Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models
aa r X i v : . [ n li n . C D ] A ug Canards Existence in FitzHugh-NagumoandHodgkin-Huxley Neuronal Models
Jean-Marc Ginoux * Laboratoire LSIS, CNRS, UMR 7296, Universit´e de Toulon,BP 20132, F-83957 La Garde cedex, France.
E-mail: [email protected] andJaume Llibre † Departament de Matem`atiques, Universitat Aut`onoma de Barcelona,08193 – Bellaterra, Barcelona, Spain.
E-mail: [email protected]
In a previous paper we have proposed a new method for proving theexistence of “canard solutions” for three and four-dimensional singularly per-turbed systems with only one fast variable which improves the methods useduntil now. The aim of this work is to extend this method to the case offour-dimensional singularly perturbed systems with two slow and two fast variables. This method enables to state a unique generic condition for theexistence of “canard solutions” for such four-dimensional singularly perturbedsystems which is based on the stability of folded singularities ( pseudo singularpoints in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is identicalto that provided in previous works. Applications of this method to the fa-mous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley modelenables to show the existence of “canard solutions” in such systems. Key Words:
Geometric singular perturbation theory, singularly perturbed dy-namical systems, canard solutions. * I would like to thank the Universitat Aut`onoma de Barcelona for its kind invitationthat allowed this paper to be written. † Partially supported by a DGES grant number PB96–1153. J.M. GINOUX AND J. LLIBRE
1. INTRODUCTION
In the beginning of the eighties, Benoˆıt and Lobry [5], Benoˆıt [6] andthen Benoˆıt [7] in his PhD-thesis studied canard solutions in R . In thearticle entitled “Syst`emes lents-rapides dans R et leurs canards,” Benoˆıt[6, p. 170] proved the existence of canards solution for three-dimensionalsingularly perturbed systems with two slow variables and one fast variablewhile using “Non-Standard Analysis”according to a theorem which statedthat canard solutions exist in such systems provided that the pseudo sin-gular point of the slow dynamics , i.e. , of the reduced vector field is of saddle type. Nearly twenty years later, Szmolyan and Wechselberger [25]extended “Geometric Singular Perturbation Theory ” to canards problemsin R and provided a “standard version” of Benoˆıt’s theorem [6]. Very re-cently, Wechselberger [39] generalized this theorem for n -dimensional sin-gularly perturbed systems with k slow variables and m fast (Eq. (1)). Themethod used by Szmolyan and Wechselberger [25] and Wechselberger [39]require to implement a “desingularization procedure” which can be sum-marized as follows: first, they compute the normal form of such singularlyperturbed systems which is expressed according to some coefficients ( a and b for dimension three and ˜ a , ˜ b and ˜ c j for dimension four) depending on thefunctions defining the original vector field and their partial derivatives withrespect to the variables. Secondly, they project the “desingularized vectorfield” (originally called “normalized slow dynamics” by Eric Benoˆıt [6, p.166]) of such a normal form on the tangent bundle of the critical manifold.Finally, they evaluate the Jacobian of the projection of this “desingular-ized vector field” at the folded singularity (originally called pseudo singularpoints by Jos´e Arg´emi [1, p. 336]). This lead Szmolyan and Wechsel-berger [25, p. 427] and Wechselberger [39, p. 3298] to a “classificationof folded singularities ( pseudo singular points )”. Thus, they showed thatfor three-dimensional singularly perturbed systems such folded singularity is of saddle type if the following condition is satisfied: a < folded singularity is of saddle type if ˜ a <
0. Then, Szmolyan and Wechselberger [25, p. 439] andWechselberger [39, p. 3304] established their Theorem 4.1. which state that“
In the folded saddle and in the folded node case singular canards perturb tomaximal canard for sufficiently small ε ”. However, in their works neitherSzmolyan and Wechselberger [25] nor Wechselberger [39] did not provide(to our knowledge) the expression of these constants ( a and ˜ a ) which arenecessary to state the existence of canard solutions in such systems.In a previous paper entitled: “Canards Existence in Memristor’s Cir-cuits” (see Ginoux & Llibre [17]) we first provided the expression of these This concept has been originally introduced by Jos´e Arg´emi [1]. See Sec. 1.8. See Fenichel [12, 15], O’Malley [23], Jones [20] and Kaper [21]ANARDS EXISTENCE IN R normalized slow dynamics and not from the projection of the“desingularized vector field” of the normal form . This method enabled tostate a unique “generic” condition for the existence of “canard solutions”for such three and four-dimensional singularly perturbed systems which isbased on the stability of folded singularities of the normalized slow dynam-ics deduced from a well-known property of linear algebra. This uniquecondition which is completely identical to that provided by Benoˆıt [6] andthen by Szmolyan and Wechselberger [25] and finally by Wechselberger [39]is “generic” since it is exactly the same for singularly perturbed systems ofdimension three and four with only one fast variable.The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with k = 2 slow and m = 2 fast variables. Since the dimension of the system is m = k + m , such problemis known as “canards existence in R ”. Moreover, in this particular casewhere k = m = 2, the folded singularities of Wechselberger [39, p. 3298]are nothing else but the pseudo singular points of the late Jos´e Arg´emi [1]as we will see below. Following the previous works, we show that for suchfour-dimensional singularly perturbed systems pseudo singular points areof saddle type if ˜ a <
0. Then, according Theorem 4.1. of Wechselberger[39, p. 3304] we provide the expression of this constant ˜ a which is neces-sary to establish the existence of canard solutions in such systems. So, wecan state that the condition ˜ a < R is“generic” since it is exactly the same for singularly perturbed systems ofdimension three and four with only one fast variable.The outline of this paper is as follows. In Sec. 1, definitions of singularlyperturbed system, critical manifold, reduced system, “constrained system”,canard cycles, folded singularities and pseudo singular points are recalled.The method proposed in this article is presented in Sec. 2 for the caseof four-dimensional singularly perturbed systems with two fast variables.In Sec. 3, applications of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables to show theexistence of “canard solutions” in such systems.
