Theory and applications of random Poincaré maps

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Nils Berglund nils.berglund@univ-orleans.

fr/mapmo/membres/berglund/ Random Dynamical

December 6 2017 Joint works with Manon Baudel

1 Nils Berglund Random Dynamical Systems Theory and applications of random Poincaré maps Nils Berglund MAPMO, Université d'orléans, France Lorentz Center, Leiden, December Joint works with Manon Baudel (Orléans), Barbara Gentz (Bielefeld), Christian Kuehn (Munich) and Damien Landon

2 Theory and applications of random Poincaré maps December 6, /13 (15) Deterministic Poincaré maps ODE ż = f (z) z R n Flow: z t = ϕ t (z 0 ) Σ R n : (n 1)-dimensional manifold Poincaré map (or rst-return map): T : Σ Σ T (z) = ϕ τ (z) where τ = inf{t > 0: ϕ t (z) Σ} T (z) z Σ

3 Theory and applications of random Poincaré maps December 6, /13 (15) Deterministic Poincaré maps ODE ż = f (z) z R n Flow: z t = ϕ t (z 0 ) Σ R n : (n 1)-dimensional manifold Poincaré map (or rst-return map): T : Σ Σ T (z) = ϕ τ (z) where τ = inf{t > 0: ϕ t (z) Σ} T (z) z Σ Benets: 1. Dimension reduction: T is an (n 1)-dimensional map 2. Stability of periodic orbits: no neutral direction 3. Bifurcations of periodic orbits easier to study (period doubling, Hopf,... ) Question: how about SDEs dz t = f (z t ) dt + σg(z t ) dw t?

4 Random Poincaré maps Theory and applications of random Poincaré maps December 6, /13 (15) dz t = f (z t ) dt + σg(z t ) dw t Sample path (Z z 0 t (ω)) t 0 D Γ1 Γ2 X0 Zτ X0 = X1 Σ

5 Random Poincaré maps Theory and applications of random Poincaré maps December 6, /13 (15) dz t = f (z t ) dt + σg(z t ) dw t Sample path (Z z 0 t (ω)) t 0 D Γ1 Γ2 X0 Zτ X0 = X1 Σ z 0 = X 0 Σ inf{t > 0: Z X 0 t Σ} = 0

6 Random Poincaré maps dz t = f (z t ) dt + σg(z t ) dw t Sample path (Z z 0 t (ω)) t 0 D Σ Γ1 Γ2 X0 Zτ X0 = X1 Σ z 0 = X 0 Σ inf{t > 0: Z X 0 t Σ} = 0 Solution: τ 0 = 0, τ n+1 = inf{t > τ n : Z X 0 t Σ } Solution: τ 0 = 0, τ n+1 = inf{t > τ n+1 : Z X 0 t Σ} X n = Z X 0 τ n Σ (X n ) n 0 is a Markov chain K(x, A) := P x {X 1 A} (X n, ω) X n+1 : random Poincaré map [J. Weiss, E. Knobloch, 1990], [P. Hitczenko, G. Medvedev, 2009] Theory and applications of random Poincaré maps December 6, /13 (15)

7 Application: Stochastic FitzHughNagumo eq. Theory and applications of random Poincaré maps December 6, /13 (15) dx t = 1 ε [x t x 3 t + y t ] dt + σ 1 ε dw (1) t neuron membrane potential dy t = [a x t by t ] dt + σ 2 dw (2) t open ion channels b = 0 for simplicity in this talk, bifurcation parameter δ := 3a2 1 2

8 Application: Stochastic FitzHughNagumo eq. Theory and applications of random Poincaré maps December 6, /13 (15) dx t = 1 ε [x t x 3 t + y t ] dt + σ 1 ε dw (1) t neuron membrane potential dy t = [a x t by t ] dt + σ 2 dw (2) t open ion channels b = 0 for simplicity in this talk, bifurcation parameter δ := 3a2 1 2 W (1) t, W (2) t : independent Wiener processes 0 < σ 1, σ 2 1, σ = σ1 2 + σ2 2

