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Controlling excitable wave behaviors through the tuning of three parameters

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Abstract

Excitable systems are a class of dynamical systems that can generate self-sustaining waves of activity. These waves are known to manifest differently under diverse conditions, whereas some travel as planar or radial waves, and others evolve into rotating spirals. Excitable systems can also form stationary stable patterns through standing waves. Under certain conditions, these waves are also known to be reflected at no-flux boundaries. Here, we review the basic characteristics of these four entities: traveling, rotating, standing and reflected waves. By studying their mechanisms of formation, we show how through manipulation of three critical parameters: time-scale separation, space-scale separation and threshold, we can interchangeably control the formation of all the aforementioned wave types.

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References

  1. Bement WM, Leda M, Moe AM, Kita AM, Larson ME, Golding AE, Pfeuti C, Su KC, Miller AL, Goryachev AB, von Dassow G (2015) Activator-inhibitor coupling between rho signalling and actin assembly makes the cell cortex an excitable medium. Nat Cell Biol 17(11):1471–83. https://doi.org/10.1038/ncb3251

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  2. Bhattacharya S, Iglesias PA (2016) The regulation of cell motility through an excitable network. IFAC-PapersOnLine 49(26):357–363

    Article  Google Scholar 

  3. Braune M, Engel H (1993) Compound rotation of spiral waves in active media with periodically modulated excitability. Chem Phys Lett 211(6):534–540

    Article  Google Scholar 

  4. Breña-Medina V, Champneys A (2014) Subcritical Turing bifurcation and the morphogenesis of localized patterns. Phys Rev E 90(3):032923

    Article  Google Scholar 

  5. Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol Cybern 95(1):1–19

    Article  CAS  Google Scholar 

  6. Cheng H, Lederer MR, Lederer W, Cannell M (1996) Calcium sparks and [Ca\(^{2+}\)]\(_{\rm i}\) waves in cardiac myocytes. Am J Physiol Cell Physiol 270(1):C148–C159

    Article  CAS  Google Scholar 

  7. Dahlem MA, Müller SC (1997) Self-induced splitting of spiral-shaped spreading depression waves in chicken retina. Exp Brain Res 115(2):319–324

    Article  CAS  Google Scholar 

  8. Davidenko JM, Pertsov AV, Salomonsz R, Baxter W, Jalife J (1992) Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature 355(6358):349–351

    Article  CAS  Google Scholar 

  9. Deneke VE, Di Talia S (2018) Chemical waves in cell and developmental biology. J Cell Biol 217(4):1193–1204

    Article  CAS  Google Scholar 

  10. Devreotes PN, Bhattacharya S, Edwards M, Iglesias PA, Lampert T, Miao Y (2017) Excitable signal transduction networks in directed cell migration. Ann Rev Cell Dev Biol 33:103–125

    Article  CAS  Google Scholar 

  11. Dockery J, Keener J (1989) Diffusive effects on dispersion in excitable media. SIAM J Appl Math 49(2):539–566

    Article  Google Scholar 

  12. Dockery JD (1992) Existence of standing pulse solutions for an excitable activator-inhibitory system. J Dyn Diff Equat 4(2):231–257

    Article  Google Scholar 

  13. Ermentrout G, Hastings S, Troy W (1984) Large amplitude stationary waves in an excitable lateral-inhibitory medium. SIAM J Appl Math 44(6):1133–1149

    Article  Google Scholar 

  14. Ermentrout GB, Rinzel J (1996) Reflected waves in an inhomogeneous excitable medium. SIAM J Appl Math 56(4):1107–1128

    Article  Google Scholar 

  15. Fife PC (1984) Propagator-controller systems and chemical patterns. In: Vidal C, Pacault A (eds) Non-equilibrium dynamics in chemical systems, pp 76–88. Springer

  16. Fife PC (2013) Mathematical aspects of reacting and diffusing systems, vol 28. Springer, Berlin

    Google Scholar 

  17. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–66

    Article  CAS  Google Scholar 

  18. Franci A, Sepulchre R (2016) A three-scale model of spatio-temporal bursting. SIAM J Appl Dyn Syst 15(4):2143–2175

    Article  Google Scholar 

  19. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    Article  CAS  Google Scholar 

  20. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–44

    Article  CAS  Google Scholar 

  21. Holmes WR, Carlsson AE, Edelstein-Keshet L (2012) Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour. Phys Biol 9(4):046005

    Article  CAS  Google Scholar 

  22. Iglesias P (2013) Excitable systems in cell motility. In: 2013 IEEE 52nd annual conference on decision and control (CDC), pp 757–762. Florence, Italy. https://doi.org/10.1109/CDC.2013.6759973

  23. Iglesias PA, Devreotes PN (2012) Biased excitable networks: how cells direct motion in response to gradients. Curr Opin Cell Biol 24(2):245–53. https://doi.org/10.1016/j.ceb.2011.11.009

