Traveling pulses in Class-I excitable media
Andreu Arinyo-i-Prats, Pablo Moreno-Spiegelberg, Manuel A. Matías, Dami? Gomila
aa r X i v : . [ n li n . PS ] J a n Traveling pulses in Class-I excitable media
Andreu Arinyo-i-Prats , , Pablo Moreno-Spiegelberg , Manuel A. Mat´ıas , and Dami`a Gomila ∗ IFISC (CSIC-UIB), Instituto de F´ısica Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain Institute of Computer Science, Czech Academy of Sciences, 182 07 Praha 8, Czech Republic (Dated: January 5, 2021)We study Class-I excitable 1-dimensional media showing the appearance of propagating travelingpulses. We consider a general model exhibiting Class-I excitability mediated by two different scenar-ios: a homoclinic (saddle-loop) and a SNIC (Saddle-Node on the Invariant Circle) bifurcations. Thedistinct properties of Class-I with respect to Class-II excitability infer unique properties to travelingpulses in Class-I excitable media. We show how the pulse shape inherit the infinite period of thehomoclinic and SNIC bifurcations at threshold, exhibiting scaling behaviors in the spatial thicknessof the pulses that are equivalent to the scaling behaviors of characteristic times in the temporalcase.
Excitability is a nonlinear dynamical regime that isubiquitous in Nature. Excitable systems, having a sta-tionary dynamics, are characterized by their response toexternal stimuli with respect to a threshold. Thus, stim-uli below the threshold exhibit linear damping in theirreturn to the fixed point, while stimuli exceeding thethreshold are characterized by a nontrivial trajectory inphase space before returning to the fixed point. Thisdifferent response to external perturbations confers ex-citable systems with unique information processing ca-pabilities, and also the possibility of filtering noise belowthe threshold level. Moreover, excitable media, that arespatially extended systems that locally exhibit excitabledynamics, can propagate information, as happens in neu-ronal fibers [1] or the heart tissue [2].From a dynamical systems point of view, excitabilityis typically associated to the sudden creation (or destruc-tion) of a limit cycle, whose remnant traces in phasespace constitute the excitable excursion [3]. The route(i.e. bifurcation) through which a limit cycle is createdor destroyed, leads to differences in the excitable dynam-ics. A basic classification of excitability is in two classes(types), depending on the response to external pertur-bations [3]. Class-I excitable systems are characterizedby a frequency response that starts from zero, leading toa (theoretically) infinite response time at the threshold.On the other hand, Class-II excitable systems are charac-terized by a frequency response that occurs in a relativelynarrow interval, and thus the response time is bounded.Regarding the bifurcations that originate these two typesof excitable systems, Class-I excitability occurs in certainbifurcations that involve a saddle fixed point when cre-ating/destroying a limit cycle, as are the cases of a ho-moclinic (saddle-loop) or SNIC (Saddle Node on the In-variant Circle), also known as SNIPER (Saddle-Node In-finite Period), bifurcations. In turn, Class-II excitabilityis mediated by transitions involving a Hopf bifurcationsuch that in relatively narrow parameter range a largeamplitude cycle is created, as is the case of subcriticalHopf bifurcations (typically close to the transition from ∗ damia@ifisc.uib-csic.es sub- to supercritical Hopf bifurcation) and also the caseof a supercritical Hopf bifurcation followed by a canard,i.e, a sudden growth of the cycle happening sometimes infast-slow systems. This excludes the regular supercriticalHopf bifurcation, characterized by a gentle growth of thelimit cycle amplitude.Excitable media, obtained by coupling spatially dy-namical systems that are locally excitable, show differentregimes in which local excitations exceeding a thresholdcan propagate across the medium [4, 5]. Many studieshave been carried out in Class-II excitable media, butmuch less is know about pulse propagation in the Class-I case. Excitable regimes are relevant in a number ofphysical, chemical and biological systems, namely semi-conductor lasers, chemical clocks, the heart, and signaltransmission in neural fibers. 