New Developments in Dynamical Systems Arising from the Biosciences. The Effect of Noise on Mixed-Mode Oscillations

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1 Barbara Gent gent New Developments in Dynamical Systems Arising from the Biosciences Mathematical Biosciences Institute, March 2011 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent University of Bielefeld, Germany Joint work with: Nils Berglund (University of Orleans, France) Christian Kuehn (MPI for the Physics of Complex Systems, Dresden, Germany)

2 Mixed-Mode Oscillations (MMOs) Belousov Zhabotinsky reaction Hudson, Hart, and Marinko: Belousov-Zhabotinskii reaction a sufficient number of the infin earlier studies, some experim presented that indicates that m Cii in the oscillatory range of the 120 system. This includes two osc '0 > oscillatory state and a steady s = conditions. Except for some e E no such phenomena were obser iv (In these early studies several oscillatory state to another we -c: -CII cluded changes from one perio 80 also changes from periodic to inverse. However, these phen ducible and subsequent improv system parameters eliminated Time Iminutesj concentrations employed by M FIG. 12. Recording from bromide ion electrode; T = 25 C; quite different than those empl Recording from bromide ion electrode; T=25 flow rate = C; flow rate = 3.99 ml/min; Ce +3 catalyst [Hudson, Hart, Marinko 79] ml/min; Ce+ 3 catalyst. there is undoubtedly also a dif concentration in the feedstream and it is known that bromide io results of Figs. 12 and 13 in various ways. We jumped a significant effect on the beha to these conditions from various starting points. We Notation:... L s j 1 the calculations of Bar-Eli and j 1 Ls j j Ls j+1 also j+1... (here perturbed the Ls = 2 2 ) two peak oscillations normally occurring at these flow rates using the techniques described tion. are continuing on the effect of above. These attempts were not successful. It seems unlikely then that bistability occurs under these condi- Mathematical analyses, such The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March thermal 2011 convection and 1 / Rossle 22

3 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Layer II Stellate Cells MMOs in Biology FIG. D: 1. subthreshold Basic electrophysiological membrane potential profileoscillations entorhinal (1 and cortex 2) and (EC) spike layerclustering II stellate(3) cells develop (SCs) at under increasingly whole cell depolaried current-clamp membrane potential levels recording positive conditions. to about A: 55 digitied mv. Autocorrelation photomicrograph function demonstrating (inset 1) thedemonstrates visualiationthe of rhythmicity a patched SC. of the - - subthreshold -, approximate oscillations border [Dickson et al 00] between layers I and II. B: V-I relationship of the SC in A demonstrating robust time-dependent inward rectification in the hyperpolariing direction. C: action potential from (D2) (*) at an expanded time and voltage scale. Note the fast-afterhyperpolariation/depolariing afterpotential/medium afterhyperpolariation (fast-ahp/dap/medium-ahp) sequence characteristic of these cells. D: subthreshold membrane potential oscillations (1 and 2) and spike clustering (3) develop at increasingly depolaried membrane potential levels positive to about 55 mv. Autocorrelation function (inset in 1) demonstrates the rhythmicity of the subthreshold oscillations. Questions: Origin of small-amplitude oscillations? Source of irregularity in pattern?

4 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 MMOs & Slow Fast Systems MMOs can be observed in slow fast systems undergoing a folded-node bifurcation (1 fast, 2 slow variables) Normal form of folded-node [Benoît, Lobry 82; Smolyan, Wechselberger 01] ɛẋ = y x 2 ẏ = (µ + 1)x ż = µ 2 Questions: Dynamics for small ε > 0? Effect of noise? First step: General results for slow fast systems (deterministic / subject to noise)

5 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 General Slow Fast Systems: Singular Limits In slow time t εẋ = f (x, y) ẏ = g(x, y) t s In fast time s = t/ε x = f (x, y) y = εg(x, y) ε 0 ε 0 Slow subsystem 0 = f (x, y) ẏ = g(x, y) / Fast subsystem x = f (x, y) y = 0 Study slow variable y on slow or critical manifold f (x, y) = 0 Study fast variable x for froen slow variable y

6 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Slow (or Critical) Manifolds C 0 = {(x, y) R n R m : f (x, y) = 0} Definition C 0 is normally hyperbolic at (x, y) C 0 if x f (x, y) has only eigenvalues λ j = λ j (x, y) with Re λ j 0 C 0 is asymptotically stable or attracting at (x, y) C 0 if Re λ j (x, y) < 0 for all j C 0 is unstable at (x, y) C 0 if Re λ j (x, y) > 0 for at least one j

7 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Fenichel s Theorem: Adiabatic Manifolds Theorem [Tihonov 52; Fenichel 79] Assume C 0 is normally hyperbolic. adiabatic manifold C ε s.t. C ε is locally invariant C ε = C 0 + O(ε) If C 0 is uniformly attracting, i.e., Re(λ j (x, y)) δ < 0 (x, y) then C ε attracts nearby solutions exponentially fast C 0 x y 1 y 2 C ε

