Variability sustained pattern formation in subexcitable media

Transcription

1 Variability sstained pattern formation in sbexcitable media Erik Glatt, Martin Gassel, and Friedemann Kaiser Institte of Applied Physics, Darmstadt University of Technology, Darmstadt, Germany (Dated: Janary 12, 2007) Starting with a sbexcitable net of FitzHgh-Nagmo elements it is shown that parameter variability (diversity) is able to indce pattern formation. These patterns are most coherent for an intermediate variability strength. This effect is similar to the well known spatiotemporal stochastic resonance generated by additive noise in sbexcitable media. Frthermore variability is able to indce a transition to an excitable behavior of the net. This transition strongly depends on the copling strength between the elements and it is only fond for strong copling. For weaker copling one observes a life time lengthening of waves propagating throgh the medim, bt the net stays sbexcitable. PACS nmbers: a,05.70.Fh I. INTRODUCTION Spatiotemporal pattern formation in excitable and sbexcitable media has been investigated since many years. Already in 1995 Jng and Mayer-Kress stdied the emergence of waves in an excitable celllar-atomatalike system nder the inflence of additive white noise [1]. The existence of noise sstained waves has been confirmed experimentally in slices of hippocampal astrocytes [2]. Frther stdies on pattern formation in sbexcitable media nder the inflence of additive noise have been performed, e.g. in the Barkley system [3]. The additive noise introdces a non-zero probability for each element in the medim to become excited. These excitations spread ot and pattern formation (wave propagation) can be observed. Similar to the effect of coherence resonance [4] one gets the most coherent system response (patterns) for an intermediate noise strength. This pattern formation is a stochastic resonance effect, known as spatiotemporal stochastic resonance (STSR). Noise indced pattern formation is also stdied, theoretically and experimentally, in presence of mltiplicative noise instead of additive noise [5, 6]. Mltiplicative noise can be handled analytically by applying Novikov s theorem [7]. The reslting deterministic eqation describes the systematic inflence of the noise on the system dynamics. This systematic effect can indce phase transitions in spatially extended systems [8 10]. In contrast to noise, internal variability (diversity) denotes static stochastic differences between the otherwise eqal elements of a net. Sch differences can inflence the spatiotemporal dynamics. The inflence of variability on the synchronization of copled oscillators was investigated by Winfree [11] and Kramoto [12]. These reslts are the basis for most of the stdies on synchronization even today [13]. Variability inflences spatiotemporal chaos [14] and can play an important role for pattern formation in a net of biochemical oscillators [15]. Frthermore variability may have a systematic effect, which can indce transitions between different dynamical regimes [16]. Recently it was demonstrated, that diversity can case resonance-like phenomena in networks of nonlinear elements [17]. In the present paper the inflence of variability on sbexcitable nets of FitzHgh-Nagmo (FHN) elements is discssed [18]. It is shown that variability can indce pattern formation in sch nets. These patterns are most coherent for a intermediate variability strength. It is demonstrated, that variability can indce a transition from the sbexcitable to the excitable net dynamics for large copling strengths. For a weaker copling no transition is observed, bt the life time of waves in the medim can be enlarged by an appropriate amont of variability. II. THE MODEL The system nder consideration is a net of N N copled FHN elements in the presence of variability in the parameters e and c = 1 ǫ [ (1 )( a) v + d] + qk v = c v + e, (1) where K and q denote the copling-fnction and -strength, respectively. In this minimal model of neral dynamics (t) represents the fast relaxing membrane potential, while v (t) mimics the slow potassim gating variable. The more realistic Hodgkin-Hxley model [19] has a more complicated nonlinear voltage dependency in the corresponding variables. In the FHN model, which describes essential parts of the Hodgkin-Hxley model qite well, this nonlinearity in is approximated by a cbic fnction and v is linearized in the relevant region of the parameter space. The parameters e and c in the FHN model are reslts of this linearization and have no direct conterpart in the Hodgkin-Hxley model. Other minimal models of neronal dynamics are also of relevance, e.g. a direct redction of the for variable Hodgkin-Hxley model [19]. The integration time is specified in time nits (t..) and the time scales are separated by the small parameter ǫ = 1. The eqations are integrated on a discrete spatiotemporal grid sing the Hen method ( t =

