Giant Squid - Hidden Canard: the 3D Geometry of the Hodgkin-Huxley Model

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1 Noname manuscript No. (will be inserted by te editor) Jonatan Rubin Martin Wecselberger Giant Squid - Hidden Canard: te 3D Geometry of te Hodgkin-Huxley Model te date of receipt and acceptance sould be inserted later Abstract Tis work is motiated by te obseration of remarkably slow firing in te uncoupled Hodgkin- Huxley model, depending on parameters τ,τ n tat scale te rates of cange of te gating ariables. After reducing te model to an appropriate nondimensionalized form featuring one fast and two slow ariables, we use geometric singular perturbation teory to analyze te model s dynamics under systematic ariation of te parameters τ,τ n, and applied current I. As expected, we find tat for fixed (τ,τ n ), te model undergoes a transition from excitable, wit a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO s), featuring slow action potential generation, arise for an intermediate range of I alues, if τ or τ n is sufficiently large. Our analysis explains in detail te geometric mecanisms underlying tese results, wic depend crucially on te presence of two slow ariables, and allows for te quantitatie estimation of transitional parameter alues, in te singular limit. In particular, we sow tat te subtresold oscillations in te obsered MMO patterns arise troug a generalized canard penomenon. Finally, we discuss te relation of results obtained in te singular limit to te beaior obsered away from, but near, tis limit. 1 Introduction Te Hodgkin-Huxley (HH) model (Hodgkin and Huxley 1952) for te action potential of te spaceclamped squid giant axon is defined by te following 4D ector field: C dv dt = I I Na I K I L dm dt = φ[α m(v )(1 m) β m (V )m] (1.1) d dt = φ[α (V )(1 ) β (V )] dn dt = φ[α n(v )(1 n) β n (V )n]. We use modern conentions suc tat te spikes of action potentials are positie, and te oltage V of te original HH model (Hodgkin and Huxley 1952) is sifted relatie to te oltage V of tis model by V = (V + 65). Te first equation is obtained by applying Kircoff s law to te space-clamped neuron, i.e. te transmembrane current is equal to te sum of intrinsic currents. C is te capacitance density in µf/cm 2, V is te membrane potential in mv and t is te in ms. Te ionic currents on te rigt and side are gien by I Na = g na m 3 (V E Na ), I K = g k n 4 (V E K ), I L = g l (V E L ) (1.2) Department of Matematics and Center for te Neural Basis of Cognition, Uniersity of Pittsburg, PA, USA; rubin@mat.pitt.edu, pone: , fax: Scool of Matematics and Statistics, Uniersity of Sydney, NSW, Australia Address(es) of autor(s) sould be gien

2 4 2 Voltage (mv) Time (ms) Fig. 1 Action potential generated by te HH model wit applied current I = 9.6 at 6.3 C. wit a fast sodium current I Na, a delayed rectifier potassium current I K and a small leak current I L, wic consists mainly of cloride current. Te current densities I x (x = Na,K,L) are measured in µa/cm 2 and te conductance densities g x in ms/cm 2. Te parameter I represents current injected into te space-clamped axon and E x are te equilibrium potentials or Nernst potentials in mv for te arious ions. Te parameters are gien by g Na = 12, g K = 36, g L =.3, E Na = 5, E K = 77, E L = 54.4, C = 1. (1.3) Te conductances of te ionic currents are regulated by oltage dependent actiation and inactiation ariables called gating ariables. Teir dynamics are described by te oter tree equations (1.1), were m denotes te actiation of te sodium current, te inactiation of te sodium current, and n te actiation of te potassium current. Eac of tese equations features a temperature scaling factor φ = (Q 1 ) (T T)/1, were Q 1 is a constant, T is temperature, and T = 6.3, bot in degrees celsius. Te gating ariables are dimensionless wit teir ranges in te interal [,1]. Te specific functions α z and β z (z = m,,n) on te rigt and sides are, in units of (ms) 1, (V + 4)/1 α m (V ) = 1 exp( (V + 4)/1), β m(v ) = 4 exp( (V + 65)/18) α (V ) =.7 exp( (V + 65)/2), β (V ) = 1/(1 + exp( (V + 35)/1)) (1.4) (V + 55)/1 α n (V ) = 1 exp( (V + 55)/1), β n(v ) =.125 exp( (V + 65)/8). Figure 1 sows an action potential at 6.3 C simulated by te HH model, wic is in good agreement wit measured action potential data of te squid giant axon (Hodgkin and Huxley 1952). FitzHug (FitzHug 196) as gien an elegant qualitatie description of te HH equations, based on te fact tat te model ariables (V,m) ae fast kinetics, wile (,n) ae slow kinetics. Tis allows te full 4D pase space to be broken into smaller pieces (2D subspaces) by fixing te slow ariables and considering te beaiour of te model as a function of te fast ariables. Tis idea proides a useful way to study te process of excitation (see also Nagumo et al. 1962). Based on FitzHug s analysis, anoter model reduction was proposed (see e.g. Rinzel 1985), wic keeps te slow-fast structure of te equations, but reduces te system to a 2D model. Te reductions are based on te following obserations: Te actiation of sodium cannels m is (ery) fast. Terefore m will reac its equilibrium almost instantaneously and m = α m (V )/(α m (V ) + β m (V )) =: m (V ) can be assumed. As FitzHug already noticed (FitzHug 196), in te course of an action potential tere appears to be an approximately linear relation between and n. Tus n can be approximated by a linear function n = n(). 2

