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1、arXiv:nlin/0406047v1 nlin.CD 21 Jun 2004 Controlling spatiotemporal chaos in excitable media by local biphasic stimulation Johannes Breuerand Sitabhra Sinha The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai - 600 113, India. Controlling spatiotemporal chaos in excitable medi
2、a by applying low-amplitude perturbations locally is of immediate ap- plicability, e.g., in treating ventricular fi brillation, a fatal disturbance in the normal rhythmic functioning of the heart. We look at a mechanism of control by the local application of a series of biphasic pulses, i.e., involv
3、ing both positive and negative stimulation. This results in faster recovery of the medium, making it possible to overdrive the chaos by generating waves with frequency higher than that possible with only positive pulses. This provides the simplest and most general understanding of the eff ectiveness
4、 of biphasic stimulation in controlling fi brillation and allows designing optimal waveshapes for controlling spatiotemporal chaos. PACS numbers: 87.19.Hh, 05.45.Gg, 05.45.Jn, 82.40.Ck A characteristic feature of excitable media is the for- mation of spiral waves and their subsequent breakup into sp
5、atiotemporal chaos. Examples include catalysis of CO on Pt(110) surface 1, cAMP waves during slime mold aggregation 2, etc.Another example of obvious im- portance is the propagation of waves of electrical exci- tation along the heart wall, initiating the muscular con- tractions that enable the heart
6、 to pump blood. In fact, spiral turbulence has been identifi ed by several investi- gators as the underlying cause of certain arrhythmias, i.e., abnormal cardiac rhythms, including ventricular fi b- rillation (VF) 3, a potentially fatal condition in which diff erent regions of the heart are no longe
7、r activated co- herently. Current methods of defi brillatory treatment in- volve applying large electrical shocks to the entire heart in an attempt to drive it to the normal state. However, this is not only painful but also dangerous, as the re- sulting damage to heart tissue can form scars that act
8、 as substrates for future cardiac arrhythmias. Devising a low-amplitude control mechanism for spatiotemporal chaos in excitable media is therefore not only an excit- ing theoretical challenge but of potential signifi cance for the treatment of VF. In this paper, we propose using low-amplitude biphas
9、ic pacing, i.e., applying a sequence of alternating positive and negative pulses locally, as a robust control method for such chaos. Most of the methods proposed for controlling spa- tiotemporal chaos in excitable media involve applying perturbations either globally or over a spatially extended syst
10、em of control points covering a signifi cant proportion of the entire system 47. However, in most real situa- tions this may not be a feasible option. Further, in the specifi c context of controlling VF, a local control scheme has the advantage that it can be readily implemented with existing hardwa
11、re of the Implantable Cardioverter- Defi brillator (ICD). This is a device implanted into pa- tients at high risk from VF that monitors the heart rhythm and applies electrical treatment, when necessary, through electrodes placed on the heart wall.A low- energy control method involving ICDs should th
12、erefore aim towards achieving control of spatiotemporal chaos by applying small perturbations from a few local sources. For most of the simulations reported in this paper we have used the modifi ed Fitzhugh-Nagumo equations pro- posed by Panfi lov as a model for ventricular activation 8. For simplic
13、ity we assume an isotropic medium; in this case the model is defi ned by the two equations governing the excitability e and recovery g variables, e/t = 2e f(e) g, g/t = (e,g)(ke g). (1) The function f(e), which specifi es fast processes (e.g., the initiation of excitation) is piecewise linear: f(e)
14、= C1e, for e e2. The function (e,g), which determines the dynamics of the recovery variable, is (e,g) = 1for e e2, and (e,g) = 3for e e1and g 1 ref. This is almost impossible with purely excitatory stimuli as reported in Ref. 6; the eff ect of locally applying such perturbations is essentially limit
15、ed by refractoriness to the immediate neighborhood of the stimulation point. A simple argument shows why a negative rectangu- lar pulse decreases the refractory period for the Panfi lov model in the absence of the diff usion term. The stimu- lation vertically displaces the e-nullcline of Eq. (1) and
16、 therefore, the maximum value of g that can be attained is reduced. Consequently, the system will recover faster from the refractory state. To illustrate this, let us assume that the stimulation is applied when e e2. Then, the dynamics reduces to e = C3(e 1) g, g = 2(ke g). In this region of the (e,
17、g)-plane, for suffi ciently high g, the trajectory will be along the e-nullcline, i.e., e 0. If a pulse stimulation of amplitude A is initiated at t = 0 (say), when e = e(0),g = g(0), at a subsequent time t, e(t) = 1 + Ag(t) C3 , and g(t) = a b a b g(0)exp(bt), where, a = 2k(1+ A C3),b = 21+(k/C3).
