Conduction velocity restitution in models of electrical excitation in the heart

Transcription

1 Conduction velocity restitution in models of electrical excitation in the heart Who? From? When? School of Mathematics and Statistics University of Glasgow December 7, 2011 / Liverpool

2 References and credits Publications Credits Simitev, R., Biktashev, V.N., Asymptotics of conduction velocity restitution in models of electrical excitation in the heart, Bull. Math. Biol., 73(1), pp , Biktashev, V.N., Suckley, R., Elkin, Y.E, Simitev, R., Asymptotic analysis and analytical solutions of a model of cardiac excitation, Bull. Math. Biol., 70(2), pp , Vadim N. Biktashev, Dept of Mathematical Sciences, University of Liverpool

3 Restitution curves Restitution curves What is a restitution curve? Two types of restitution curves: APD restitution curve is the dependence of the duration of an action potential (APD) on the duration of the preceding diastolic interval (DI). 40 a b 0 E -40 Elev -80 APD DI t CV restitution curve is the dependence of conduction velocity (CV) on the distance to the preceding action potential (P). Click to play ANIMATION Why are restitution curves important? Integral quantity - Restitution is relatively easy to measure and gives a good summary of the overall state of the medium. Restitution hypothesis - A slope of the APD restitution curve greater than one indicates instability of a sequence of action potentials (Nolasco & Dahlen, 1968).

4 Restitution curves Restitution curves What is a restitution curve? Two types of restitution curves: APD restitution curve is the dependence of the duration of an action potential (APD) on the duration of the preceding diastolic interval (DI). 40 a b 0 E -40 Elev -80 APD DI t CV restitution curve is the dependence of conduction velocity (CV) on the distance to the preceding action potential (P). Click to play ANIMATION Why are restitution curves important? Integral quantity - Restitution is relatively easy to measure and gives a good summary of the overall state of the medium. Restitution hypothesis - A slope of the APD restitution curve greater than one indicates instability of a sequence of action potentials (Nolasco & Dahlen, 1968).

5 Restitution curves Mathematical description Typical electrophysiological models of cardiac excitation In a one-dimensional, homogeneous and isotropic medium: E t = X l I l (E,y) + D 2 E x 2, y t = Fy(E,y). (1a) (1b) Direct numerical computation of restitution CV restitution curves are typically computed by direct numerical simulation of the reaction-diffusion PDEs (1) following a particular protocol. Drawback: Protocol dependent Drawback: Unstable states cannot be found Drawback: Numerically expensive The equations are very stiff The functional dependences in I l (E,y) are very messy Drawback: Lack of conceptual understanding

6 Restitution curves Mathematical description Typical electrophysiological models of cardiac excitation In a one-dimensional, homogeneous and isotropic medium: E t = X l I l (E,y) + D 2 E x 2, y t = Fy(E,y). (1a) (1b) Direct numerical computation of restitution CV restitution curves are typically computed by direct numerical simulation of the reaction-diffusion PDEs (1) following a particular protocol. Drawback: Protocol dependent Drawback: Unstable states cannot be found Drawback: Numerically expensive The equations are very stiff The functional dependences in I l (E,y) are very messy Drawback: Lack of conceptual understanding

7 Illustration Noble model (1962) Example: The simplest physiological model where de dt = g1(e) m3 (E) θ(e Em) h + g2(e) n4 + G(E) + 2 E x 2, dh dt = f h(e) (h (E) h), dn = fn(e) (n (E) n), dt (2c) G(E) = g 1(E)W(E) + g 3(E), W(E) = m 3 (E)h (E) g 1(E) = C M 1 g Na (E Na E), g 2(E) = C M 1 g K (E K E), g 3(E) = C M 1 ˆg Na1 (E Na E) + g K1 (E)(E K E), g K1 (E) = 1.2 exp (( E 90)/50) exp((e + 90)/60), y (E) = α y(e)/ (α y(e) + β y(e)), y = h, n, m, f y(e) = α y(e) + β y(e), y = h, n, α m(e) = 0.1 ( E 48) exp(( E 48)/15) 1, βm(e) = 0.12 (E + 8) exp((e + 8)/5) 1, α h (E) = 0.17 exp(( E 90)/20), β h (E) = α n(e) = 1 exp(( E 42)/10) + 1, ( E 50), βn(e) = exp (( E 90)/80), exp(( E 50)/10) 1 C M = 12, g Na = 400, g K = 1.2, g Na1 = 0.14, E K = 110 E Na = 40. (2a) (2b)