1-D pulse propagation isalso behind nerve impulse transmission.In one spatial dimension both pulse propagation andperiodic wave train regimes are found in Class-II [1, 4],and their instabilities have been characterized for rep-resentative models [6, 7]. In 2 and 3 spatial dimensionsfurther regimes are reported in Class-II, like spiral waves,including spiral breakup leading to spatiotemporal chaos[8] and Winfree turbulence [9]. These transitions arerelevant in the study of excitable waves in heart tissue[2, 10, 11], where they are associated to certain patolo-gies.Class-I excitability is much less studied and appearsin models of population [12, 13] or neural [14] dynam-ics, and evidence of pulse propagation has been foundin seagrasses [15]. Class-I pulse propagation has also re-cently been studied in arrays of coupled semiconductorlasers [16]. The different properties of Class-I and Class-II excitability, specially the divergence of the period atthreshold, can significantly modify the properties of spa-tiotemporal structures in excitable media. In this Letterwe characterize traveling pulses in Class-I excitable me-dia and show their distinct properties and instabilities.To address this problem we propose a general modelbased on the normal form of a codimension-3 bifurcation[17] which is the simplest continuous model one can writewith Class-I excitable behavior that can be accessed ei-ther through an homoclinic (saddle-loop) or a SNIC bi- −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 μ μ CuspSNSL − DL − L − L + SNSNIC
FIG. 1. Simplified phase diagram of the temporal system( ∂ xx u = ∂ xx v = 0) in the parameter space ( µ , µ ). The di-agram is organized by three main codimension-2 points: i)a Cusp (blue dot), where the Saddle-Node bifurcation lines(blue) meet, ii) a Saddle-Node Separatrix Loop bifurcation(SNSL − , red dot) on which a SNIC (blue dot-dashed line)and iii) a homoclinic of a stable cycle ( L − , red dashed line)bifurcation lines join, and ii) a bifurcation (DL − , red triangle)in which L − meets with a homoclinic bifurcation of an unsta-ble cycle ( L + , red solid line). The dotted end of L + indicatesthat the line continues (not shown) but it is not relevant forthis work. furcation. To this normal form we add 1-D diffusion tostudy spatial propagation: ∂ t u = v + D ∂ xx u (1) ∂ t v = ε u + µ u + µ + v ( ν + bu − ε u ) + D ∂ xx v . We choose ε = ε = − ν = 1, b = 2 . D = D = 1, considering µ and µ ascontrol parameters.The dynamical regimes exhibited by the temporal sys-tem (local dynamics), which is the system that describesthe time evolution of homogeneous solutions ( ∂ xx u = ∂ xx v = 0), are shown in Fig. 1, where only the mostrelevant transitions to our study are shown. The fixedpoints of this temporal system have v ⋆ = 0 and u ⋆ beingdetermined by the solutions to the cubic equation formedby the first 3 terms of the second equation in (1), that cor-responds to the normal form of the cusp codimension-2bifurcation. The two blue lines are saddle node bifurca-tions that mark the boundary between the inside regionwith 3 real roots and the outside region with 1 real plusa pair of complex conjugate roots, these 2 lines joining ata cusp point in µ = µ = 0 (blue circle in Fig. 1) with atriple degenerate root.There are other codimension-2 points, in addition tothe cusp, that organize the scenario of interest to ourstudy. One of them is the SNSL − point (Saddle-Node Separatrix Loop) [18], (red circle in Fig. 1), that is char-acterized by a nascent (i.e. with a zero eigenvalue) ho-moclinic (saddle loop) bifurcation. It is precisely fromthis SNSL − point that come up the two principal bound-aries of the Class-I excitability region: a SNIC line (bluedot-dashed line), emerging upwards, and a homoclinic(saddle loop) line (red dashed line), L − downwards.Another relevant codimension-2 point is the DL − (redtriangle upside down), representing a homoclinic (saddle-loop) bifurcation to a neutral (resonant) saddle [19], im-plying a transition between L − (that involves a stablecycle) and L + (red line, that involves an unstable cycle),and that leads also to the emergence of a fold of cyclesbifurcation line, not shown in Fig. 1 for simplicity. Theleft SN line, both saddle-loop bifurcations ( L − and L + )and the SNIC curve delimit the Class-I excitable region(shaded grey area in Fig. 1). In this region a perturba-tion around the lower fixed point (black dot in Fig. 2c)that crosses the threshold, i.e. the stable manifold ofthe middle fixed point (cross in Fig. 2c), will trigger anexcitable trajectory that comes back to the lower fixedpoint [panels a) and c) in Fig. 2].The two above mentioned bifurcation lines, SNIC and L − are the ones of special interest to our study, as theymediate two different ways of entering the Class-I ex-citable region, cf. [20, 21], and it will be reflected in thebehavior of the pulses to be considered below. Further-more, there are other codimension-2 points not relevantfor our analysis [22]. u a) −10−505 v −2 0 2u−10−505 v c) 0 6 12 18 24ξ−1012 u b)−1 0 1 2u−202 v d) −202 v FIG. 2. Comparison of the temporal dynamics of an excitableexcursion and the spatial dynamics of a 1-D pulse sustained bythis excitable dynamics. a) temporal excitable trajectory of u and v (spatially homogeneous system) starting from an initialcondition just