3 Voltage (ms) 2 4 Voltage (mv) Time (ms) Time (ms) Fig. 2 Simulation of modified HH model wit applied current I = 9.6. Left: τ = 1 yields a firing frequency of approximately 7Hz. Rigt: τ = 2 yields a firing frequency of approximately 7Hz. Te first reduction can be matematically justified by a center manifold reduction (see Teorem 1), wic reduces te HH model to a 1 fast, 2 slow ariable model. Te second reduction is a purely empirical obseration and as no matematical justification. Tere is no a priori argument as to wy te nonlinear ariables (n, ) sould ae a linear relation. Noneteless, te reduced HH model resulting from applying tis relationsip can describe te action potential (Figure 1) ery well and can be analyzed in te corresponding 2D pase space. It as become conentional wisdom tat te qualitatie properties of te Hodgkin-Huxley model can be reduced to a 2D flow suc as tat described by Rinzel (Rinzel 1985). But tese reductions do not capture te full dynamics of te full HH model. Rinzel and Miller (198) as well as Guckeneimer and Olia (22) ae gien eidence for caos in te HH model. Tis clearly points out tat a rigorous reduction to a 2D model is not possible, as caos requires models wit pase spaces of at least tree dimensions. Anoter interesting obseration was made in a ariation of te HH model by Doi et al. (Doi and Kumagai 21, Doi et al. 21, Doi et al. 24, Doi and Kumagai 25), wo replaced φ in (1.1) wit tree independent constants τ m,τ,τ n : C dv dt = I g nam 3 (V E Na ) g k n 4 (V E K ) g l (V E L ) dm dt = 1 (α m (V )(1 m) β m (V )m) (1.5) τ m d dt = 1 (α (V )(1 ) β (V )) τ dn dt = 1 (α n (V )(1 n) β n (V )n), τ n were τ m = τ = τ n = 1 corresponds to te classical HH model at 6.3 degrees celsius. In teir work tey obsered a dramatic slowing of te firing rate wen tey increased eiter τ or τ n 1-fold. Actually, suc a big cange in te constants is not needed to obsere tis beaiour. If e.g. τ is canged from 1 to 2, ten te firing rate of action potentials due to applied current I = 9.6 slows down dramatically, from approximately 7Hz to 7Hz (see Figure 2). Tis represents a 1-fold decrease in firing rate, altoug te constant was just increased 2-fold. Note te sub-tresold oscillations in te interspike interals for τ = 2, wic do not exist for τ = 1. Furtermore, Doi et al. (Doi and Kumagai 21, Doi et al. 21, Doi et al. 24, Doi and Kumagai 25) also obsered caotic beaiour witin tis modified model (not sown ere), wic indicates again tat te classical 2D reduction is not appropriate to capture te full dynamics of te classical HH model. An interesting obseration is tat modifying te speed of actiation and inactiation of te ion cannels leaes te monotonic steady-state current-oltage relation of te model neuron uncanged. Terefore te modified HH model is still classified as a Type II neuron, as is te classical HH model (Rinzel and 3

4 Ermentrout 1989). Preiously, it ad been belieed tat slow firing rates in single neuron models could be acieed only in Type I neurons, wic ae an N-saped current oltage relation, as found for neurons wit A-type potassium cannels. Te wide range of firing rates seen tere is due to a omoclinic bifurcation in te 2D pase space. Te 3D analysis we present ere explains a dynamic mecanism by wic Type II model neurons can also ae a wide range of firing rates. Te main question we address is te following: How does a slow firing rate emerge from te geometry of te HH model? A similar obseration of significant slowing of firing rates, as in Figure 2, as been made by Droer et al. (Droer et al. 24, Rubin 25) in a network of (Type II) HH model neurons coupled wit excitatory synapses. Tis network syncronizes ery quickly after synaptic excitation is actiated and te firing rate of te network slows down dramatically, compared to te single neuron firing rate wit constant current injection. Te analysis of tis network can be reduced to a 3D model. Te key to understanding te obsered actiity is te so called canard penomenon (Benoit 1983, Szmolyan and Wecselberger 21), wic traps te solution for a significant amount of near te expected action potential tresold before it can fire again. Wecselberger (25a) as sown tat te extreme delay is due to canards of folded node type (Szmolyan and Wecselberger 21). Te ortex structure described in Droer et al. (24) can be rigorously understood in terms of inariant manifolds analysed in Wecselberger (25a), wic form a multi-layered trapping region. Te solutions wit significant delays tat we ae described aboe consist of a certain number of subtresold oscillations combined wit a relaxation oscillation type action potential, as sown in Figure 2 for τ = 2. Suc solutions are called mixed-mode-oscillations (MMO s), and teir relation to te canard penomenon was first demonstrated by Milik et al. (1998). A more detailed analysis of MMO s and generalization of te canard penomenon was done by Brøns et al. (26). In tis paper, we apply geometric singular perturbation tecniques (Szmolyan and Wecselberger 21, Szmolyan and Wecselberger 24, Wecselberger 25a, Brøns et al. 26) suitable for te analysis of te single HH model neuron. We explore te geometry of te uncoupled HH neuron carefully, explain ow a significant slowing of te firing rate may occur, and explain a mecanism troug wic complex oscillatory patterns may arise in tis system. Te outline is as follows: In Section 2 we reduce te HH model to a 3D model tat captures all te qualitatie features obsered in te full model. In Section 3 we gie an oeriew of results on relaxation oscillations and MMO s in general, using results from geometric singular perturbation teory. In Section 4 we apply tese results to te reduced 3D HH model. Tis enables us to explain te mecanism underlying te obsered oscillatory penomena and to predict wat forms of solutions will arise as τ,τ n, and I are aried. Finally, in Section 5 we conclude wit a discussion. 2 HH model reduction We will apply geometric singular perturbation tecniques for te analysis of te HH equation, formulated to include te modification proposed by Doi et al. (Doi and Kumagai 21, Doi et al. 21, Doi et al. 24, Doi and Kumagai 25: C dv dt = I g nam 3 (V E Na ) g k n 4 (V E K ) g l (V E L ) dm dt = 1 τ mˆτ m (V ) (m (V ) m) (2.6) d dt = 1 τ ˆτ (V ) ( (V ) ) dn dt = 1 τ nˆτ n (V ) (n (V ) n) were ˆτ x (V ) (in ms) and x (V ) (dimensionless), wit x = m,,n, are defined as follows: ˆτ x (V ) = 1 α x (V ) + β x (V ), x α x (V ) (V ) = α x (V ) + β x (V ). (2.7) 4