9 Application: Stochastic FitzHughNagumo eq. Theory and applications of random Poincaré maps December 6, /13 (15) dx t = 1 ε [x t x 3 t + y t ] dt + σ 1 ε dw (1) t neuron membrane potential dy t = [a x t by t ] dt + σ 2 dw (2) t open ion channels b = 0 for simplicity in this talk, bifurcation parameter δ := 3a2 1 2 W (1) t, W (2) t : independent Wiener processes 0 < σ 1, σ 2 1, σ = σ1 2 + σ ε = 0.1 δ = 0.02 σ 1 = σ 2 = 0.03

10 Mixed-mode oscillations (MMOs) Time series t x t for ε = 0.01, δ = , σ = ,..., Theory and applications of random Poincaré maps December 6, /13 (15)

11 Random Poincaré map Theory and applications of random Poincaré maps December 6, /13 (15) y nullclines D P x separatrix Σ X 0 X 1 X 0, X 1,... substochastic Markov chain describing process killed on D Number of small oscillations: N = inf{n 1: X n Σ}

12 Random Poincaré map Theory and applications of random Poincaré maps December 6, /13 (15) y nullclines D P x separatrix Σ X 0 X 1 X 0, X 1,... substochastic Markov chain describing process killed on D Number of small oscillations: N = inf{n 1: X n Σ} Theorem 1 [B & Landon, Nonlinearity 25: , 2012] N is asymptotically geometric: lim P{N = n + 1 N > n} = 1 λ 0 n where λ 0 R + : principal eigenvalue of the kernel K, λ 0 < 1 if σ > 0 Proof: follows from existence of spectral gap Details

13 Histograms of distribution of N (1000 spikes) Theory and applications of random Poincaré maps December 6, /13 (15) σ=ε= δ= δ= δ= δ=

14 Weak-noise regime Theory and applications of random Poincaré maps December 6, /13 (15) Theorem 2 [B & Landon, Nonlinearity 25: , 2012] Assume ε and δ/ ε suciently small There exists κ > 0 s.t. for σ 2 (ε 1/4 δ) 2 / log( ε/δ) Principal eigenvalue: 1 λ 0 exp { κ (ε1/4 δ) 2 } σ 2 Expected number of small oscillations: E µ 0 [N] C(µ 0 ) exp {κ (ε1/4 δ) 2 } σ 2 where C(µ 0 ) = probability of starting on Σ above separatrix

15 Weak-noise regime Theory and applications of random Poincaré maps December 6, /13 (15) Theorem 2 [B & Landon, Nonlinearity 25: , 2012] Assume ε and δ/ ε suciently small There exists κ > 0 s.t. for σ 2 (ε 1/4 δ) 2 / log( ε/δ) Principal eigenvalue: 1 λ 0 exp { κ (ε1/4 δ) 2 } σ 2 Expected number of small oscillations: E µ 0 [N] C(µ 0 ) exp {κ (ε1/4 δ) 2 } σ 2 where C(µ 0 ) = probability of starting on Σ above separatrix Proof: Construct A Σ such that K(x, A) exponentially close to 1 for all x A, λ 0 π 0 (A) = Σ π 0(dx)K(x, A) A π 0(dx)K(x, A) λ 0 inf x A K(x, A)

16 Weak-noise regime Theory and applications of random Poincaré maps December 6, /13 (15) Theorem 2 [B & Landon, Nonlinearity 25: , 2012] Assume ε and δ/ ε suciently small There exists κ > 0 s.t. for σ 2 (ε 1/4 δ) 2 / log( ε/δ) Principal eigenvalue: 1 λ 0 exp { κ (ε1/4 δ) 2 } σ 2 Expected number of small oscillations: E µ 0 [N] C(µ 0 ) exp {κ (ε1/4 δ) 2 } σ 2 where C(µ 0 ) = probability of starting on Σ above separatrix Proof: Construct A Σ such that K(x, A) exponentially close to 1 for all x A, λ 0 π 0 (A) = Σ π 0(dx)K(x, A) A π 0(dx)K(x, A) λ 0 inf x A K(x, A) Linear approximation around separatrix provides good description for dynamics with stronger noise

