Stochastic Nerve Axon Equations

Transcription

1 Stochastic Nerve Axo Equatios Wilhelm Staat Istitut fu r Mathematik, Fakulta t II TU Berli Berstei Ceter for Computatioal Neurosciece Berli Liz, December 13, 216

2 Stochastic processes i Neurosciece Modellig impact of oise i eural systems o all scales - microscopic - e.g., i io chael dyamics - mezoscopic - o observables i sigle euros, e.g. membrae potetial - macroscopic - e.g., i eural populatios Aalysis - mathematical framework for cotiuum limits (bridgig scales) - multiscale aalysis, w.r.t. coheret structures - model reductio, w.r.t. observables, mea field theories Numerical approximatio - strog ad weak approximatio errors - robust estimatio

3 Hodgki-Huxley Equatios (1952) math. descriptio for the geeratio of Actio Potetials (AP) τ tv = λ 2 2 xxv g Na m 3 h(v E Na ) g K 4 (v E K ) g L (v E L ) + I dp dt = αp(v)(1 p) βp(v)p p {m,, h} where v membrae potetial, v = v(t, x), t, x [, L] m,, h gatig variables, m,, h 1 τ resp. λ specific time resp. space costats gna, g K, g L coductaces ENa, E K, E L restig potetials αp(v) = ap 1 v+ap 1 e a2 p (v+ap ), βp(v) = 2 b1 p e b p (v+bp ) typical shape of v

4 Hodgki-Huxley Equatios (1952) math. descriptio for the geeratio of Actio Potetials (AP) τ tv = λ 2 2 xxv g Na m 3 h(v E Na ) g K 4 (v E K ) g L (v E L ) + I dp dt = αp(v)(1 p) βp(v)p, p {m,, h} biophysiological relevat feature: I < I + excitable I < I oscillatory I (I, I +) ihibitory I > I +

5 I reality more like this... [Höpfer, Math. Bioscieces, 27] due to fluctuatios betwee ope ad closed states of io chaels regulatig v

6 Chael oise Illustratio: measuremets of sigle Na-chael i the giat axo of squid (cosidered by Hodgki-Huxley) [Vadeberg, Bezailla, Biophys. J., 1991]

7 Chael oise impact o APs spotaeous spikig (due to radom opeeig of sufficiet umbers of Na-chaels) time jitter - spike time distributio icreases with time APs ca split up or aihilate propagatio failure places limits o the axo diameter (aroud.1µm), hece also o the wirig desity e.g., [White, et al., Treds Neurosci. 2, Faisal, et al., Curret Biology 25, Faisal, et al., PLOS 27]

8 Addig oise to Hodgki-Huxley equatios Addig chael oise yields a stochastic partial differetial equatio: Curret oise τ tv = λ 2 xxv g Na m 3 h(v E Na ) g K 4 (v E K ) g L (v E L ) + I + σ tξ(t, x) dp dt = αp(v)(1 p) βp(v)p, p {m,, h} (1) σ =.2 σ =.35 σ =.6

9 Addig oise to Hodgki-Huxley equatios addig chael oise yields a stochastic pde τ tv = λ 2 xxv g Na m 3 h(v E Na ) g K 4 (v E K ) g L (v E L ) + I + σ tξ(t, x) dp dt = αp(v)(1 p) βp(v)p, p {m,, h} (2) 1 u u x features: subthreshold excitability (well-kow already i the poit euro) due to spatial extesio: spotaeous spikig, backpropagatio, aihilatio, propagatio failure

10 Illustratio - subthreshold excitability (already kow from the poit euro case) I = 6., σ =. I = 6., σ =.25

11 Illustratio - spotaeous activatio, backpropagatio I = 6., σ =.25 I = 2., σ =.36

12 Subuit oise (classical) diffusio approximatio for the Markov chai dyamics p(t) modellig io chael dyamics with p(t) = p() + t α p(v(s))(1 p(s)) β p(v(s))p(s) ds + M (Np) t E ( ( M (Np) t ) ) 2 = 1 t E (α p(v(s))(1 p(s)) + β p(v(s))p(s)) ds N p τ t v = λ 2 xx v + g L (v E L ) g Na (X)(v E Na ) g K (X)(v E Na ) + ξ v t X = α(v)(1 X) β(v)x + σ(v, X)ξ X Rem represetatio i terms of Wieer oise drive sde causes troubles at reflectig boudaries {, 1}

13 Coductace oise leads to pde with radom coefficiets... τ t v = λ 2 xx v + g L (v E L ) g Na (X, ξ Na )(v E Na ) g K (X, ξ K )(v E Na ) + ξ v t X = α(v)(1 X) β(v)x + σ(v, X)ξ X compariso ad validatio of differet types, except i case studies, largely ope also derivatio from first priciples