5 Fig. 3 Functions 1/ˆτ m() (solid), 1/ˆτ () (das-dotted), and 1/ˆτ n() (dased); all in (ms) Dimensionless ersion of HH model As a starting point we nondimensionalize system (2.6) and identify a small perturbation parameter ε suc tat we can apply singular perturbation tecniques. Te following table sows te units of te ariables and parameters in system (2.6): ariable units parameter units V mv E x mv m 1 g x ms/cm 2 1 C µf/cm 2 n 1 I µa/cm 2 t ms τ x 1 To make te ariables (V,t) dimensionless, we ae to identify a typical oltage scale k and a typical scale k t, and define new dimensionless ariables (,τ) suc tat V = k, t = k t τ. (2.8) Using tis transformation, te dimensionless HH system is ten gien by d dτ = k t g C [Ī ḡ nam 3 ( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] dm dτ = k t τ mˆτ m () (m () m) (2.9) d dτ = k t τ ˆτ () ( () ) dn dτ = k t τ nˆτ n () (n () n) wit dimensionless parameters Ēx = E x /k, ḡ x = g x /g and Ī = I/(k g). Te Nernst potentials E x set a natural range for te obsered action potentials as E K V E Na. Terefore te maximum ariation of te membrane potential is 127mV in our problem, and we coose k = 1mV as a typical scale for te potential V. We furter coose g = g na as a reference conductance since it is te maximum conductance in tis problem. Note tat under tis coice all terms in te square bracket of te rigt and side of te first equation in (2.9) are bounded (in absolute alues) by one. Terefore te caracteristic scale of tis rigt and side is gien by (k t g Na )/C. Next, let us ceck te rigt and sides of te gating equations in (2.9). We ae x 1, x () 1 and terefore x () x 1. Te only differences in te orders of magnitude of te gating equations may 5

6 arise from te functions ˆτ x (). Te functions ˆτ x () are gien in ms and terefore include caracteristic scales. Recall from (2.7) tat 1/ˆτ x () = α x () + β x (). (2.1) Figure 3 sows a plot of te functions 1/ˆτ x () oer te pysiological range [.77,.5]. Tis figure sows tat max [.77,.5] (1/ˆτ m ()) is of an order of magnitude bigger tan 1/ˆτ () and 1/ˆτ n (), wic are of comparable size. We define 1/ˆτ x () = ˆT x /ˆt x () were ˆT x = max [.77,.5] (1/ˆτ x ()). Note tat ˆT x as dimension (ms) 1 wile ˆt x () is now dimensionless and 1/ˆt x () 1 for [.77,.5]. Te alues of te scaling factors are approximately ˆT m 1(ms) 1 wile ˆT ˆT n 1(ms) 1 (see Figure 3). From (2.9), we obtain te following system C d k t g Na dτ = [Ī ḡ nam 3 ( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] 1 dm ˆT m k t dτ = 1 τ mˆt m () (m () m) (2.11) 1 d ˆT k t dτ = 1 τ ˆt () ( () ) 1 dn ˆT n k t dτ = 1 τ nˆt n () (n () n). Note tat (C/g Na ) and (1/ ˆT m ) are fast reference s (.1ms) wile (1/ ˆT ) and (1/ ˆT n ) are slow reference s ( 1ms). Our aim is to understand te long delays of action potentials, so we coose te gien slow scale 1/ ˆT 1/ ˆT n 1ms as a reference and set k t = 1ms. Wit tat setting, te two dimensionless parameters C/(k t g Na ) and 1/( ˆT m k t ) on te left and side are small. Since te actiation of te sodium cannel m is directly related to te dynamics of te membrane (action) potential, we assume tat (,m) eole on te same fast scale and set ε := C k t g na 1, 1 := ε 1. (2.12) ˆT m k t T m Furtermore we define ˆT k t =: T and ˆT n k t =: T n so tat eac T x is a dimensionless parameter. Wit tese definitions we obtain finally te HH equations in dimensionless form and as a singularly perturbed system ε d dτ = [Ī m3 ( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] ε dm dτ = 1 τ m t m () (m () m) (2.13) d dτ = 1 τ t () ( () ) dn dτ = 1 τ n t n () (n () n) wit (,m) as fast ariables, (,n) as slow ariables and t x () := ˆt x ()/T x. Tis reflects exactly te assumptions made in te pioneering work of FitzHug (196). Te significantly faster actiation of te sodium cannel m tan its inactiation and te actiation of te potassium cannel n makes te creation of action potentials possible. Remark 1 A misleading statement about te HH system is often found in te literature, namely tat te gate m eoles on te fastest scale in tis system. Te correct statement is tat m eoles faster tan te oter two gating ariables, wic is essential for te creation of action potentials, but m actually eoles slower tan te membrane potential V (i.e., te parameter T m < 1). One could argue tat te HH system eoles on tree different scales: V fast, m intermediate and (,n) slow. But to apply classical singular perturbation tecniques, wic allow for just two different scales, we group (V,m) as fast and (,n) as slow, based on (2.12), as described aboe. 6

7 n Fig. 4 Cubic saped critical manifold S of te dimensionless HH system sown in (,, n) space, I = Reduction to 3D model By setting ε = in te 4D singularly perturbed system (2.13) we obtain te reduced system (also called te slow subsystem). Tis system is a differential algebraic system describing te eolution of te slow ariables (n,) constrained to a 2D manifold S, called te critical manifold, wic is defined by te two equations n 4 (,m,) = Ī m3 ( ĒNa) ḡ l ( ĒL) ḡ k ( ĒK), m(,n,) = m (). If we project S into te (,n,) space by using te identity m = m (), ten S is defined by n 4 (,) = Ī m () 3 ( ĒNa) ḡ l ( ĒL) ḡ k ( ĒK). (2.14) Tis critical manifold S is a cubic saped surface as sown in Figure 4, a typical feature of relaxation oscillators in general. Te slow dynamics on te critical manifold describes e.g. te slow depolarization towards te action potential tresold sown in Figure 1. Weneer te neuron fires an action potential, it canges to a fast dynamics were te slow ariables are (almost) constant but te fast ariables canges rapidly. Tis beaiour is described by te layer problem (or fast subsystem) d = [Ī dτ m3 ( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] =: f(,m) (2.15) 1 dm 1 = dτ 1 τ m t m () (m () m) =: g(,m), wic is obtained by canging to te fast τ 1 = τ/ε and taking te limit ε. Te slow ariables = and n = n are now constants. Te critical manifold S is te manifold of equilibria for te layer problem, and trajectories of te layer problem eole along one-dimensional sets (,,n ), called fast fibers, near tis manifold S. If we linearize te layer problem at S we obtain information about te transient beaiour of solutions along tese fast fibers, in te neigborood of tis manifold. It is well known tat solutions are quickly attracted along te fast fibers to one of te outer two attracting brances of te critical manifold and, to leading order, follow te reduced flow towards te associated fold cure. In te neigbourood of te fold cure te dynamics canges significantly and te layer problem will eentually cause a fast transient beaiour towards te oter attracting branc obsered e.g. as an upstroke in te action potential. 7