2 2 01 t..) [8] and the Forward Time Centered Space scheme ( h = ) in time and space, respectively. The integration in space is performed sing periodic bondary conditions. The grid points are labeled by the indices 1 i, j N. In order to stdy pattern formation and signal transmission throgh the net, the fnction K is chosen as diffsive nearest-neighbor copling, sing a 9- point Laplacian for radial symmetry e (O1) (E1) (E2) (O2) (E2) (B) (O3) (O4) (O5) K = 2, 0.4 (E1) 2 = 1 6 [ i+1,j+1 + i+1,j 1 + i 1,j+1 + i 1,j 1 +4( i+1,j + i 1,j + i,j+1 + i,j 1 ) 20 i,j ]. (2) The stability analysis of a single FHN element (-th element) is presented in Fig. 1. The most important parameter regimes (E1), (O1) and (B) are well known and often discssed in the literatre. Additionally one finds other dynamical regimes in small regions of the parameter space. Starting from the oscillatory regime (O1) with one stable limit cycle and one nstable focs increasing c for e = 0.1 leads to a stabilization of the focs. Now the regime (O3) is realized. Both attractors are separated by an nstable limit cycle. A frther increase of c enlarges the area of attraction of the focs and at the border to the excitable regime (E1) the limit cycles disappear. Once in the regime (E1) increasing c leads to a bifrcation, where a saddle point and an nstable focs arise in addition to the stable focs. This regime is called (E2). A frther increase of c stabilizes the nstable focs and the bistable regime (B) emerges. It is also possible to find a dynamical regime with three nstable fixed points, two nstable foci and a saddle point, all of them inside a stable limit cycle. This regime is called (O2). It shold be possible to nderstand the other dynamical regimes with the informations given. In the present contribtion only variability in the slow variables v (t) is considered. De to the variability the vales (c and e ) of the parameters c and e can change from element to element. The parameter vales are Gassian distribted nmbers, with e = E, (e E)(e kl E) = σ 2 vaδ,kl, c = C, (c C)(c kl C) = σ 2 vm δ,kl. (3) In the following σvm 2, the variance of the Gassian distribtion P(c, σ vm ), and σva, 2 the variance of the Gassian distribtion P(e, σ va ), are denoted as variability intensities, whereas σ vm and σ va are denominated as variability strengths. In the following the variability in the parameter e is called additive variability, becase this parameter is an additive term in the differential eqation [Eq. (1)]. This is different for the parameter c, which is mltiplied with the system variables v. Hence the variability in this parameter is called mltiplicative variability. Throghot this text the following set of parameters is sed (a, d, E, N) = (1, 6, 0.1, 256), while the c FIG. 1: Stability analysis of one FitzHgh-Nagmo element [Eq. (1)] dependent on its parameter vales c and e, (a, d) = (1, 6). ( ) bondaries of the dynamical regimes, (- -) c = 2.93, (- -) c = 3.6, ( ) e = 0.1. (E1) One stable fixed point (excitable regime). (E2) like (E1), additionally two nstable fixed points. (O1) one nstable fixed point and one stable limit cycle (oscillatory regime). (O2) like (O1), additionally two nstable fixed points. (O3) one stable fixed point and one stable limit cycle. (O4) like (O3), additionally two nstable fixed points. (B) two stable fixed points and one nstable fixed point (bistable regime). (O5) like (B) additionally one stable limit cycle. Unstable limit cycles are not mentioned. variability strengths σ va and σ vm, the parameter C and the copling strength q are varied. In the following a net is called excitable, if a spatially niform temporally constant soltion is stable and a small excitation can indce an excitation wave, which is sstained in the net. If the wave dies ot after some time, the net is called sbexcitable. In this paper these expressions are also sed for finite nets in presence of variability, even if in these cases small flctations arond the ideal homogeneos soltion occr. III. ADDITIVE VARIABILITY In this chapter we se the fixed parameter vales q = 20, C = 2.93 and σ vm = 0. Only the additive variability strength σ va is varied. For the chosen parameters and withot variability each FHN