8 Restitution curves Idealised Boundary-Value-Problem formulation Traveling waves To eliminate the drawback of protocol dependence. Assume solutions in the form of waves travelling with a constant velocity c > 0 and a fixed shape. Introduce a travelling wave ansatz for the dynamical variables F = E,y F(z) = F(x ct) Periodic BVP Then the equations become a set of ODEs: d 2 E dz 2 + cde dz + X l with periodic boundary conditions E(0) = E(c P), I l (E,y) = 0, c dy + Fy(E,y) = 0, dz (3b) de dz = de, z=0 dz z=c P y(0) = y(c P), E(0) = E 0. (3a)

9 Restitution curves Idealised Boundary-Value-Problem formulation Traveling waves To eliminate the drawback of protocol dependence. Assume solutions in the form of waves travelling with a constant velocity c > 0 and a fixed shape. Introduce a travelling wave ansatz for the dynamical variables F = E,y F(z) = F(x ct) Periodic BVP Then the equations become a set of ODEs: d 2 E dz 2 + cde dz + X l with periodic boundary conditions E(0) = E(c P), I l (E,y) = 0, c dy + Fy(E,y) = 0, dz (3b) de dz = de, z=0 dz z=c P y(0) = y(c P), E(0) = E 0. (3a)

10 Restitution curves Structure of the BVP Integration constants Available constraints Expected result Construction of the CV-restitution curve Order of ODEs (dim(y) + 2) Two additional unknown parameters + c and P Total = (dim(y) + 4) Periodic boundary conditions (dim(y) + 2) One phase condition + E(0) = E 0 Total = (dim(y) + 3) One-parameter family of solutions. The ideal dynamic CV restitution curve describing the dependence of the wave speed c on the wave period P is the projection of this family onto the (P, c) plane. Challenge This method is applicable to any cardiac model but it is difficult to apply due to 1. the overwhelmingly messy functional form, 2. the inherent stiffness of cardiac equations. Suggested resolution Asymptotics.

11 Restitution curves Structure of the BVP Integration constants Available constraints Expected result Construction of the CV-restitution curve Order of ODEs (dim(y) + 2) Two additional unknown parameters + c and P Total = (dim(y) + 4) Periodic boundary conditions (dim(y) + 2) One phase condition + E(0) = E 0 Total = (dim(y) + 3) One-parameter family of solutions. The ideal dynamic CV restitution curve describing the dependence of the wave speed c on the wave period P is the projection of this family onto the (P, c) plane. Challenge This method is applicable to any cardiac model but it is difficult to apply due to 1. the overwhelmingly messy functional form, 2. the inherent stiffness of cardiac equations. Suggested resolution Asymptotics.

12 Restitution curves Structure of the BVP Integration constants Available constraints Expected result Construction of the CV-restitution curve Order of ODEs (dim(y) + 2) Two additional unknown parameters + c and P Total = (dim(y) + 4) Periodic boundary conditions (dim(y) + 2) One phase condition + E(0) = E 0 Total = (dim(y) + 3) One-parameter family of solutions. The ideal dynamic CV restitution curve describing the dependence of the wave speed c on the wave period P is the projection of this family onto the (P, c) plane. Challenge This method is applicable to any cardiac model but it is difficult to apply due to 1. the overwhelmingly messy functional form, 2. the inherent stiffness of cardiac equations. Suggested resolution Asymptotics.

