Noise-induced bursting Georgi Medvedev Department of Mathematics, Dreel University September 16, 27
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.)
citable systems under stochastic forcing.2.15.1 1.2 1.8.5.6.4.5.1.15.2.2.2.4.6.8 1 1.2.2.2.4 1 2 3 4 5 6 7 { 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
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 &
The deterministic model 1 1 t t ẋ = f(, y), ẏ = ɛg(, y), = ( 1, 2 ) T IR 2, y IR 1,
Fast subsystem ẋ = f(, y), ẏ =, (ɛ = ) imit cycles: (y) = { = φ(s, y) : s < T (y)} quilibria: = { = ψ(y) : y > }
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)
The randomly perturbed system 1 Type I: y o t 1 Type II: y o t
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.
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)
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.
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}.
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..
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.
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
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)
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
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.
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.
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
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 ).
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
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
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.
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 6 1 3 2.5 5 2 4 1.5 3 1 2 1.5 Type I 1 2 3 4 5 6 7 8 9 1 2 4 6 8 1 12 14 16 18 2 6 1 3 2.5 5 2 4 1.5 3 1 2 1.5 Type II 5 1 15 2 25 2 4 6 8 1 12 14 16 18 2
A numerical eample: the NaP + K + KM model.1.2.3.4.5 y c.6.7.5.1.15.2.25.3.35.15.12.1.1.8.6.5.4.2 5 1 15 2 25 3 35 4 45 5 55 5 1 15 2 25 3 35 4 45 5 55 Type I Type II
lectrically coupled cells 1 15 2 25 3 35 4 45 5 55 6.35.4 15 2 25 3 35 4 45 5 1 2 3 4 5 6 1 15 2 25 3 35 4 45 5 55 6.35.4 15 2 25 3 35 4 45 5 55 1 2 3 4 5 6 v i =... + d(v i+1 2v i + v i 1 ) + σẇ (i) t
lectrically coupled cells 1 15 2 25 3 35 4 45 5 55 6.35.4 15 2 25 3 35 4 45 5 55 1 2 3 4 5 6 1 15 2 25 3 35 4 45 5 55 6.35.4 15 2 25 3 35 4 45 5 55 1 2 3 4 5 6
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 417624