14 Illustratio: Stochastic Hodgki-Huxley equatios u u u u u x u x realizatios of propagatio failure (resp. spotaeous activity) i more realistic models, see Sauer, S., J Comput Neurosci 216

15 Propagatio failure - umerical studies Tuckwell, et al. (28,21,211) - umerical study of P ( Propagatio failure ) w.r.t. σ from [Tuckwell, Neural Computatio, 28]

16 Propagatio failure - computatioal approach detectig propagatio failure Φ(v) := L v(x) v dx, v = restig potetial failure occurs w.r.t. give threshold θ if Φ(v(t)) < θ for some t [T, T ] hece iterested i computig ( ) p σ := P σ mi Φ(v(t)) < θ T t T

17 Spotaeous activity - computatioal approach detectig spotaeous activity usig Φ(v) = L v(x) v dx, v = restig potetial w.r.t. give threshold θ if Φ(v(t)) > θ for some t [T, T ] leads to the probability ( ) s σ := P σ mi Φ(v(t)) > θ T t T

18 Numerical Illustratios 1 pσ sσ θ =.5.5 θ =.2 θ = p ref σ σ 1.5 θ =.2 θ =.4 θ =.5 θ = σ typical plots for p σ vs. σ (resp. s σ vs. σ)

19 Model reductio w.r.t. Φ Assumig the AP ˆv is loc. exp. attractig with rate κ, hece implies where d(v ˆv) κ (v ˆv) dt + σ dξ(t) dφ(v(t)) κ (c Φ(v(t))) dt + σdβ(t) c = L ˆv(t, x) v dx idep. of time, σ2 = σ 2 1 t Var ( L W(t, x) dx ) (β(t)) - 1-dim BM ad reduces the problem to computig first passage-time probabilities of 1-dim. OU-processes d Φ = κ (c Φ) dt + σdβ(t)

20 Numerical Illustratios 1 sσ θ =.2.5 s σ θ =.3 s σ σ 1 pσ θ =.7.5 p σ θ =.5 p σ σ compariso of p σ (resp. s σ) for the full spde with the 1-dim ou typical plots for p σ vs. σ (resp. s σ vs. σ)

21 A better fit i the case of FHN parameters for FHN-system take from Tuckwell, op.cit. OU-Approximatio: κ =.2, c = 8.6 θ = 5

22 Aalysis ad Numerical Approximatio joit with Marti Sauer (TU Berli) realizatio of (2) as (desity cotrolled) (stochastic) evolutio equatio τ tv = λ 2 2 xxv g Na m 3 h(v E Na ) g K 4 (v E K ) g L (v E L ) + I dp dt = αp(v)(1 p) βp(v)p, p {m,, h} crucial properties eq is either Lipschitz or oe-sided Lipschitz joitly i (v, m,, h) coditio o io-chael cocetratios X = (m,, h), first part is liear w.r.t. v (oe-sided Lipschitz will be sufficiet for the geeral theory) coditio o v, secod part is a forward Kolmogorov eq, i particular, p (x) [, 1] implies p(t, x) [, 1]

23 Abstract settig Mathematical modellig as stochastic evolutio equatio o the fuctio space H = L 2 (, 1) dv(t) = v(t) + f (v(t), X(t)) dt + BdW (t), dx i (t) = f i (v(t), X i (t)) dt + B i (v(t), X i (t)) dw i (t). Assumptio A Local Lipschitz cotiuity ad coditioal mootoicity ( f (v, x), f (v, x) L 1 + v r 1) (1 + ρ(x)) for some r [2, 4] f i (v, x i ), f i (v, x i ) L (1 + ρ i ( v )) (1 + x i ) for ρ i s.th. ρ i (v) e α v v f (v, x) L(1 + ρ(x)) xi f i (v, x i ) L

24 Abstract settig, ctd. Mathematical modellig as stochastic evolutio equatio o the fuctio space H = L 2 (, 1) dv(t) = v(t) + f (v(t), X(t)) dt + BdW (t), dx i (t) = f i (v(t), X i (t)) dt + B i (v(t), X i (t)) dw i (t). Assumptio B Recurrece for voltage & ivariace of [, 1] for gatig variables v f (v, x) κ K v > K, x 1 f i (v, x i ) f i (v, x i ) x i, v R x i 1, v R

25 Abstract settig, ctd. dv(t) = v(t) + f (v(t), X(t)) dt + BdW (t), dx i (t) = f i (v(t), X i (t)) dt + B i (v(t), X i (t)) dw i (t). Assumptio C B L 2(H, H 1 ), i.e., (Bv)(x) = 1 B i : H H d L 2(H) with itegral kerels b(x, y)v(y) dy b H 1 ([, 1] 2 ) 1 (B i (v, x)u)(x) = 1 { x 1} b i (v(x), x(x), x, y)u(y) dy b i (v, x) L 2 ([, 1] 2 ) beig Lipschitz w.r.t. v ad x with liear growth b i (v 1, x 1, x, y) b i (v 2, x 2, x, y) L( v 1 v 2 + x 1 x 2 ) B i (v, x) L2 (H,H 1 ) b i (v, x) H 1 ([,1] 2 ) L( v H 1 + x H 1 d )