13 Restitution curves Structure of the BVP Integration constants Available constraints Expected result Construction of the CV-restitution curve Order of ODEs (dim(y) + 2) Two additional unknown parameters + c and P Total = (dim(y) + 4) Periodic boundary conditions (dim(y) + 2) One phase condition + E(0) = E 0 Total = (dim(y) + 3) One-parameter family of solutions. The ideal dynamic CV restitution curve describing the dependence of the wave speed c on the wave period P is the projection of this family onto the (P, c) plane. Challenge This method is applicable to any cardiac model but it is difficult to apply due to 1. the overwhelmingly messy functional form, 2. the inherent stiffness of cardiac equations. Suggested resolution Asymptotics.

14 Parameter embeddings and asymptotic analysis Parameter embeddings Rationale We introduce auxiliary parameters to achieve a well-defined mathematical approach to asymptotics. Definition Parameter embedding - P1 A parametric embedding with parameter ǫ of a function f(x) is any function f(x; ǫ) such that f(x, 1) = f(x). In the context of ǫ 0 we say asymptotic embedding. Aim: To simplify the functional form of the model. Justification: Complicated functional form is not always essential, especially for conceptual understanding. Explicitly: Replace f(x) by f(x; ǫ) = ǫf(x) + (1 ǫ)g(x), ǫ [0, 1]. Example: Click to play ANIMATION Parameter embedding - P2 Aim: To take into account intrinsic scale separation of the model. Justification: Well-tested mathematical approach. Explicitly: Replace f(x) by ( ǫf(x) f(x; ǫ) = ǫ 1 f(x) if f(x) relatively SMALL, if f(x) relatively LARGE, ǫ [0, 1].

15 Parameter embeddings and asymptotic analysis Parameter embeddings Rationale We introduce auxiliary parameters to achieve a well-defined mathematical approach to asymptotics. Definition Parameter embedding - P1 A parametric embedding with parameter ǫ of a function f(x) is any function f(x; ǫ) such that f(x, 1) = f(x). In the context of ǫ 0 we say asymptotic embedding. Aim: To simplify the functional form of the model. Justification: Complicated functional form is not always essential, especially for conceptual understanding. Explicitly: Replace f(x) by f(x; ǫ) = ǫf(x) + (1 ǫ)g(x), ǫ [0, 1]. Example: Click to play ANIMATION Parameter embedding - P2 Aim: To take into account intrinsic scale separation of the model. Justification: Well-tested mathematical approach. Explicitly: Replace f(x) by ( ǫf(x) f(x; ǫ) = ǫ 1 f(x) if f(x) relatively SMALL, if f(x) relatively LARGE, ǫ [0, 1].

16 Parameter embeddings and asymptotic analysis Parameter embeddings Rationale We introduce auxiliary parameters to achieve a well-defined mathematical approach to asymptotics. Definition Parameter embedding - P1 A parametric embedding with parameter ǫ of a function f(x) is any function f(x; ǫ) such that f(x, 1) = f(x). In the context of ǫ 0 we say asymptotic embedding. Aim: To simplify the functional form of the model. Justification: Complicated functional form is not always essential, especially for conceptual understanding. Explicitly: Replace f(x) by f(x; ǫ) = ǫf(x) + (1 ǫ)g(x), ǫ [0, 1]. Example: Click to play ANIMATION Parameter embedding - P2 Aim: To take into account intrinsic scale separation of the model. Justification: Well-tested mathematical approach. Explicitly: Replace f(x) by ( ǫf(x) f(x; ǫ) = ǫ 1 f(x) if f(x) relatively SMALL, if f(x) relatively LARGE, ǫ [0, 1].

17 Parameter embeddings and asymptotic analysis Example of param embeddings: Noble model Selecting useful embeddings A black art based on observations drawn from numerical experiments.