8 Te following result sows tat te transient beaiour near te fold-line is described by a 3D ector-field representing te flow on a 3D center manifold of (2.13); for a more general result see (Brøns et al. 26). Teorem 1 Te ector field (2.13) on te fast scale τ 1 = τ/ε possesses a tree dimensional center manifold M along te fold cure, wic is exponentially attracting. Te ector field (2.13) reduced to M is gien by: ε d dτ = [Ī m3 ()( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] =: F(,n,) d dτ = 1 τ t () ( () ) =: H(,) (2.16) dn dτ = 1 τ n t n () (n () n) =: N(,n). Proof: Introducing a new ariable m = m m () in system (2.13) on te fast scale τ 1 = τ/ε gies te layer problem d = [Ī dτ (m () + m) 3 ( ĒNa) ḡ k n 4 ( ĒK) ḡ l ( ĒL)] =: f(, m) 1 d m 1 = dτ 1 τ m t m () m m ()f(, m) =: ḡ(, m) (2.17) Te critical manifold S is defined by {f(, m) =, m = }. Hence ḡ = m () f S. S Furtermore ḡ 1 m = S τ m t m () m () f m <, S since f m = 3m 2 ()( ĒNa) > S and m >. It follows tat te Jacobian ( f f ḡ m ḡ m as a single zero eigenalue weneer ( f/ ) S =, wic appens along te fold cure. In tat case te Jacobian is gien by ( ) f m ḡ m Terefore te eigenector for te zero eigenalue is gien by (, m) = (1, ) and te center manifold is approximately gien by m =. Te statement follows. ) S Remark 2 Tis local center manifold reduction near te fold cure is one of te classical global reduction steps (i.e., setting m = m ()) in te literature. Te global reduction is basically justified by te fact tat te dynamics away from te fold cures is slaed to te reduced flow of te two attracting brances of te critical manifold. Te center manifold reduction resembles an instantaneous approac of te gating ariable m to its equilibrium state m = m (V ). Te actiation speed of m is fast compared to tat of te oter two gating ariables (n,), but it is actually modest compared to te dynamics of te membrane potential. So we ae to expect quantitatie canges in te reduced 3D model (2.16) compared to te full HH model (2.13). Indeed, we obsere for te classical case (τ x = 1,x = m,n,) te onset of firing of action potentials in te full HH model for an injected current of I 6.3 wile in te 3D model (2.16) we obsere te onset already for I 4.. Tis earlier onset reflects te increased speed of actiation of te sodium cannels. Remark 3 Note tat we are arying te original applied current I, wic canges te dimensionless parameter Î = I/(k g Na ) in systems (2.13) and (2.16). Tis is done for easier comparison wit results on te original system (1.1). 8

9 Remark 4 Te standard 2D reduction, wic includes te center manifold (m = m ()), uses te relation =.8 n, but tis was obtained for 18.5 degrees celsius, using a different reduction approac (Rinzel 1985). Te best empirical linear fit to te silent pase part of an action potential from te original 4D system is = n, wile te best linear fit to te 3D system (2.16) is =.9 1.5n. Interestingly, te reduction using =.8 n restores te onset of action potentials to I 6.3. As far as we are aware, tis is coincidental. Te 2D critical manifold S of te 3D singularly perturbed system (2.16) is gien by (2.14) and sown in Figure 4. Clearly, te 1D fast ector field is tangent at folds, i.e. te eigenalue of te 1D layer problem is zero tere. Te folds represent saddle-node bifurcations of te layer problem. Te outer brances of S are attracting wile te inner branc is repelling. Te reduced 3D HH system (2.16) is a singularly perturbed system in a form suitable for a geometric analysis. 3 Geometric analysis of singularly perturbed systems Te cubic sape of te critical manifold S (2.14) allows system (2.16) to exibit relaxation oscillation type solutions. Tere is te possibility of (classical) relaxation oscillations as sown in Figure 2 (left), or more complicated dynamics like mixed mode oscillations sown in Figure 2 (rigt), just to name two possibilities. Te main difference in te dynamics between tese two cases occurs near te lower fold of te critical manifold, were te flow eiter jumps immediately to te upper branc and creates an action potential or stays longer near te fold and produces subtresold oscillations before jumping. In tis section, we sow ow one deduces tese oscillatory beaiours from properties of te singular limit systems obtained from (2.16). 3.1 Relaxation oscillations Relaxation oscillations in 3D systems wit 1 fast and 2 slow ariables and a structure like te reduced HH system (2.16) were studied by Szmolyan, Wecselberger (24), Brøns et al. (26) and Guckeneimer et al. (25), based on geometric singular perturbation teory (Fenicel 1979, Jones 1995, for an oeriew see Wecselberger 25b). Te basic assumption in te geometric singular perturbation analysis is tat te critical manifold is cubic saped, as is true for te HH model (2.16). Let S := {(,,n) R 3 : F(,,n) = }. Assumption 1 Te manifold S := {(,,n) S : [,1]} is cubic-saped, i.e. S = S a L S r L + S + a wit attracting upper and lower brances S ± a, S + a S a := {(,,n) S : F (,,n) < }, a repelling branc S r := {(,,n) S : F (,,n) > } and fold cures L ±, L + L := {(,,n) S : F (,,n) =, F (,,n) }. We would like to describe relaxation oscillations in teir singular limit. Note tat for sufficiently small alues of te perturbation parameter, < ε 1, fast jumps are executed near te lower fold cure. In te singular limit, we describe tese jumps as projections along te fast fibers of te layer problem onto te oter attracting branc of te critical manifold. After te jump, te trajectory follows te reduced flow until it reaces te oter fold cure, were it gets projected back along anoter fast fiber onto te first attracting branc of te critical manifold. Let P(L ± ) S a be te projection along te fast fibers of te fold cure L ± onto te opposite attracting branc S a. Definition 1 A singular periodic orbit Γ of system (2.16) is a piecewise smoot closed cure Γ = Γ a Γ f Γ + a Γ + f consisting of solutions Γ ± a S ± a of te reduced system connecting points of te projection-cures P(L ) S ± a and te fold cures L ±, were tese slow solutions are connected by fast fibers Γ ± f from L± to P(L ± ). Assumption 2 Tere exists a singular periodic orbit Γ for system (2.16). We sow in Section 4 tat tis assumption is usually fulfilled for sufficiently large injected current I. A sketc of a singular periodic orbit is sown in Figure 5. Te existence of suc a singular periodic orbit can be sown in te following way: 9