element is in the excitable regime (Fig. 1), reslting in a spatially niform temporally constant soltion for Eq. 1 and any initial conditions. Waves are not sstained in the absence of variability and ths the net is sbexcitable. Introdcing additive variability in the net the parameter vales e change from element to element. A change in this parameter shifts the linear nllcline of a FHN element (Fig. 2). Variability in parameter d, which is also additive variability, wold shift the cbic nllcline instead of the linear one. For the observed phenomena it is not important which nllcline is shifted and all the reslts presented in this chapter can also be fond for variability in parameter d. The shift of the nllcline has

3 3 v (a) FIG. 2: Nllclines for one FitzHgh-Nagmo element [Eq. (1)], ( ) cbic nllcline, (- -) linear nllcline for e = E. (a) ( ) linear nllclines for e = E ± 0.15, (b) ( ) linear nllclines for e = E ± Other parameters: c = P os (b) σ va FIG. 3: Probability P os for one FitzHgh-Nagmo element [Eq. (1)] to be in the oscillatory regime ( > e > 0.403), dependent on the variability strength σ va. Other parameters: c = a strong inflence on its dynamics. This is demonstrated by the stability analysis in Fig. 1. For e > the element has only one stable fixed point (excitable regime). In this case a small excitation can stimlate a large loop in the phase space (single spike) before the element goes back to the fixed point. For the parameter range > e > the element has one nstable fixed point srronded by a stable limit cycle (oscillatory regime). Other dynamical regimes are also possible for according vales of e. This discssion shows that the net [Eq. (1)] with variability in parameter e may consist of elements in different dynamical regimes. The probability of an element to be in a certain regime depends on σ va, this is shown for the oscillatory regime in Fig. 3. In a net of Hodgkin-Hxley elements [20] the additive variability, stdied in the present contribtion, wold mean that the nllclines n (V ) of the gating variable n differ from element to element. Those nllclines are shifted to different vales of the voltage V, while their shape is conserved. Variability in parameter d, what means additive variability in the fast variable, wold be variability in the parameter I of the Hodgkin-Hxley model, bt one has to notice that changing I also changes the shape of the nllcline. The inflence of additive noise on pattern formation was stdied in detail (e.g. [1 3]). For weak noise strengths the probability of an element to get excited by the noise is almost zero and no pattern formation can be observed. This behavior changes abrptly if a certain threshold of the noise strength is reached. Now there is a mch higher probability for each element in the medim to become excited. These excitations spread ot in the net and pattern formation (wave propagation) can be observed. The most coherent patterns are observed for an intermediate noise strength (STSR), becase for higher noise strengths the coherent patterns are destroyed. Since in the present contribtion the additive noise is replaced by variability in parameter e one has instead of a stochastic external forcing a diverse net. The inflence of the variability on the observed pattern formation is shown in Fig. 4. For a small variability strength (σ va < 0.12) no pattern formation can be observed [Fig. 4(b)], whereas for a slightly larger variability strength one observes the development of wave fronts, which spread throgh the net [Fig. 4(c)]. A frther increase of the variability strength leads to a destrction of the large coherent strctres [Fig. 4(d)]. To qantify the coherence of the variability indced patterns the spatial cross correlation S of the variables is sed [3]. It is defined as the space and time averaged nearest-neighbor amplitde-distance of all elements K(t) (spatial atocovariance) normalized by the total spatial amplitde variance V (t), which is defined as V (t) = 1 N 2 ( ū) 2, (4) where ū denotes the mean vale of the fast variable at time t for all elements (mean field). The spatial atocovariance of the nearest-neighbors can be written as K(t) = 1 N 2 1 N b N ( ū)(b ū), (5) with b consisting of all N = 4 elements of a von Nemann neighborhood at each lattice site. Using different local neighborhoods for b does not change the reslts qalitatively. The spatial cross correlation is given by K(t) S =, (6) V (t) where.. T denotes the time average over the total integration time T. S is a measre for the relative change of the order of a spatially extended system. If the net is completely synchronized in space and time one gets the maximm vale S = 1. For a complitely asynchronos net dynamics one receives S = 0 and for a strong anticorrelated dynamics the spatial cross correlation takes p a minimm vale of mins one. Applying the spatial cross correlation measre S [Fig. 5] one clearly sees that the coherence of the observed pattern formation shows a resonance-like behavior with increasing variability strength σ va. One can clearly discern a resonance crve with a maximm at σ va 2. At the first view this phenomenon looks similar to STSR T

