Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of FitzHugh-Nagumo type. NJIT CAMS Technical Report

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1 Canard-like explosion of limit cycles in to-dimensional pieceise-linear models of FitzHugh-Nagumo type. NJIT CAMS Technical Report 2-4 Horacio G. Rotstein, Stephen Coombes 2, Ana Maria Gheorghe 2, Department of Mathematical Sciences Ne Jersey Institute of Technology Neark, NJ 72, USA 2 School of Mathematical Sciences Uniersity of Nottingham Nottingham NG7 2RD, UK Abstract We inestigate the mechanism of abrupt transition beteen small and large amplitude oscillations in fast-slo pieceise-linear (PWL) models of FitzHugh-Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes) respectiely. The minimal model that shos such phenomenon has a cubic-like nullcline (for the fast equation) ith to or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models ith only one linear piece in the middle branch or a discontinuity beteen the left and right branches (McKean model) sho a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model, e inestigate the bifurcation structure, e describe the mechanism that leads to the abrupt, canard-like transition beteen subthreshold oscillations and spikes, and e proide an Also, Center for Applied Mathematics and Statistics, Ne Jersey Institute of Technology. horacio@njit.edu

2 analytical ay of predicting the amplitude regime of a gien limit cycle trajectory hich includes the approximation of the canard critical control parameter. We extend our results to models ith a larger number of linear pieces. Our results for PWL-FHN type models are consistent ith similar results for smooth FHN type models. In addition, e deelop tools that are amenable for the inestigation of a ariety of related, and more complex, problems including forced, stochastic and coupled oscillators. Introduction In a to-dimensional relaxation oscillator, the canard phenomenon (or canard explosion) refers to the abrupt increase in the amplitude of the limit cycle created in a Hopf bifurcation as a control parameter crosses a ery small critical range hich is exponentially small in the parameter defining the slo time scale [, 2, 3, 4, 5, 6, 7]. Depending on hether the Hopf bifurcation is supercritical or subcritical, the small amplitude limit cycles are either stable or unstable respectiely. The large amplitude, relaxation-type limit cycles are alays stable. We illustrate the canard phenomenon for both cases in Fig. for a smooth, fast-slo system of FitzHugh-Nagumo (FHN) type of the form { = f, = ǫ [α λ ], () here α, λ and ǫ are constants, < ǫ, α >, and f is a cubic-like function. We used ǫ =. and the prototypical cubic function f = h 3 +a 2 ith h = 2 and a = 3 hose maximum and minimum occur at (, ) and (, ) respectiely. This canonical choice ensures that the large amplitude oscillations are O(). The parameters α and λ control the slope of the -nullcline N = α λ and its position relatie to the -nullcline N = f respectiely. In the context of neuroscience, the ariables and correspond to a dimensionless oltage and a recoery ariable respectiely, and the parameter λ can be thought of as a dimensionless externally applied (DC) current, after redefining +λ. In Appendix A e proide a technical discussion about the Hopf bifurcation and canard phenomenon for these systems. For λ = the - and -nullclines in Fig. (N and N respectiely) intersect at the minimum of the cubic nullcline N. For the parameters e used, the Hopf bifurcation is supercritical (subcritical) for α > 3 (α < 3) (see Appendix A). Consequently, the Hopf bifurcation is supercritical in Figs. -A (α = 4) and subcritical in Figs. -B (α = 2). The left panels sho sketches of the graphs of the limit cycle amplitude ersus the control parameter λ (A = indicates a fixed point). In both cases, the Hopf bifurcation occurs as λ crosses λ H = O(ǫ) > (see Appendix A). For alues of λ < λ H (λ > λ H ) the fixed-points are stable (unstable). In the supercritical case (Fig. -A), the fixed-point is the only attractor for λ < λ H and the system is excitable [8, 9,, ]. In the subcritical case (Fig. -B), there is bistability for a range of alues of λ here large amplitude oscillations and a fixed point coexist. The small amplitude limit cycles created at λ = λ H are stable in the supercritical case (Fig. -A) and unstable in the subcritical case (Fig. -B). Their amplitude increases sloly for alues of λ close to λ H. In the supercritical case (Fig. -A) this happens as λ increases. As λ crosses from 2

3 A supercritical α = 4 λ =.78 α = 4 λ =.79 stable unstable.4.2 N N Tr.4.2 N N Tr.8.8 A λ H λ c λ B subcritical α = 2 λ =.27 α = 2 λ =.28 stable unstable.4.2 N N Tr.4.2 N N Tr.8.8 A λ c λ H λ Figure : Supercritial and subcritical canard phenomenon (canard explosion) for the FHN model () ith f = , ǫ =. and arious alues of α and λ. Left panels: bifurcation diagrams. A = corresponds to fixed-points, A > corresponds to the amplitude of limit cycles, λ H and λ c indicate the Hopf bifurcation and canard critical points respectiely. Middle and right panels: phase-planes for representatie alues of λ. N and N represent the - and -nullclines respectiely and Tr represents the trajectory. A Supercritical case: λ H.25 and λ c.78. Small amplitude limit cycles are stable. B Subcritical case: λ H.35 and λ c.278. Small amplitude limit cycles are unstable. The trajectory shon in the middle panel displays damped oscillations and conerges to the stable fixed-point. 3