2. DEFINITIONS2.1. Singularly perturbed systems
According to Tikhonov [37], Jones [20] and Kaper [21] singularly per-turbed systems are defined as: ~x ′ = ε ~f ( ~x, ~y, ε ) ,~y ′ = ~g ( ~x, ~y, ε ) . (1) J.M. GINOUX AND J. LLIBRE where ~x ∈ R k , ~y ∈ R m , ε ∈ R + , and the prime denotes differentiationwith respect to the independent variable t ′ . The functions ~f and ~g areassumed to be C ∞ functions of ~x , ~y and ε in U × I , where U is an opensubset of R k × R m and I is an open interval containing ε = 0.In the case when 0 < ε ≪ i.e. ε is a small positive number, thevariable ~x is called slow variable, and ~y is called fast variable. UsingLandau’s notation: O ( ε p ) represents a function f of u and ε such that f ( u, ε ) /ε p is bounded for positive ε going to zero, uniformly for u in thegiven domain.In general we consider that ~x evolves at an O ( ε ) rate; while ~y evolvesat an O (1) slow rate. Reformulating system (1) in terms of the rescaledvariable t = εt ′ , we obtain ˙ ~x = ~f ( ~x, ~y, ε ) ,ε ˙ ~y = ~g ( ~x, ~y, ε ) . (2)The dot represents the derivative with respect to the new independentvariable t .The independent variables t ′ and t are referred to the fast and slow times, respectively, and (1) and (2) are called the fast and slow systems,respectively. These systems are equivalent whenever ε = 0, and they arelabeled singular perturbation problems when 0 < ε ≪
1. The label “singu-lar” stems in part from the discontinuous limiting behavior in system (1)as ε → In such case system (2) leads to a differential-algebraic system (D.A.E.)called reduced slow system whose dimension decreases from k + m = n to m . Then, the slow variable ~x ∈ R k partially evolves in the submanifold M called the critical manifold . The reduced slow system is˙ ~x = ~f ( ~x, ~y, ε ) ,~ ~g ( ~x, ~y, ε ) . (3) The critical manifold is defined by M := n ( ~x, ~y ) : ~g ( ~x, ~y,
0) = ~ o . (4) In certain applications these functions will be supposed to be C r , r > It represents the approximation of the slow invariant manifold, with an error of O ( ε ).ANARDS EXISTENCE IN R reduced slowsystem (3) persists as a locally invariant slow manifold of the full problem(1) for ε sufficiently small. This locally slow invariant manifold is O ( ε )close to the critical manifold .When D ~x ~f is invertible, thanks to the Implicit Function Theorem, M is given by the graph of a C ∞ function ~x = ~G ( ~y ) for ~y ∈ D , where D ⊆ R k is a compact, simply connected domain and the boundary of D isa ( k − C ∞ submanifold .According to Fenichel [12, 15] theory if 0 < ε ≪ ~G ( ~y, ε ) defined on D such that the manifold M ε := n ( ~x, ~y ) : ~x = ~G ( ~y, ε ) o , (5)is locally invariant under the flow of system (1). Moreover, there existperturbed local stable (or attracting) M a and unstable (or repelling) M r branches of the slow invariant manifold M ε . Thus, normal hyperbolicityof M ε is lost via a saddle-node bifurcation of the reduced slow system (3).Then, it gives rise to solutions of “canard” type. A canard is a solution of a singularly perturbed dynamical system (1)following the attracting branch M a of the slow invariant manifold , passingnear a bifurcation point located on the fold of this slow invariant manifold ,and then following the repelling branch M r of the slow invariant manifold .A singular canard is a solution of a reduced slow system (3) following the attracting branch M a, of the critical manifold , passing near a bifurcationpoint located on the fold of this critical manifold , and then following the repelling branch M r, of the critical manifold .A maximal canard corresponds to the intersection of the attracting andrepelling branches M a,ε ∩ M r,ε of the slow manifold in the vicinity of anon-hyperbolic point.According to Wechselberger [39, p. 3302]:“Such a maximal canard defines a family of canards nearby which areexponentially close to the maximal canard, i.e. a family of solutions of (1)that follow an attracting branch M a,ε of the slow manifold and then follow,rather surprisingly, a repelling/saddle branch M r,ε of the slow manifold fora considerable amount of slow time. The existence of this family of canardsis a consequence of the non-uniqueness of M a,ε and M r,ε . However, in thesingular limit ε →
0, such a family of canards is represented by a uniquesingular canard.” The set D is overflowing invariant with respect to (2) when ε = 0. See Kaper [21]and Jones [20]. J.M. GINOUX AND J. LLIBRE
Canards are a special class of solution of singularly perturbed dynami-cal systems for which normal hyperbolicity is lost. Canards in singularlyperturbed systems with two or more slow variables ( ~x ∈ R k , k >
2) andone fast variable ( ~y ∈ R m , m = 1) are robust, since maximal canardsgenerically persist under small parameter changes . In order to characterize the “slow dynamics”, i.e. the slow trajectoryof the reduced slow system (3) (obtained by setting ε = 0 in (2)), FlorisTakens [28] introduced the “constrained system” defined as follows:˙ ~x = ~f ( ~x, ~y, ,D ~y ~g. ˙ ~y = − ( D ~x ~g. ~f ) ( ~x, ~y, ,~ ~g ( ~x, ~y, . (6)Since, according to Fenichel [12, 15], the critical manifold ~g ( ~x, ~y,
0) maybe considered as locally invariant under the flow of system (1), we have: d~gdt ( ~x, ~y,
0) = 0 ⇐⇒ D ~x ~g. ˙ ~x + D ~y ~g. ˙ ~y = ~ . By replacing ˙ ~x by ~f ( ~x, ~y,
0) leads to: D ~x ~g. ~f ( ~x, ~y,
0) + D ~y ~g. ˙ ~y = ~ . This justifies the introduction of the constrained system .Now, let adj ( D ~y ~g ) denote the adjoint of the matrix D ~y ~g which is thetranspose of the co-factor matrix D ~y ~g , then while multiplying the left handside of (6) by the inverse matrix ( D ~y ~g ) − obtained by the adjoint methodwe have: ˙ ~x = ~f ( ~x, ~y, ,det ( D ~y ~g ) ˙ ~y = − ( adj ( D ~y ~g ) .D ~x ~g. ~f ) ( ~x, ~y, ,~ ~g ( ~x, ~y, . (7) Then, by rescaling the time by setting t = − det ( D ~y ~g ) τ we obtain the fol-lowing system which has been called by Eric Benoˆıt [6, p. 166] “normalizedslow dynamics”: See Benoˆıt [6, 9], Szmolyan and Wechselberger [25] and Wechselberger [38, 39].ANARDS EXISTENCE IN R ~x = − det ( D ~y ~g ) ~f ( ~x, ~y, , ˙ ~y = ( adj ( D ~y ~g ) .D ~x ~g. ~f ) ( ~x, ~y, ,~ ~g ( ~x, ~y, . (8)where the overdot now denotes the time derivation with respect to τ .Let’s notice that Jos´e Arg´emi [1] proposed to rescale time by setting t = − det ( D ~y ~g ) sgn ( det ( D ~y ~g )) τ in order to keep the same flow direction in(8) as in (7). By application of the Implicit Function Theorem, let suppose that wecan explicitly express from Eq. (4), say without loss of generality, x as a function φ of the other variables. This implies that M is locallythe graph of a function φ : R k → R m over the base U = ( ~χ, ~y ) where ~χ = ( x , x , ..., x k ). Thus, we can span the “normalized slow dynamics”on the tangent bundle at the critical manifold M at the pseudo singularpoint . This leads to the so-called desingularized vector field :˙ ~χ = − det ( D ~y ~g ) ~f ( ~χ, ~y, , ˙ ~y = ( adj ( D ~y ~g ) .D ~x ~g. ~f ) ( ~χ, ~y, . (9) As recalled by Guckenheimer and Haiduc [18, p. 91], pseudo-singularpoints have been introduced by the late Jos´e Arg´emi [1] for low-dimensionalsingularly perturbed systems and are defined as singular points of the “nor-malized slow dynamics” (8). Twenty-three years later, Szmolyan and Wech-selberger [25, p. 428] called such pseudo singular points , folded singularities .In a recent publication entitled “A propos de canards” Wechselberger [39,p. 3295] proposed to define such singularities for n -dimensional singularlyperturbed systems with k slow variables and m fast as the solutions of thefollowing system: det ( D ~y ~g ) = 0 , ( adj ( D ~y ~g ) .D ~x ~g. ~f ) ( ~x, ~y,
0) = ~ ,~g ( ~x, ~y,
0) = ~ . (10)Thus, for dimensions higher than three, his concept encompasses that ofArg´emi. Moreover, Wechselberger [39, p. 3296] proved that folded singu-larities form a ( k − k = 2 the foldedsingularities are nothing else than the pseudo singular points defined by J.M. GINOUX AND J. LLIBRE
Arg´emi [1]. While for k > folded singularities are no more pointsbut a ( k − n = k + m variables and on the otherhand, that the system (10) comprises p = 2 m + 1 equations. However, inthe particular case k = m = 2, two equations of the system (10) are linearlydependent. So, such system only comprises p = 2 m = 2 k equations. So, allthe variables (unknowns) of system (10) can be determined. The solutionsof this system are called pseudo singular points . We will see in the next Sec.2 that the stability analysis of these pseudo singular points will give rise toa condition for the existence of canard solutions in the original system (1).