10 L + Γ S + a P(L ) S r P(L + ) L S a Fig. 5 Scematic illustration of a critical manifold S and singular periodic orbit Γ leading to classical relaxation oscillations. Bot fold points of te singular orbit Γ are jump points. Sow te existence of subsets N ± P(L ) wit te property tat all trajectories of te reduced flow wit initial conditions in N ± reac te fold cure L ± (in finite ). It follows tat te associated maps Π ± : N ± P(L ) L ± are well defined. If te return map Π := P Π + P Π : N P(L + ) is also well defined and as te property tat Π(N ) N ten, by Brouwer s fixed point teorem, te existence of a singular periodic orbit follows, i.e. a fixed point of te return map exists. Uniqueness of te fixed point would follow if Π is a contraction. Te key to finding singular periodic orbits Γ is to calculate te reduced flow on te critical manifold S and to find solutions Γ ± a. Recall tat, based on (2.14), S is gien as a grap n(,), [,1], R, along wic F(,n,) =. Tus, we define a projection of te reduced system onto te (,)-plane. Implicitly differentiating F(,n,) = wit respect to gies te relationsip F = (F H +F n N) and we obtain ( )(ḣ ) ( 1 = F H F n N + F H ). (3.18) Te equation for is singular along te fold cures, F =. Terefore we rescale to obtain te desingularized reduced flow on te critical manifold. Using ḣ, to represent differentiation wit respect to rescaled, tis system takes te form (ḣ ) ( ) F = H. (3.19) F n N + F H Tis system as te same pase portrait as te reduced system (3.18), but te orientation of trajectories is reersed on S r, were F >. Te local dynamics near te fold cures L ± can be completely understood from analysis of (3.19). Typically, fold points p L ± are jump points and are defined by te normal switcing condition (F n N + F H) p L ±. (3.2) Under tis condition te reduced flow (3.18) becomes unbounded along te fold cure L ±. Tus trajectories of system (2.16) reacing te icinity of L ± subsequently jump away from te fold. Tis jumping beaiour near L ± is part of te mecanism leading to (classical) relaxation oscillations in system (2.16) sown in Figure 2 (left). Te following assumptions guarantee te existence of suc (periodic) relaxation oscillations. Assumption 3 Te two fold points p L ± of te singular periodic orbit Γ are jump points, i.e. (3.2) is fulfilled. Assumption 4 Te singular periodic orbit is transersal to te cures P(L ) on S ± a. Teorem 2 (Szmolyan and Wecselberger 24) Gien system (2.16) under Assumptions 1-4, tere exists generically a periodic relaxation orbit for sufficiently small ε. 1

11 Tis teorem sows under wic conditions trains of classical action potentials can be found (Figure 2, left). Obiously, if te singular periodic orbit is obtained by te contraction mapping principle as described aboe ten te periodic relaxation orbit is a (local) attractor. If te periodic orbit follows from Brouwer s fixed point teorem, ten we also know tat tere exists a periodic relaxation orbit tat is a local attractor but additional periodic orbits could also exist. For more details on relaxation oscillations we refer to (Szmolyan and Wecselberger 24). 3.2 Excitable state Te approac described aboe for te calculation of a singular periodic relaxation oscillation assumes tat tere are no equilibrium points of te reduced flow on S ± a between te subsets N ± P(L ) and te fold cures L ±. If tere exists an equilibrium of te reduced flow, e.g. on te lower branc S a, ten tis equilibrium is usually stable and its basin of attraction includes a subset Ñ P(L + ). For suc initial conditions te map Π is not defined and we expect te system to be in an excitable state. Proposition 1 Consider system (2.16) under Assumption 1. If tere exists a stable equilibrium on te lower attracting branc S a tat is a local (global) attractor of te reduced flow, ten tere exists a local (global) attractor for sufficiently small ε. If te conditions described in Proposition 1 old, ten te system is said to be locally (globally) in an excitable state. Te strategy to sow excitability is as follows: Identify te subset Ñ P(L + ) tat lies in te basin of attraction of te equilibrium on te lower attracting branc. If Ñ = P(L + ), ten te equilibrium is a global attractor. Oterwise, take te complementary subset (Ñ ) c P(L + ) and ceck weter Π((Ñ ) c ) Ñ, i.e. weter te return map Π maps te complementary subset into te basin of attraction of te equilibrium. If tis condition olds, ten te equilibrium is a global attractor. If not, ten te equilibrium is just a local attractor and may co-exist wit (an)oter local attractor(s), e.g. singular periodic relaxation oscillations. Wic local attractor a trajectory will approac ten depends on te corresponding initial condition of system (2.16). 3.3 MMO s and canards MMO s consist of L large amplitude (relaxation) oscillations followed by s small amplitude (sub-tresold) oscillations, and te symbol L s is assigned to tis pattern. Te subtresold oscillations of suc MMO patterns can be explained by folded singularities of te reduced flow as described in (Milik 1998, Wecselberger 25a, Brøns et al. 26). Typically, a folded singularity is an isolated point p L ± wic iolates te normal switcing condition (3.2). Since F = on L ±, a folded singularity is an equilibrium of te desingularized flow (3.19). Definition 2 We call p L ± a folded node, folded saddle, or folded saddle-node if, as an equilibrium of (3.19), it is a node, a saddle, or a saddle-node. For MMO s to exist, folded nodes (or, in te limiting case, folded saddle-nodes) are required. A typical pase portrait of te reduced flow near a folded node is sown in Figure 6. Note tat tere exists a wole sector of solutions (sadowed region) tat is funnelled troug te folded node singularity to te repelling branc S r of te critical manifold. Solutions wit suc a property are called singular canards and are a direct consequence of te existence of a folded singularity. Te sector of singular canards is called te funnel of te folded node singularity. For a detailed introduction to canard solutions we refer to (Szmolyan and Wecselberger 21, Wecselberger 25a, Brøns et al. 26) and references terein. Te borders of te funnel are gien by te fold cure (F in Figure 6) and te so called primary strong canard. Tis primary strong canard is te solution of te reduced flow (3.19) tat corresponds to te unique strong eigendirection of te folded node singularity. All oter singular canards are tangent to te so called primary weak canard corresponding to te weak eigendirection of te folded node. Te following assumption is needed for te existence of MMO s as sown e.g. in Figure 2, rigt: Assumption 5 Te fold point p L of te singular periodic orbit Γ is a folded node (folded saddlenode) singularity, wile te fold point p + L + is a jump point. 11