above the saddle point; c) representation of thesame excitable excursion on the ( u, v ) phase space; b) stable 1-D pulse as a function of the spatial coordinate in the movingreference frame; d) the same pulse in the ( u, v ) (sub)phasespace. Dot, cross, and circle in c) and d) indicate lower,middle and upper fixed points respectively. Here µ = 0 . µ = 1 . By initializing the system with a strong enough local-ized perturbation around the lower homogeneous solutiona pair of solitary (or traveling) pulses that propagate withfixed shape and constant and opposite velocities are gen-erated. One of such pulses, the one moving to the left,is shown in Fig. 2b). A convenient way of characterizingthis pulse is using a moving reference frame, ξ = x − ct ,where c is the velocity of the pulse yet to be determined.In this coordinate system the partial differential equa-tions (1) become ordinary differential equations, and inour case we get, du/dξ = u ξ ; dv/dξ = v ξ (2) du ξ /dξ = − ( v + c u ξ ) dv ξ /dξ = u − µ u − µ − v (1 + bu + u ) − c v ξ . Trajectories of this system describe stationary solutionsof (1) in the reference frame moving with velocity c [23].Only bounded trajectories have a physical meaning. Inparticular, excitable pulses are represented in this sys-tem as homoclinic trajectories originating from the lowerfixed point (panels b and d in Fig. 2). c is computednumerically simultaneously with the field profiles, and itvaries weakly with parameters in the excitable region. FIG. 3. Phase diagram showing the bifurcation lines of 1-Dpulses. Traveling pulses are stable in the green region. Themain instabilities discussed in this work are the HeteroclinicI (black line) and the SNIC (blue dot-dashed line). The otherlines that bound the stability region are the Heteroclinic IIand the Hopf of pulses (not discussed in this work). The SN, L − and L + lines are shown in order to compare the diagramwith respect to Fig. 1. The × , ∗ and + symbols mark theparameter values studied in Figs. 2, 4, and 6 respectively. Although the dynamical system describing temporaldynamics for homogeneous solutions and the spatial dy-namical system (2) are different, one may observe im-portant similarities in their solutions. Roughly speak-ing, a traveling pulse somehow transcribes the temporaldynamics in space, such that the spatial profile of thepulse resembles the excitable trajectory in time. Thus,Fig. 2(a) shows a excitable (open) trajectory in the tem-poral dynamics, while Fig. 2(b) shows a excitable pulse in the spatial dynamics (a homoclinic orbit), for the sameparameter values. In Figs. 2c) and 2d) the trajectoriesfrom Fig. 2a) and 2b) are represented in the ( u, v ) phasespace respectively. The similarity of panels c) and d) ofFig. 2 anticipates the results presented in this work.Next we analyze how the infinite period bifurcationsleading to Class-I excitability in the temporal system,namely the homoclinic and SNIC bifurcations, affect theshape of the traveling pulses. To do so we study thedomain of stability of pulses in the ( µ , µ ) parameterspace, shown in Fig. 3, where the cusp and saddle-nodelines of Fig. 1 are also included in the diagram for com-parison. This domain is delimited by several bifurcationsat which the pulse is destroyed or made unstable: Het-eroclinics I and II, SNIC, and Hopf of pulses. Here wefocus on the Heteroclinic I and SNIC bifurcations, whichare connected to the L − (homoclinic) and the SNIC bi-furcations of the temporal system respectively.Let us first consider the Heteroclinic I curve, repre-sented as a black line in Fig. 3. Approaching this bi-furcation the pulse shape changes drastically, generatinga plateau at the value of the middle (saddle) homoge-neous solution (Fig. 4b). As the spatial trajectory ap-proaches the saddle point (through its stable manifold)there is a slowing down of the spatial dynamics, inher-ited from the temporal homoclinic (Fig. 4a), that man-ifests as a plateau in the spatial profile. The plateau ismore clear as one is very close to the bifurcation (blackline). Fig. 4c) shows the temporal excitable excursionin the ( u, v ) (sub)phase space, where it can be seen thatthe trajectory gets closer and closer to the saddle point,marked with a cross. The spatial counterpart (Fig. 4d)behaves analogously, leading to the formation of a dou-ble heteroclinic at threshold, where the size of the plateaudiverges.This slowing down has a characteristic logarithmicscaling law in the width of the plateau with respect tothe parameter distance to the bifurcation [24], as shownin Fig. 5a). The red line represents the expected scal-ing slope from theory, that depends on the logarithmicparameter distance divided by the (independently ob-tained) leading unstable eigenvalue of the saddle point,and we can see that the agreement is