8 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Folded-Node Bifurcation: Slow Manifold ɛẋ = y x 2 ẏ = (µ + 1)x y C r 0 ż = µ 2 C a 0 x L Slow manifold has a decomposition C 0 = {(x, y, ) R 3 : y = x 2 } = C a 0 L C r 0

9 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Folded-Node: Adiabatic Manifolds and Canard Solutions Assume ε sufficiently small µ (0, 1), µ 1 N Theorem [Benoît, Lobry 82; Smolyan, Wechselberger 01; Wechselberger 05; Brøns, Krupa, Wechselberger 06] [Desroches et al 11 (to appear)]

10 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Folded-Node: Adiabatic Manifolds and Canard Solutions Assume ε sufficiently small µ (0, 1), µ 1 N Theorem Existence of strong and weak (maximal) canard γ s,w ε 2k + 1 < µ 1 < 2k + 3: k secondary canards γ j ε γ j ε makes (2j + 1)/2 oscillations around γ w ε [Desroches et al 11 (to appear)]

11 Folded-Node: Canard Spacing F GEOMETRY OF SLOW MANIFOLDS NEAR A FOLDED NODE 1153 x. y (a) γs η1 η2 η3 γw 3 3 S 1.5 γw η η 2 0 F η 1 γ s 0 F [Desroches, Krauskopf, Osinga 08] 3 γw 1.5 Lemma 0 η3 η2 x 1.5 (b) 3 y x γ s γ w η 1 η 2 η 3 (c) y 5 Figure 9. Spiraling behavior of the secondary canards of (1.2) with µ =8.5. Panel (a) shows 3 canards γs and γw (black) and the three secondary canards η1, η2, and η3; also shown for orient parabolic cylinder S with its fold curve F. Panels (b) and (c) show the projections onto the (x, )- planes, respectively. 1.5 For = 0: Distance between canards γε k and γε k+1 is O(e c0(2k+1)2µ ) We remark here that the oscillations of u (1) near a fixed canard increase as µ is η1 γs F Hence, canards 0 η i for large i can be detected reliably for larger µ and then contin into the range of lower values of µ. The Effect 1.5 of Noise on Mixed-Mode Oscillations 1.5 Barbaraγs Gent MBI, 25 March / 22

12 Folded-Node: Canard Spacing GEOMETRY OF SLOW MANIFOLDS NEAR A FOLDED NODE x (a) γs η1 η2 η3 γw 3 3 S γw η η2 η1 γs F [Desroches, Krauskopf, Osinga 08] 3 x γw η3 1.5 Lemma 1.5 η2 For = 0: Distance between η1 γs canards γ F ε k and γ k The Effect 1.5 of Noise on Mixed-Mode Oscillations 1.5 Barbaraγs Gent MBI, 25 March / 22 y (b) x 5 3 Figure 9. Spiraling behavior of the secondary canards canards γs and γw (black) and the three secondary canar parabolic cylinder S with its fold curve F. Panels (b) and planes, respectively. We remark here that the oscillations of u (1) ne Hence, canards η i for large i can be detected re into the range of lower values of µ. ε is O(e c0(2k+1)2µ ) 5.2. Spiraling behavior of the secondary c ondary canards around the weak canard, we con

13 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Random Perturbations of General Slow Fast Systems dx t = 1 ε f (x t, y t ) dt + σ ε F (x t, y t ) dw t dy t = g(x t, y t ) dt + σ G(x t, y t ) dw t {W t } t 0 k-dimensional (standard) Brownian motion adiabatic parameter ε > 0 (no quasistatic approach) noise intensities σ = σ(ε) > 0, σ = σ (ε) 0 with σ (ε)/σ(ε) = ϱ(ε) 1 Timescales: We are interested in the regime T relax = O(ε) T driving = 1 T Kramers = ε e V /σ2 (in slow time) Assumption: C 0 is uniformly attracting (for the deterministic system)

14 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Deviation from the Adiabatic Manifold due to Noise Main idea Consider deterministic process (xt det, yt det ) C ε (using invariance of C ε) Linearie SDE for deviation ξ t := x t xt det from adiabatic manifold where A(y det t Key observation ) = xf (x det dξt 0 = 1 det A(yt )ξt 0 dt + σ F 0 (y det ε ε t ) dw t, y det ) and F 0 is 0th-order approximation to F t t Resulting process ξt 0 is Gaussian 1 σ 2 Cov ξ0 t is a particular solution of the deterministic slow fast system det εẋ (t) = A(yt )X (t) + X (t)a(yt det ) T + F 0 (yt det )F 0 (yt det ) T ẏ det t = g(xt det, yt det ) System admits an adiabatic manifold {(X (y, ε), y): y D 0 }

15 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Defining Typical Neighbourhoods of Adiabatic Manifolds Typical neighbourhoods B(h) = { (x, y): [ x x(y, ε) ], X (y, ε) 1[ x x(y, ε) ] < h 2} where C ε = {( x(y, ε), y): y D 0 } C ε (xt det, yt det ) First-exit times B(h) τ D0 = inf{s > 0: y s / D 0 } τ B(h) = inf{s > 0: (x s, y s ) / B(h)}