12 A x B x F z F z C S r x F y z S a Fig. 6 Scematic illustration of te reduced flow near a folded node singularity. A) Te desingularized flow near a node singularity. Te bold cure corresponds to te strong stable eigendirection of te node wile te dased cure corresponds to te weak eigendirection of te node. Te fold F lies on te z-axis. B) Te reduced flow near te folded node singularity obtained from A by reersing te flow on S r (x > ). All trajectories on S a (x < ) witin te sadowed region are funelled troug te folded node singularity to S r (x > ). Tese trajectories are called singular canards. Te bold trajectory is called te primary strong canard wile te dased trajectory is called te primary weak canard. C) 3D representation of te reduced flow on te critical manifold. Teorem 3 (Brøns et al. 26) Suppose tat system (2.16) satisfies Assumptions 1-2,4-5. If te segment Γ a of te singular periodic orbit Γ is in te interior of te singular funnel, ten for sufficiently small ε, tere exists a stable periodic orbit of MMO type 1 s for some s >. Actually, it is possible to calculate te (maximal) number of small oscillations s of tis 1 s MMO pattern. Define µ := λ 1 /λ 2 < 1 as te ratio of te eigenalues λ 1/2 ( λ 1 λ 2 ) corresponding to te node singularity of te desingularized flow (3.19). Ten te number of maximal subtresold oscillations is gien by (Wecselbeger 25a) [ ] 1 µ s = s(µ) = (3.21) 2µ were te rigt and side denotes te greatest integer less tan or equal to (1 µ)/(2µ). Different MMO patterns L s wit L 1 and s < s can just be obtained under te ariation of an additional parameter in system (2.16) tat canges te global return mecanism. Teorem 4 (Brøns et al. 26) Suppose tat system (2.16) satisfies Assumptions 1-2,4-5. Assume tat tere exist a parameter β in system (2.16) and a alue β suc tat for β = β, te segment Γa of te singular periodic orbit Γ consists of a segment of te primary strong canard. Ten te following olds, proided ε is sufficiently small: For eac 1 s < s, tere exists an interal J s wit lengt of order O(ε 1 µ 2 ) suc tat if β J s, ten a stable 1 s MMO pattern exists. A sketc of a singular periodic orbit Γ leading to MMO s is gien in Figure 7. Te existence of suc a singular periodic orbit can be sown as follows: 12

13 L + Γ S + a P(L ) S r P(L + ) L S a Fig. 7 Critical manifold S and singular periodic orbit Γ leading to MMO s. Te fold point in te silent pase of te singular orbit Γ is a canard point (folded node/folded saddle-node singularity), wile te fold point in te actie pase is a jump point. Calculate te return map Π for a single initial condition in P(L + ) tat lies witin te basin of attraction of te folded node resp. folded saddle-node singularity (te funnel) and sow tat tis initial condition is mapped back into te funnel by Π. Tis guarantees immediately te existence of a unique singular periodic orbit fulfilling Assumption 5, since all trajectories witin te singular funnel are contracted to te folded singularity. Te small (subtresold) oscillations obsered in a MMO pattern occur in te neigbourood of te folded node singularity. Te reason can be rougly explained as follows: Note tat existence and uniqueness of solutions of system (2.16) for small ε is guaranteed. In te singular limit, oweer, uniqueness is lost for te reduced flow along te fold cure. In particular, at te folded node singularity we ae a continuum of solutions, te singular canards, passing from te attracting branc S a troug one single point, te folded node singularity, to te repelling branc S r as described aboe. Wecselberger (25a) sowed tat a discrete number of canard solutions persist under small perturbations < ε 1, i.e. all tese canard solutions connect from te attracting branc S a,ε to te repelling branc S r,ε. Te existence of inariant manifolds S a,ε and S r,ε away from te fold, wic are O(ε) perturbations of S a and S r, are guaranteed by Fenicel teory (Fenicel 1979, Jones 1995). Furtermore, solutions of (2.16) on S a,ε and S r,ε will approximately follow te reduced flow on S a and S r. By a pure topological argument it follows tat te only way for canard solutions to connect tese manifolds S a,ε and S r,ε witout iolating te uniqueness of oter solutions nearby is gien by rotations of te manifolds S a,ε and S r,ε near te fold cure. For a more detailed explanation of tis geometric structure we refer to (Guckeneimer and Haiduc 25, Wecselberger 25a). Teorem 4 states tat under te ariation of an additional parameter β in system (2.16), wic canges te global return mecanism, MMO patterns of type 1 s wit 1 s < s are realized. Combinations of adjacent MMO patterns 1 s and 1 s +1 are usually obsered in te transition from one stable MMO pattern 1 s +1 to anoter stable MMO pattern 1 s under ariation of β. Witin te transition from a 1 1 MMO pattern to relaxation oscillations (1 MMO pattern) one may also obsere L 1 patterns, L > 1, as well as combinations of patterns. More complex L s MMO patterns as well as related combinations are found for µ, suc tat s(µ), and/or for larger ε. Tese complex patterns are not well coered by te current teory and te deelopment of suc a teory will be te focus of future work. 4 Analysis of te 3D HH system (2.16) In te following we sow under wic conditions on te parameters of system (2.16) we obtain eiter an excitable system, relaxation oscillations or MMO s. 13