    Article  CAS  PubMed  Google Scholar 

  24. Jalife J (2003) Rotors and spiral waves in atrial fibrillation. J Cardiovasc Electrophysiol 14(7):776–780

    Article  Google Scholar 

  25. Jones CK, Rubin JE (1998) Existence of standing pulse solutions to an inhomogeneous reaction-diffusion system. J Dyn Differ Equ 10(1):1–35

    Article  Google Scholar 

  26. Keener JP (1980) Waves in excitable media. SIAM J Appl Math 39(3):528–548

    Article  Google Scholar 

  27. Keener JP (1986) A geometrical theory for spiral waves in excitable media. SIAM J Appl Math 46(6):1039–1056

    Article  Google Scholar 

  28. Keener JP, Sneyd J (1998) Mathematical physiology, vol 1. Springer, New York

    Book  Google Scholar 

  29. Kondo S, Asai R (1995) A reaction-diffusion wave on the skin of the marine angelfish Pocamanthus. Nature 376(6543):765–768

    Article  CAS  Google Scholar 

  30. Kosek J, Marek M (1995) Collision-stable waves in excitable reaction-diffusion systems. Phys Rev Lett 74(11):2134

    Article  CAS  Google Scholar 

  31. Lapique L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarization. J Physiol Pathol Gen 9:620–635

    Google Scholar 

  32. Leppänen T (2005) The theory of Turing pattern formation. In: Barrio RA, Kaski KK (eds) Current topics in physics: in honor of Sir Roger J Elliott, pp 199–227. World Scientific

  33. Li G, Ouyang Q, Petrov V, Swinney HL (1996) Transition from simple rotating chemical spirals to meandering and traveling spirals. Phys Rev Lett 77(10):2105

    Article  CAS  Google Scholar 

  34. Li JH, Haim M, Movassaghi B, Mendel JB, Chaudhry GM, Haffajee CI, Orlov MV (2009) Segmentation and registration of three-dimensional rotational angiogram on live fluoroscopy to guide atrial fibrillation ablation: a new online imaging tool. Heart Rhythm 6(2):231–237

    Article  Google Scholar 

  35. Luther S, Fenton FH, Kornreich BG, Squires A, Bittihn P, Hornung D, Zabel M, Flanders J, Gladuli A, Campoy L et al (2011) Low-energy control of electrical turbulence in the heart. Nature 475(7355):235

    Article  CAS  Google Scholar 

  36. Meinhardt H (1993) A model for pattern formation of hypostome, tentacles, and foot in hydra: how to form structures close to each other, how to form them at a distance. Dev Biol 157(2):321–333

    Article  CAS  Google Scholar 

  37. Meinhardt H, de Boer PA (2001) Pattern formation in Escherichia coli: a model for the pole-to-pole oscillations of Min proteins and the localization of the division site. Proc Natl Acad Sci USA 98(25):14202–14207

    Article  CAS  Google Scholar 

  38. Miao Y, Bhattacharya S, Edwards M, Cai H, Inoue T, Iglesias PA, Devreotes PN (2017) Altering the threshold of an excitable signal transduction network changes cell migratory modes. Nat Cell Biol 19(4):329–340

    Article  CAS  Google Scholar 

  39. Mikhailov A, Davydov V, Zykov V (1994) Complex dynamics of spiral waves and motion of curves. Physica D 70(1–2):1–39

    Article  Google Scholar 

  40. Mikhailov AS, Zykov VS (1995) Spiral waves in weakly excitable media. In: Kapral R, Showalter K (eds) Chemical waves and patterns, pp 119–162. Springer

  41. Mori Y, Jilkine A, Edelstein-Keshet L (2008) Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys J 94(9):3684–3697

    Article  CAS  Google Scholar 

  42. Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35(1):193–213

    Article  CAS  Google Scholar 

  43. Muller L, Piantoni G, Koller D, Cash SS, Halgren E, Sejnowski TJ (2016) Rotating waves during human sleep spindles organize global patterns of activity that repeat precisely through the night. Elife 5(e17):267

    Google Scholar 

  44. Murray JD (2001) Mathematical biology. II spatial models and biomedical applications (Interdisciplinary applied mathematics), vol 18. Springer, New York

    Google Scholar 

  45. Murray JD (2002) Mathematical biology. I. (Interdisciplinary applied mathematics), vol 17. Springer, New York

    Google Scholar 

  46. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070

    Article  Google Scholar 

  47. Narayan SM, Krummen DE, Shivkumar K, Clopton P, Rappel WJ, Miller JM (2012) Treatment of atrial fibrillation by the ablation of localized sources: confirm (conventional ablation for atrial fibrillation with or without focal impulse and rotor modulation) trial. J Am Coll Cardiol 60(7):628–636