3. FOUR-DIMENSIONAL SINGULARLY PERTURBEDSYSTEMS WITH TWO FAST VARIABLES
A four-dimensional singularly perturbed dynamical system (2) with k = 2 slow variables and m = 2 fast may be written as:˙ x = f ( x , x , y , y ) , (11a)˙ x = f ( x , x , y , y ) , (11b) ε ˙ y = g ( x , x , y , y ) , (11c) ε ˙ y = g ( x , x , y , y ) , (11d)where ~x = ( x , x ) t ∈ R , ~y = ( y , y ) ∈ R , 0 < ε ≪ f i and g i are assumed to be C functions of ( x , x , y , y ). The critical manifold equation of system (11) is defined by setting ε = 0in Eqs. (11c & 11d). Thus, we obtain: g ( x , x , y , y ) = 0 , (12a) g ( x , x , y , y ) = 0 . (12b)By application of the Implicit Function Theorem, let suppose that we canexplicitly express from Eqs. (12a & 12b), say without loss of generality, x and y as functions of the others variables: x = φ ( x , y , y ) , (13a) y = φ ( x , x , y ) . (13b) ANARDS EXISTENCE IN R The constrained system is obtained by equating to zero the time deriva-tive of g , ( x , x , y , y ): dg dt = ∂g ∂x ˙ x + ∂g ∂x ˙ x + ∂g ∂y ˙ y + ∂g ∂y ˙ y = 0 (14a) dg dt = ∂g ∂x ˙ x + ∂g ∂x ˙ x + ∂g ∂y ˙ y + ∂g ∂y ˙ y = 0 (14b)Eqs. (14a & 14b) may be written as: ∂g ∂y ˙ y + ∂g ∂y ˙ y = − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) (15a) ∂g ∂y ˙ y + ∂g ∂y ˙ y = − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) (15b)By solving the system of two equations (15a & 15b) with two unknowns( ˙ y , ˙ y ) we find:˙ y = − (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y + (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y det (cid:2) J ( y ,y ) (cid:3) , (16a)˙ y = − (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y + (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y det (cid:2) J ( y ,y ) (cid:3) . (16b)So, we have the following constrained system:˙ x = f ( x , x , y , y ) , ˙ x = f ( x , x , y , y ) , ˙ y = − (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y + (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y det (cid:2) J ( y ,y ) (cid:3) , ˙ y = − (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y + (cid:16) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:17) ∂g ∂y det (cid:2) J ( y ,y ) (cid:3) , g ( x , x , y , y ) , g ( x , x , y , y ) . (17)0 J.M. GINOUX AND J. LLIBRE
By rescaling the time by setting t = − det (cid:2) J ( y ,y ) (cid:3) τ we obtain the “nor-malized slow dynamics”:˙ x = − f ( x , x , y , y ) det (cid:2) J ( y ,y ) (cid:3) = F ( x , x , x , y ) , ˙ x = − f ( x , x , y , y ) det (cid:2) J ( y ,y ) (cid:3) = F ( x , x , x , y ) , ˙ y = (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y = G ( x , x , x , y ) , ˙ y = (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y = G ( x , x , x , y ) , g ( x , x , y , y ) , g ( x , x , y , y ) . (18)where the overdot now denotes the time derivation with respect to τ . Then, since we have supposed that x and y may be explicitly expressedas functions of the others variables (13a & 13b), they can be used to projectthe normalized slow dynamics (18) on the tangent bundle of the criticalmanifold. So, we have:˙ x = − f ( x , x , y , y ) det (cid:2) J ( y ,y ) (cid:3) = F ( x , y ) , ˙ y = ( ∂g ∂x ˙ x + ∂g ∂x ˙ x ) ∂g ∂y − ( ∂g ∂x ˙ x + ∂g ∂x ˙ x ) ∂g ∂y = G ( x , y ) . (19) Pseudo-singular points are defined as singular points of the “normalizedslow dynamics”, i.e. as the set of points for which we have: det (cid:2) J ( y ,y ) (cid:3) = 0 , (20a) (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y = 0 , (20b) (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y = 0 , (20c) g ( x , x , y , y ) = 0 , (20d) g ( x , x , y , y ) = 0 . (20e) ANARDS EXISTENCE IN R Remark 1.
Let’s notice on the one hand that Eqs. (20b) & (20c) arelinearly dependent and on the other hand that contrary to previous workswe don’t use the “desingularized vector field” (19) but the “normalizedslow dynamics” (18).The Jacobian matrix of system (18) reads: J ( F ,F ,G ,G ) = ∂F ∂x ∂F ∂x ∂F ∂y ∂F ∂y ∂F ∂x ∂F ∂x ∂F ∂y ∂F ∂y ∂G ∂x ∂G ∂x ∂G ∂y ∂G ∂y ∂G ∂x ∂G ∂x ∂G ∂y ∂G ∂y (21) Without loss of generality, it seems reasonable to extend Benoˆıt’s generichypotheses introduced for the three-dimensional case to the four-dimensionalcase. So, first, let’s suppose that by a “standard translation” the pseudosingular point can be shifted at the origin O (0 , , ,
0) and that by a “stan-dard rotation” of y -axis that the slow manifold is tangent to ( x , x , y )-hyperplane, so we have f (0 , , ,
0) = g (0 , , ,
0) = 0 ∂g ∂x (cid:12)(cid:12)(cid:12)(cid:12) (0 , , , = ∂g ∂x (cid:12)(cid:12)(cid:12)(cid:12) (0 , , , = ∂g ∂y (cid:12)(cid:12)(cid:12)(cid:12) (0 , , , = 0 (22)Then, let’s make the following assumptions for the non-degeneracy of the folded singularity : f (0 , , , = 0 ; ∂g ∂x (cid:12)(cid:12)(cid:12)(cid:12) (0 , , , = 0 ; ∂ g ∂y (cid:12)(cid:12)(cid:12)(cid:12) (0 , , , = 0 . (23)According to these generic hypotheses Eqs. (22-23), the Jacobian matrix(21) reads:2 J.M. GINOUX AND J. LLIBRE J ( F ,F ,G ,G ) = − f ∂P∂x − f ∂P∂x − f ∂P∂y − f ∂P∂y a a a a a a a a (24)where P = det (cid:2) J ( y ,y ) (cid:3) ,a i = − f ∂g ∂x ∂ g ∂y ∂x i + ∂g ∂y (cid:18) f ∂ g ∂x ∂x i + ∂g ∂x ∂f ∂x i (cid:19) for i = 1 , ,a i = − f ∂g ∂x ∂ g ∂y ∂y i + ∂g ∂y (cid:18) f ∂ g ∂x ∂y i + ∂g ∂x ∂f ∂y i (cid:19) for i = 3 , ,a i = f ∂g ∂x ∂ g ∂y ∂x i − ∂g ∂y (cid:18) f ∂ g ∂x ∂x i + ∂g ∂x ∂f ∂x i (cid:19) for i = 1 , ,a i = f ∂g ∂x ∂ g ∂y ∂y i − ∂g ∂y (cid:18) f ∂ g ∂x ∂y i + ∂g ∂x ∂f ∂y i (cid:19) for i = 3 , . Thus, we have the following Cayley-Hamilton eigenpolynomial associatedwith such a Jacobian matrix (24) evaluated at the pseudo singular point , i.e. , at the origin: λ − σ λ + σ λ − σ λ + σ = 0 (25)where σ = T r ( J ) is the sum of all first-order diagonal minors of J , i.e. ,the the trace of the Jacobian matrix J , σ represents the sum of all second-order diagonal minors of J and σ represents the sum of all third-orderdiagonal minors of J . It appears that σ = | J | = 0 since one row of theJacobian matrix (24) is null. So, the eigenpolynomial reduces to: λ (cid:0) λ − σ λ + σ λ − σ (cid:1) = 0 (26)But, according to Wechselberger [39], σ vanishes at a pseudo singularpoint as it’s easy to prove it. So, the eigenpolynomial (26) is reduced to λ (cid:0) λ − σ λ + σ (cid:1) = 0 (27)Let λ i be the eigenvalues of the eigenpolynomial (27) and let’s denoteby λ , = 0 the obvious double root of this polynomial. We have: ANARDS EXISTENCE IN R σ = T r ( J ( F ,F ,G ,G ) ) = λ + λ = ∂g ∂x ∂g ∂y ∂f ∂y ,σ = X i =1 (cid:12)(cid:12)(cid:12) J ii ( F ,F ,G ,G ) (cid:12)(cid:12)(cid:12) = λ λ = (cid:18) ∂g ∂y (cid:19) " f ∂ g ∂x ∂ g ∂y − (cid:18) ∂ g ∂x ∂y (cid:19) ! + f ∂g ∂x (cid:18) ∂ g ∂y ∂f ∂x − ∂ g ∂x ∂y ∂f ∂y (cid:19)(cid:21) (28)where σ = T r ( J ( F ,F ,G ,G ) ) = p is is the sum of all first-order diagonalminors of J ( F ,F ,G ,G ) , i.e. the trace of the Jacobian matrix J ( F ,F ,G ,G ) and σ = P i =1 (cid:12)(cid:12)(cid:12) J ii ( F ,F ,G ,G ) (cid:12)(cid:12)(cid:12) = q represents the sum of all second-orderdiagonal minors of J ( F ,F ,G ,G ) . Thus, the pseudo singular point is ofsaddle-type iff the following conditions C and C are verified: C : ∆ = p − q > ,C : q < . (29)Condition C is systematically satisfied provided that condition C isverified. Thus, the pseudo singular point is of saddle-type iff q < R Following the works of Wechselberger [39] it can be stated, while using astandard polynomial change of variables, that any n -dimensional singularlyperturbed systems with k slow variables ( k >
2) and m fast ( m >
1) (1)can be transformed into the following “normal form”:˙ x = ˜ ax + ˜ by + O (cid:0) x , ǫ, x , x y , y (cid:1) , ˙ x = 1 + O ( x , x , y , ǫ ) ,ǫ ˙ y = ˜ cy + O (cid:0) ǫx , ǫx , ǫy , x , x , y , x y (cid:1) ,ǫ ˙ y = − (cid:0) x + y (cid:1) + O (cid:0) ǫx , ǫx , ǫy , ǫ , x y , y , x x y (cid:1) . (30)We establish in Appendix A for any four-dimensional singularly per-turbed systems (11) with k = 2 slow and m = 2 fast variables that4 J.M. GINOUX AND J. LLIBRE ˜ a = 12 " f ∂ g ∂x ∂ g ∂y − (cid:18) ∂ g ∂x ∂y (cid:19) ! + f ∂g ∂x (cid:18) ∂ g ∂y ∂f ∂x − ∂ g ∂x ∂y ∂f ∂y (cid:19) ˜ b = − ∂g ∂x ∂f ∂y , ˜ c = ∂g ∂y . Thus, in his paper Wechselberger [39, p. 3304] provided in the frameworkof “standard analysis” a generalization of Benoˆıt’s theorem [6] for any n -dimensional singularly perturbed systems with k slow variables ( k > m fast ( m > Theorem 2.
In the folded saddle case of system (30) singular canards perturb to maxi-mal canards solutions for sufficiently small ε ≪ .Proof. See Wechselberger [39].Since our method doesn’t use the “desingularized vector field” (19) butthe “normalized slow dynamics” (18), we have the following proposition:
Proposition 3.
If the normalized slow dynamics (18) has a pseudo singular point of saddletype, i.e. if the sum σ of all second-order diagonal minors of the Jacobianmatrix of the normalized slow dynamics (18) evaluated at the pseudo sin-gular point is negative, i.e. if σ < then, according to Theorem 2, system (11) exhibits a canard solution which evolves from the attractive part of theslow manifold towards its repelling part.Proof. By making some smooth changes of time and smooth changes of co-ordinates (see Appendix A) we brought the system (11) to the following“normal form”:˙ x = ˜ ax + ˜ by + O (cid:0) x , ǫ, x , x y , y (cid:1) , ˙ x = 1 + O ( x , x , y , ǫ ) ,ǫ ˙ y = ˜ cy + O (cid:0) ǫx , ǫx , ǫy , x , x , y , x y (cid:1) ,ǫ ˙ y = − (cid:0) x + y (cid:1) + O (cid:0) ǫx , ǫx , ǫy , ǫ , x y , y , x x y (cid:1) , Then, we deduce that the condition for the pseudo singular point to beof saddle type is ˜ a <
0. According to Eqs. (29) it is easy to verify that
ANARDS EXISTENCE IN R σ = T r ( J ( F ,F ,G ,G ) ) = λ + λ = − ˜ b ˜ c,σ = X i =1 (cid:12)(cid:12)(cid:12) J ii ( F ,F ,G ,G ) (cid:12)(cid:12)(cid:12) = λ λ = 2˜ a ˜ c . So, the condition for which the pseudo singular point is of saddle type, i.e. σ < i.e. ˜ a < slow and two fast variables will enable to prove,as many previous works such as those of Tchizawa & Campbell [30] andTchizawa [30, 31, 32, 33, 34, 35], the existence of “canard solutions” insuch system. According to Tchizawa [36], it is very important to notice, onthe one hand that the fast equation has 2-dimensional in the system R and, on the other hand that the fast system can give attractive, repulsive orattractive-repulsive at each pseudo singular point . Then, Tchizawa [36] hasestablished that the jumping direction can be shown using the eigenvectors.In the same way we will find again the results of Rubin et al. [24] concern-ing the existence of “canard solutions” in the Hodgkin-Huxley model butwith a set of more realistic parameters used in Chua et al. [10, 11].6 J.M. GINOUX AND J. LLIBRE