18 Ò½ Parameter embeddings and asymptotic analysis Example of param embeddings: Noble model Selecting useful embeddings Example of parameter embedding P1 A black art based on observations drawn from numerical experiments. µ µ ½º½ ¼º¼½ ¾ Ò½ ½ ¼º Ò ¼º¼¼ ¼ Ò ¼º ¼º¼¼ ¹½ ¹¼º½ ¹½¼¼ ¹ ¼ ¼ ¼º¼¼½ ¹½¼¼ Ý ¹ ¼ ¼ ¹¾ Figure: (a b) The caricature approximations (thin lines) of the right-hand side functions of the Noble model (2) (thick lines).

19 ÙÒ ÓÒ Ò½ Parameter embeddings and asymptotic analysis Example of param embeddings: Noble model Selecting useful embeddings Example of parameter embedding P1 A black art based on observations drawn from numerical experiments. µ µ ½º½ ¼º¼½ ¾ Ò½ ½ ¼º Ò ¼º¼¼ ¼ Ò ¼º ¼º¼¼ ¹½ ¹¼º½ ¹½¼¼ ¹ ¼ ¼ ¼º¼¼½ ¹½¼¼ Ý ¹ ¼ ¼ ¹¾ Figure: (a b) The caricature approximations (thin lines) of the right-hand side functions of the Noble model (2) (thick lines). Method To estimate relative magnitude define scaling functions τ l (w 1,...) df l /dw l for a set of ODEs dw l /dt = F l (w 1,... w N ). Example of parameter µ µ embedding P2 ½ ¼º¼¾ Ñ ½ ½ ¼º ¼º ¼º ½ Ñ ½ Ò ¾Ï ¾¼Ë ¼º¼½ ¼º¼½ ¼º¾ ¼º¼¼ ¼ ¹½¼¼ ¹ ¼ Ñ ¼ ¹½¼¼ ¹ ¼ ¼ ¼ Figure: Main functions of E which determine the asymptotic properties of the Noble model.

20 Illustration - the caricature Noble model Final result: The caricature Noble model P1 Embed a first parameter µ to simplify functional form as shown above. Take the limit µ 0. The functions of the caricature Noble model become: g 2(E) = g 21θ(E E) + g 22θ(E E ), g 21 = 2, g 22 = 9, 8 < k 1(E 1 E), E (, E ), G(E) = k 2(E E 2), E [E :, E ), k 3(E 3 E), E [E, + ), k 1 = 3/40, k 2 = 1/25, k 3 = 1/10, E 1 = 280/3, E 2 = (k 1/k 2 + 1)E E 1k 1/k 2 = 55, E 3 = (k 2/k 3 + 1)E E 2k 2/k 3 = 1, F h = 1/2, F n = 1/270, E Na = 40, (4) E = 80, E = 15, G Na = 100/3, P2 Embed a second parameter ǫ to split scales of the problem. The equations of the caricature Noble model become E t = 1 ǫ G Na (E Na E) θ(e E ) h + g 2(E) n 4 + G(E) + ǫ 2 E x 2, h t = 1 ǫ F h `θ(e E) h, n t = `θ(e Fn E ) n, (5a) (5b) (5c)

21 Illustration - the caricature Noble model Final result: The caricature Noble model P1 Embed a first parameter µ to simplify functional form as shown above. Take the limit µ 0. The functions of the caricature Noble model become: g 2(E) = g 21θ(E E) + g 22θ(E E ), g 21 = 2, g 22 = 9, 8 < k 1(E 1 E), E (, E ), G(E) = k 2(E E 2), E [E :, E ), k 3(E 3 E), E [E, + ), k 1 = 3/40, k 2 = 1/25, k 3 = 1/10, E 1 = 280/3, E 2 = (k 1/k 2 + 1)E E 1k 1/k 2 = 55, E 3 = (k 2/k 3 + 1)E E 2k 2/k 3 = 1, F h = 1/2, F n = 1/270, E Na = 40, (4) E = 80, E = 15, G Na = 100/3, P2 Embed a second parameter ǫ to split scales of the problem. The equations of the caricature Noble model become E t = 1 ǫ G Na (E Na E) θ(e E ) h + g 2(E) n 4 + G(E) + ǫ 2 E x 2, h t = 1 ǫ F h `θ(e E) h, n t = `θ(e Fn E ) n, (5a) (5b) (5c)