4 4 σva (a) FIG. 4: Snapshots of (t) from Eq. (1), random initial conditions. (a) dependent on the variability strength σva after t = 40 t... (b)-(d) dependent on t, (b) σva = 0.1, (c) σva = 0.18, (d) σva = 0.5. Other parameters: C = 2.93, σvm = 0. t (t..) (b) (c) (d) (a) (b) 5 v S σva FIG. 5: The cross correlation measre S [Eq. (6)] for a net of FitzHgh-Nagmo elements [Eq. (1)] averaged over ten realizations of the net, dependent on the variability strength σva, total integration time T = 40 t... Other parameters: C = 2.93, σvm = 0. indced by additive white noise, bt the development of the patterns is different and so are the observed strctres. In the case of noise indced STSR the stochastic external driver randomly excites elements in the net. The excited elements act as excitation centers from which the waves spread throgh the medim. The excitation centers are not fixed and ths the patterns change their shape strongly in time. In the case withot noise bt with additive variability one increases the diversity between the elements inside the net with growing σva. For σva < 3 nevertheless nearly all elements are excitable [Fig. 3]. This changes for higher variability strengths, where a reasonable amont of elements becomes oscillatory. These elements are however bond to the spatiotemporally niform soltion of the net for σva For higher variability strengths the diversity in the net is large enogh to destroy the spatially niform temporally constant net dynamics. Single elements or small clsters of elements in the net start to oscillate. These oscillating elements now act as excitation centers from which the waves spread throgh the otherwise sbexcitable net. If the additive variability strength is frther increased a weak copling is not able to conserve an ordered net FIG. 6: Nllclines for one FitzHgh-Nagmo element [Eq. (1)], ( ) cbic nllcline, (- -) linear nllcline for c = C. (a) ( ) linear nllclines for c = C ±1.5, (b) ( ) linear nllclines for c = C ± 3.0. Other parameters: C = 3.6, e = 0.1. dynamics anymore and the dynamics of the single elements become more and more dominant. The coherent strctres get destroyed [Fig. 4(d)]. For a sbexcitable net with larger copling strengths (q > 20) the observed phenomena are shifted to larger variability strengths. IV. MULTIPLICATIVE VARIABILITY In this chapter we se the parameter vales C = 3.6 and σva = 0. The variability strength σvm and the copling strength q are varied. For the chosen parameters and withot mltiplicative variability each FHN element again has only one stable fixed point (Fig. 1), reslting in a spatially niform temporally constant soltion of the net and any initial conditions. Again the net is sbexcitable, bt not as close to the transition to the excitable dynamics as in the chapter before, becase of a different vale of C. In contrast to variability in parameter e mltiplicative variability changes the slope of the linear nllcline (Fig. 6), this again has an inflence on the dynamics of a single element (Fig. 1). For 1.58 c 6.32 the element is in the excitable regime. For other vales of c other dynamical regimes are realized. For c < 0 the slope of the linear nllkline is negative and the dynamics of the element completely changes. Conseqently the parameter vales

5 <c> σ vm FIG. 7: Effective mean vale c [Eq. (7)] for a net of FHN elements [Eq. (1)], dependent on the variability strength σ vm. (- -) c = 2.92 expected transition to an excitable net for q = 20. Other parameters: C = 3.6. c < 0 have to be exclded by setting the probability distribtion P(c, σ vm ) to zero for corresponding vales of c. For the largest variability strength sed in this paper this wold affect 10% of the random vales generated with a normal Gassian distribtion. Similar to the net with variability in parameter e the net [Eq. (1)] with variability in parameter c consists of elements in different dynamical regimes. In a net of Hodgkin-Hxley elements the mltiplicative variability wold not only case a shift of