4. COUPLED FITZHUGH-NAGUMO EQUATIONS
The FitzHugh-Nagumo model [16, 22] is a simplified version of the Hodgkin-Huxley model [19] which models in a detailed manner activation and deac-tivation dynamics of a spiking neuron. By coupling two FitzHugh-Nagumomodels Tchizawa & Campbell [29] and Tchizawa [30, 35] obtained the fol-lowing four-dimensional singularly perturbed system with two slow andtwo fast variables: dx dt = 1 c ( y + bx ) , (31a) dx dt = 1 c ( y + bx ) , (31b) ε dy dt = x − y y , (31c) ε dy dt = x − y y . (31d)where 0 < ε ≪ b is the “canard parameter” or “duck parameter”while c is a scale factor. The slow manifold equation of system (31) is defined by setting ε = 0 inEqs. (31c & 31d). Thus, we obtain: dx dt = 1 c ( y + bx ) ,dx dt = 1 c ( y + bx ) ,dy dt = − c ( y + bx ) + y c ( y + bx ) y y − ,dy dt = − c ( y + bx ) + y c ( y + bx ) y y − , x − y y , x − y y . (32) ANARDS EXISTENCE IN R Then, by rescaling the time by setting t = − det (cid:2) J ( y ,y ) (cid:3) τ = − ( y y − dx dt = − c ( y + bx ) (cid:0) y y − (cid:1) = F ( x , x , y , y ) ,dx dt = − c ( y + bx ) (cid:0) y y − (cid:1) = F ( x , x , y , y ) ,dy dt = 1 c ( y + bx ) + y c ( y + bx ) = G ( x , x , y , y ) ,dy dt = 1 c ( y + bx ) + y c ( y + bx ) G ( x , x , y , y ) , x − y y , x − y y . (33) From Eqs. (20), the pseudo-singular points of system (31) are definedby: det (cid:2) J ( y ,y ) (cid:3) = y y − , (34a) (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y (34b)= 1 c ( y + bx ) + y c ( y + bx ) = 0 , (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y − (cid:18) ∂g ∂x ˙ x + ∂g ∂x ˙ x (cid:19) ∂g ∂y (34c)= 1 c ( y + bx ) + y c ( y + bx ) = 0 ,g ( x , x , y , y ) = x − y y = 0 , (34d) g ( x , x , y , y ) = x − y y = 0 . (34e)According to Tchizawa & Campbell [29] and Tchizawa [30, 31], there aresix pseudo singular points , the last four are depending on the parameter b .8 J.M. GINOUX AND J. LLIBRE (˜ x , ˜ x , ˜ y , ˜ y ) = (cid:18) ± , ∓ , ± , ∓ (cid:19) , (35a)(˜ x , ˜ x , ˜ y , ˜ y ) = ± q −√ − b b (cid:0) √ − b (cid:1) √ b , ∓ q −√ − b b (cid:0) − b + 3 √ − b (cid:1) √ b , ∓ s − √ − b b , ∓ √ b p − √ − b ! , (35b)(˜ x , ˜ x , ˜ y , ˜ y ) = ± (cid:0) − √ − b (cid:1) q √ − b b √ b , ∓ q √ − b b (cid:0) − b − √ − b (cid:1) √ b , ∓ s − √ − b b , ∓ √ b p − √ − b ! . (35c) The Jacobian matrix of system (33) evaluated at the pseudo singularpoints (35a) reads: J ( F ,F ,G ,G ) = b )3 c − b )3 c − b )3 c b )3 cbc bc c − b c bc bc − b c c (36) Remark 4.
Although the pseudo singular points have not been shiftedat the origin extension of Benoˆıt’s generic hypotheses (22-23) are satisfied.In other words, we have σ = σ = 0.According to Eqs. (28) we find that: ANARDS EXISTENCE IN R p = σ = T r ( J ) = + 2 c ,q = σ = − b (3 + 4 b )9 c (37)Thus, according to Prop. 3, the pseudo singular points are of saddle-typeif and only if: − b (3 + 4 b )9 c < C and C : C : ∆ = 4(3 + 8 b ) c > ,C : q = − b (3 + 4 b )9 c < . (38)Let’s choose arbitrarily b as the “canard parameter” or “duck parame-ter”. Obviously, it appears that the condition C is still satisfied. Finally,the pseudo singular points are of saddle-type if and only if we have: b > b < − . (39) Remark 5.
Let’s notice that the pseudo singular points are of node-type if − < b < J ( F ,F ,G ,G ) of system (33) evaluated at the pseudosingular points (35b) reads: − √ − b c − ( − b +3 √ − b ) bc ( − b +3 √ − b ) bc √ − b c √ − b c bc √ − b bc − √ − b cbc −√ − b c √ − b c −√ − b bc (40) Remark 6.
Although, the pseudo singular points have not been shiftedat the origin extension of Benoˆıt’s generic hypotheses (22-23) are satisfied.In other words, we have σ = σ = 0.0 J.M. GINOUX AND J. LLIBRE
According to Eqs. (28) we find that: p = σ = T r ( J ) = + 3 bc ,q = σ = 16 (cid:0) − b (cid:1) c (41)Thus, according to Prop. 3, the pseudo singular points are of saddle-typeif and only if: 16 (cid:0) − b (cid:1) c <
0∆ = p − q > q < . So, we have the following conditions C and C : C : ∆ = (cid:18) bc (cid:19) − (cid:0) − b (cid:1) c > ,C : q = 16 (cid:0) − b (cid:1) c < . (42)Let’s choose arbitrarily b as the “canard parameter” or “duck param-eter”. Obviously, it appears that if the condition C is verified then thecondition C is de facto satisfied. Finally, the pseudo singular points areof saddle-type if and only if we have: b >
32 or b < − . (43) Remark 7.
Because of the symmetry of this coupled FitzHugh-Nagumoequations, the Jacobian matrix of system (33) evaluated at the pseudosingular points (35c) provides the same result as just above.
ANARDS EXISTENCE IN R
5. HODGKIN-HUXLEY MODEL
The original Hodgkin-Huxley model [19] is described by the followingsystem of four nonlinear ordinary differential equations: dVdt = 1 C M (cid:2) I − ¯ g K n ( V − V K ) − ¯ g Na m h ( V − V Na ) − ¯ g L ( V − V L ) (cid:3) (44a) dndt = α n ( V )(1 − n ) − β n ( V ) n (44b) dmdt = α m ( V )(1 − m ) − β m ( V ) m (44c) dhdt = α h ( V )(1 − h ) − β h ( V ) h (44d)where: α n ( V ) = 0 . V + 10) / (cid:18) exp V + 1010 − (cid:19) , (45a) β n ( V ) = 0 .
125 exp (
V / , (45b) α m ( V ) = 0 . V + 25) / (cid:18) exp V + 2510 − (cid:19) , (45c) β m ( V ) = 4 exp( V / , (45d) α h ( V ) = 0 .