    Article  Google Scholar 

  48. Panfilov AV, Keener JP (1993) Effects of high frequency stimulation on cardiac tissue with an inexcitable obstacle. J Theor Biol 163(4):439–448

    Article  CAS  Google Scholar 

  49. Pertsov AM, Davidenko JM, Salomonsz R, Baxter WT, Jalife J (1993) Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. Circ Res 72(3):631–650

    Article  CAS  Google Scholar 

  50. Petrov V, Scott SK, Showalter K (1994) Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system. Philos Trans R Soc Lond A Math Phys Eng Sci 347(1685):631–642

    Article  CAS  Google Scholar 

  51. Quail T, Shrier A, Glass L (2014) Spatial symmetry breaking determines spiral wave chirality. Phys Rev Lett 113(15):158101

    Article  Google Scholar 

  52. Rinzel J, Terman D (1982) Propagation phenomena in a bistable reaction-diffusion system. SIAM J Appl Math 42(5):1111–1137

    Article  Google Scholar 

  53. Showalter K, Tyson JJ (1987) Luther’s 1906 discovery and analysis of chemical waves. J Chem Educ 64(9):742

    Article  CAS  Google Scholar 

  54. Steinbock O, Zykov V, Müller SC (1993) Control of spiral-wave dynamics in active media by periodic modulation of excitability. Nature 366(6453):322–324

    Article  CAS  Google Scholar 

  55. Takahashi K, Saleh M, Penn RD, Hatsopoulos NG (2011) Propagating waves in human motor cortex. Front Hum Neurosci 5:40

    Article  Google Scholar 

  56. Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci 237(641):37–72

    Article  Google Scholar 

  57. Tyson JJ, Keener JP (1988) Singular perturbation theory of traveling waves in excitable media (a review). Physica D 32(3):327–361

    Article  Google Scholar 

  58. Vanag VK, Epstein IR (2001) Pattern formation in a tunable medium: the Belousov-Zhabotinsky reaction in an aerosol ot microemulsion. Phys Rev Lett 87(22):228301

    Article  CAS  Google Scholar 

  59. Verschueren N, Champneys A (2017) A model for cell polarization without mass conservation. SIAM J Appl Dyn Syst 16(4):1797–1830

    Article  Google Scholar 

  60. Weiner OD, Marganski WA, Wu LF, Altschuler SJ, Kirschner MW (2007) An actin-based wave generator organizes cell motility. PLoS Biol 5(9):e221. https://doi.org/10.1371/journal.pbio.0050221

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. Wiener N, Rosenblueth A (1946) The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch Inst Cardiol Mex 16(3):205–265

    CAS  PubMed  Google Scholar 

  62. Wilson D, Moehlis J (2016) Toward a more efficient implementation of antifibrillation pacing. PloS ONE 11(7):e0158239

    Article  Google Scholar 

  63. Winfree AT (1972) Spiral waves of chemical activity. Science 175(4022):634–636

    Article  CAS  Google Scholar 

  64. Winfree AT (2001) The geometry of biological time. In: Interdisciplinary applied mathematics, vol 12. Springer, Berlin

  65. Zhabotinsky AM, Eager MD, Epstein IR (1993) Refraction and reflection of chemical waves. Phys Rev Lett 71(10):1526

    Article  CAS  Google Scholar 

  66. Zykov V (1980) Analytic evaluation of the relationship between the speed of a wave of excitation in a two-dimensional excitable medium and the curvature of its front. Biophysics 25(5):888–892

    CAS  Google Scholar 

  67. Zykov V, Krekhov A, Bodenschatz E (2017) Fast propagation regions cause self-sustained reentry in excitable media. Proc Nat Acad Sci 114(6):1281–1286

    Article  CAS  Google Scholar 

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Acknowledgements

This review arose as a consequence of a workshop on the control of cellular and molecular systems at the Mathematical Biosciences Institute at The Ohio State University, which receives major funding from the National Science Foundation Division of Mathematical Sciences. We are grateful to Yuchuan Miao, Marc Edwards and Peter Devreotes for fruitful conversations and collaboration on wave propagation in migratory cells.

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  1. Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, USA

    Sayak Bhattacharya & Pablo A. Iglesias

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  1. Sayak Bhattacharya
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  2. Pablo A. Iglesias
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Correspondence to Pablo A. Iglesias.

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Communicated by Peter J. Thomas.

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This work was supported by DARPA under contract number: HR0011-16-C-0139.

This article belongs to the Special Issue on Control Theory in Biology and Medicine. It derived from a workshop at the Mathematical Biosciences Institute, Ohio State University, Columbus, OH, USA.

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Bhattacharya, S., Iglesias, P.A. Controlling excitable wave behaviors through the tuning of three parameters. Biol Cybern 113, 61–70 (2019). https://doi.org/10.1007/s00422-018-0771-0

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