4 λ =.78 to λ =.79 (middle to right panels), the limit cycle explodes. The small(exponentially small in ǫ) range of alues of the control parameter λ oer hich this canard explosion occurs can be approximated by the canard critical alue λ c =.782. In the subcritical case (Fig. -B), the unstable small amplitude limit cycle (not shon) explodes as λ decreases belo λ c =.278. The alues of λ c ere computed using the formulas gien in Appendix A. One feature of the canard phenomenon is that trajectories eole in close icinities of the unstable (middle) branch of the - nullcline for a significant amount of time before moing either to the left or to the right to generate small and large amplitude oscillations respectiely (see Fig. -A for example). The canard phenomenon for smooth to-dimensional systems has been inestigated by arious authors. It as discoered by Benoit et al. [4] for the Van der Pol (VDP) oscillator here the - nullcline is a ertical line (see Appendix A.3). They used non-standard analysis techniques. Eckhaus [2] and Baer et al. [3] inestigated the canard phenomenon using asymptotic techniques. In particular they found expressions for λ c for VDP- and FHN-type equations. The canard phenomenon has also been studied by Dumortier and Roussarie [] and by Krupa and Szmolyan [5, 6] for more general to-dimensional slo-fast systems. These results assumed a set of non-degeneracy conditions hich in particular imply that the -nullcline has to be locally parabolic at its minimum. For future use, e sketch some of these results in Appendix A. The goal of this paper is to inestigate the mechanism leading to the canard explosion in todimensional, pieceise linear (PWL) systems of the form () ith f substituted by a cubic-like PWL function. PWL caricatures of nonlinear models hae been fruitfully used in a number of different branches of the applied sciences ranging from biology to mechanics as a ay to proide ne insights into the dynamics of smooth models or as a conenient ay to explicitely analyze them hen a general set of mathematical techniques is not aailable. For example in neuroscience the McKean model [2] may be regarded as a ariant of the FitzHugh-Nagumo model [3] that proides a planar model of an excitable cell in hich the dynamics is broken into simpler linear pieces. This has alloed for a number of results about the existence and stability of periodic orbits and traeling aes to be obtained [4, 5, 6, 7, 8, 9] and insight into action potential generation and response to stimulation to be obtained [2]. An extension of this approach to other single neuron models, including the Morris-Lecar model, has recently been pursued by Tonnelier and Gerstner [2] and Coombes [22]. In a mechanical setting PWL modeling has helped to shed light on the motion of rocking blocks [23] and models of suspension bridges [24], and indeed impacting systems in general (see the recent book by di Bernardo et al. [25]). Many of the techniques for analysing mechanical systems hae been taken oer to the study of oscillatory electronic circuits and in particular the analysis of non-smooth bifurcation such as those of border-collision type [26]. In the systems biology context techniques originally deeloped by Filippo [27] (for ordinary differential equations ith a discontinuous ector field) hae been applied to gene regulatory netorks ith sitch-like interactions (arising in the limit of a steep sigmoidal nonlinearity) [28, 29]. From a more mathematical perspectie PWL systems hae alloed for the analysis of limit cycles [3] and their bifurcations [3, 32] and hae been shon numerically to support canard solutions [33]. The inestigation of the dynamics of PWL oscillators, and in particular the abrupt transition beteen limit cycle amplitude regimes proides complementary information to preious studies on 4

5 analogous smooth systems. These studies hae focused on the existence of the so called canard solutions (including the so called maximal canards hich exist for alues of the control parameter in the small transition range beteen small and large amplitude limit cycles) and the conditions under hich canard explosions occur, and less attention has been paid to the actual eolution of trajectories for alues of λ near the critical alue for the abrupt transition (λ c ). Understanding this is important not only for the understanding of the dynamics of single oscillators but also for the understanding of the effects that external inputs (e.g., pulsatile, sinusoidal, noisy and synaptic-like) exert on single oscillators and the dynamics of oscillatory netorks. For small enough alues of ǫ, trajectories corresponding to single oscillators can be approximated oer most of their period either by the stable branches of the -nullclines, hich approximate the slo manifolds, or by the horizontal fast fibers. These phases of the oscillation are typically robust to external perturbations. By contrast, perturbations in a range of phases in hich the trajectory is eoling in a close icinity of the unstable branch of the -nullcline can lead to disparate effects here the resulting amplitude regime (small or large) of the perturbed system is different from that of the unperturbed one. More specifically, using neuroscience terminology, spiking can be created or suppressed, or delayed / adanced by a significant amount of time ith respect to the unperturbed oscillation. The canard critical alue λ c (see Appendix A) proides information on hether the limit cycle trajectory of an unperturbed oscillator crosses the unstable branch of the -nullcline (small amplitude oscillations) or not (large amplitude oscillations). Hoeer, hen the system is perturbed the relatie position beteen nullclines changes and so do the fixed-point and the canard-critical alue. For time-dependent perturbations, the effects of these changes in eoling trajectories are difficult to predict as are the resulting amplitude regimes of perturbed trajectories. In particular, the canard critical alue, that becomes time-dependent, is no longer an appropriate tool for this purpose. Asymptotic techniques can be used to proide a description of trajectories for small alues of ǫ. Hoeer, een for single oscillators [2, 3] their description becomes complicated and proides little intuition on their eolution. In addition, as the alue of ǫ increases to intermediate alues still smaller than O() these approximations become less accurate. In Section 2, e introduce some notation on PWL systems of FHN type, here the function f in () is substituted by a cubic-like PWL function, and e oerie, for future use, the solutions to the linear regimes corresponding to each one of the linear pieces in a PWL system. In Section 3, e inestigate the dynamics of PWL systems haing to linear pieces in the middle branch of the cubiclike function and one linear piece in the other to. We refer to them as PWL,2, systems. These are the simplest PWL models for hich canard explosions occur. In linear systems ith only one linear piece in the middle branch, limit cycles hae a single (large) amplitude regime. We first present the bifurcation structure of PWL,2, systems. Then, e inestigate the mechanism leading to the canardlike explosion of limit cycles. Although solutions for PWL systems can be computed analytically, the insight they proide into the underlying dynamics is poor. We use dynamical systems tools to explain these dynamics and anser arious releant questions about the abrupt transition beteen limit cycle amplitude regimes. In addition, e proide a ay of calculating analytical approximations to the canard critical alue λ c and discuss the accuracy of these approximations and the dependence of λ c on other releant model parameters. Finally, e sho that, although the amplitude of the small limit 5