07 exp(
V / , (45e) β n ( V ) = 1 / (cid:18) exp V + 3010 + 1 (cid:19) (45f)The first equation (44a) results from the application of Kirchhoff’s lawto the space clamped squid giant axon. Thus, the total membrane current C M dV /dt for which C M represents the specific membrane capacity and V the displacement of the membrane potential from its resting value, is equalto the sum of the following intrinsic currents: I K = ¯ g K n ( V − V K ) I Na = ¯ g Na m h ( V − V Na ) I L = ¯ g L ( V − V L )where I K is a delayed rectifier potassium current, I Na is fast sodiumcurrent and I L is the “leakage current”. The parameter I is the total2 J.M. GINOUX AND J. LLIBRE membrane current density, inward positive, i.e. the total current injectedinto the space clamped squid giant axon and V K , V Na and V L are the equi-librium potentials of potassium, sodium and “leakage current” respectively.The maximal specific conductances of the ionic currents are denoted ¯ g K ,¯ g Na and ¯ g L respectively. Functions α n,m,h and β n,m,h are gates’ openingand closing rates depending on V . Variable m denotes the activation ofthe sodium current, variable h the inactivation of the sodium current andvariable n the activation of the potassium current. These dimensionlessgating variables vary between [0 , V , and the referencedirection of the current I are defined as the negative of the voltages andcurrents. We have opted to adopt the reference assumption in Hodgkin &Huxley [19] for ease in comparison of our results with those from Hodgkinand Huxley . The parameter values are exactly those chosen in the originalHodgkin-Huxley [19] works: C M = 1 . µF/cm V Na = − mVV K = 12 mVV L = − . mV ¯ g Na = 120 mS/cm ¯ g K = 36 mS/cm ¯ g L = 0 . mS/cm According to Suckley and Biktashev [26] and Suckley [27], dimensionlessfunctions ¯ n , ¯ h and ¯ m called gates’ instant equilibrium values, i.e. , steady-state relation for gating variable n , h and m respectively as well as τ n , τ h and τ m called gates dynamics time scales in ms , i.e. , time constant forgating variable n , h and m respectively may be defined as follows: For more details see Chua et al. [10, 11]ANARDS EXISTENCE IN R n ( V ) = α n ( V ) α n ( V ) + β n ( V ) (46a)¯ h ( V ) = α h ( V ) α h ( V ) + β h ( V ) (46b)¯ m ( V ) = α m ( V ) α m ( V ) + β m ( V ) (46c) τ n ( V ) = 1 α n ( V ) + β n ( V ) (46d) τ h ( V ) = 1 α h ( V ) + β h ( V ) (46e) τ m ( V ) = 1 α m ( V ) + β m ( V ) (46f)By using Eqs. 46, the original Hodgkin-Huxley model [19] reads: dVdt = 1 C M (cid:2) I − ¯ g K n ( V − V K ) − ¯ g Na m h ( V − V Na ) − ¯ g L ( V − V L ) (cid:3) (47a) dndt = ¯ n − nτ n (47b) dhdt = ¯ h − hτ h (47c) dmdt = ¯ m − mτ m (47d)Now, in order to apply the singular perturbation method to the Hodgkin-Huxley model, two small multiplicative parameters ε ≪ n, h ) and ( m, V ) enables to justify such anintroduction. So, in order to differentiate slow variables from fast variables,Suckley and Biktashev [26], Suckley [27] and Rubin and Wechselberger [24]have plotted the inverse of “time constant for gating variable i ”, i.e. , τ i − according to V with i = n, h, m . In Fig. 1, they have been plotted forthe original functions α i and β i (Eqs. 45a). However, let’s notice that thisplot is exactly the same as those presented by Rubin and Wechselberger[24] (Fig. 1) for a nondimensionalized three-dimensional Hodgkin-Huxleysingularly perturbed system obtained after the following variable changes: V → − V and ¯ I → − ¯ I , then V → V + 65 and finally V → V /
J.M. GINOUX AND J. LLIBRE Τ n - Τ h - Τ m - - - - - - -
20 0 200246810 V Τ i - FIG. 1.
Graph of 1 /τ i (ms − ) against V ( mV ).ANARDS EXISTENCE IN R τ i − according to V with i = n, h, m over the physiological range. We observe that τ m − is of an order ofmagnitude bigger than τ h − and τ n − , which are of comparable size. In-deed, we can deduce that the values of times scales are approximately τ m − ≈ ms − while τ n − ≈ τ h − ≈ ms − . Then, it appears that m corresponds to the fast variable while n and h correspond to slow vari-ables. Moreover, since the activation of the sodium channel m is directlyrelated to the dynamics of the membrane (action) potential V , Rubin andWechselberger [24] consider that m and V evolve on the same fast timescale. So, the Hodgkin-Huxley model may be transformed into a singularlyperturbed system with two time scales in which the slow variables are ( n, h )and the fast variables are ( m, V ).So, according to Awiszus et al. [2], Suckley and Biktashev [26], Suckley[27] and Rubin and Wechselberger [24] small multiplicative parameters 0 <ε ≪ g Na and set:¯ g Na → ¯ g Na ¯ g Na = 1, ¯ g K → ¯ g K ¯ g Na = 0 .
3, ¯ g L → ¯ g L ¯ g Na = 0 . . other parameters are kept as for the original Hodgkin-Huxley model [19]: C M = 1 . µF/cm , ¯ V Na = − mV , ¯ V K = 12 mV , ¯ V L = − . mV. Then, by posing ¯ I → ¯ I ¯ g Na , ε = C M ¯ g Na = 1120 and ( n, h, m, V ) = ( x , x , y , y )to consistent with the notations of Sec. 3, we obtain: dx dt = ¯ x − x τ = f ( x , x , y , y ) (48a) dx dt = ¯ x − x τ = f ( x , x , y , y ) (48b) ε dy dt = ¯ y − y τ = g ( x , x , y , y ) (48c) ε dy dt = ¯ I − ¯ g K x ( y − V K ) − ¯ g Na y x ( y − V Na ) − ¯ g L ( y − V L )= g ( x , x , y , y ) (48d)6 J.M. GINOUX AND J. LLIBRE where (¯ x , ¯ x , ¯ y ) = (¯ n, ¯ h, ¯ m ) and τ , , = τ n,h,m .Let’s notice that the multiplicative parameter ε has been introduced ar-tificially in Eq. (48c). This is due to the fact that it has been stated abovethat the time scale of variable m , i.e. , y is tenth times greater than thetime scale of variables n and h , i.e. of variables x and x . Moreover,this parameter is identical to those use in Eq. (48d) since it has been alsoconsidered that m and V , i.e. , y and y evolve on the same fast time scale.According to the Geometric Singular Perturbation Theory , the zero-orderapproximation in ε of the slow manifold associated with the Hodgkin-Huxley model (48) is obtained by posing ε = 0 in Eqs. (48c & 48d).So, the slow manifold is given by: x = ¯ I − ¯ g K x ( y − V K ) − ¯ g L ( y − V L )¯ g Na ¯ y ( y − V Na ) (49a) y = ¯ y ( y ) (49b)Then, the fast foliation is within the