14 4.1 Singularities of te reduced flow To analyse te reduced flow (3.18) on te critical manifold S, we calculate te desingularized system (3.19) as described in Section 3 for te 3D HH system (2.16) and ceck under wic conditions we find singularities. Te rigt and side functions of (3.19) are gien by F = (m 3 () + ḡ k n 4 (,) + ḡ l + 3m 2 ()( ĒNa)m ()) F = (m ()( ĒNa)) F n = (4ḡ k n 3 (,)( ĒK)) and n(,) is defined in (2.14). Recall tat te folds L ± are defined by F =. Te functions H and N depend on te constants τ and τ n. Eac of te functions F, F and F n depends on te parameter I. Tere are two classes of equilibria of system (3.19). Te first occurs were H =, corresponding to = (), and N =, corresponding to n(,) = n (), were te function n(,) depends on I but not on τ and τ n. Putting tis togeter yields te condition n(, ()) = n (), wic as a solution tat is independent of τ and τ n but depends on I, called (I) below. Te second class of equilibria, folded singularities, arises were F = (along te fold line) and F n N + F H =. Tese points depend on τ, τ n and I. Tere are no equilibria wit H = and N, because F n. Remark 5 Bot folded and regular singularities are equilibrium points of system (3.19), but only te latter are actually equilibrium points of (3.18), as pointed out in Section 3. A bifurcation occurs as I aries, for fixed τ, τ n, wen te te cure of equilibria (I) intersects te fold-line F =. Tis occurs at te special alue of I at wic F ((I),n((I), ((I))), ((I))) =. But since te functions (I), (), n(,) are independent of τ and τ n, te location of tis bifurcation is independent of τ and τ n as well. Parameter continuation by using XPPAUT (Ermentrout 22) sows tat tis bifurcation alue is gien by I c 4.8. Figures 8-9 sow tis bifurcation for different alues of τ and τ n, wit I taken as te bifurcation parameter. Clearly, te bifurcation point is independent of τ and τ n as claimed, and te bifurcation appens on te fold cure. We will discuss furter details about tese diagrams below. Proposition 2 Generically, te reduced system (3.18) possesses, for I = I c 4.8, a folded saddle-node on te fold cure L (independent of τ and τ n ). For nearby alues I < I c, system (3.18) possesses a folded saddle and a stable node singularity. For nearby alues I > I c, system (3.18) possesses a folded node and a saddle singularity. Terefore, te bifurcation of te ordinary and te folded singularity near I = I c resembles a transcritical bifurcation. Remark 6 Tere exist two different types of folded saddle-node (FSN) singularities. Te FSN type I corresponds to a true saddle-node bifurcation of folded singularities, wile a FSN type II corresponds to a transcritical bifurcation of a folded singularity wit an ordinary singularity (Szmolyan and Wecselberger 21). Terefore, te folded singularity at I = I c described in Proposition 2 is a FSN type II. In general, te proposition states tat tere exists a folded node singularity for certain parameter alues. In tese cases MMO s are possible, depending on te global return mecanism, as described in Section 3. In te following we will split te analysis into 3 different cases: te classical case (τ = 1,τ n = 1), te case (τ > 1,τ n = 1) and te case (τ = 1,τ n > 1). Classical case (τ = 1,τ n = 1) In tat case, system (3.18) as a folded saddle for < I < I c, a folded saddle-node type II for I = I c, and a folded node for I > I c on te lower attracting branc (in te silent pase). Furtermore, te lower attracting branc Sa as a stable node for < I < I c, wic bifurcates ia te folded saddle-node singularity to te repelling middle branc S r, were it becomes a saddle for I > I c (see Proposition 2). Figure 1 illustrates tese singularities using te nullclines of te desingularized system (3.19) wit I = 1 < I c and I = 1 > I c, respectiely. In tese as well as analogous pase plane pictures below, te 14

15 .59.6 saddles.61 folded nodes.62 folded saddles nodes I Fig. 8 Bifurcation of equilibria (black: regular, oter colors: folded) for te reduced flow for different alues of τ (τ = 1 (red), τ = 15 (dark blue)). Solid cures are asymptotically stable (nodes, folded nodes), wile dased are unstable (saddles, folded saddles). Furter details are gien in te text τ n =3 τ n =4.75 τ n = τ n = I Fig. 9 Bifurcation of equilibria (black: regular, all oter colors: folded) for te reduced flow for different alues of τ n. Solid cures are asymptotically stable (nodes), wile dased are unstable (saddles). Furter details are gien in te text. cyan cures are te solutions of F H =, one corresponding to F = (i.e., te lower fold L ) and te oter corresponding to H =, wile te black cure consists of solutions of F n N + F H =. Folded equilibria are indicated by red symbols, wile regular equilibria are indicated by blue symbols. Finally, nodes are marked wit circles, wile saddles are marked wit triangles. So, in Figure 1 wit I = 1 (left), we see a stable node and a folded saddle, wile in Figure 1 wit I = 1 (rigt), we see a folded node and a saddle. Remark 7 Tere exist two oter folded singularities of te reduced flow wic ae no influence on te dynamics of te classical case, but will become more important in te oter two cases (τ > 1,τ n = 1) and (τ = 1,τ n > 1). First, tere is anoter folded saddle in te silent pase for 1. Tis folded saddle will moe to te rigt for τ n > 1 (see corresponding case study (τ = 1,τ n > 1) below). Second, tere 15

16 Fig. 1 Pase plane for system (3.19) for te classical case (τ = 1, τ n = 1) wit I = 1 < I c (left) and I = 1 > I c (rigt) τ Fig. 11 Te coordinate for te folded focus of system (3.19) depends on τ (solid cure). Te dotted lines demarcate = 1 and τ = 1.4 and illustrate tat te focus becomes non-pysiological for τ < τ f 1.4. is also a folded focus in te actie pase for > 1. Tis folded focus will moe into te pysiologically significant range < < 1 for τ > 1 (see corresponding case study (τ > 1,τ n = 1) below). Case (τ > 1,τ n = 1) Tis case is similar to te classical case. Again, system (3.18) as a folded saddle for < I < I c, a folded saddle-node type II for I = I c, and a folded node for I > I c on te lower attracting branc (in te silent pase), for eac fixed τ > 1. As in te classical case, te lower attracting branc S a as a stable node for < I < I c, wic bifurcates ia te folded saddle-node singularity to te repelling middle branc S r, were it becomes a saddle for I > I c (see Proposition 2). In addition to sowing te cure of regular equilibria generated by arying I, Figure 8 sows examples of ow te folded singularities depend on I for τ = 1 and τ = 15, te former of wic is closer to te cure of regular equilibria, illustrating tat teir dependence on τ is quite weak. Te only difference between τ = 1 and τ > 1 wit respect to singularities is tat tere exists a folded singularity of focus type in te actie pase for τ > τ f witin te pysiological significant domain < < 1. Figure 11 sows te alue of te folded focus for different alues of τ. At τ f 1.4, te alue is approximately one, wile it is less tan 1 for τ > τ f. Folded foci do not support canard solutions. Te main influence of folded foci on te reduced flow is tat tey direct te flow a certain way. 16

