Skip to main content
SpringerLink
Log in

An elastic rod model for anguilliform swimming

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We employ a eometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor’s assumption of prescribed shape to fail, leading to time-periodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz M., Stegun I.A.(1965). Handbook of Mathematical Functions. Dover Publications, New York

    Google Scholar 

  2. Alexander R.McN.(2003). Principles of Animal Locomotion. Princeton University Press, Princeton NJ

    Google Scholar 

  3. Antman S.S.(1995). Nonlinear Problems of Elasticity. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  4. Ashby M.F., Gibson L.J., Wegst U., Olive R.(1995). The mechanical properties of natural materials. I Material property charts. Proc. R. Soc. Lond. A 450: 123-140

    Google Scholar 

  5. Bowtell G., Williams T.(1991). Anguilliform body dynamics: modelling the interaction between muscle activation and body curvature. Phil. Trans. R. Soc. Lond. B 334: 385-390

    Google Scholar 

  6. Bowtell G., Williams T.(1994). Anguilliform body dynamics: a continuum model for the interaction between muscle activation and body curvature. J. Math. Biol. 32: 83-91

    Article  MATH  Google Scholar 

  7. Buchanan T.(1992). Neural network simulations of coupled locomotor oscillators in the lamprey spinal cord. Biol. Cybern. 66, 367-374

    Article  Google Scholar 

  8. Byrd P.F., Friedman M.D.(1971). Handbook of Elliptic Integrals for Scientists and Engineers. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  9. Carling J.C., Bowtell G., Williams T.L.(1994). Swimming in the lamprey: modelling the neural pattern generation, the body dynamics and the fluid mechanics. In: Maddock L., Bone Q., Rayner J.M.V. (eds.), Mechanics and Physiology of Animal Swimming. Cambridge University Press, Cambridge, pp 119-132

    Google Scholar 

  10. Carling J.C., Williams T.L., Bowtell G.(1998). Self-propelled anguilliform swimming: Simultaneous solution of the two-dimensional Navier-Stokes equations and newton’s laws of motion. J. Exp. Biol. 201, 3143-3166

    Google Scholar 

  11. Cheng J.Y., Blickhan R.(1994). Bending moment distribution along swimming fish. J. Theor. Biol. 168, 337-348

    Article  Google Scholar 

  12. Cheng J.Y., Pedley T.J., Altringham J.D. (1998). A continuous dynamic beam model for swimming fish. Phil. Trans. R. Soc. Lond. B 353(1371): 981-997

    Article  Google Scholar 

  13. Cohen, A.H.: Personal communication (2006)

  14. Cohen A.H., Holmes P., Rand R.H.(1982). The nature of coupling between segmental oscillators of the lamprey spinal generator for locomotion: A model. J. Math Biol. 13, 345-369

    Article  MATH  MathSciNet  Google Scholar 

  15. Cohen A.H., Rossignol S., Grillner S. (eds).(1988). Neural Control of Rhythmic Movements in Vertebrates. Wiley, New York

    Google Scholar 

  16. Cohen A.H., Wallén P.(1980). The neuronal correlate of locomotion in fish. “Fictive swimming” induced in an in vitro preparation of the lamprey spinal cord. Exp. Brain. Res. 41, 11-18

    Article  Google Scholar 

  17. Coleman B.D., Dill E.H.(1992). Flexure waves in elastic rods. J. Acoustical Soc. Amer. 91, 2663-2673

    Article  MathSciNet  Google Scholar 

  18. Coleman B.D., Dill E.H., Lembo M., Lu Z., Tobias I.(1993). On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch. Rational Mech. Anal. 121, 339-359

    Article  MathSciNet  Google Scholar 

  19. Cortez R., Fauci L., Cowen N., Dillon R.(2004). Simulation of swimming organisms: Coupling internal mechanics with external fluid dynamics. Computing in Science and Engineering. 6(3): 38-45