17 Summary: Parameter regimes Theory and applications of random Poincaré maps December 6, /13 (15) σ ε 3/4 III σ = (δε) 1/2 II σ = δε 1/4 I ε 1/2 σ = δ 3/2 Regime I: rare isolated spikes Theorem 2 applies (δ ε 1/2 ) Interspike interval exponential Regime II: clusters of spikes # interspike osc asympt geometric σ = (δε) 1/2 : geom(1/2) Regime III: repeated spikes P{N = 1} 1 Interspike interval constant δ σ 1 = σ 2 : ( P{N = 1} Φ Φ(x) = 1 2π x see also (πε)1/4 (δ σ 2 /ε) σ e y 2 /2 dy [Muratov & Vanden Eijnden '08] )

18 Theory and applications of random Poincaré maps December 6, /13 (15) Spectral theory of random Poincaré maps X n = Z X 0 τ n Σ (X n ) n 0 Markov chain, of kernel K(x, A) = P{X n+1 A X n = x} Under appropriate conditions K(x, A) = k(x, y) dy A D Σ X0 Zτ X0 = X1 Γ1 Γ2

19 Theory and applications of random Poincaré maps December 6, /13 (15) Spectral theory of random Poincaré maps X n = Z X 0 τ n Σ (X n ) n 0 Markov chain, of kernel K(x, A) = P{X n+1 A X n = x} Under appropriate conditions K(x, A) = k(x, y) dy A Markov semigroups: (Kϕ)(x) = (µk)(y) = Σ Σ k(x, y)ϕ(y) dy = E x [ϕ(x 1 )] K is compact operator Fredholm theory D Σ X0 Zτ X0 = X1 Γ1 ϕ L µ(x)k(x, y) dx = P µ {X 1 dy} µ L 1 Γ2

20 Theory and applications of random Poincaré maps December 6, /13 (15) Spectral theory of random Poincaré maps X n = Z X 0 τ n Σ (X n ) n 0 Markov chain, of kernel K(x, A) = P{X n+1 A X n = x} Under appropriate conditions K(x, A) = k(x, y) dy A Markov semigroups: (Kϕ)(x) = (µk)(y) = Σ Σ k(x, y)ϕ(y) dy = E x [ϕ(x 1 )] K is compact operator Fredholm theory D Σ X0 Zτ X0 = X1 Γ1 ϕ L µ(x)k(x, y) dx = P µ {X 1 dy} µ L 1 Γ2 Spectral decomposition: Kφ i = λ i φ i π i K = λ i π i π i, φ j = δ ij n k(x, y) = λ n 0φ 0 (x)π 0 (y) + λ n 1φ 1 (x)π 1 (y) +...

21 Theory and applications of random Poincaré maps December 6, /13 (15) Spectral theory of random Poincaré maps X n = Z X 0 τ n Σ (X n ) n 0 Markov chain, of kernel K(x, A) = P{X n+1 A X n = x} Under appropriate conditions K(x, A) = k(x, y) dy A Markov semigroups: (Kϕ)(x) = (µk)(y) = Σ Σ k(x, y)ϕ(y) dy = E x [ϕ(x 1 )] K is compact operator Fredholm theory D Σ X0 Zτ X0 = X1 Γ1 ϕ L µ(x)k(x, y) dx = P µ {X 1 dy} µ L 1 Γ2 Spectral decomposition: Kφ i = λ i φ i π i K = λ i π i π i, φ j = δ ij k n (x, y) = λ n 0φ 0 (x)π 0 (y) + λ n 1φ 1 (x)π 1 (y) +...