22 Illustration - the caricature Noble model Exact analytical solution of the Caricature model Success! Solution procedure Functionall form is so simple that exact analytical solution in the case 2 xe = 0 can now be found! 1. The equations for h and n are separable find h(t) and n(t). 2. Upon substitution of h(t) and n(t) into the voltage equation it becomes a first-order linear ODE - solution easy to find. 3. Initial conditions E(0) = E α > E, h(0) = 1, n(0) = Natural continuity conditions at E and E. ( 1 exp( Fnt), t [0, t ] Solution n(t) = (6a) `exp(fnt ) 1 exp( F nt), t [t, ] ( exp( Fh t/ǫ), t [0, t ] h(t) = 1 `1 + exp(f h t /ǫ) exp( F h t/ǫ), t [t, ] (6b) 8 1 E(t) = exp G Na exp F «! " ht k 3 t E α exp G «Na F h ǫ F h 4X «k 3 E 3 u( k 3, t) g 0 2 ( 1) l 4 u ((4 l) F n k 3, t) >< l l=0 E(t) = G «NaE Na Fh u ǫ ǫ k 3, t, t [0, t ] 2 E(t) = (E w(t )) exp (k 2 (t t )) + w(t), t [t, t ] >: 3 E(t) = (E E 1 ) exp ` k 1 (t t ) + E 1, t [t, ] (6c)

23 Ò Illustration - the caricature Noble model Deviation of the exact solution from the original Noble model Solution (cont.) u(κ, t) ǫ GNa «κǫ " F h Γ κǫ, G! Na Γ κǫ, G Na exp F h t «!#, F h F h F h F h F h F h ǫ w(t) E 2 g2 0 4X ( 1) l 4 «exp ( l Fnt), l=0 l k 2 + l Fn and Γ(a, x) is the upper incomplete gamma function. Measure of deviation Advantage: The deviation of the parametric embedding (caricature model) from the original model (Noble) can be measured exactly via continuation in ǫ. ¼ µ Ø ØÝ µ ½º½ ½º½ ¼ ¹ ¼ ¼º ¼º ¾ Ò ¼º ¼º ¹½¼¼ ¼ ¹¼º½ ¹½ ¾¼¼ ¼¼ ¼ Ø ¼º¾ ¼º ¼º ¼º ½ ¹¼º½ Ì Figure: The numerical solution of the original Noble model (thick lines) in comparison with the analytical solution (6) of the caricature model (5) (corresponding thin lines) (a) in slow time t [0, 600], for ǫ = 1,, and (b) in fast time T [0, 1], for ǫ = 10 3.

24 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Slow-time subsystem Derivation 1. Take the limit ǫ The h-equation implies h θ(e E). 3. Hence the first term of the E-equation is proportional to θ(e E )θ(e E) = 0 and vanishes in the limit ǫ 0 despite the large factor ǫ 1 in front of it. 4. The diffusion term 2 xe vanishes in the same limit. Slow-time equations Then the equations reduce to de dη = g 2(E) n 4 + G(E), dn dη = `θ(e Fn E ) n, (8a) (8b) where η = t x/c is the traveling wave coordinate. Description of The slow-time subsystem describes the plateau and the recovery stages of the traveling action potentials.

25 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Slow-time subsystem Derivation 1. Take the limit ǫ The h-equation implies h θ(e E). 3. Hence the first term of the E-equation is proportional to θ(e E )θ(e E) = 0 and vanishes in the limit ǫ 0 despite the large factor ǫ 1 in front of it. 4. The diffusion term 2 xe vanishes in the same limit. Slow-time equations Then the equations reduce to de dη = g 2(E) n 4 + G(E), dn dη = `θ(e Fn E ) n, (8a) (8b) where η = t x/c is the traveling wave coordinate. Description of The slow-time subsystem describes the plateau and the recovery stages of the traveling action potentials.