17 We sall see in te next two subsections tat te folded focus as no significant impact on te solutions tat we are studying. Case (τ = 1,τ n > 1) As can be seen in Figure 9, te bifurcation diagram for system (3.19), wit bifurcation parameter I, depends muc more strongly on τ n tan on τ. Te black cure in Figure 9, wic switces from solid (for I < I c 4.8) to dased (for I > I c ), is te cure of regular equilibria, wic is independent of τ n, as noted earlier. As τ n increases from 1, te cure of folded equilibria deelops two folds, wic correspond to saddle-node bifurcations of folded equilibria in te parameter I for eac fixed τ n. Figure 9 sows examples of te cures of folded equilibria for I > for a ariety of alues of τ n, including τ n = 1 (red), wic is sown for comparison. We define seeral critical alues of τ n. First, tere exists a alue τn c 4.75 suc tat for all τ n < τn, c tere is a saddle-node bifurcation of folded equilibria at some I = I + SN > I c, lying aboe te cure of regular equilibria (e.g. τ n = 3 in Figure 9). Examples of te corresponding pase planes of (3.19) are sown in Figure 12 for I = 1 < I c, I + SN > I = 7 > I c and I = 1 > I + SN, wic illustrate te following: Tere exist tree singularities in te domain of interest, two folded saddles and one (ordinary) node, for I = 1. Te folded saddle to te rigt of te node bifurcates ia a transcritical bifurcation at I = I c to a folded node to te left of te saddle. For I = I + SN te folded node and te oter folded saddle (to te left) anniilate eac oter in a saddle-node bifurcation and we are only left wit a saddle singularity on te repelling side of te critical manifold. Note tat te folded saddle-node singularity at I = I c is of type II, wile te folded saddle-node singularity at I = I + SN is of type I. For τn c < τ n < τn (τn 1.5), te saddle-node bifurcation at I + SN > I c lies below te cure of regular equilibria, as sown for τ n = 7 in Figure 9. Figure 13 sows pase planes for τn > τ n = 7 > τn c for I = 1 < I c, for I = 6 and I = 7, bot aboe I c and below I + SN, and for I = 1 > I+ SN. In tis case, te folded saddle to te left of te node bifurcates ia a transcritical bifurcation at I = I c to a folded node to te rigt of te saddle. As I increases towards I + SN, a folded saddle from te rigt ( > 1) approaces te folded node. Tey finally anniilate eac oter for I = I + SN in a saddle-node bifurcation. Again, we are left wit only a saddle singularity on te repelling side of te critical manifold for I > I + SN. In bot cases, τ n < τ c n and τ n > τ n > τ c n, a folded node emerges for I > I c ia a transcritical bifurcation and persists up to te saddle-node bifurcation at I = I + SN. In te limit τ n = τ c n tese two bifurcations merge to a single pitcfork bifurcation at I = I c. Tis yields a cure of folded saddle equilibria, sown as te τ n = 4.75 cure in Figure 9. Te alue τ n = τ c n is te unique alue for wic te bifurcation at I = I c is not transcritical and does not create a cure of folded nodes. As can be seen in Figure 9 for τ n = 7, tere is also a lower (wit respect to I) saddle-node bifurcation of folded equilibria, wit a folded node at relatiely negatie alues for I aboe tis bifurcation. Let I SN denote te I alue were tis lower saddle-node bifurcation occurs. In fact, te interal of τ n alues oer wic tis lower saddle-node exists is gien by (τn,τn), were τn 4.2. Howeer, te saddle-node exists for I SN < for τ n < τ n < τn 5.75, wic explains wy tis bifurcation is not isible in Figure 9 for τn c = Figure 14 sows bot te upper (at I = I + SN, solid) and lower (at I = I SN, dased) cures of saddle-node bifurcation points, as functions of τ n. Note tat te two bifurcations come togeter in a cusp at τn, wic is approximately gien by τn 1.5. Furter, as τ n decreases, te lower saddle-node occurs at progressiely larger and ence becomes pysiologically irreleant. In general, te lower folded nodes ae no influence on te dynamics witin te pysiological releant boundaries. Terefore, we will not consider tem in te following analysis of relaxation oscillations. Note also tat te upper saddle-node occurs at progressiely larger alues I = I + SN as τ n decreases towards τ n = 1, wic is wy it is not obsered in te τ n = 1 bifurcation cures in Figures 8-9. Finally, if τ n > τn, aboe te cusp of saddle-node bifurcations, ten te bifurcation diagram wit respect to I becomes more like te classical case again, wit a single transcritical bifurcation at I = I c (witin te pysiological releant domain). To summarize: In all tree cases tere exists a node singularity for I < I c on te lower attracting branc (silent pase). In te cases (τ 1,τ n = 1) tere exists on L a (pysiologically releant) folded node 17

18 Fig. 12 Pase planes for system (3.19) for τ = 1, τ n = 3: I = 1 < I c (upper left), I + SN rigt) and I = 1 > I + SN (lower). > I = 7 > Ic (upper singularity for I > I c. In te case (τ = 1,τ n > 1) tere exists on L a (pysiologically releant) folded node singularity for I + SN > I > I c, were I + SN approaces I c in te (degenerate) limit τ n τ c n, but I + SN > I c for τ n τ c n. 4.2 Transersality of reduced flow at P(L ± ) To apply Teorems 2-4 (relaxation oscillations or MMO s) we ae to sow tat te associated singular periodic orbit is transersal to te projections of te fold cures P(L ± ) (Assumption 4). Instead of erifying Assumption 4 for eac specific example, we gie eidence tat te reduced flow is transersal to P(L ± ) and directed towards te fold cures in te wole domain of interest. Tis more general transersality argument gies important insigt into te nature of te reduced flow, since it sows tat te reduced flow (Π,Π + ) and associated projections presere te orientation of trajectories. Tis orientation preseration property will elp us to calculate te return map Π defined in Section 3 in order to find singular periodic orbits. Proposition 3 Te flow of te reduced system (3.18) is transersal to P(L ± ) and directed towards L witin te pysiologically releant domain of ( < < 1 but not 1) and te parameter range under study (I,τ 1,τ n 1). Moreoer, in tis parameter range, te return map Π, were it is defined, preseres te orientation of trajectories, in te sense tat if p 1 = ( 1, 1 ),p 2 = ( 2, 2 ) P(L + ) wit 1 < 2, and Π(p 1 ) = (ĥ1, ˆ 1 ),Π(p 2 ) = (ĥ2, ˆ 2 ), ten ĥ1 < ĥ2. As wit te propositions in Section 4.3 below, we use numerical obserations to support tis proposition. In te discussion below, we omit ery small alues of ( 1) and consider te rest of te pysiological range of ( < < 1). All of te statements we make old independent of te alues of I,τ,τ n, witin 18