    Article  Google Scholar 

  20. Van Dyke M.(1982). An Album of Fluid Motion. Parabolic Press, Stanford CA

    Google Scholar 

  21. Ekeberg Ö.(1993). A combined neuronal and mechanical model of fish swimming. Biol. Cybern. 69, 363-374

    MATH  Google Scholar 

  22. Ekeberg Ö., Grillner S.(1999). Simulations of neuromuscular control in lamprey swimming. Phil. Trans. R. Soc. Lond. B 354, 895-902

    Article  Google Scholar 

  23. Fauci L.J., Peskin C.S.(1988). A computational model of aquatic animal locomotion. J. Comput. Phys. 77, 85-108

    Article  MATH  MathSciNet  Google Scholar 

  24. Ghigliazza R.M., Holmes P.(2005). Towards a neuromechanical model for insect locomotion: Hybrid dynamical systems. Regul. Chaotic Dynam. 10(2): 193-225

    Article  MATH  MathSciNet  Google Scholar 

  25. Gillis G.B.(1998). Neuromuscular control of anguilliform locomotion: Patterns of red and white muscle activity during swimming in the American eel Anguilla rostrata. J. Exp. Biol. 201, 3245-3256

    Google Scholar 

  26. Grillner S., Wallén P.(2002). Cellular basis of a vertebrate locomotor system – steering, intersegmental and segmental co-ordination and sensory control. Brain Research Review 40, 92-106

    Article  Google Scholar 

  27. Grillner S., Wallén P., Brodin L., Lansner A.(1991). Neuronal network generating locomotor behavior in lamprey. Ann. Rev. Neurosci. 14, 169-199

    Article  Google Scholar 

  28. Guckenheimer J., Holmes P.(1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  29. Hagedorn P.(1981). Non-linear Oscillations. Oxford University Press, Oxford UK

    MATH  Google Scholar 

  30. Hatze H.(1977). A myocybernetic control model of skeletal muscle. Biol. Cybern. 25, 103-119

    Article  MATH  Google Scholar 

  31. Hatze H.(1978). General myocybernetic control model of skeletal-muscle. Biol. Cybern. 28, 143-157

    Article  MATH  Google Scholar 

  32. Hellgren J., Grillner S., Lansner A.(1992). Computer simulation of the segmented neural network generating locomotion in lamprey by using populations of network interneurons. Biol. Cybern. 68, 1-13

    Article  Google Scholar 

  33. Hill A.V.(1938). The heat of shortening and the dynamic constants of muscle. Philos. Trans. Roy. Soc. Lond. B 126, 136-195

    Google Scholar 

  34. Huxley A.F.(1974). Review lecture: muscular contraction. J. Physiology (London) 243, 1

    Google Scholar 

  35. Jung R., Kiemel T., Cohen A.H.(1996). Dynamic behavior of a neural network model of locomotor control in the lamprey. J. Neurophys. 75(3): 1074-1086

    Google Scholar 

  36. Kanso E., Marsden J.E., Rowley C.W., Melli-Huber J.(2005). Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15, 255-289

    Article  MathSciNet  MATH  Google Scholar 

  37. Keener J., Sneyd J.(1998). Mathematical Physiology. Springer-Verlag, New York

    MATH  Google Scholar 

  38. Lamb, H.: Hydrodynamics (sixth edn.; reprinted by Dover Publications Inc., New York). Cambridge University Press, Cambridge, UK (1932)

  39. Lighthill M.J.(1960). Note on the swimming of slender fish. J. Fluid Mech. 9, 305-317

    Article  MathSciNet  Google Scholar 

  40. Lighthill M.J.(1969). Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413-446

    Article  Google Scholar 

  41. Pedley T.J., Hill S.J.(1999) Large-amplitude undulatory fish swimming: fluid mechanics coupled to internal mechanics. J. Exp. Biol. 202, 3431-3438