26 A priori solutio sets X := { X Ft -adapted : X(t, x) 1 P a.s. a.e. for all t [, T ] } V := v F t -adapted : v(t) L R t P-a.s. for all t [, T ] for some R t with Gaussia momets E [ exp ( α 2 R2 T )] < for some α >

27 2 step fixed poit iteratio give X v X Step 1 solve v +1 X i V Step 2 coversely solve X +1 X v +1 set + 1

28 Step 1 dv +1 (t) = v +1 (t) dt + f (v +1 (t), X (t)) dt + B dw (t) Decompose v +1 (t) = Z(t) + Y (t), where dy (t) = Y (t) dt + B dw (t) is a Orstei-Uhlebeck process, ad t Z(t) = Z(t) + f (Z(t) + Y (t), X (t)) solves a radom pde

29 Step 1, ctd. verify v +1 V usig cutoff via ormal cotractio φ(v) := mi / max{, v ± R ± R Y t } with R Y t := sup Y (s) L s [,t]

30 Step 2 dx +1 i (t) = f i (v +1 (t), X +1 i (t)) dt + B i (v +1, X +1 (t)) dw i (t) verify X +1 X usig cutoff via ormal cotractio φ(x i ) := mi{, x i } φ(x i ) := max{1, x i } i

31 Mai result: Existece & Uiqueess Theorem 1 [Sauer, S, Math Comp 216] p max{2(r 1), 4} v L p (Ω, F, P, H), with Gaussia momets (w.r.t. L ) x L p (Ω, F, P; H d ) with x 1 P-a.s. for a.e. x The there exists a uique variatioal solutio (v, X) with v V ad X X

32 Spatial approximatio No previous results for this settig Neither weak or strog rates restrict to fiite differece approximatio, but geeral fiite elemet method (w.r.t. H 1 ) should work as well v 3 v v1 v2 v

33 Spatial approximatio, ctd. Well-kow ad simple fiite differece method v3 v v 1 v2 v (D + v) k = (v k+1 v k ) (D v) k = (v k v k 1 ) ( v) k = 2 (v k+1 2v k + v k 1 )

34 Spatial approximatio, ctd. Well-kow ad simple fiite differece method v3 v v 1 v2 v ṽ H 1 (, 1) liear iterpolatio Dṽ(x) = k ṽ(x) = k (v k+1 v k )1 [ ) k (x) H, k+1 2 (v k+1 2v k + v k 1 )δ k (x) H 1 (, 1)

35 Spatial approximatio, ctd. Discretizatio of oise part without additioal assumptios β k (t) := W (t), I k 1 2 Ik H bk,l := ( I k I l ) 1 I k I l b(x, y) dy dx BW (t) bk,l W (t), 1 I l H = bk,l I l 1 2 β l (t) l l =: (B P W (t)) k

36 Mai result: Numerical Approximatio Key observatio: X X, ṽ V with the same uiform boud R t as for v i.e., uiform bouds for the supremum i terms of the Orstei-Uhlebeck process Y Theorem 2 [Sauer, S, Math Comp 216] Defie the error as E (t) := (v(t) ṽ (t), X(t) X (t)), the there exists a costat C ad a process G t < P-a.s. such that [ ] ] E sup e Gt E (t) 2 H d+1 2E [ E() 2 Hd+1 + C t [,T ] 2.

37 Mai result: Numerical Approximatio, ctd. Theorem 2 [Sauer, S, Math Comp 216] Defie the error as E (t) := (v(t) ṽ (t), X(t) X (t)), the there exists a costat C ad a process G t < P-a.s. such that [ ] ] E sup e Gt E (t) 2 H d+1 2E [ E() 2 Hd+1 + C t [,T ] 2. Corollary For all ε (, 1) there exists a r.v. C ε, P-a.s. fiite, such that sup E (t) 2 H d+1 C ε t [,T ] 1 ε.

38 Strog covergece rates NB itegrability of exp(g t) yields strog covergece rates Assumptio D Let f ad f = (f i ) i satisfy [ ] [ ] f (v1, x 1 ) f (v 2, x 2 ) v1 v 2 L( v f(v 1, x 1 ) f(v 2, x 2 ) x 1 x 1 v x 1 x 2 2 ) 2 (e.g. FitzHugh-Nagumo) Theorem 3 [Sauer, S, Math Comp 216] Uder the additioal Assumptio D there exists a costat C such that [ ] ] E sup E (t) 2 H d+1 2e LT E [ E() 2 Hd+1 + C t [,T ] 2.