18、The negative stimulation has to be kept on till the system crosses into the region where e 0, after which no further increase of g can occur, as dictated by the dynamics of Eq. (1). Now, the time required by the system to enter this re- gion is 1 bln a/bg(0) a/b , where = C3(1e2)+A. Therefore, this
19、time is reduced when A fc, is smaller than at higher frequencies. We also compare the performance of sinusoidal waves with rectangular pulses, adjusting the amplitudes so that they have the same energy. The for- mer is much less eff ective than the latter at lower stimu- lation frequencies, which is
20、 the preferred operating region for the control method. The eff ectiveness of overdrive control is limited by the size of the system sought to be controlled. As shown in Fig. 3, away from the control site, the generated waves are blocked by refractory regions, with the probability of block increasin
21、g as a function of distance from the site of stimulation 13. To see whether the control method is eff ective in reasonably large systems, we used it to termi- nate chaos in the two-dimensional Panfi lov model, with L = 500 14. Fig. 4 shows a sequence of images illus- trating the evolution of chaos c
22、ontrol when a sequence of biphasic rectangular pulses are applied at the center. The time necessary to achieve the controlled state, when 2 the waves from the stimulation point pervade the entire system, depends slightly on the initial state of the sys- tem when the control is switched on. Not surpr
23、isingly, we fi nd that the stimulation frequency used to impose control in Fig. 4 belongs to a range for which feff fc. Although most of the simulations were performed with the Panfi lov model , the arguments involving phase plane analysis apply in general to excitable media having a cubic-type nonl
24、inearity. To ensure that our explanation is not sensitively model dependent we obtained similar stimulation response diagrams for the Karma model 15. Some local control schemes envisage stimulating at special locations, e.g., close to the tip of the spiral wave, thereby driving the spiral wave towar
25、ds the edges of the system where they are absorbed 16.However, aside from the fact that spatiotemporal chaos involves a large number of coexisting spirals, in a practical situation it may not be possible to have a choice regarding the loca- tion of the stimulation point. We should therefore look for
26、 a robust control method which is not critically sensi- tive to the position of the control point in the medium. There have been some proposals to use periodic stimula- tion for controlling spatiotemporal chaos. For example, recently Zhang et al 12 have controlled some excitable media models by appl
27、ying sinusoidal stimulation at the center of the simulation domain. Looking in detail into the mechanism of this type of control, we have come to the conclusion that the key feature is the alternation be- tween positive and negative stimulation , i.e., biphasic pacing, and it is, therefore, a specia
28、l case of the general scheme presented here. Previous explanations of why biphasic stimulation is better than purely excitatory stimulation (that use only positive pulses), have concentrated on the response to very large amplitude electrical shocks typically used in conventional defi brillation 17,1
29、8 and have involved de- tails of cardiac cell ion channels 19.To the best of our knowledge the present paper gives the simplest and most general picture for understanding the effi cacy of the biphasic scheme using very low amplitude perturbation, as it does not depend on the details of ion channels
30、re- sponsible for cellular excitation. There are some limitations to achieving control over a large spatial domain in an excitable medium by pacing at a particular point. Under some parameter regimes, the circular waves propagating from this point may them- selves become unstable and undergo conduct
31、ion block at a distance from the origin (similar to the process out- lined in Ref. 20).In addition, the control requires a slightly higher amplitude and has to be kept on for peri- ods much longer than spatially extended control methods 6. However, these drawbacks may be overcome if we use multiple
32、stimulation points arranged so that their regions of infl uence cover the entire simulation domain. In conclusion, we have proposed a simple explanation of the effi cacy of low-amplitude biphasic stimulation in controlling spatiotemporal chaos in excitable media (e.g., VF). It is based on the compet