perfect.The fact that the Heteroclinic I bifurcation curve fol-lows closely the L − and L + lines of the temporal dy-namics (Fig. 1) and that the quantitative scaling for thewidth of the pulse has the same form of that of a homo-clinic bifurcation in time indicate how the bifurcations ofthe temporal dynamics permeate the spatial dynamicaldescription of the pulse, even though the connection isnot straight forward from the equations.The second instability of pulses that we consider is theSNIC bifurcation (blue dash-dotted line in Fig. 3). Atthis bifurcation a cycle is reconstructed when a saddleand a node collide, namely the lower and middle fixedpoints. As the pulse approaches the SNIC, there is aslowing down in the approach to stable fixed point in thespatial dynamics, as the spatial eigenvalue tends to zero
20 40 60t−202 u a)a)a) 0 30 60 90ξ−101 u b)b)b)−0.5 −0.4 −0.3 −0.2u−0.10.00.1 v c)c)c) −0.5 −0.4 −0.3u−0.10.00.1 v d)d)d) FIG. 4. a) Divergence of the duration of the excitable ex-cursion in the temporal system approaching the Homoclinicbifurcation, and b) divergence of the plateau in the pulses ap-proaching the Heteroclinic I bifurcation. Here µ = 0 . µ = µ c − ∆ µ where the homoclinic bifurcation occurs at µ c = 0 . µ c = 0 . µ = 10 − ( grey ), ∆ µ = 10 − ( green ), ∆ µ = 10 − ( black ) in all panels. Panels c) and d) show a zoom in of themost relevant region of the phase space ( u, v ) for the temporaland spatial dynamics respectively.FIG. 5. (a) Scaling of the pulse width, γ , approaching theHeteroclinic bifurcation. γ is defined as the distance since thepulse separates 10 − from the stable homogeneous solutionuntil it comes back to the same distance. The expected scalingis γ = λ log ( | µ − µ c | ) (shown in red in the plot), where λ is the closest eigenvalue to zero of the middle fixed point inthe spatial dynamics. b) Scaling of the eigenvalue, λ , thatbecomes 0 at the SNIC as a function of the parameter distanceto the SNIC bifurcation. The expected scaling is a power lawwith exponent 1 / λ ∝ p | µ − µ c | (red line). too.In the temporal case the power law manifests in thedivergence of the characteristic time to reach the stablefixed point [21]. Analogously, one would expect a power-law scaling of the pulse thickness, that would diverge atthe onset of bifurcation. However, due to the exponentialapproach to the saddle close to the bifurcation, the thick-ness of the pulse is not well defined. So we have turnedto measure the approach rate, that is proportional to the leading eigenvalue, that becomes zero at the bifurcation.
10 20t02 a)a)a) 24 48 (cid:1) u - u b)b)b) (cid:2) (cid:0) (cid:3) v c)c)c) 0 2u (cid:4) v d)d)d) u - u FIG. 6. Same as in Fig. 4 for the SNIC bifurcation. u − u isplotted in panels a) and b), where u is the bottom stable fixedpoint. Here µ = 2 . µ c = 1 . µ = − − , − , − for black , green and grey curvesrespectively. This scaling is shown in Fig. 6b) and, as expected in asaddle-node bifurcation, it follows a power-law with ex-ponent 1 /
2. The behavior of the system at the other sideof the bifurcation corresponds to a wave train, that inthis case is the analog of a temporal periodic behavior,the SNIC marking, thus, a transition from wave trains topulses.In conclusion, we have shown the existence of 1-Dpulses in a model with excitable behavior correspondingto Class-I excitability mediated by two different bifur-cations, SNIC and homoclinic (saddle-loop). We havecharacterized the region in parameter space in whichthe pulse is stable and two specific instability regimesof pulses: a heteroclinic that occurs for parameter val-ues close to the temporal homoclinic and the SNIC, bothdefining Class-I excitability. These instabilities exhibit atthe transition the same scaling behaviors for the widthof the pulses than those found in the period of the os-cillations of the temporal case, namely logarithmic andpower-law for the heteroclinic and SNIC bifurcations re-spectively, unveiling a profound relation between tempo-ral systems and the spatiotemporal structures of partialdifferential equations. Further instabilities of pulses inthis system, marked as Heteroclinic II and a Hopf bifur-cations in Fig. 3, are beyond the scope of the presentwork and will be studied elsewhere [22].This work sets the ground to the study and explorespatio-temporal structures in Class-I excitable media,both 1- and 2-dimensional, a field unexplored hitherto.We acknowledge financial support fromFEDER/Ministerio de Ciencia, Innovaci´on y Uni-versidades - Agencia Estatal de Investigaci´on throughthe SuMaEco project (RTI2018-095441-B-C22) and theMar´ıa de Maeztu Program for Units of Excellence inR&D (No. MDM-2017-0711). AAiP and PMS havecontributed equally to this work. [1] J. Keener and J. Sneyd,
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