Stochastic differential equations in neuroscience
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1 Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans Simona Mancini, MAPMO-Orléans IGK Seminar Bielefeld, June 18, 2010
2 Plan 1. Deterministic Modeling neurons Slow fast dynamical systems Excitability : Types I and II 2. Stochastic Mathematical tools Sample-path approach Application to excitable systems
3 Excitable systems The structure of a neuron Single neuron communicates by generating action potential Excitable: small change in parameters yields spike generation 1
4 ODE models for action potential generation Hodgkin Huxley model (1952) Morris Lecar model (1982) C v= g Ca m (v)(v v Ca ) g K w(v v K ) g L (v v L ) + I(t) τ w (v)ẇ= (w w (v)) m (v) = 1+tanh((v v 1)/v 2 ) 2, τ w (v) = w (v) = 1+tanh((v v 3)/v 4 ) 2 Fitzhugh Nagumo model (1962) C g v= v v 3 + w + I(t) τẇ= α βv γw τ cosh((v v 3 )/v 4 )), For C/g τ: slow fast systems of the form ε v= f(v, w) ẇ= g(v, w) 2
5 Deterministic slow fast systems εẋ= f(x, y) x : fast variable ẏ= g(x, y) y : slow variable ε 1: Singular perturbation theory Qualitative analysis: nullclines f = 0 and g = 0 x f < 0 g < 0 f > 0 g < 0 f < 0 g > 0 f > 0 g > 0 y 3
6 Quantitative results Stable slow manifold: f = 0, x f < 0 Tikhonov (1952) / Fenichel (1979): Orbits converge to ε-neighbourhood of stable slow manifold Dynamic bifurcations: f = 0, x f = 0 local analysis x x x ε 2/3 ε 1/3 y ε 1/2 ε 1/2 y y Saddle node f(x, y) = x 2 y +... Transcritical f(x, y) = x 2 + y Pitchfork f(x, y) = yx x
7 Excitability of type I Stable equilibrium point at intersection of f = 0 and g = 0 Close to a saddle node-on-invariant-circle (SNIC) bifurcation At bifurcation, periodic solutions appear Period diverges at bifurcation point Example: Morris Lecar model 5
8 Excitability of type II Stable equilibrium point at intersection of f = 0 and g = 0 Close to a Hopf bifurcation At bifurcation, periodic solutions appear Period converges at bifurcation point Canard (french duck) phenomenon Example: Fitzhugh Nagumo model 6
9 Adding noise dx t = 1 ε f(x t, y t ) dt + σ ε dw t dy t = g(x t, y t ) dt + σ dw t W t, W t : Brownian motions (independent) Ẇ t, Ẇ t : white noises Different mathematical methods : PDEs evolution of probability density, exit from domain Large deviations rare events, exit from domain Stochastic analysis sample-path properties... 7
10 Noise and partial differential equations dx t = f(x t ) dt + σ dw t Generator: Lϕ = f ϕ σ2 ϕ Adjoint: L ϕ = (fϕ) σ2 ϕ x R n Kolmogorov forward or Fokker Planck equation: t µ = L µ where µ(x, t) = probability density of x t Exit problem: Given D R n, characterise τ D = inf{t > 0: x t D} Fact: u(x) = E x {τ D } satisfies Lu(x) = 1 u(x) = 0 x D x D Similar boundary value problems give distribution of exit time and exit location 8
11 Noise and partial differential equations dx t = f(x t ) dt + σ dw t x R n Generator: Lϕ = f ϕ σ2 ϕ Adjoint: L ϕ = (fϕ) σ2 ϕ Kolmogorov forward or Fokker Planck equation: t µ = L µ where µ(x, t) = probability density of x t Exit problem: Given D R n, characterise x τd τ D = inf{t > 0: x t D} Fact: u(x) = E x {τ D } satisfies Lu(x) = 1 u(x) = 0 x D x D D Similar boundary value problems give distribution of exit time and exit location 8-a
12 Noise and large deviations dx t = f(x t ) dt + σ dw t x R n Large deviation principle: Probability of sample path x t being close to given curve ϕ : [0, T ] R n behaves like e I(ϕ)/σ2 Rate function: (or action functional or cost functional) I [0,T ] (ϕ) = 1 2 T 0 ϕ t f(ϕ t ) 2 dt 9
13 Noise and large deviations dx t = f(x t ) dt + σ dw t x R n Large deviation principle: Probability of sample path x t being close to given curve ϕ : [0, T ] R n behaves like e I(ϕ)/σ2 Rate function: (or action functional or cost functional) I [0,T ] (ϕ) = 1 2 T 0 ϕ t f(ϕ t ) 2 dt Application to exit problem: (Wentzell, Freidlin 1969) Assume D contains unique equilibrium point x Cost to reach y D: V (y) = inf inf{i [0,T ] (ϕ): ϕ 0 = x, ϕ T = y} T >0 Gradient case: f(x) = V (x) V (y) = 2(V (y) V (x )) { } 1 Mean first-exit time: E[τ D ] exp inf V (y) y D σ 2 9-a