6 cycle changes ith the model parameters, the period is independent of λ. In Section 4, e extend our results to PWL systems haing three linear pieces in the middle branch and one linear piece in the other to. We discuss the implications of our results for more general and complex models in Section 5. 2 Pieceise linear models of FitzHugh-Nagumo type We consider the folloing pieceise linear (PWL) models of FitzHugh-Nagumo (FHN) type here the cubic-like smooth function f in system () is substituted by a PWL caricature f pl : { = f pl, = ǫ [α λ ]. (2) We use prototypical PWL functions f pl ith minimum ( min, min ) = (,) and maximum ( max, max ) = (,). As in the smooth case discussed aboe, this choice ensures that large amplitude oscillations are O(). Each branch of f pl consists of one or more linear pieces L j indexed by j. The - and -nullclines of model (2) are gien by respectiely. N = f pl and N = α λ (3) 2. Construction of pieceise linear functions We refer to a cubic-like PWL function haing M, M l and M r linear pieces in the middle, left and right branches respectiely as a PWL Ml,M,M r function. If the number of linear pieces in all branches of f is equal (M l = M r = M), then e refer to this function as PWL M. We use an analogous terminology for the corresponding models. We illustrate this in Fig. 2 for a PWL,2, model In the process of building the PWL function f pl e use the folloing notation: We partition the interal [ min, max ] into M segments ith end-points j, for j =,...,M, here = min, M = max and j < j. We call L j (j =,...,M) the linear piece that has endpoints ( j, j ) and ( j, j ). We call η j, j =,...,M the slope of the linear piece L j. Gien the alues,..., M and the first M slopes (η,...,η M ) e calculate j = η j ( j j )+ j for j =,...,M. The slope of the last piece in the middle branch is gien by η M = ( M M )/( M M ). We proceed in a similar manner ith the left and right branches. In the left branch j = M l,..., and in the right branch, j = M,...,M +M r. Alternatiely, e use the notation L l,j and L r,j as illustrated in Fig. 2. 6

7 2.2 PWL approximations of smooth ector fields In order to link the cubic-like PWL functions f pl in eq. (2) to the corresponding smooth functions f in eq. () e use M = M l = M r and proceed as follos: We call = ( max min )/M and j = j for j = M l,...,m +M r ith M = M l = M r. We call j = f( j ) for j = M l,...,m r. We call η j = ( j j )/( j j ) for j = M l +,...,M +M r. As M increases, f pl approaches the corresponding smooth function f. Fig. 3 illustrates the approximation of the solutions to the corresponding PWL M systems (2) to the smooth system () as M increases. In both Figs. 3-A and -B, the top-left panels sho the phase-plane diagrams corresponding to the smooth ector fields for to different alues of λ hich are close to the canard critical point λ c. The parameters α and ǫ are as in Fig. -A (supercritical case). In Fig. 3-A, λ > λ c and the smooth system exhibit large amplitude oscillations. In Fig. 3-B, λ < λ c and the smooth system exhibits small amplitude oscillations The remaining panels in both Figs. 3-A and -B correspond to the PWL approximations for a number of linear pieces that increases from the top-middle to the bottom-right panels. As M increases, the dynamics of the PWL system approach the dynamics of the corresponding smooth systems. For λ > λ c (Fig. 3-A), the limit cycle is in the large amplitude regime for all alues of M. In contrast, for λ < λ c (Fig. 3-B), the limit cycle is in the large amplitude regime for small alues of M ( 9) and the transition to the small amplitude oscillations regime occurs for M =. We explain the corresponding mechanism in the folloing sections. 2.3 Dynamics of the basic linear components In order to inestigate the dynamics of a PWL system such as (2) it is conenient to consider it diided into a number of linear regimes R j associated to the linear pieces L j. A linear regime R j is a strip in the phase-plane bounded by the ertical lines crossing the end-points of the linear piece L j, = ˆ j and = ˆ j respectiely (see Fig. 2). More specifically, a point (,) in phase-plane belongs to R j if ˆ j ˆ j here ˆ j and ˆ j are the -coordinates of the left and right endpoints of the linear piece L j. Note that to regimes corresponding to linear pieces ith a joint endpoint interesect at the ertical line containing this point. The dynamics of each linear regime R j are goerned by a linear system of the form { = η j ( ˆ j )+ŵ j, = ǫ [α λ ], here (ˆ j,ŵ j ) and η j are the left end-point and slope of the corresponding linear piece L j respectiely. The initial conditions,j and,j in each regime are equal to the alues of and at the end of the preious one. More specifically, if R i is the regime prior to R j (either R j or R j+ ), and (4) 7

8 the solution in R i arries at i (-coordinate of the joint point beteen L i and L j ) at a time t i, then the initial conditions for system (4) corresponding to the linear regime R j are = i and = (t i ). The solutions to (4) can be calculated using standard methods [34]. We present them belo and illustrate them in Section 3 tied to our explanation of the canard explosion. For simplicity, e drop the indices from the endpoints (ˆ,ŵ), the slope η and the initial conditions (, ). The fixed point for system (4) is gien by = The eigenalues are gien by λ ηˆ +ŵ α η = λη αηˆ +αŵ. (5) α η r,2 = η ǫ ± (η +ǫ) 2 4ǫα. (6) 2 The fixed-point (, ) is stable (unstable) for η < ǫ (η > ǫ). There are to critical slopes gien by η + cr = ǫ+2 ǫα and η cr = ǫ 2 ǫα (7) such that the eigenalues are complex for η (η + cr,η cr) and real otherise; i.e., the fixed point (, ) is a focus for η (η cr,η + cr) and a node otherise. We sho the corresponding stability diagram for ǫ =. in Fig. 4-A. For a fixed alue of α, as the slope of the linear piece η increases, the fixed point (, ) changes from a stable node through stable and unstable spirals, to an unstable node. The horizontal line η = ǫ corresponds to the Hopf bifurcation points. As ǫ increases ( from red to blue in Fig. 4-B) the Hopf bifurcation line moes upards and the interal of alues of η (range of slopes of the linear pieces) corresponding to the spiraling behaior idens. The solution to system (4) for real eigenalues r and r 2 ((η +ǫ) 2 4ǫα > ) is gien by ith [ ] = c [ r 2 +ǫ ] e r t + c 2 [ r +ǫ c = ( )(r +ǫ) ( ) r r 2 and c 2 = ( )(r 2 +ǫ)+( ) r r 2. (9) In Fig. 4-C e sho graphs of the real roots (r and r 2 ) as a function of the slope of the linear pieces η for arious alues of α and ǫ =.. For η < (left and right branches of the cubic-like PWL function) the corresponding fixed-point is stable (both eigenalues are negatie) hile for η > (middle branch of the cubic-like PWL function) the fixed-point is unstable (r 2 may become negatie as η increases for small enough alues of α). For small alues of ǫ and large enough alues of η there is a time scale separation (fast-slo system) hich dissappears as η decreases or α increases. The solution to system (5) for complex eigenalues r and r 2 ((η +ǫ) 2 4ǫα < ) is gien by [ ] = c [( β ) cosµt + ( µ ) ] e r 2t sinµt ] + [ e rt + ] (8) 8