26 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Fast-time subsystem Derivation 1. Stretch time and space, T = t/ǫ, X = x/ǫ. 2. Take the limit ǫ 0, 3. The n-equation decouples from the rest - neglect. 4. Non-dimensionalize: v = V E E Na E, ξ = X p F h, τ = F h T, g = GNa F h, C = c/ p F h. Fast-time equations Then the equations reduce to dv dζ = 1 d 2 v C 2 + g (1 v) θ(v) H, dζ2 (9a) dh dζ = θ(v v) H, (9b) Description of The fast-time subsystem describes the wave front of the traveling action potentials.

27 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Fast-time subsystem Derivation 1. Stretch time and space, T = t/ǫ, X = x/ǫ. 2. Take the limit ǫ 0, 3. The n-equation decouples from the rest - neglect. 4. Non-dimensionalize: v = V E E Na E, ξ = X p F h, τ = F h T, g = GNa F h, C = c/ p F h. Fast-time equations Then the equations reduce to dv dζ = 1 d 2 v C 2 + g (1 v) θ(v) H, dζ2 (9a) dh dζ = θ(v v) H, (9b) Description of The fast-time subsystem describes the wave front of the traveling action potentials.

28 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Coupling via boundary conditions Rationale Boundary conditions In a periodic wave train, a front propagates in the tail of the preceding wave, so slow pieces described by (8) and fast pieces described by (9) alternate. There are one slow piece and one fast piece per period. This situation is summarized by the following set of boundary conditions For the fast-time subsystem v( ) = v α, v( ) = v ω, For the slow-time subsystem dv = 0, H( ) = 1, v(0) = 0, dζ ζ E(0) = E + v ω (E Na E ), E(P) = E + v α (E Na E ), (10) n(0) = n(p), Parameters Structure where C (0, ), P (0, ), v α (, v ) v ω (0, 1) are parameters to be found. Expect a one-parameter family of solutions. Problem 5-th order + 4 integration constants = 9 unknowns. Constraints 8 constraints available.

29 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Coupling via boundary conditions Rationale Boundary conditions In a periodic wave train, a front propagates in the tail of the preceding wave, so slow pieces described by (8) and fast pieces described by (9) alternate. There are one slow piece and one fast piece per period. This situation is summarized by the following set of boundary conditions For the fast-time subsystem v( ) = v α, v( ) = v ω, For the slow-time subsystem dv = 0, H( ) = 1, v(0) = 0, dζ ζ E(0) = E + v ω (E Na E ), E(P) = E + v α (E Na E ), (10) n(0) = n(p), Parameters Structure where C (0, ), P (0, ), v α (, v ) v ω (0, 1) are parameters to be found. Expect a one-parameter family of solutions. Problem 5-th order + 4 integration constants = 9 unknowns. Constraints 8 constraints available.

30 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Coupling via boundary conditions Rationale Boundary conditions In a periodic wave train, a front propagates in the tail of the preceding wave, so slow pieces described by (8) and fast pieces described by (9) alternate. There are one slow piece and one fast piece per period. This situation is summarized by the following set of boundary conditions For the fast-time subsystem v( ) = v α, v( ) = v ω, For the slow-time subsystem dv = 0, H( ) = 1, v(0) = 0, dζ ζ E(0) = E + v ω (E Na E ), E(P) = E + v α (E Na E ), (10) n(0) = n(p), Parameters Structure where C (0, ), P (0, ), v α (, v ) v ω (0, 1) are parameters to be found. Expect a one-parameter family of solutions. Problem 5-th order + 4 integration constants = 9 unknowns. Constraints 8 constraints available.