22 Theory and applications of random Poincaré maps December 6, /13 (15) Assumptions dz t = f (z t ) dt + σg(z t ) dw t z t D 0 R d+1 Assumption 1: Domain f C 2 (D 0, R d+1 ), D D 0 positively invariant under deterministic ow

23 Theory and applications of random Poincaré maps December 6, /13 (15) Assumptions dz t = f (z t ) dt + σg(z t ) dw t z t D 0 R d+1 Assumption 1: Domain f C 2 (D 0, R d+1 ), D D 0 positively invariant under deterministic ow Assumption 2: Deterministic α- and ω-limit sets N 2 asymptotically stable periodic orbits Γ 1,..., Γ N in D Other limit sets are unstable points or orbits, no heteroclinic connections

24 Theory and applications of random Poincaré maps December 6, /13 (15) Assumptions dz t = f (z t ) dt + σg(z t ) dw t z t D 0 R d+1 Assumption 1: Domain f C 2 (D 0, R d+1 ), D D 0 positively invariant under deterministic ow Assumption 2: Deterministic α- and ω-limit sets N 2 asymptotically stable periodic orbits Γ 1,..., Γ N in D Other limit sets are unstable points or orbits, no heteroclinic connections Assumption 3: Ellipticity g C 1 (D 0, R (d+1) k ) 0 < c ξ 2 ξ, g(z)g(z) T ξ c + ξ 2 z D ξ R d+1 \ {0}

25 Assumptions dz t = f (z t ) dt + σg(z t ) dw t z t D 0 R d+1 Assumption 1: Domain f C 2 (D 0, R d+1 ), D D 0 positively invariant under deterministic ow Assumption 2: Deterministic α- and ω-limit sets N 2 asymptotically stable periodic orbits Γ 1,..., Γ N in D Other limit sets are unstable points or orbits, no heteroclinic connections Assumption 3: Ellipticity g C 1 (D 0, R (d+1) k ) 0 < c ξ 2 ξ, g(z)g(z) T ξ c + ξ 2 z D ξ R d+1 \ {0} Assumption 4: Connement Lyapunov function V C 2 (D 0, R + ), V (z) as z D 0 (LV )(z) cv + d 1lz D, c > 0, d 0 L: generator of diusion Theory and applications of random Poincaré maps December 6, /13 (15)

26 Theory and applications of random Poincaré maps December 6, /13 (15) Metastable hierarchy FreidlinWentzell theory: Rate function: I [0,T ] (γ) = 1 T ( γ s f (γ s )) T [gg T (γ s )] 1 ( γ s f (γ s )) ds 2 0 Large-deviation principle: P { (z t ) 0 t T Λ } e inf γ Λ I [0,T ] (γ)/σ 2

27 Theory and applications of random Poincaré maps December 6, /13 (15) Metastable hierarchy FreidlinWentzell theory: Rate function: I [0,T ] (γ) = 1 T ( γ s f (γ s )) T [gg T (γ s )] 1 ( γ s f (γ s )) ds 2 0 Large-deviation principle: P { (z t ) 0 t T Λ } e inf γ Λ I [0,T ] (γ)/σ 2 Quasipotential between periodic orbits: H(i, j) = inf T >0 inf I [0,T ] (γ) γ:γ i Γ j

28 Theory and applications of random Poincaré maps December 6, /13 (15) Metastable hierarchy FreidlinWentzell theory: Rate function: I [0,T ] (γ) = 1 T ( γ s f (γ s )) T [gg T (γ s )] 1 ( γ s f (γ s )) ds 2 0 Large-deviation principle: P { (z t ) 0 t T Λ } e inf γ Λ I [0,T ] (γ)/σ 2 Quasipotential between periodic orbits: H(i, j) = inf Assumption 5: Metastable hierarchy θ > 0 s.t. 2 k N T >0 inf I [0,T ] (γ) γ:γ i Γ j min l<k H(k, l) min H(i, j) θ i<k j i H(4, {1, 2, 3}) x 4 H(3, {1, 2}) x 3 H(2, 1) x 2 x 1