9 + c 2 [( β ) sinµt ( µ ) cosµt ] e rt + [ ] () ith here and c =, µ = c 2 = ( )(η +ǫ) 2( ) 2µ () 4αǫ (η +ǫ) 2 / 2. (2) r = η ǫ 2, β = η +ǫ 2. (3) In Fig. 4-D e sho graphs of the natural frequency µ as a function of the slope of the linear piece for arious alues of α and ǫ =.. 3 The dynamics of PWL,2, models of FHN-type Here e inestigate the mechanisms of generation of both small and large amplitude limit cycles and the abrupt transition beteen them (canard-like explosion) in PWL,2, models of the form (2) here the cubic-like function f pl is as illustrated in Fig. 2. We sho an example in Fig. 5 (panels A and B). The middle branch of f pl has to linear pieces, L and L 2, ith slopes η and η 2 respectiely. We considered arious representatie alues of the lengths and slopes of these linear pieces. The maximum and minimum of f pl are located at (,) and (,) respectiely. As mentioned aboe, this canonical choice ensures that large amplitude oscillations are O(). The left and right branches haeonelinearpieceeach, L l, andl r,, ithslopesη l, andη r, respectiely. Weusedη l, = η r, =. As e ill see, the number of pieces and the choice of slopes in both the left and right branches hae little effect on the canard-like explosion for small enough alues of ǫ. In a PWL,2, model, the to linear pieces in the middle branch (L and L 2 ) join at the point (ˆ,ŵ ). For the example shon in Fig. 2, (ˆ,ŵ ) = (.3,.9). From eqs. (5), if the intersection beteen the to nullclines, N = α λ and N = f pl, occurs on the linear piece L, then the fixed point (, ) is gien by = λ α η and = λη α η (4) here η, ˆ and ŵ hae been substituted by η, ˆ = and ŵ = in eqs. (5). For λ =, (, ) = (,). As λ increases (decreases), the -nullcline N = α λ moes to the right (left), and so does the fixed point (, ). This remains true if the intersection beteen the to nullclines occurs on the linear piece L 2. In this case, eq. (4) has to be modified accordingly. 9

10 R l, R R 2 R r, L l, L r, L 2.2 L l, N N Figure 2: Nullclines (N and N ), linear pieces and linear regimes for a PWL,2, cubic-like model (2) ith α = 4 and λ =. The slope of the linear pieces in N are (from left to right): η l, = (left branch), η =.3 and η 2 =.3 (middle branch), and η r, = (right branch). The linear pieces in the middle branch join at (ˆ,ŵ ) = (.3,.9). (ˆ,ŵ) η r r 2 r µ set L l, (,) S - N I L r, S - N I L (.3,.9) U - F II L 2 (,) U - N II L (.4,.2) U - F III L 2 (,) U - N III L (.9,.27) U - F IV L 2 (,) U - N IV Table : PWL,2, models of FHN type for α = 4 and ǫ =.. The parameters correspond to three models (sets I/II, sets I/III and sets I/IV) ith the same left and right branches ith a single linear piece each (set I). For each linear piece L j, the table shos the right endpoints (ˆ,ŵ), the slope η, the eigenalues of the corresponding linear regime (real eigenalues r and r 2, or real and imaginary parts, r = (η ǫ)/2 and µ respectiely if the eigenalues are complex). The transition from unstable foci (U-F) to unstable nodes (U-N) occurs at η cr + =.39. For the parameters e used, the transition from stable foci (S-F) to stable nodes (S-N) occur at ηcr =.4.

11 A.5 α = 4 λ =.8 N.5 M = α = 4 λ =.8 N.5 M = 3 α = 4 λ =.8 N N N N Tr Tr Tr M = α = 4 λ =.8 N.5 M = α = 4 λ =.8 N.5 M = α = 4 λ =.8 N N N N Tr Tr Tr B.5 α = 4 λ =.7 N.5 M = α = 4 λ =.7 N.5 M = 3 α = 4 λ =.7 N N N N Tr Tr Tr M = 9 α = 4 λ =.7 N.5 M = α = 4 λ =.7 N.5 M = α = 4 λ =.7 N N N N Tr Tr Tr Figure3: Solutions to PWL M models approximate the corresponding smooth FHN model ith f = as the number of linear pieces M increases. A: λ =.8. B: λ =.7. The top-left panels in A and B sho the phase-planes for the smooth FHN model in the relaxation oscillations and small amplitude oscillations regimes respectiely. The corresponding alues of λ are close to the canard critical point λ c.78. As the number of linear pieces M increases (from the top-middle panel to the bottom-right panel) in both A and B, the solutions to the PWL M models approximate the solution to the smooth FHN model (shon in the top-left panel).