16 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Concentration of Sample Paths near Adiabatic Manifolds Theorem [Berglund & G 03] Assume non-degeneracy of noise term: X (y, ε) and X (y, ε) 1 uniformly bounded in D 0 Then ε 0 > 0 h 0 > 0 ε ε 0 h h 0 P { τ B(h) < min(t, τ D0 ) } } C n,m (t) exp { h2 [ ] 1 O(h) O(ε) 2σ 2 where C n,m (t) = [ C m + h n]( 1 + t ) ε 2 Remarks Bound is sharp: Similar lower bound If initial condition not on C ε : additional transitional phase On longer time scales: Behaviour of slow variables becomes crucial ( Assumptions on g)

17 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Special Case: Slowly Driven Systems Typical neighbourhood dx t = 1 ε f (x t, t) dt + σ ε F (x t, t) dw t B(h) = { (x, t): [ x x det t ) ], X (t, ε) 1[ x x det ] < h 2 } t Estimate for all admissible t P { τ B(h) < t } C n,m (t) exp x 1 { h2 x 2 y det t = t } [ ] 1 O(h) O(ε) 2σ 2

18 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Stochastic Folded Nodes: Rescaling dx t = 1 ε (y t x 2 t ) dt + σ ε dw (1) t dy t = [ (µ + 1)x t t ] dt + σ dw (2) t d t = µ 2 dt Rescaling (blow-up transformation): (x, y,, t) = ( ε x, εȳ, ε, ε t) dx t = (y t xt 2 ) dt + σ (1) dw ε3/4 t dy t = [ (µ + 1)x t t ] dt + σ ε d t = µ 2 dt (2) dw 3/4 t Rescale noise intensities: (σ, σ ) = (ε 3/4 σ, ε 3/4 σ ) and consider as time

19 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Stochastic Folded Nodes: Final Reduction Step Deviation (ξ, η ) = (x x det, y y det ) satisfies dξ = 2 µ (η ξ 2 2x det ξ ) d + 2σ µ dw (1) dη = 2 µ (µ + 1)ξ d + 2σ µ dw (2) We re in business... (almost) For small µ: Slowly driven system with two fast variables Calculate asymptotic covariance matrix Use Neishtadt s theorem on delayed Hopf bifurcations to obtain the correct asymptotic behaviour of the sie of the covariance tube Use general result on concentration of sample paths

20 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Stochastic Folded Nodes: Concentration of Sample Paths Theorem [Berglund, G & Kuehn 10 (submitted)] P { τ B(h) < } } C( 0, ) exp { κ h2 2σ 2 [ 0, µ] Recall: For = 0 Distance between canards γ k ε and γ k+1 ε is O(e c0(2k+1)2µ ) Section of B(h) is close to circular with radius µ 1/4 h Noisy canards become indistinguishable when typical radius µ 1/4 σ distance

21 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Canards or Pasta...? γ s ε γ 1 ε γ 2 ε γ 3 ε γ 4 ε γ 5 ε γ w ε

22 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Noisy Small-Amplitude Oscillations Theorem [Berglund, G & Kuehn 10 (submitted)] Canards with 2k+1 2 oscillations become indistinguishable from noisy fluctuations for σ > σ k (µ) = µ 1/4 e (2k+1)2 µ 0.6 σ 1 σ 0(µ) (a) σ Zoom (b) σ 2(µ) 2 σ 1(µ) σ 2(µ) µ 1 σ 4(µ) σ 3(µ) µ

23 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Early Escape Model allowing for global returns Consider > µ Σ J (a) S 0 = neighbourhood of γ w, growing like Theorem [Berglund, G & Kuehn 10] κ, κ 1, κ 2, C > 0 s.t. for σ log σ κ1 µ 3/4 y p 1, x = 0.3 pdet p(y, ) x (b) 200 P { τ S0 > } C log σ κ2 e κ(2 µ)/(µ log σ ) Remark r.h.s. small for 0.05 y µ log σ /κ

24 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 Mixed-Mode Oscillations in the Presence of Noise x y t x Observations Noise smears out small-amplitude oscillations Early transitions modify the mixed-mode pattern

25 The Effect of Noise on Mixed-Mode Oscillations Barbara Gent MBI, 25 March / 22 MMOs with Noise References Nils Berglund, Barbara Gent and Christian Kuehn, Hunting French ducks in a noisy environment, preprint, submitted to J. Differential Equations (2010) Slow Fast Systems with Noise Nils Berglund, Barbara Gent, Geometric singular perturbation theory for stochastic differential equations, J. Differential Equations 191, 1 54 (2003), Noise-Induced Phenomena in Slow Fast Dynamical Systems. A Sample-Paths Approach, Springer, London (2005) Introduction to Noise in Slowly-Driven Systems, Beyond the Fokker Planck equation: pathwise control of noisy bistable systems, J. Phys. A 35, (2002), Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations, Stoch. Dyn. 2, (2002)