14 Noise and stochastic analysis dx t = f(x t ) dt + σ(x) dw t x R n Integral form for solution: t x t = x 0 + f(x s) ds + σ(x s) dw s 0 0 where the second integral is the Itô integral t 10
15 Noise and stochastic analysis dx t = f(x t ) dt + σ(x) dw t x R n Integral form for solution: t x t = x 0 + f(x s) ds + σ(x s) dw s 0 0 where the second integral is the Itô integral Application to the exit problem: The Itô integral is a martingale its maximum can be controlled in terms of variance at endpoint (Doob) : P { t sup t [0,T ] σ(x s) dw s 0 δ } 1 δ 2E [( T t 0 σ(x s) dw s ) 2 ] Itô isometry: E [( T 0 σ(x s) dw s ) 2 ] = T 0 E[σ(x s) 2 ] ds 10-a
16 Application to slow fast systems dx t = 1 ε f(x t, y t ) dt + σ ε dw t dy t = g(x t, y t ) dt + σ dw t Use different methods Near stable slow manifold (f = 0, x f < 0) Near bifurcation points (f = 0, x f = 0) Far from slow manifold (f 0) 11
17 Near stable slow manifold Slow fast system with y t = t dx t = 1 ε f(x t, t) dt + σ ε dw t If stable slow manif: f(x (t), t) = 0, If stable slow manif.: a (t) = x f(x (t), t) a 0 then adiabatic solution: x(t, ε) = x (t) + O(ε) of εẋ = f(x, t) 12
18 Near stable slow manifold Slow fast system with y t = t dx t = 1 ε f(x t, t) dt + σ ε dw t If stable slow manif: f(x (t), t) = 0, If stable slow manif.: a (t) = x f(x (t), t) a 0 then adiabatic solution: x(t, ε) = x (t) + O(ε) of εẋ = f(x, t) Observation: Let ā(t, ε) = x f( x(t, ε), t) = a (t) + O(ε) Consider linearised equation at x(t, ε): dξ t = 1 ε ā(t, ε)ξ t dt + σ dw t ε ξ t : gaussian process with variance σ 2 v(t), s.t. ε v = 2ā(t, ε)v + 1 Asymptotically, v(t) v (t) = 1/2 ā(t, ε) B(h): strip of width h v (t, ε) around x(t, ε) 12-a
19 Near stable slow manifold dx t = 1 ε f(x t, t) dt + σ ε dw t Theorem: [B. & Gentz, PTRF 2002] C(t, ε)e κ h 2 /2σ 2 P { leaving B(h) before time t } C(t, ε)e κ +h 2 /2σ 2 κ ± = 1 O(h) C(t, ε) = 2 1 π ε t 0 ā(s, ε) ds h σ [ 1 + error of order e h 2 /σ 2 t/ε ] x (t) x t B(h) x(t, ε) 12-b
20 Saddle node bifurcation e.g. f(x, y) = y x 2 σ σ c = ε 1/2 x B(h) σ σ c = ε 1/2 x y y Deterministic case σ = 0: Solutions stay at distance ε 1/3 above bifurcation point until time ε 2/3 after bifurcation. Theorem: [B. & Gentz, Nonlinearity 2002] 1. If σ σ c : Paths likely to stay in B(h) until time ε 2/3 after bifurcation, maximal spreading σ/ε 1/6. 2. If σ σ c : Transition typically for t σ 4/3 transition probability 1 e cσ2 /ε log σ 13
21 Excitability of type I Near bifurcation point: σ δ 3/4 σ δ 3/4 dx t = 1 ε (y t x 2 t ) dt + σ ε dw t dy t = (δ y t ) dt Global behaviour: 14
22 Excitability of type I Time series of x t : σ δ 3/4 : rare spikes, times between spikes exponentially distributed, mean waiting time of order e δ3/2 /σ 2 Poisson point process σ δ 3/4 : frequent spikes, more regularly spaced, waiting time of order log σ 15
23 Excitability of type II Near bifurcation point: dx t = 1 ε (y t x 2 t ) dt + σ ε dw t dy t = (δ x t ) dt δ > ε: equilibrium (δ, δ 2 ) is a node Similar behaviour as before, crossover at σ δ 3/2 δ < ε: equilibrium (δ, δ 2 ) is a focus. Two-dimensional problem 16
24 Excitability of type II Time series of x t : Muratov and Vanden Eijnden (2007): σ δε 1/4 : rare spikes δε 1/4 σ (δε) 1/2 : rare sequences of spikes (MMOs) σ (δε) 1/2 : more frequent and regularly spaced spikes 17
25 References N. B. & B. Gentz, Pathwise description of dynamic pitchfork bifurcations with additive noise, Probab. Theory Related Fields 122, (2002), A sample-paths approach to noiseinduced synchronization: Stochastic resonance in a double-well potential, Ann. Appl. Probab. 12, (2002), The effect of additive noise on dynamical hysteresis, Nonlinearity 15, (2002), Noise-Induced Phenomena in Slow- Fast Dynamical Systems. A Sample-Paths Approach. Springer, Probability and its Applications, 276+xvi pages (2006), Stochastic dynamic bifurcations and excitability, in C. Laing and G. Lord, (Eds.) Stochastic Methods in Neuroscience, Oxford University Press (2009) 18
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