12 A B ε =. unstable nodes unstable spirals (η + ε) 2 4 ε α = ε =. ε =.5 ε =. η η ε = η.2 stable spirals stable nodes α α C D α = ε =. r r ε =. α = α = 4 α=.2.25 µ α = α = η.5.5 η Figure 4: Stability diagrams and eigenalues for the basic linear components in a PWL model of FHN type. A and B. Stability diagrams. For a fixed alue of α, as η increases the fixed-point (, ) changes from a stable node, through a stable and unstable focus, to an unstable node. C. Real eigenalues (r and r 2 ). They correspond to alues of α and η satisfying (η +ǫ) 2 4αǫ >. D. Natural frequencies µ for complex eigenalues. They correspond to alues of α and η satisfying (η +ǫ) 2 4αǫ <. 2

13 A B α = 4 λ =.29 α = 4 λ =.3 N N N N C α = 4 N N λ=.293 λ=.2935 L i α = N N λ=.293 λ=.2935 L i Figure 5: Canard phenomenon in a PWL,2, model of FHN type. A. Small amplitude limit cycle for λ =.29. B. Large amplitude limit cycle for λ =.3. C. Phase-plane diagram shoing the inflection line L (or = 2 ) separating beteen a small amplitude limit cycle (λ =.293) and a large amplitude limit cycle (λ =.2935). The inflection line is defined for alues of [.3,]. For clarity, e extended it beyond this domain. The right panel is a magnification of the left one. We used (, ) = (.3,.9), η =.3, ǫ =. and α = 4. (See also Table, sets I and II). 3

14 For each linear piece, the eigenalues are gien by (6) ith η substituted by the corresponding alue of η j. Eigenalues and eigenectors depend on the linear piece on hich the fixed-point (, ) is located but not on the precise location of the fixed-point on that linear piece; i.e., as λ increases, the eigenalues and eigenectors remain unchanged as long as the fixed point remains on the same linear piece. For future use, e present in Table (sets I and II) the eigenalues (r and r 2 ), or their real and imaginary parts (r and µ) if they are complex, for the fixed-points in Fig. 2. Fixed-points located on the left and right branches are stable nodes (r < and r 2 < ), fixed-points located on the linear piece L are unstable foci (r > and µ ), and fixed-points located on the linear piece L 2 are unstable nodes (r > and r 2 > ). 3. Assumptions and properties of the linear regimes In PWL,2, systems, the linear regimes R and R 2 satisfy some geometric and dynamic constraints. First, byconstruction, theslopesofl andl 2 (linearpiecesinthemiddlebranch)andthe-coordinate ˆ of the point joining them are related by η 2 = η ˆ ˆ. (5) The exact form of this equation depends on our canonical choice of the maximum and minimum of f pl. Similarexpressionscanbefoundforotheraluesof( min, min )and( max, max )(seesection 2.). Secondly, from (5), if η <, then η 2 >. This also follos from geometric considerations. The slope of the line L joining the minimum (,) and maximum (,) of the cubic like PWL,2, function is η =. (This line ould be the middle branch of a cubic-like PWL,, function as the ones shon in Figs. 3-A and -B (top-left panels)). The to linear pieces L and L 2 in Fig. 2 can be thought of as resulting from the line L breaking into to linear pieces ith slopes smaller and larger than (the slope of L) respectiely. Finally, the to regimes R and R 2 hae fixed-points ith different stability properties. More specifically, if R has an unstable focus ( < η < η cr + ith η cr + ), then R 2 has an unstable node (η 2 > η cr), + and iceersa. (The critical slope η cr + is gien by the first equation in (7)). This is true since, by assuming the contrary (η 2 η cr + ), substituting in (5) and rearranging terms e arrie to η η cr +, contradicting our preious assumption. The conerse is also true and follos from similar arguments by noting that eq. (5) holds if η and η 2 are exchanged. We sho in Section 3.6 that PWL,2, systems do not display small amplitude oscillations if R has an unstable node; i.e., a necessary (but not sufficient) condition for the occurrence of small amplitude oscillations is that R has a focus. Since the occurrence of the canard phenomenon requires the existence of small amplitude oscillations, here e consider parameters α, ǫ and η satisfying η < η cr +. We also require that η < α in order to allo for the intersection beteen the linear piece L and the -nullcline (N = α λ) to occur for appropriate alues of λ. In addition, e assume that the system has only one fixed-point. Note that these assumptions are analogous to the requirement that the smooth system () undergoes a supercritical Hopf bifurcation. Consider the cubic function f = h 3 + a 2 used in Fig. 4

15 (h = 3 and a = 2). As shon in Appendix A.4, for fixed alues of h and a, there exists a critical alue α crit = 2a 2 (3h) such that the Hopf bifurcation is supercritical for α > α crit and subcritical for α < α crit. As α decreases belo α crit, the smooth system loses its ability to generate stable small amplitude limit cycles. (For the choice of parameters in Fig., α crit = 3). For a PWL,2, and a fixed alue of η, since η + cr is a decreasing function of α, a large enough decrease in the alue of α causes η + cr to decrease belo η thus changing the stability properties of the corresponding fixed-point of the linear regime R from an unstable focus to an unstable node and hence small amplitude oscillations are no longer possible (see Section 3.6). 3.2 Bifurcation structure In Fig. 6-A e present the limit cycle amplitude diagram for the PWL,2, system corresponding to Fig. 2 and ǫ =. as a function of the control parameter λ. Linear stability properties of fixed-points ere determined as in section 2.3. Periodic solutions are constructed using the explicit formulas for trajectories gien by the solutions in Section 2.3, matching continuously across sections here = ˆ j, and enforcing periodicity of solutions. These constraints generate a set of nonlinear algebraic equations hich e sole numerically as in the earlier ork by Coombes [22] to determine the period of solution, T, and the maximum and minimum alues of the orbit. For λ <, there is a stable node lying on the left branch of the -nullcline (linear piece L l, ). As λ increases, the fixed-point (4) moes to the right and crosses the minimum of the -nullcline hen λ =. For < λ <.29 the system has small amplitude limit cycles qualitatiely similar to the ones shon in Fig. 5-A. The amplitudes of these limit cycles increase ith λ. For λ =.3 the system has a large amplitude, relaxation type limit cycle (Fig. 5-B). A major difference beteen these small and large amplitude limit cycles is that trajectories cross the middle branch of the -nullcline in the former hile they don t, and moe aay from R, in the latter. This is analogous to the smooth case. With increasing alues of ǫ it is possible for the branch of small amplitude oscillations to deelop a fold so that stable small and large amplitude oscillations can coexist (and preclude the canard phenomenon). We illustrate this in Fig. 6-B. This phenomenon is not obsered in smooth systems for the same parameter alues (α = 4 and ǫ =.). A similar folding in the PWL system occurs for other parameter sets as e illustrate in Fig. 7-A for α = 2 and ǫ =.. Hoeer, differently from the α = 4 case, for the α = 2 case, the folding phenomenon persists in the corresponding smooth system Fig. 7-B. The period of a small amplitude limit cycle is independent of λ Here e sho that for a periodic orbit like that shon in Fig. 5-A hich isits only to distinct regimes R and R 2 in the middle branch the period of the orbit is independent of λ. The pieceise dynamical system (4) may be ritten in the form z j = A jz j +b j, z j = [ ], (,) R j, (6) 5

