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Biological Cybernetics

, Volume 105, Issue 2, pp 89–119 | Cite as

Revealing non-analytic kinematic shifts in smooth goal-directed behaviour

  • M. K. WeirEmail author
  • A. P. Wale
Original Paper

Abstract

How do biological agents plan and organise a smooth accurate path to shift from one smooth mode of behaviour to another as part of graceful movement that is both plastic and controlled? This paper addresses the question in conducting a novel shape analysis of approach and adjustment phases in rapid voluntary target aiming and 2-D reaching hand actions. A number of mode changing experiments are reported that investigate these actions under a range of goals and conditions. After a typically roughly aimed approach, regular projective adjustment is observed that has height and velocity kinematic profiles that are scaled copies of one another. This empirical property is encapsulated as a novel self-similar shift function. The mathematics shows that the biological shifts consist of continual deviation from their full Taylor series everywhere throughout their interval, which is a deep form of plasticity not described before. The experimental results find the same approach and adjustment strategy to occur with behavioural trajectories over the full and varied range of tested goals and conditions. The trajectory shapes have a large degree of predictability through using the shift function to handle extensive variation in the trajectories’ adjustment across individual behaviours and subjects. We provide connections between the behavioural features and results and various neural studies to show how the methodology may be exploited. The conclusion is that a roughly aimed approach followed by a specific highly plastic shift adjustment can provide a regular basis for fast and accurate goal-directed motion in a simple and generalisable way.