33、ition between the fre- quency of the applied stimulation, and the eff ective fre- quency (obtained from the dominant timescale) of chaos. The former can be increased relative to the latter only by decreasing the refractory period which is achieved by a negative stimulus prior to applying the excitat
34、ory posi- tive stimulation. Our analysis makes it possible to design pacing waveforms for maximum effi ciency in controlling chaos. We thank Sudeshna Sinha for helpful comments. J.B. would like to thank DAAD for fi nancial support. 1 M. Hildebrand, M. B ar and M. Eiswirth, Phys. Rev. Lett. 75, 1503
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38、). 11 V. Krinsky and K. I. Agladze, Physica D 8, 50 (1983). 12 H. Zhang, B. Hu and G. Hu, Phys. Rev. E 68, 026134 (2003). 13 The profi le of the stimulated wave changes as it propa- gates along the medium, from biphasic at the stimula- tion source to gradually becoming indistinguishable from a purel
39、y excitatory stimulation. As a result, far away from the source of stimulation, the response cannot have a frequency higher than 1 ref. 14 The initial condition used for this purpose is a broken plane wave which is allowed to evolve for 5000 time units into a state displaying spiral turbulence. 15 A
40、. Karma, Phys. Rev. Lett. 71, 1103 (1993). 16 V. Krinsky, F. Plaza and V. Voignier, Phys. Rev. E 52, 2458 (1995). 17 J. P. Keener and T. J. Lewis , J. Theoret. Biol. 200, 1 (1999). 18 C. Anderson and N. A. Trayanova, Math. Biosci. 174, 91 (2001). 19 J. L. Jones and O. H. Tovar, Crit. Care Med. 28 (S
41、uppl.), N219 (2000). 20 J. J. Fox, R. F. Gilmour and E. Bodenschatz, Phys. Rev. Lett. 89, 198101 (2002). 3 0.050.10.150.20.250.3 0 0.01 0.02 0.03 0.04 Stimulation frequency, f Effective frequency, feff 0102030 10 20 30 t neg ref 1:1 2:1 3:1 4:1 FIG. 1. Stimulation response diagram for one-dimensiona
42、l Panfi lov model (L = 40) for diff erent stimulation frequencies f. The dotted and broken curves represent purely excitatory pulses of amplitude A = 5, pulse duration = 0.05f1, and A = 10, = 0.1f1, respectively, while the solid curve repre- sents biphasic pulses of amplitude A = 10 and pulse durati
43、on = 0.1f1(as shown in the bottom left corner). Note that, the highest eff ective frequencies fefffor the the three cases are very diff erent. The ratio f : feff is shown for the fi rst four peaks. The inset shows the decrease in refractory period in absence of the diff usion term when a negative st
44、imulation is applied at diff erent times (tneg) after the initial excitation. 0.050.10.150.20.250.3 0.01 0.02 0.03 0.04 0.05 Stimulation frequency, f Effective frequency, feff fc 00.040.080.12 0 10 20 PSD frequency FIG. 2. Stimulation response diagram for two-dimensional Panfi lov model (L = 26) sho
45、wing relative performance of dif- ferent waveforms.The dash-dotted line represents a sinus wave (A = 6) and the solid curve represents a wave of bipha- sic rectangular pulses (A = 18.9), as shown in the bottom left corner, such that they have the same total energy. The inset shows the power spectra
46、of spatiotemporal chaos in the 2-D Panfi lov model (L = 500). The e variable was recorded for 3300 time units and the resulting power spectral density was averaged over 32 points. The peak occurs at the characteristic frequency fc 0.0427 which is indicated in the main fi gure by the broken line. Dis
47、tance from stimulation source, x Stimulation frequency, f 51015202530 0.04 0.06 0.08 0.1 0.12 0.14 1:1 2:1 3:2 3:1 4:1 6:1 fc 2 fc 3 fc FIG. 3. Distance dependence of stimulus response for dif- ferent stimulation frequencies f in the two-dimensional Pan- fi lov model. Biphasic rectangular pulses (A
48、= 18.9) having duration = 0.1f1are applied, which elicit a response hav- ing an eff ective frequency feffat a particular location. The fi rst three cells (x = 1,2,3) are within the region subject to direct stimulation. The shaded regions represent diff erent re- sponse ratios f : feff. Integral mult
49、iples of the characteristic frequency fcare indicated on the f-axis. FIG. 4. Control of spatiotemporal chaos in the two- di- mensional Panfi lov model (L = 500) by applying biphasic pulses with amplitude A = 18.9 and frequency f = 0.13 at the center of the simulation domain. The pulse shape is rect- angular, having a duration of 0.77 time units. Snapshots are shown for (top left) t = 0, (top right) t = 1000, (bottom left) t = 2700 and (bottom right) t = 3800 time units. The excitation wavefronts are shown in white, black marks the re- covered regions ready