16 A α = 4 ε=. stable fixed point unstable fixed point stable limit cycle α = 4 ε=. stable fixed point unstable fixed point stable limit cycle B λ 2 α =4 ε= λ.5 α =4 ε= stable fixed point.5 unstable fixed point stable limit cycle unstable limit cycle λ.5 stable fixed point unstable fixed point stable limit cycle unstable limit cycle λ Figure 6: Bifurcation diagrams for the PWL,2, model corresponding to Fig. 2 ith ǫ =. (A) and ǫ =. (B). For fixed-points e present their -coordinate. For periodic orbits e present the maximum and minimum alues of their -coordinate. The right panels are magnifications of the left ones. 6

17 A 2 α =2 ε=..5 α =2 ε= B stable fixed point.5 unstable fixed point stable limit cycle unstable limit cycle λ 2 α =2 ε=..5 stable fixed point unstable fixed point stable limit cycle unstable limit cycle λ.3 α =2 ε= stable fixed point unstable fixed point stable limit cycle unstable limit cycle λ.2.3 stable fixed point unstable fixed point stable limit cycle unstable limit cycle λ Figure 7: Comparison beteen the bifurcation diagrams for the PWL,2, model (A) and the corresponding smooth FHN model (B) for α = 2 and ǫ =.. For fixed-points e present their -coordinate. For periodic orbits e present the maximum and minimum alues of their -coordinate. The right panels are magnification of the left ones. 7

18 here e use the index j to distinguish the dynamics of each linear regime. Here [ ] [ ] ηj ηjˆ A j = ǫα ǫ, j +ŵ j b j =. (7) ǫλ The solution to (6) may be ritten using matrix exponentials as z j (t) = G j (t)z j ()+K j (t)b j, G j (t) = e A jt, K j (t) = t G j (s)ds. (8) Since the phase space is diided naturally into only to pieces e may ithout loss of generality set (ˆ,ŵ ) = (,). We construct a periodic orbit by eoling a trajectory according to (8), ith initial data (,) = (,()) (ith () as yet undetermined), until = is reached again for the first time at T >. We then eole the trajectory ith ne initial data (,(T )) until meeting = again a time T 2 later. The times-of-flight T j are determined by soling the threshold crossing conditions (T ) = = (T +T 2 ). A periodic solution can then be found by soling (T +T 2 ) = () for (), yielding the period T = T +T 2. For the special case that the trajectory only isits R and R 2, b j is independent of j and gien by b j = ǫλ(,) T. Eolution from initial data (,()) to (,(T )) gies the pair of equations G 2 (T ) [ () ] ǫλk 2 (T ) [ ] = [ (T ) ]. (9) Diiding by ǫλ means that e may sole for this pair of equations (say using Cramer s rule) in the form () ǫλ = F (T ) (T ), ǫλ = F 2(T ), (2) for some explicit functions F,2. A similar argument, ith eolution of a trajectory from (,(T )) to (,()), gies [ ] [ ] [ ] G (T 2 ) ǫλk (T 2 ) =. (2) (T ) () Similarly e may sole for the pair ((),(T )) as () ǫλ = F 3(T 2 ), (T ) ǫλ = F 4(T 2 ), (22) for some explicit functions F 3,4. Equating (2) and (22) gies to simultaneous equations for the pair (T,T 2 ): F (T ) = F 3 (T 2 ), F 2 (T ) = F 4 (T 2 ). (23) hich are independent of λ. Hence T = T +T 2 is independent of λ. 3.3 The mechanism of generation of small and large amplitude limit cycles and the abrupt transition beteen them We begin by explaining the mechanisms that goern the generation of small and large amplitude limit cycles and the abrupt transition beteen both limit cycle amplitude regimes in the context of the 8