29 Theory and applications of random Poincaré maps December 6, /13 (15) Metastable hierarchy FreidlinWentzell theory: Rate function: I [0,T ] (γ) = 1 T ( γ s f (γ s )) T [gg T (γ s )] 1 ( γ s f (γ s )) ds 2 0 Large-deviation principle: P { (z t ) 0 t T Λ } e inf γ Λ I [0,T ] (γ)/σ 2 Quasipotential between periodic orbits: H(i, j) = inf Assumption 5: Metastable hierarchy θ > 0 s.t. 2 k N T >0 inf I [0,T ] (γ) γ:γ i Γ j min l<k H(k, l) min H(i, j) θ i<k j i H(4, {1, 2, 3}) x 4 H(3, {1, 2}) x 3 H(2, 1) x 2 Remark: Using Doob's h-transform, one may replace Assumption 4 by Assumption 4': Connement θ > 0 such that min i H(i, D) max H(i, j) + θ i j x 1

30 Theory and applications of random Poincaré maps December 6, /13 (15) Main results [Baudel & B, SIAM J. Math. Anal. 49: , 2017] B k Σ: nbh of Γ k Σ, M k = j k B k, τ A, τ + A Theorem 1: Eigenvalues 1st-passage/return time to A The N largest eigenvalues of K are real and positive. θ k, c > 0 s.t. λ 0 = 1 λ k = 1 P πk+1 0 {τ + M k < τ + B k+1 }[1 + O(e θ k/σ 2 )] 1 k N 1 c λ k < 1 log(σ 1 ) =: ρ k N π k+1 0 : QSD on B k+1 and P πk+1 0 {τ + M k < τ + B k+1 } e H(k+1,{1,...,k})/σ2

31 Theory and applications of random Poincaré maps December 6, /13 (15) Main results [Baudel & B, SIAM J. Math. Anal. 49: , 2017] B k Σ: nbh of Γ k Σ, M k = j k B k, τ A, τ + A Theorem 1: Eigenvalues 1st-passage/return time to A The N largest eigenvalues of K are real and positive. θ k, c > 0 s.t. λ 0 = 1 λ k = 1 P πk+1 0 {τ + M k < τ + B k+1 }[1 + O(e θ k/σ 2 )] 1 k N 1 c λ k < 1 log(σ 1 ) =: ρ k N π k+1 0 : QSD on B k+1 and P πk+1 0 {τ + M k < τ + B k+1 } e H(k+1,{1,...,k})/σ2 Theorem 2: Eigenfunctions φ 0 (x) = 1 φ k (x) = P x {τ Bk+1 < τ Mk }[1 + O(e θ/σ 2 )] + O(e θ k /σ 2 ) π k (B k+1 ) = 1 O(e κ/σ2 ) π k (B j ) = O(e θ/σ 2 ) 0 k < j < N N 1 k n (x, y) = π 0 (y) + λ n i φ i (x)π i (y) + O(ρ n ) n log(ρ 1 ) Proofs i=1