planes x = constant and x = constant .The fold curve is defined as the location of the points where g ( x , x , y , y ) =0, g ( x , x , y , y ) = 0 and det (cid:2) J ( g ,g ) (cid:3) = 0. For the Hodgkin-Huxleymodel (48), the fold curve is thus given by Eqs. (49a & 49b) and by thedeterminant of the Jacobian matrix of the following fast foliation : dy dt = ¯ y − y τ = g ( x , x , y , y ) (50a) dy dt = ¯ I − ¯ g K x ( y − V K ) − ¯ g Na y x ( y − V Na ) − ¯ g L ( y − V L )= g ( x , x , y , y ) (50b)The Jacobian matrix of the fast foliation (50) reads: J ( g ,g ) = ¯ y ′ τ − τ ′ ( ¯ y − y ) τ − τ − (¯ g K x + ¯ g Na y x + ¯ g L ) − g Na y x ( y − V Na ) (51)where the ( ′ ) denotes the derivative with respect to y . Then, takinginto account Eqs. (49b), i.e. , y = ¯ y we have: ANARDS EXISTENCE IN R J ( g ,g ) = ¯ y ′ τ − τ − (¯ g K x + ¯ g Na ¯ y x + ¯ g L ) − g Na ¯ y x ( y − V Na ) (52)So, the determinant of the Jacobian matrix of the fast foliation (50) is: det (cid:0) J ( g ,g ) (cid:1) = − τ (cid:2) ¯ g K x + ¯ g Na ¯ y x + ¯ g L + 3¯ g Na ¯ y ′ ¯ y x ( y − V Na ) (cid:3) (53)Thus, the condition for the fold curve is det (cid:0) J ( g ,g ) (cid:1) = 0, which gives:¯ g K x + ¯ g Na ¯ y x + ¯ g L + 3¯ g Na ¯ y ′ ¯ y x ( y − V Na ) = 0 (54)Therefore: x = − ¯ g K x + ¯ g L ¯ g Na ¯ y (¯ y + 3 ¯ y ′ ( y − V Na )) (55)By subtracting Eq. (49a) from Eq. (55) we obtain x : x = x = (cid:20) − ¯ I [¯ y + 3 ¯ y ′ ( y − V Na )] + ¯ g L ( V Na − V L )¯ y + 3 ¯ y ′ ( y − V Na )( y − V L )¯ g K [( V K − V Na )¯ y − y ′ ( y − V Na )( y − V K )] (cid:21) / (56)Plugging this value of x (56) into Eq. (55) provides: x = x = ¯ I + ¯ g L ( V K − V L )¯ g Na ¯ y [( V Na − V K )¯ y + 3 ¯ y ′ ( y − V Na )( y − V K )] (57)So, the fold curve is given by the set of parametric equations (56-57) interms of y .The pseudo singular points are given by Eqs. (20) which reads for theHodgkin-Huxley model (48):8 J.M. GINOUX AND J. LLIBRE ¯ y − y τ = 0 , (58a)¯ I − ¯ g K x ( y − V K ) − ¯ g Na y x ( y − V Na ) − ¯ g L ( y − V L ) = 0 , (58b) (cid:20) g K x ( y − V K )( x − ¯ x ) τ + ¯ g Na y ( y − V Na )( x − ¯ x ) τ (cid:21) = 0 , (58c) (cid:20) g K x ( y − V K )( x − ¯ x ) τ + ¯ g Na y ( y − V Na )( x − ¯ x ) τ (cid:21) τ = 0 , (58d) τ (cid:0) ¯ g K x + ¯ g Na y x + ¯ g L (cid:1) + 3¯ g Na y x ( y − V Na )( τ y ′ + ( y − ¯ y ) τ ′ ) = 0 . (58e)Let’s notice that Eqs. (58c) and (58d) are identical. Moreover, thedefinition of τ (46f) enables to simplify the above system (58). Thus, wehave:¯ I − ¯ g K x ( y − V K ) − ¯ g Na ¯ y x ( y − V Na ) − ¯ g L ( y − V L ) = 0 , (59a)4¯ g K x ( y − V K )( x − ¯ x ) τ + ¯ g Na ¯ y ( y − V Na )( x − ¯ x ) τ = 0 , (59b)¯ g K x + ¯ g Na ¯ y x + ¯ g L + 3¯ g Na ¯ y ¯ y ′ x ( y − V Na ) = 0 . (59c)Moreover, Eqs. (59a) and (59c) indicate that the pseudo singular point belongs to the slow manifold and to the fold curve . So, let’s replace in Eq.(59b) the variables x and x by the variables x and x given by Eq.(56) and Eq. (57) respectively which represent the parametric equations of fold curve .4 g K x ( y − V K )( x − ¯ x ) τ + ¯ y ( y − V Na )( x − ¯ x ) τ = 0 . (60)Thus, it appears that Eq. (60) depends on the variable y , on the func-tions gates dynamics time scales τ ( y ) and τ ( y ) and on the bifurcationparameter ¯ I . According to Rubin and Wechselberger [24], the function y ( ¯ I ), solution of (60) is independent of time multiplicative constants k and k that one could set in factor of τ ( y ) and τ ( y ).So, following their works, let’s plot the function y ( ¯ I ) solution of (60) forvarious values of these time constants by posing successively in (60) k = 1,3, 4 .
75 and 7 and while fixing k = 1. The result is presented in Fig. 2. ANARDS EXISTENCE IN R k n = k n = k n = k n = - - - - - I - y FIG. 2.
Function y (¯ I ) for various values of parameter k n = 1 , , . , I C ≈ − . J.M. GINOUX AND J. LLIBRE
Let’s notice that this plot is exactly the same as those presented byRubin and Wechselberger [24] (Fig. 8-9) for a nondimensionalized three-dimensional Hodgkin-Huxley singularly perturbed system which had beenobtained after the following variable changes: V → − V and ¯ I → − I , then V → V + 65 and finally V → V / I C ≈ − . of Eq. (60) provides a betterapproximation of the bifurcation parameter value:¯ I C = − . µA This value corresponds to a voltage y = − . mV .For ¯ I ≈ − .
1, the coordinate of the pseudo singular point can be com-puted numerically:( x , x , y , y ) = (0 . , . , . , − . λ − σ λ + σ λ − σ λ + σ = 0for which it is easy to prove that σ = σ = 0. So, this eigenpolynomialreduces to: λ (cid:0) λ − σ λ + σ (cid:1) = 0According to Eqs. (28) we find that: p = T r ( J ) = 144 . ,q = σ = − . pseudo singular points is of saddle-type. Moreover, numerical computation of the eigenvalues of this Jacobianmatrix evaluated at the pseudo singular point provides: The function y (¯ I ) solution of (59) has been plotted with Mathematica c (cid:13) while usingthe ContourPlot function used for representing implicit function since such functioncannot be expressed explicitly. This resolution has been made while using the function FindRoot in Mathematica c (cid:13) .ANARDS EXISTENCE IN R λ , λ , λ , λ ) = ( − . , . , , pseudo singular point is of saddle-type and canard solution may occur in the four-dimensional Hodgkin-Huxley singularly perturbed system (48) for the original set of parametervalues.In Fig. 3, 4 & 5 canard solution of the four-dimensional Hodgkin-Huxleysingularly perturbed system for the “canard value” of ¯ I ≈ − . x , x , y ) phase-space and then in the ( x , y ) phase plane.The green point represents the pseudo singular point . The trajectory curve, i.e. , the canard solution has been plotted in red while the fold curve is inyellow. We observe on Fig. 3 that when the trajectory curve reaches the fold at the pseudo singular point it “jump” suddenly to the other partof the slow manifold before being reinjected towards the pseudo singularpoint .2 J.M. GINOUX AND J. LLIBRE
FIG. 3.
Phase portrait, canard solution and slow manifold of the Hodgkin-Huxleysystem (48) in the ( n, h, V ) phase space.ANARDS EXISTENCE IN R - - - - -
20 00.40.50.60.7 V n FIG. 4.