the nllclines n (V ) as for additive variability bt also change their shape (slope). In [16] it was shown that variability in parameter c has a systematic effect on the mean gradient angle α of the linear nllcline of the net. Frthermore it was demonstrated that this effective mean vale can describe a special phase transition (from the oscillatory to the excitable net dynamics) for large nets with strong copling qite well. An effective mean parameter c dependent on σ vm can be calclated: 1 c (σ vm ) = tan[ α (σ vm )], α (σ vm ) = α(c)p(c, σ vm )dc. (7) Thereby α(c) is the gradient angle of the linear nllcline, which only depends on the parameter c. The global net parameter c (σ vm ), for the parameter vales chosen in this chapter, is presented in Fig. 7. Starting from c (0) = 3.6 the effective parameter drops with increasing strength of the mltiplicative variability, reaching a vale of 2.92 for σ vm 1.6. For a net withot variability one finds the transition from the sbexcitable to the excitable net dynamics for a parameter vale c = 2.92 for q = 20 (the transition point depends on q, see also Fig. 9). If c determines the net dynamics one wold ths expect to find a transition to the excitable behavior for σ vm 1.6 and an appropriate copling strength. For different vales of q the transition point changes, bt one wold still expect to find the transition. For σ vm 2.5 the crve c (σ vm ) has a minσ vm q FIG. 8: Snapshots of (t) from Eq. (1) after t = 40 t.., special initial conditions indce two spiral waves in an excitable net, dependent on the variability strength σ vm and the copling strength q. Other parameters: C = 3.6, σ va = 0. σ vm q excitable FIG. 9: Border of the excitable regime for Eq. (1), dependent on the variability strength σ vm and the copling strength q. (- -) Transition predicted by Eq. (7). Other parameters: C = 3.6, σ va = 0. imm and is rising for larger variability strengths. This is cased by the ctoff of P(c, σ vm ) for c < 0. To stdy the effects of the variability in parameter c nmerical simlations varying the variability strength σ vm and the copling strength q are performed. Starting with random initial conditions no pattern formation is observed in the net for the examined parameter range (5 < q < 100 and 0 < σ vm < 4). In contrast to additive variability excitation waves are not indced by variability in parameter c. The spatially niform temporally constant soltion of the net stays stable. In a next step the simlations are repeated with special initial conditions, which indce two spiral waves in the net. In a sbexcitable medim these waves die ot qickly, whereas in an excitable net the indced waves are sstained and can be observed even for long integration times. The reslts after a propagation time of 40 t.. are displayed in Fig. 8. For q 20 the indced waves are not preserved in the net for any variability strength σ vm and the net stays sbexcitable. The excitable pattern forming regime is only fond for q > 20 and an appropriate variability strength σ vm

6 6 t (t..) q=30 q=20 q= σ vm FIG. 10: Life time of two spiral waves indced by special initial conditions in a net of FitzHgh-Nagmo elements [Eq. (1)], averaged over ten realizations of the net, dependent on the variability strength σ vm for three different copling strength q. (- -) Transition predicted by Eq. (7) for copling strength q = 30. Other parameters: C = 3.6, σ va = 0. The border of the excitable regime is identified sing simlations of ten different net realizations over an total integration time of T = 100 t... The reslts, dependent on the relevant parameters σ vm and q, are presented in Fig. 9. As mentioned above the transition does not occr for copling strengths q 20. For larger copling strengths the net becomes excitable for sfficiently large variability strengths. For a strong copling q 25 the q-dependent transition to excitable net dynamics can be predicted well sing Eq. (7). Ths the effective parameter c describes the net dynamics for strong copling. Once in the excitable regime a frther increase of σ vm has a disordering inflence, leading to a destrction of the patterns and the excitable net dynamics (dependent on q). The strong dependence on the copling strength q of the observed transition can be explained. The transition is generated by the systematic effect of the mltiplicative variability [Eq. (7)]. This is however not the only effect of the variability. It also has a disordering effect, which is dominant for a weak copling, whereas for larger copling strengths this disordering effect gets more and more sppressed. In