31 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Discussion Advantages Challenge Next steps The asymptotic boundary-value problem of CV restitution is essentially simpler than the full one (3): 1. Small parameters are eliminated. The resulting system is no longer stiff. 2. Simpler right-hand sides of equations. System of lower dimension at each AP stage. However, to be useful the coupled asymptotic boundary value problem must satisfy two essential requirements: 1. the coupled problems must be well-posed. 2. their asymptotic solution must provide a good approximation to the solution of the full non-asymptotic problem. This problem has a non-tikhonov asymptotic structure. For this reason, it is not obvious that the asymptotic formulation of the CV restitution problem satisfies either of these requirements. A proof of well-posedness is difficult in the general case. We take the following approach: 1. We prove the well-posedness of the archetypal Caricature Noble problem (8), (9) and (10). 2. The convergence of the asymptotic and full solutions is demonstrated numerically.

32 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Discussion Advantages Challenge Next steps The asymptotic boundary-value problem of CV restitution is essentially simpler than the full one (3): 1. Small parameters are eliminated. The resulting system is no longer stiff. 2. Simpler right-hand sides of equations. System of lower dimension at each AP stage. However, to be useful the coupled asymptotic boundary value problem must satisfy two essential requirements: 1. the coupled problems must be well-posed. 2. their asymptotic solution must provide a good approximation to the solution of the full non-asymptotic problem. This problem has a non-tikhonov asymptotic structure. For this reason, it is not obvious that the asymptotic formulation of the CV restitution problem satisfies either of these requirements. A proof of well-posedness is difficult in the general case. We take the following approach: 1. We prove the well-posedness of the archetypal Caricature Noble problem (8), (9) and (10). 2. The convergence of the asymptotic and full solutions is demonstrated numerically.

33 P2 asymptotic reduction of the periodic BVP for the caricature Noble model Discussion Advantages Challenge Next steps The asymptotic boundary-value problem of CV restitution is essentially simpler than the full one (3): 1. Small parameters are eliminated. The resulting system is no longer stiff. 2. Simpler right-hand sides of equations. System of lower dimension at each AP stage. However, to be useful the coupled asymptotic boundary value problem must satisfy two essential requirements: 1. the coupled problems must be well-posed. 2. their asymptotic solution must provide a good approximation to the solution of the full non-asymptotic problem. This problem has a non-tikhonov asymptotic structure. For this reason, it is not obvious that the asymptotic formulation of the CV restitution problem satisfies either of these requirements. A proof of well-posedness is difficult in the general case. We take the following approach: 1. We prove the well-posedness of the archetypal Caricature Noble problem (8), (9) and (10). 2. The convergence of the asymptotic and full solutions is demonstrated numerically.

34 Construction of the CV restitution curve Mathching Combining the fast- and slow-time systems we are left with one free parameter v α. From the exact solution of fast subsystem determine c = c(v α) and v ω = v ω(v α). From the exact Solution of slow subsystem determine P = P(v α). Finally the CV restitution curve has the parametric representation `P(vα), c(v α).

35 Construction of the CV restitution curve Mathching Combining the fast- and slow-time systems we are left with one free parameter v α. From the exact solution of fast subsystem determine c = c(v α) and v ω = v ω(v α). From the exact Solution of slow subsystem determine P = P(v α). Finally the CV restitution curve has the parametric representation `P(vα), c(v α). Caricature Noble CV restitution curve c [su/ms] c [su/ms] ǫ= 1 ǫ= 0.7 ǫ= 0.3 ǫ= 0 asymptotic tip position (a) P [ms] (b) P [ms] Figure: Restitution curves of the Caricature Noble model, (a) in Cartesian and (b) logarithmic coordinates. Insets in panel (a) show selected features magnified. Lines in all plots as described by the legend in panel (b).

36 Construction of the CV restitution curve Conclusion Some results Some future problems 1. Proposed a consistent procedure for reduction of cardiac electrophysiological models. 2. Found out that cardiac models have non-tikhonov asymptotic structure, so care is needed. 3. Successfully applied to the problem of CV-restitution curves. 1. Initiation of propagating action potentials 2. Efficient numerical methods based on asymptotic reduction 3. Higher dimensions 4. More realistic models