su; Deville, Vanden-ijnden, & Muratov, 25; Berglund & Gentz, 26

4 Bursting neuron models under stochastic forcing 1 y c t 1 y o t Approaches: Gentz) ẋ = f(, y), ẏ = ɛg(, y), = ( 1, 2 ) T IR 2, y IR 1, action functional (cf, Freidlin & Wentzell), sample-path (cf, Berglund &

5 The deterministic model 1 1 t t ẋ = f(, y), ẏ = ɛg(, y), = ( 1, 2 ) T IR 2, y IR 1,

6 Fast subsystem ẋ = f(, y), ẏ =, (ɛ = ) imit cycles: (y) = { = φ(s, y) : s < T (y)} quilibria: = { = ψ(y) : y > }

7 Slow dynamics y c ẏ = ɛg(ψ(y), y), = ψ(y) ẏ = ɛg(y), G(y) = 1 T (y) T (y) g (φ(s), y) ds. Bursting: G(y) > (Averaging) Spiking: G(y c ) = (Pontriagin & Rodygin, 1961)

8 The randomly perturbed system 1 Type I: y o t 1 Type II: y o t

9 Type I & type II models y o y o Type I { ẋt = f ( t, y t ) + σpẇ t, ẏ t = ɛg( t, y t ), Type II { ẋt = f ( t, y t ), ẏ t = ɛg( t, y t ) + σqẇ t.

10 Probabilistic techniques { ẋt = f ( t, y t ) + σpẇ t, ẏ t = ɛg( t, y t ), Asymptotic epansions (Freidlin & Wentzell, 1979) Randomly perturbed slow fast systems (Berglund & Gentz, 26) Stochastic difference equations (Kesten, 1973; Goldie 1991; Vervaat, 1979)

11 Geometric random variables Definition: Recall that Y is a geometric random variable with parameter p, < p < 1 if IP (Y = k) = p(1 p) k 1, k 1. emma: et Y be a random variable with values in the set of positive integers. Y is a geometric random variable with parameter p, < p < 1, iff IP(Y = n) IP(Y n) = p, n 1. Definition: We say that Y is asymptotically geometric with parameter p( < p < 1) if lim n IP (Y = n) IP(Y n) = p.

12 The randomly perturbed map: additive perturbation h Consider r 1, r 2,... are iid copies of N(, 1) and Y IR. Y n = λy n 1 + ςr n, n 1 λ = 1 ε, < ε < 1 For a given h >, let τ = inf{k 1 : Y k > h}.

13 The randomly perturbed map: additive perturbation Y n = λy n 1 + ςr n, n 1 Theorem: et ε (, 1), λ = 1 ε, β 2 = ς2 ε(2 ε), and h Y >. Then for sufficiently small ς >, τ is asymptotically geometric r.v. parameter p = 1 { } β 2π hφ(h/β) ep h2 2β 2, with where Φ() = 1 2π e t2 /2 dt, is the distribution function of an N(, 1) r.v..

14 The randomly perturbed map: random slope Y n = λ(1 + σr 1,n )Y n 1 + σr 2,n, n 1, where (r 1,n, r 2,n ) n=1 are i.i.d. copies of a 2D random vector (r 1, r 2 ) Theorem: et h and λ (, 1) are both of order 1 and < σ 1 so that γ := λi 1 + σr 1 < 1. Then τ is asymptotically geometric r.v. with parameter σ c 2π e 2σ 2, where a positive constant c depends on h and µ but not on σ. In addition, min 1, (1 λ)h 1 + (λh) 2 c 2 c 1 + h 2.

15 Diffusive escape Y n = λy n 1 + ςr n, n 1, λ = 1 ε, < ε ς, (i.e.,λ 1) Theorem: et λ = 1. Then where IP(τ n) Ψ a (n), a = h ς 2. Ψ a () = 2 ( 1 Φ ( )) a, a >..6 h.2 4 8

16 The Poincare map Σ 2 Σ y c τ ν 1 Σ - a section transverse to I.C.: { dt = f ( t, y t ) dt + σp( t )dw t, dy t = ɛg( t, y t )dt = ( 1, 2 ) T Σ, y < y bp O(σ) y t = y + O(ɛ) d t = f ( t, y ) dt + σp( t, y )dw t + O(ɛ)

17 The Poincare map for the fast subsystem: coordinate transformation 2 Σ τ ν 1 (y ) = { = φ(θ, y ) : θ [, T (y ))}, Moving coordinates: (θ, ξ), = φ(θ) + ξν(θ), θ [, T ), where ν(θ) is normal to

18 In new coordinates (cf, Hale, ODs), 2 Σ τ ν 1 dθ t = (1 + b 1 (θ t )ξ t + O(ξ 2 t ))dt + σh 1(θ t, ξ t ) ( 1 + b 2 (θ t )ξ t + O(ξ 2 t )) dw t, dξ t = ( a(θ)ξ t + O(ξ 2 t )) dt + σh 2 (θ t, ξ t )dw t, < µ := ep ( T a(θ)dθ ) = ep ( T divf (φ(θ)) ) < 1, h 1 (θ, ξ) = < p, τ > < τ, τ > = p1 f 1 + p 2 f 2 f 2, h 2 (θ, ξ) = < p, ν > < τ, τ > = p2 f 1 p 1 f 2 f 2.