19 example presented in Fig. 5 as λ changes from.29 to.3. The parameters e used (η =.3, α = 4 and ǫ =. ith η + cr =.39) correspond to the supercritical canard phenomenon illustrated in Fig. -A for the smooth FHN system. As e progress in our discussion, e explain ho changes in the alues of these parameters affect the resulting dynamics. We ill focus on changes in the relatie lengths of L and L 2, ǫ and α. The dynamics of a PWL,2, system is diided into four linear regimes R j corresponding to the four linear pieces L j, and indexed accordingly (see Section 2.3) as shon in Fig. 2. The dynamics in each of these regimes are goerned by a linear system of the form (4). The initial conditions for each regime R j are equal to the alues of and at the end of the preious regime (either R j or R j+ ) as described in Section 2. Fig. 8-A shos the to trajectories in Fig. 5-A and -B (λ =.29 and λ =.3) in the same phase-plane. (The -nullcline is independent of λ. The - nullclines for both alues of λ are indistinguishable. We plotted the -nullcline corresponding to Fig. 5-A.) In these figures e hae plotted the limit cycle trajectories after transients disappeared. These trajectories hae been computed using the analytical solutions presented in Section 2.3. Alternatiely, they can be computed numerically. Geometric and dynamic information corresponding to the four linear regimes are summarized in Table. We ill refer to this table in our explanation. As e mentioned in Section, the analytical solutions for PWL systems proide a poor insight into the models dynamics. Belo e use dynamical systems tools to explain these dynamics and anser arious releant questions about the abrupt transition beteen limit cycle amplitude regimes. In our explanation e inestigate ho limit cycle trajectories initially in the linear regime R l,, near the left branch of the -nullcline L l,, return to this regime after a cycle. One difficulty in inestigating these limit cycle trajectories is that it is not possible to consider an initial point exactly on the limit cycle. We oercome this difficulty by computing an asymptotic approximation (for small alues of ǫ) to the limit cycle trajectory in the linear regime R l, (see Appendix B.4). This is the trajectory e follo in our explanation and e ill refer to it as the limit cycle trajectory although it is only an asymptotic approximation to the real limit cycle trajectory. Linear regimes: irtual and actual fixed-points A distinctie feature of PWL systems is that in each regime R j dynamics are organized around a irtual fixed-point ( j, j ) hich results from the intersection beteen the -nullcline N = α λ and the linear piece L j or its extension beyond the boundaries of R j, [ˆ j,ˆ j ]. In the latter case, irtual fixed-points are located outside the corresponding regime; i.e. j / [ˆ j,ˆ j ]. We illustrate this in Fig. 8-B for the linear regime R l,. The ertical dashed lines separate beteen the linear regimes R l, and R. The actual fixed-point (, ) of the PWL,2, system (blue dot) is located on the intersection beteen the - and -nullclines (solid-red and -green lines respectiely). The irtual fixed-point ( l,, l, ) for the linear regime R l, is located on the intersection beteen the extension of the linear piece L l, (dotted-red line) and the -nullcline hich occurs in the linear regime R (and not in the linear regime R l, ). Clearly, the actual fixed-points (, ) of a PWL system is also the irtual fixed-points for the regime here it is are located (see Fig. 9-B). Within the boundaries of each regime, trajectories eole as if the corresponding linear system, 9

20 hich goerns their dynamics as long as the trajectory is in that regime, goern their dynamics globally (for all alues of t). In other ords, ithin the boundaries of each regime trajectories eole according to the linear dynamics defined in that regime and they do not feel that the dynamics goerning their eolution ill change at a future time hen the trajectory moes to a different regime. The dynamics in the linear regime R l, In Fig. 8-B e consider to trajectories initially located in the linear regime R l,, close to the -nullcline. The blue one is the limit cycle trajectory hich eoles according to the PWL,2, system. It eoles in a small neighborhood of the -nullcline due to the fast-slo nature of the system (ǫ =. ). The dotted-cyan trajectory eoles according to the dynamics of the linear regime R l,. It heads toards, and eentually conerges to, the irtual fixed-point ( l,, l, ) (cyan dot) hich is a stable node. (This follos from the analytic solution presented in Section 2.3. Since ǫ, it also follos from the asymptotic approximation to the slo manifold computed in Appendix B.4). For as long as the blue trajectory is in the linear regime R l, it eoles as the cyan one; i..e, as if it ere attracted to the irtual fixed-point (cyan dot). This irtual fixed-point ceases to be the blue trajectory s target once this trajectory crosses the boundary beteen the linear regimes R l, and R since its dynamics is no longer goerned by the linear regime R l, but by the linear regime R. The dynamics in the linear regime R ThedynamicsofthetrajectoryinthelinearregimeR isshoninfig. 9. Thisfigureisamagnification of Fig. 8-A. The linear piece L has slope η =.3 < η + cr =.39. The dashed-red lines represent the extension of the linear piece L beyond the boundaries of the linear regime R. The corresponding fixed-point (blue dot) is an unstable focus (see Table ) located on the intersection beteen L (solidred line) and the -nullcline (green line) and is located in the linear regime R. We sho three trajectories in the right panel. The solid-blue and dashed-blue trajectories correspond to the small and large amplitude limit cycles shon in Fig. 8-A for λ =.29 and λ =.3 respectiely. The solid-blue trajectory crosses thelinear piece L almost at the boundarybeteen thelinear regimes R and R 2 and neer enters R 2. The dashed-blue trajectory crosses this boundary and moes into the linear regime R 2. We address the dynamics of this trajectory in next section. The dashed-cyan trajectory initially coincides ith the solid-blue trajectory and its eolution is goerned (globally) by the linear regime R. The left panel shos this cyan trajectory in the absence of the blue ones. Since the fixed-point is a focus, the cyan trajectory spirals out. (We only sho the trajectory for the releant time interal.) Once it enters the linear regime R, the solid-blue trajectory also spirals out as if its dynamics ere globally goerned by this regime. The solid-blue and cyan trajectories split hen they cross from R to R l,. This has no consequences for the cyan trajectories hich continues to spiral out. The dynamics of the blue trajectory, instead, ceases to be goerned by that of R and returns to be goerned by the linear regime R l,. Consequently, it crosses the -nullcline (solid-red line). The reminder of the dynamics are as explained aboe for the linear regime R l, (see Fig. 8). 2

21 A B N N trajectories x 3 8 R l, R Figure 8: Canard-like phenomenon in a PWL,2, model ith α = 4 and ǫ =.. Dynamics in the linear regime R l,. The nullclines are as in Fig. 2. The ertical (thin) dashed-lines separate beteen linear regimes and intersect the -nullcline N at the joint points beteen the corresponding linear pieces. The actual fixed-point of the PWL,2, system is located in the linear regime R and indicated ith a blue dot (panel B) on the intersection beteen the to nullclines. A. Super-imposed trajectories for the small (solid) and large (dashed) amplitude limit cycles shoed in Figs. 5-A and -B for λ =.29 and λ =.3 respectiely. B. Dynamics of the trajectory in the linear regime R l,. The dotted-red line indicates the continuation of the linear piece L l, beyond the boundaries of the linear regime. The irtual fixed-point ( l,, l, ) = (.58,.58) for R l, is located in R and indicated ith a cyan dot (intersection beteen the dotted-red line and the -nullcline N ). The dotted-cyan cure corresponds to a trajectory hose global dynamics are goerned by the linear regime R l, and shos that the irtual fixed-point is a stable node. Initially, this trajectory is located near the trajectory of the PWL,2, system (solid-blue). The slope of the linear pieces in N (from left to right) are: η l, = (left branch), η =.3 and η 2 =.3 (middle branch), and η r, = (right branch). The linear pieces in the middle branch join at (ˆ,ŵ ) = (.3,.9). 2