the case of a strong copling the elements are strongly bond to each other and the effective mean vale c determines the global net dynamics, for not too large variability strengths. Ths the transition to excitable dynamics can only be observed for a sfficiently large copling. For smaller copling strengths the net stays sbexcitable, bt the mltiplicative variability may nevertheless have systematic effects on the observed net dynamics. One net property, which is systematically inflenced by the variability in parameter c, is the life time of the initialized waves (Fig. 8). This life time is presented in Fig. 10 for three different vales of the copling strength q. For q = 10 no effect on the life time of the waves is observed. This is also tre for additive variability which has no systematic effect on the net (not presented in the figre). For the strong copling strength q = 30 one observes a monoton grows of the life time with increasing variability strength σ vm. At the border to the excitable net dynamics the life time tends to infinity. More interesting is the intermediate copling strength q = 20. For this vale one observes a strong increase in the life time with growing variability strength starting from σ vm = 0, bt the life time does not diverge (no transition). After a sharp maximm the life time is lessened with a frther increase of σ vm. This means that for an intermediate copling strength and an intermediate variability strength σ vm the life time of waves in a sbexcitable medim is strongly enhanced. V. SUMMARY In this contribtion it cold be shown that additive variability is able to indce pattern formation in a sbexcitable net of FHN elements. This variability-indced strctre formation exhibits a resonance-like dependence on the variability strength σ va. The response of the net to mltiplicative variability is very different. In this case no pattern formation is indced, bt one observes a systematic inflence on the net dynamics. This effect cases a transition to excitable dynamics for a copling strength q > 20 and it enhances the life time of patterns in the sbexcitable net for an intermediate copling strength. For a strong copling the effective net parameter c (σ vm ) determines the net dynamics and predicts the transition qite accrate. These reslts demonstrate that variability may play an important role in biophysical systems. Variability cold be essential for pattern formation mechanisms or stochastic resonance effects. At least it may inflence noise indced effects, sch as temporal or spatiotemporal stochastic resonance. Mltiplicative variability has in addition a systematic effect, which cold case a wide range of interesting phenomena, like variability indced phase transitions. The effects of variability have been stdied sing the paradigmatic FHN model in a rather general framework. We hope that these theoretical findings will contribte to the theory of extended systems inflenced by variability (diversity) and noise [8]. Beyond this we hope that this work may foster experimental research in this field. [1] P. Jng and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130 (1995). [2] P. Jng, A. Cornell-Bell, F. Moss, S. Kádár, J. Wang, and K. Showalter, Chaos 8, 567 (1998). [3] H. Bsch and F.Kaiser, Phys. Rev. E 67, (2003). [4] A. S. Pikovsky and J. Krths, Phys. Rev. Lett. 78, 775

7 7 (1997). [5] S. Kádár, J. Wangi, and K. Showalter, Natre 391, 770 (1998). [6] S. Alonso, I. Sendina-Nadal, V. Perez-Mnzri, J. M. Sancho, and F. Sages, Phys. Rev. Lett. 87, (2001). [7] E. A. Novikov, Sov. Phys. 20, 1920 (1964). [8] J. García-Ojalvo and J. M. Sancho, Noise in Spatially Extended Systems (Springer, Berlin, 1999). [9] E. Ullner, A. Zaikin, J. García-Ojalvo, and J. Krths, Phys. Rev. Lett. 91, (2003). [10] E. Glatt, H. Bsch, A. Zaikin, and F. Kaiser, Phys. Rev. E 73, (2006). [11] A. T. Winfree, J. Theor. Biol. 16, 15 (1967). [12] Y. Kramoto, Chemical oscillations, waves and trblence (Springer, Berlin, 1984). [13] A. Pikovsky, M. Rosenblm, and J. Krths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001). [14] Y. Braiman, J. F. Lindner, and W. L. Ditto, Natre 378, 465 (1995). [15] M.-T. Hütt, H. Bsch, and F. Kaiser, Nova Acta Leopoldina NF , 381 (2003). [16] E. Glatt, M. Gassel, and F. Kaiser, Phys. Rev. E 73, (2006). [17] C. J. Tessone, C. R. Mirasso, R. Toral, and J. D. Gnton, Phys. Rev. Lett. 97, (2006). [18] J. S. Nagmo and S. Yoshizawa, Proc. Inst. Radio Engineers 50, 2061 (1962). [19] J. Moehlis, J. Math. Biol. 52, 141 (2006). [20] A. L. Hodgkin and A. F. Hxley, J. Physiol. 117, 500 (1952).