    Google Scholar 

  42. Peskin C.S.(2002). The immersed boundary method. Acta Numerica 11, 479-517

    Article  MathSciNet  MATH  Google Scholar 

  43. Riener R., Quintern J.(1997). A physiologically based model of muscle activation verified by electrical stimulation. Bioelectrochem. Bioenerget. 43, 257-264

    Article  Google Scholar 

  44. Taylor G.(1952). Analysis of the swimming of long and narrow animals. Proc. Roy. Proc. Lond. A 214(1117): 158-183

    Article  MATH  Google Scholar 

  45. Thompson, W.T.: Vibration Theory and Applications. George Allen and Unwin (1965)

  46. Tytell E.D.(2004). The hydrodynamics of eel swimming. I. Wake structure. J. Exp. Biol. 207, 1825-1841

    Article  Google Scholar 

  47. Tytell E.D. (2004). The hydrodynamics of eel swimming. II. Effect of swimming speed. J. Exp. Biol. 207, 3265-3279

    Article  Google Scholar 

  48. Tytell E.D.(2004). Kinematics and hydrodynamics of linear acceleration in eels Anguilla rostrata. Proc. Roy. Proc. Lond. B 271, 2535-2541

    Article  Google Scholar 

  49. Tytell, E.D.: Personal communication (2005)

  50. ideler J.J., Hess F. (1984). Fast continuous swimming of two pelagic predators, saithe (Pollachius virens) and mackerel (Scomber scombrus): a kinematic analysis. J. Exp. Biol. 109, 209-228

    Google Scholar 

  51. Wallen P., Williams T.L.(1984). Fictive locomotion in the lamprey spinal cord in vitro compared with swimming in the intact and spinal animal. J. Physiol. 347(1): 225-239

    Google Scholar 

  52. Ward A.B., Azizi E.(2004). Convergent evolution of the head retraction escape response in elongate fishes and amphibians. Zoology 107, 205-217

    Article  Google Scholar 

  53. Williams T.L.(1992). Phase coupling by synaptic spread in chains of coupled neuronal oscillators. Science 258, 662-665

    Article  Google Scholar 

  54. Williams T.L., Bowtell G., Carling J.C., Sigvardt K.A., Curtin N.A.(1995). Interactions between muscle activation, body curvature and the water in the swimming lamprey. Soc. Exp. Biol. Symp. 49, 49-59

    Google Scholar 

  55. Williams T.L., Bowtell G., Curtin N.A.(1998). Predicting force generation by lamprey muscle during applied sinusiodal movement using a simple dynamic model. J. Exp. Biol. 201, 869-875

    Google Scholar 

  56. Williams T.L., Grillner S., Smoljaninov V.V., Wallen P., Rossignol S. (1989). Locomotion in lamprey and trout: The relative timing of activation and movement. J. Exp. Biol. 143, 559-566

    Google Scholar 

  57. Wu T.Y-T.(1961). Swimming of a waving plate. J. Fluid Mech.10, 321-344

    Article  MATH  MathSciNet  Google Scholar 

  58. Wu T.Y-T.(1971). Hydrodynamics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46, 337-355

    Article  MATH  Google Scholar 

  59. Zajac F.E. (1989). Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. CRC Crit. Rev. Lett. Biomed. Eng. 17, 359-411

    Google Scholar 

Download references

Author information

Author notes

  1. T. McMillen

    Present address: Department of Mathematics, California State University of Fullerton, Fullerton, CA, 92384, USA

Authors and Affiliations

  1. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    T. McMillen & P. Holmes

  2. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

    P. Holmes

Authors
  1. T. McMillen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Holmes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. McMillen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McMillen, T., Holmes, P. An elastic rod model for anguilliform swimming. J. Math. Biol. 53, 843–886 (2006). https://doi.org/10.1007/s00285-006-0036-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-006-0036-8

Keywords

Search

Navigation