32 Theory and applications of random Poincaré maps December 6, /13 (15) Main results [Baudel & B, SIAM J. Math. Anal. 49: , 2017] B k Σ: nbh of Γ k Σ, M k = j k B k, τ A, τ + A Theorem 1: Eigenvalues 1st-passage/return time to A The N largest eigenvalues of K are real and positive. θ k, c > 0 s.t. λ 0 = 1 λ k = 1 P πk+1 0 {τ + M k < τ + B k+1 }[1 + O(e θ k/σ 2 )] 1 k N 1 c λ k < 1 log(σ 1 ) =: ρ k N π k+1 0 : QSD on B k+1 and P πk+1 0 {τ + M k < τ + B k+1 } e H(k+1,{1,...,k})/σ2 Theorem 2: Eigenfunctions φ 0 (x) = 1 φ k (x) = P x {τ Bk+1 < τ Mk }[1 + O(e θ/σ 2 )] + O(e θ k /σ 2 ) π k (B k+1 ) = 1 O(e κ/σ2 ) π k (B j ) = O(e θ/σ 2 ) 0 k < j < N Theorem 3: Expected hitting times E x [τ Mk ] = [1 λ k ] 1 [1 + O(e κ/σ2 )] x B k+1, 1 k N 1 Proofs

33 Theory and applications of random Poincaré maps December 6, /13 (15) Outlook In progress: cases without metastable hierarchy (eigenvalue crossings) In progress: applications (Hawkes, LotkaVolterra, MorrisLecar,... ) In progress: numerical algorithms (based on QSDs, adaptive multilevel splitting) Related: exit through an unstable periodic orbit [3.] Related: MMOs in fast-slow systems with folded node singularity [4.] Open: asymptotics beyond large deviations References: 1. N. B. & Damien Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHughNagumo model, Nonlinearity 25, (2012) 2. Manon Baudel & N. B., Spectral theory for random Poincaré maps, SIAM J. Math. Analysis 49, (2017)

34 Theory and applications of random Poincaré maps December 6, /13 (15) Outlook In progress: cases without metastable hierarchy (eigenvalue crossings) In progress: applications (Hawkes, LotkaVolterra, MorrisLecar,... ) In progress: numerical algorithms (based on QSDs, adaptive multilevel splitting) Related: exit through an unstable periodic orbit [3.] Related: MMOs in fast-slow systems with folded node singularity [4.] Open: asymptotics beyond large deviations References: 1. N. B. & Damien Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHughNagumo model, Nonlinearity 25, (2012) 2. Manon Baudel & N. B., Spectral theory for random Poincaré maps, SIAM J. Math. Analysis 49, (2017) 3. N. B. & Barbara Gentz, On the noise-induced passage through an unstable periodic orbit II: General case, SIAM J. Math. Analysis 46, (2014) 4. N. B., Barbara Gentz & Christian Kuehn, From random Poincaré maps to stochastic mixed-mode-oscillation patterns, J. Dynam. Di. Eq. 27, (2015)

35 Proof of asymptotically geometric distribution Theory and applications of random Poincaré maps December 6, /13 (15) Theorem 1 [B & Landon, Nonlinearity 25: , 2012] N is asymptotically geometric: lim P{N = n + 1 N > n} = 1 λ 0 n where λ 0 (0, 1) if σ > 0 is principal eigenvalue of the kernel K Proof: Markov chain on Σ, kernel K with density k [Ben Arous, Kusuoka, Stroock '84] λ 0 sup K(x, Σ) < 1 by ellipticity (k bounded below) x Σ P µ0 {N > n} = P µ0 {X n Σ} = Σ µ 0(dx)K n (x, Σ) P µ0 {N > n} = P µ0 {X n Σ} = Σ µ 0(dx)λ n 0 φ 0(x)[1 + O(( λ 1 /λ 0 ) n )] P µ0 {N > n} = P µ0 {X n Σ} = λ n 0 µ 0, φ 0 [1 + O(( λ 1 /λ 0 ) n )] P µ0 {N = n + 1} = Σ Σ µ 0(dx)K n (x, dy)[1 K(y, Σ)] P µ0 {N = n + 1} = λ n 0 (1 λ 0) µ 0, φ 0 [1 + O(( λ 1 /λ 0 ) n )] Existence of spectral gap follows from positivity condition [Birkho '57] Back