19 The Poincare map for the fast subsystem: definition 2 Σ τ ν et θ = and ξ = σρ. 1 By T denote the first return time for the unperturbed system. Note T T. Definition. The time of the first return T σ : Definition. The first return map: θ T = T and lim σ T σ = T ξ = P (ξ ), where ξ = ξ Tσ.

20 The Poincare map for the fast subsystem: construction emma. On a finite time interval t [, t], the following epansions hold θ t = t + σθ (1) t + O(σ 2 ), ξ t = ξ A(t) + σ θ (1) t t = ρ b 1 (s)a(s)ds + t b 1 (s) s ( t ) A(t, s) := ep a(u)du s t A(t, s)h 2(s)dw s + O(σ 2 ), A(s, u)h 2 (u)dw u ds + emma. T = T + σt (1) + o(σ), T (1) = θ (1) T. t and A(t) = A(t, ). h 1 (s)dw s, emma. ξ = µξ (1 + σr 1 ) + σr 2 + o(σ), r 1,2 are Gaussian r.v. r 1 = a() { T b 1 (s) s A(s, u)h 2 (u)dw u ds + T h 1 (s)dw s }, r 2 = T A(s)h 2 (s)dw s

21 The first return map for the slow variable P(w).3 h 2 Σ.15 τ y c.15.3 w ν 1 I.C.: < y bp y = O(1), = φ() + ξ ν() Σ, and ξ = O(σ). The first return map: ȳ = P (y, ξ ), where P (y, ξ ) = y T, emma: inearize around y c : P (y, ξ) = y + ɛg(y) + ɛσr 3 + ɛaξ + o(ɛσ), G(y) = T g (φ(s), y) ds P (η, ξ) = λη + ɛσr 3 + ɛa 2 ξ +..., λ = 1 ɛg (y c ).

22 The first return map for the slow variable Σ 2 Σ y c τ ν 1.3 h P(w).15 h.15.3 w The Poincare map: { ξn+1 = µξ n (1 + σr 1,n ) + σr 2,n +..., η n+1 = λη n + ɛσr 3,n + ɛa 2 ξ n +..., n =, 1, 2,..., τ ξ = inf n> {ξ n > h ξ } and τ η = inf n> {η n > h η }, τ = min{τ ξ, τ η }, The distribution of τ approimates that of the number of spikes in one burst

23 The distribution of τ Σ 2 Σ y c τ ν 1 { ξn+1 = ( ) µξ n 1 + σr1,n + σr2,n, η n+1 = λη n + ɛσr 3,n + ɛa 2 ξ n, n =, 1, 2,..., We show that τ ξ and τ η are asymptotically geometric with parameters: p ξ c 1 σ e σ 2 and p η ɛσc 3 e c 4 ɛσ 2 c 1 2π Thus, p η p ξ τ τ ξ

24 Type II models Σ 2 Σ y c τ ν 1 { ẋt = f ( t, y t ), ẏ t = ɛg( t, y t ) + σqẇ t. The Poincare map: The same argument as above yields, ξ n+1 = µξ n (1 + ɛσr 1,n ) + ɛσr 2,n, η n+1 = λη n + ɛσr 3,n + ɛa 2 ξ n, n =, 1, 2,..., p ξ ɛσc 5 e C 6 ɛ 2 σ 2, p η ɛσc 7 e C 8 ɛσ 2. In contrast to the type I model, p ξ p η τ = τ η.

25 A numerical eample: the NaP + K + KM model C v = g NaP m (v)(v NaP ) g K n(v K ) g KM y(v K ) g (v ) + I, τ n ṅ = n (v) n, τ y ẏ = y (v) y Type I Type II

26 A numerical eample: the NaP + K + KM model y c Type I Type II

27 lectrically coupled cells v i =... + d(v i+1 2v i + v i 1 ) + σẇ (i) t

28 lectrically coupled cells

29 Conclusions In the presence of noise, spiking neuron models that are close to transition to bursting ehibit irregular bursting. Goal: To study the statistical properties of the emergent bursting patterns. Techniques: Poincare map for the randomly perturbed system. Results: the distributions of the number of spikes within one burst and the interspike time intervals; the dependence on these distributions on small and control parameters present in the model. Support: NSF grant IOB