Phase portrait, canard solution and slow manifold of the Hodgkin-Huxleysystem (48) in the (
V, n ) phase plane. J.M. GINOUX AND J. LLIBRE - - - - -
20 00.10.20.30.40.5 V h FIG. 5.
Phase portrait, canard solution and slow manifold of the Hodgkin-Huxleysystem (48) in the (
V, h ) phase plane.ANARDS EXISTENCE IN R
6. DISCUSSION
In a previous paper entitled: “Canards Existence in Memristor’s Cir-cuits” (see Ginoux & Llibre [17]) we have proposed a new method forproving the existence of “canard solutions” for three and four-dimensionalsingularly perturbed systems with only one fast variable which improves themethods used until now. This method enabled to state a unique “generic”condition for the existence of “canard solutions” for such three and four-dimensional singularly perturbed systems which is based on the stabil-ity of folded singularities of the normalized slow dynamics deduced froma well-known property of linear algebra. This unique condition which iscompletely identical to that provided by Benoˆıt [6] and then by Szmolyanand Wechselberger [25] and finally by Wechselberger [39] was consideredas “generic” since it was exactly the same for singularly perturbed systemsof dimension three and four with only one fast variable. In this work wehave extended this new method to the case of four-dimensional singularlyperturbed systems with two slow and two fast variables and we have statedthat the condition for the existence of “canard solutions” in such systemsis exactly identical to those proposed in our previous paper. This resultconfirms the genericity of the condition ( σ <
0) we have highlighted andprovides a simple and efficient tool for testing the occurrence of “canardsolutions” in any three or four-dimensional singularly perturbed systemswith one or two fast variables. Applications of this method to the famouscoupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley modelhas enabled to show the existence of “canard solutions” in such systems.However, in this paper, only the case of pseudo singular points or foldedsingularities of saddle-type has been analyzed. Of course, the case of of pseudo singular points or folded singularities of node-type and focus-typecould be also studied with the same method.
7. ACKNOWLEDGEMENTS
We would like to thank to Ernesto P´erez Chavela for previous discus-sions related with this work. The authors are partially supported by aMINECO/FEDER grant number MTM2008-03437. The second authoris partially supported by a MICINN/FEDER grants numbers MTM2009-03437 and MTM2013-40998-P, by an AGAUR grant number 2014SGR-568, by an ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers316338 and 318999, and FEDER-UNAB10-4E-378.6
J.M. GINOUX AND J. LLIBRE
APPENDIX
Change of coordinates leading to the normal forms of four-dimensionalsingularly perturbed systems with two fast variables are given in the fol-lowing section.
Normal form of 4D singularly perturbed systemswith two fast variables
Let’s consider the four-dimensional singularly perturbed dynamical sys-tem (11) with k = 2 slow variables and m = 2 fast and let’s make thefollowing change of variables: x = α x, x = αy, y = α z, y = αu where α ≪ . (A-1)By taking into account extension of Benoˆıt’s generic hypothesis Eqs.(22,23) and while using Taylor series expansion the system (11) becomes:˙ x = ∂f ∂y y + ∂f ∂u u, ˙ y = f ( x, y, z, u ) , (cid:16) εα (cid:17) ˙ z = ∂g ∂z z + 12 ∂ g ∂y y + 12 ∂ g ∂u u + ∂ g ∂y∂u yu, (cid:16) εα (cid:17) ˙ u = ∂g ∂x x + 12 ∂ g ∂y y + 12 ∂ g ∂u u + ∂ g ∂y∂u yu. (A-2)Then, let’s make the standard polynomial change of variables: X = Ax + By ,Y = yf ,Z = Cy + Dz + Eu,U = F y + Gu. (A-3)From (A-3) we deduce that:
ANARDS EXISTENCE IN R x = X − Bf Y A ,y = f Y,z = 1 D (cid:20) Z − Cf Y − EG ( U − F f Y ) (cid:21) ,u = U − F f YG . (A-4)The time derivative of system (A-3) gives:˙ X = A ˙ x + 2 By ˙ y, ˙ Y = ˙ yf , ˙ Z = C ˙ y + D ˙ z + E ˙ u, ˙ U = F ˙ z + G ˙ u. (A-5)Then, multiplying the third and fourth equation of (A-5) by ( ε/α ) andwhile replacing in (A-5) ˙ x , ˙ y , ˙ z and ˙ u by the right-hand-side of system(A-2) leads to: ˙ X = A ˙ x + 2 By ˙ y, ˙ Y = ˙ yf , (cid:16) εα (cid:17) ˙ Z = (cid:16) εα (cid:17) C ˙ y + (cid:16) εα (cid:17) D ˙ z + (cid:16) εα (cid:17) E ˙ u, (cid:16) εα (cid:17) ˙ U = (cid:16) εα (cid:17) F ˙ y + (cid:16) εα (cid:17) G ˙ u. (A-6)Since ε/α ≪
1, the first terms of the right-hand-side of the third andfourth equation of (A-16) can be neglected. So we have:˙ X = A (cid:18) ∂f ∂y y + ∂f ∂u u (cid:19) + 2 Bf y, ˙ Y = 1 , (cid:16) εα (cid:17) ˙ Z = D (cid:18) ∂g ∂z z + 12 ∂ g ∂y y + 12 ∂ g ∂u u + ∂ g ∂y∂u yu (cid:19) + E (cid:18) ∂g ∂x x + 12 ∂ g ∂y y + 12 ∂ g ∂u u + ∂ g ∂y∂u yu (cid:19) , (cid:16) εα (cid:17) ˙ U = G (cid:18) ∂g ∂x x + 12 ∂ g ∂y y + 12 ∂ g ∂u u + ∂ g ∂y∂u yu (cid:19) . (A-7)8 J.M. GINOUX AND J. LLIBRE
Then, by replacing in (A-7) x , y , z and u by the right-hand-side of (A-4) and by identifying with the following system in which we have posed:( ε/α ) = ǫ :˙ X = ˜ aY + ˜ bU + O (cid:0) X, ǫ, Y , Y U, U (cid:1) , ˙ Y = 1 + O ( X, Y, U, ǫ ) ,ǫ ˙ Z = ˜ cZ + O (cid:0) ǫX, ǫY, ǫU, X , y U, U , Y U (cid:1) ,ǫ ˙ U = − (cid:0) X + U (cid:1) + O (cid:0) ǫX, ǫY, ǫU, ǫ , X U, U , XY U (cid:1) , (A-8)we find: ˜ a = A (cid:18) ∂f ∂y − FG ∂f ∂u (cid:19) f + 2 Bf , ˜ b = AG ∂f ∂u , ˜ c = ∂g ∂z . (A-9)where A = 12 ∂g ∂x ∂ g ∂u ,B = 14 " ∂ g ∂u ∂ g ∂y − (cid:18) ∂ g ∂y∂u (cid:19) ,G = − ∂ g ∂u . (A-10)Finally, we deduce:˜ a = 12 " f ∂ g ∂x ∂ g ∂y − (cid:18) ∂ g ∂x ∂y (cid:19) ! + f ∂g ∂x (cid:18) ∂ g ∂y ∂f ∂x − ∂ g ∂x ∂y ∂f ∂y (cid:19) ˜ b = − ∂g ∂x ∂f ∂y , ˜ c = ∂g ∂y . (A-11)This is the result we established in Sec. 2.7. ANARDS EXISTENCE IN R REFERENCES
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