22 .2.2 R l, R R 2 R l, R R Figure 9: Canard-like phenomenon in a PWL,2, model ith α = 4 and ǫ =.. Dynamics in the linear regime R. This figure is a magnification of Fig. 8-A. The nullclines are as in Fig. 2. The solid-red PWL cure and the solid-green line represent the - and -nullclines (N and N ) respectiely. The ertical (thin) dashed-lines separate beteen linear regimes and intersect N at the joint points beteen the corresponding linear pieces. The dotted-red lines represent the continuation of the linear piece L beyond the boundaries of the linear regime. The actual fixed-point (, ) = (.78,.24) of the PWL,2, system is located in the linear regime R and indicated ith a blue dot on the intersection beteen the to nullclines. The dotted-red lines indicates the continuation of the linear piece L beyond the boundaries of the linear regime. The dotted-cyan cure corresponds to a trajectory hose global dynamics are goerned by the linear regime R and illustrates that the fixed-point is an unstable focus. The cyan trajectory spirals out ith frequency µ =.26. The slope of the linear pieces in N (from left to right) are: η l, = (left branch), η =.3 and η 2 =.3 (middle branch), and η r, = (right branch). The linear pieces in the middle branch join at (ˆ,ŵ ) = (.3,.9). 22

23 The small amplitude limit cycle in Figs. 8 and 9 corresponds to a alue of λ (=.29) for hich the limit cycle trajectory intersects the -nullcline in the linear regime R and neer crosses to the linear regime R 2. Limit cycle trajectories for smaller alues of λ intersect the linear piece L at loer alues of (not shon). These alues of depend on the amplitude of the initial oscillation of the trajectory after entering the linear regimes R (see solid-blue trajectory in Fig. 9). This amplitude, in turn, depends on the distance D = D(,λ) = D(,, ) = ( ) 2 +( ) 2 (24) beteen the initial point (, ) = (, ) in R and the fixed-point (, ) R, parametrized by λ, through the constants c and c 2 () in the solution () presented in Section 2.3. (This parametrization is ell defined since e assumed that for each alue of λ there is a unique fixedpoint.) Note that in (24) = and = η. To leading order, the initial alue can be analytically approximated using the asymptotic computation in Appendix B.4. For ǫ, = O(ǫ). For λ =, D = since (, ) = (,) is a stable node. In Section 3.5 e sho that D is an increasing function of λ. For a fixed alue of the length of the linear piece L, if D is small enough, then the trajectory crosses L as it occurs in Figs. 8 and 9 for the solid-blue trajectory. On the other hand, if D, and hence λ, is large enough, then the trajectory reaches the boundary beteen the linear regimes R and R 2 ithout crossing the linear piece L as it occurs in Figs. 8 and 9 for the dashed-blue trajectory. Once the trajectory crosses this boundary its dynamics ceases to be goerend by the linear regime R 2 and turns to be goerned by the linear regime R 2. The dynamics in the linear regime R 2 The dynamics of the trajectory in the linear regime R 2 is shon in Fig.. The linear piece L 2 has slope η 2 =.3 > η cr + =.39. The corresponding irtual fixed-point (blue dot) is an unstable node (see Table ) located on the intersection beteen the extension of the linear piece L 2 (dashed-red lines) and the -nullcline (green line) and is located in the linear regime R l, (not in R 2 ). As in the preious figures, e sho three types of trajectories. The solid-blue and dashed-blue trajectories correspond to the small and large amplitude limit cycles shon in Fig. 8-A for λ =.29 and λ =.3 respectiely. Their dynamics are goerned by the PWL,2, system. Both eole ery close in the linear regimes R l, and R and bifurcate near the boundary beteen these to linear regimes. The to dashed-cyan trajectories eole according the dynamics of the linear regime R 2 for all alues of t. They are initially located near the irtual fixed-point and the dashed-blue trajectory respectiely. They moe fast along almost horizontal direction as expected from fast-slo systems. They illustrate the fact that the irtual fixed-point is an unstable node (see Table, set II). The dashed-blue trajectory eentually crosses the boundary beteen the linear regimes R 2 and R r, and its dynamics changes accordingly. The dynamics in the linear regime R r, The dynamics of the trajectory in the linear regime R r, is shon in Fig.. It is qualitatiely similar to the dynamics on the linear regime R l,. The linear piece L r, has slope η r, =. The 23

24 R R l, R 2 R r, Figure : Canard-like phenomenon in a PWL,2, model ith α = 4 and ǫ =.. Dynamics in the linear regime R 2. The nullclines are as in Fig. 2. The solid-red PWL cure and the green line represent the - and -nullclines (N and N ) respectiely. The ertical (thin) dashed-lines separate beteen linear regimes and intersect N at the joint points beteen the corresponding linear pieces. The dotted-red lines represent the continuation of the linear piece L 2 beyond the boundaries of the linear regime. The irtual fixed-point ( 2, 2 ) is located in R l, and indicated ith a blue dot (intersection beteen the dotted-red line and the -nullcline N ). The dashed-cyan cures correspond to trajectories hoseglobaldynamicsaregoernedbythelinearregimer 2 andillustratethatthefixed-pointisanunstable node. Initially, the to cyan cures are located near the irtual fixed-point and near the trajectory of the PWL,2, system (solid-blue cure) respectiely. The slope of the linear pieces in N (from left to right) are: η l, = (left branch), η =.3 and η 2 =.3 (middle branch), and η r, = (right branch). The linear pieces in the middle branch join at (ˆ,ŵ ) = (.3,.9). 24