36 Theory and applications of random Poincaré maps December 6, /13 (15) Proof of spectral decomposition FeynmanKac-type relation: For e u > sup P x {X 1 A c } { x A c (Kψ)(x) = e u ψ(x) x A c ψ(x) = E x [e uτ A φ(x τa )] ψ(x) = φ(x) x A To estimate (λ k, φ k ) choose A = M k+1 = k+1 j=1 B j

37 Theory and applications of random Poincaré maps December 6, /13 (15) Proof of spectral decomposition FeynmanKac-type relation: For e u > sup P x {X 1 A c } { x A c (Kψ)(x) = e u ψ(x) x A c ψ(x) = E x [e uτ A φ(x τa )] ψ(x) = φ(x) x A To estimate (λ k, φ k ) choose A = M k+1 = k+1 j=1 B j Restricted kernel: K u (x, dy) = E x [e u(τ + A 1) 1l{X τ + A dy}] (Kφ)(x) = e u φ(x) x Σ (K u φ)(x) = e u φ(x) x A

38 Theory and applications of random Poincaré maps December 6, /13 (15) Proof of spectral decomposition FeynmanKac-type relation: For e u > sup P x {X 1 A c } { x A c (Kψ)(x) = e u ψ(x) x A c ψ(x) = E x [e uτ A φ(x τa )] ψ(x) = φ(x) x A To estimate (λ k, φ k ) choose A = M k+1 = k+1 j=1 B j Restricted kernel: K u (x, dy) = E x [e u(τ + A 1) 1l{X τ + A dy}] (Kφ)(x) = e u φ(x) x Σ (K u φ)(x) = e u φ(x) x A 1st approximation: K u (x, dy) K 0 (x, dy) = P x {X τ + A dy} Trace process

39 Theory and applications of random Poincaré maps December 6, /13 (15) Proof of spectral decomposition FeynmanKac-type relation: For e u > sup P x {X 1 A c } { x A c (Kψ)(x) = e u ψ(x) x A c ψ(x) = E x [e uτ A φ(x τa )] ψ(x) = φ(x) x A To estimate (λ k, φ k ) choose A = M k+1 = k+1 j=1 B j Restricted kernel: K u (x, dy) = E x [e u(τ + A 1) 1l{X τ + A dy}] (Kφ)(x) = e u φ(x) x Σ (K u φ)(x) = e u φ(x) x A 1st approximation: K u (x, dy) K 0 (x, dy) = P x {X τ + dy} Trace process 2nd approximation: K 0 (x, dy) K (x, dy) = k+1 i=1 A 1l {x B i }P πi 0 {Xτ + A dy}

40 Theory and applications of random Poincaré maps December 6, /13 (15) Proof of spectral decomposition FeynmanKac-type relation: For e u > sup P x {X 1 A c } { x A c (Kψ)(x) = e u ψ(x) x A c ψ(x) = E x [e uτ A φ(x τa )] ψ(x) = φ(x) x A To estimate (λ k, φ k ) choose A = M k+1 = k+1 j=1 B j Restricted kernel: K u (x, dy) = E x [e u(τ + A 1) 1l{X τ + A dy}] (Kφ)(x) = e u φ(x) x Σ (K u φ)(x) = e u φ(x) x A 1st approximation: K u (x, dy) K 0 (x, dy) = P x {X τ + dy} Trace process 2nd approximation: K 0 (x, dy) K (x, dy) = k+1 i=1 A 1l {x B i }P πi 0 {Xτ + A Lemma: Though K not reversible disjoint A 1, A 2 Σ π 0 (x)p x {τ + A 2 < τ + A 1 } dx = π 0 (x)p x {τ + A 1 < τ + A 2 } dx A 1 A 2 dy} Back

Some results on interspike interval statistics in conductance-based models for neuron action potentials

Some results on interspike interval statistics in conductance-based models for neuron action potentials Nils Berglund MAPMO, Université d Orléans CNRS, UMR 7349 et Fédération Denis Poisson www.univ-orleans.fr/mapmo/membres/berglund