Keywords

Goal-directed Kinematics Non-analytic Plasticity Shift Smooth behaviour 

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References

  1. Abend W, Bizzi E, Morasso P (1982) Human arm trajectory formation. Brain 105: 331–348PubMedCrossRefGoogle Scholar
  2. Arbib M (2003) Mammalian motor control. In: Arbib M. (eds) Handbook of brain theory and neural networks. MIT Press, Cambridge, pp 110–113Google Scholar
  3. Arechavaleta G, Laumond JP, Hicheur H, Berthoz A (2006) The non holonomic nature of human locomotion: a modeling study. In: Proceedings of the IEEE/RAS-EMBS international conference on biomedical robots and biomechatronics. February 2006, Pisa, Italy, pp 158–163Google Scholar
  4. Atkeson CG, Hollerbach JM (1985) Kinematic features of unrestrained vertical arm movements. J Neurosci 5: 2318–2330PubMedGoogle Scholar
  5. Ben-Itzhak S, Karniel A (2007) Minimum acceleration criterion with constraints implies bang-bang control as an underlying principle for optimal trajectories of arm reaching movements. Neural Comput 20: 779–812CrossRefGoogle Scholar
  6. Bernstein N (1967) The coordination and regulation of movements. Pergamon, LondonGoogle Scholar
  7. Biess A, Liebermann DG, Flash T (2007) A computational model for redundant human three-dimensional pointing movements: integration of independent spatial and temporal motor plans simplifies movement dynamics. J Neurosci 27(48): 13045–13064PubMedCrossRefGoogle Scholar
  8. Bhushan N, Shadmehr R (1999) Computational nature of human adaptive control during learning of reaching movements in force fields. Biol Cybern 81: 39–60PubMedCrossRefGoogle Scholar
  9. Bullock D, Grossberg S (1988) Neural dynamics of planned arm movements: emergent invariants and speed-accuracy properties during trajectory formation. Psychol Rev 95: 49–90PubMedCrossRefGoogle Scholar
  10. Bullock D, Grossberg S, Guenther FH (1993) A self-organizing neural model of motor equivalent reaching and tool use by a multijoint arm. J Cogn Neurosci 5: 408–435CrossRefGoogle Scholar
  11. Bullock D (2003) Motoneuron recruitment. In: Arbib M (eds) Handbook of brain theory and neural networks. MIT Press, Cambridge, pp 683–686Google Scholar
  12. Burdet E, Milner TE (1998) Quantisation of human motions and learning of accurate movements. Biol Cybern 78: 307–318PubMedCrossRefGoogle Scholar
  13. Burdet E, Tee KP, Mareels I, Milner TE, Chew CM, Franklin DW, Osu R, Kawato M (2006) Stability and motor adaptation in human arm movements. Biol Cybern 94: 20–32PubMedCrossRefGoogle Scholar
  14. Chillingworth DRJ (1976) Differential topology with a view to applications. Pitman Publishing, LondonGoogle Scholar
  15. Choset H (2005) Principles of robot motion: theory, algorithms, and implementation. MIT Press, CambridgeGoogle Scholar
  16. de ‘Sperati C, Viviani P (1997) The relationship between curvature and velocity in two-dimensional smooth pursuit eye movements. J Neurosci 17: 3932–3945Google Scholar
  17. Dounskaia N (2007) Kinematic invariants during cyclical arm movements. Biol Cybern 96(2): 147–163PubMedCrossRefGoogle Scholar
  18. Dudek G, Jenkin M (2000) Computational principles of mobile robotics. Cambridge University Press, CambridgeGoogle Scholar
  19. Edgar GA (1990) Measure, topology and fractal geometry. Springer-Verlag, New York, NYGoogle Scholar
  20. Elliott D, Helsen WF, Chua R (2001) A century later: Woodworth’s (1899) two-component model of goal-directed aiming. Psychol Bull 127(3): 340–357CrossRefGoogle Scholar
  21. Falconer K (2006) Fractal geometry: mathematical foundations and applications, 2nd edn. Wiley, New YorkGoogle Scholar
  22. Flake GW (2000) The computational beauty of nature: computer explorations of fractals, chaos, complex systems and adaptation, Bradford book. MIT Press, CambridgeGoogle Scholar
  23. Flash T, Handzel AA (2007) Affine differential geometry analysis of human arm movements. Biol Cybern 96(6): 577–601PubMedCrossRefGoogle Scholar
  24. Flash T, Hogan N (1985) The coordination of arm movements: An experimentally confirmed mathematical model. J Neurosci 5(7): 1688–1703PubMedGoogle Scholar
  25. Georgopoulos AP, Kalaska JF, Massey JT (1981) Spatial trajectories and reaction times of aimed movements: effects of practice, uncertainty, and change in target location.. J Neurophysiol 46: 725–743PubMedGoogle Scholar
  26. Georgopoulos AP, Kettner RE, Schwartz AB (1988) Primate motor cortex and free arm movements to visual targets in three-dimensional space. ii coding of the direction of movement by a neuronal population. J Neurosci 8: 2928–2937PubMedGoogle Scholar
  27. Gielen CC, Houk JC, Marcus SL, Miller LE (1984) Viscoelastic properties of the wrist motor servo in man. Ann Biomed Eng 12: 599–620PubMedCrossRefGoogle Scholar
  28. Gordon J, Ghilardi MF, Ghez C (1994) Accuracy of planar reaching movements. I. Independence of direction and extent variability. Exp Brain Res 99: 97–111PubMedCrossRefGoogle Scholar
  29. Harris CM, Wolpert DM (1998) Signal-dependent noise determines motor planning. Nature 394: 780–784PubMedCrossRefGoogle Scholar
  30. Hermens F, Gielen S (2004) Posture-based or trajectory-based movement planning: a comparison of direct and indirect pointing movements. Exp Brain Res 159(3): 340–348PubMedCrossRefGoogle Scholar
  31. Hicheur H, Vieilledent S, Richardson MJE, Flash T (2005) Velocity and curvature in human locomotion along complex curved paths: a comparison with hand movements. Exp Brain Res 162(2): 145–154PubMedCrossRefGoogle Scholar
  32. Hogan N (1984) An organizing principle for a class of voluntary movements. J Neurosci 4: 2745–2754PubMedGoogle Scholar
  33. Jeffrey A (1996) Mathematics for engineers and scientists. Chapman & Hall, LondonGoogle Scholar
  34. Jordan MI (1990) Motor learning and the degrees of freedom problem. In: Jeannerod M, Hillsdale NJ (eds) Attention and performance XIII. Erlbaum, Mahwah, pp 796–836Google Scholar
  35. Karniel A, Mussa-Ivaldi FA (2003) Sequence, time, or state representation: how does the motor control system adapt to variable environments?. Biol Cybern 89: 10–21PubMedGoogle Scholar
  36. Kawato M, Maeda T, Uno Y, Suzuki R (1990) Trajectory formation of arm movements by cascade neural network model based on minimum torque-change criterion. Biol Cybern 62: 275–288PubMedCrossRefGoogle Scholar
  37. Kawato M (1996) Trajectory formation in arm movements. In: Zelaznik HN (eds) Advances in motor learning and control.. Human Kinetics, ChampaignGoogle Scholar
  38. Kawato M (1999) Internal models for motor control and trajectory planning. Curr Opin Neurobiol 9: 718–727PubMedCrossRefGoogle Scholar
  39. Zelaznik HN (1996) Behavioral analysis of trajectory formation: the speed-accuracy trade-off as a tool to understand strategies of motor-control. In: Zelaznik HN (eds) Advances in motor learning and control.. Human Kinetics, ChampaignGoogle Scholar
  40. Kirk DE (1970) Optimal control theory: an introduction. Prentice Hall, Upper Saddle River, NJGoogle Scholar
  41. Klein Breteler MD, Gielen SC, Meulenbroek RG (2001) End-point constraints in aiming movements: effects of approach angle and speed. Biol Cybern 85(1): 65–75PubMedCrossRefGoogle Scholar
  42. Kraizlis RJ, Stone SL (2003) Pursuit eye movements. In: The handbook of brain theory and neural networks, MIT Press, Cambridge, pp 929–934Google Scholar
  43. Lacquaniti F, Terzuolo CA, Viviani P (1983) The law relating kinematic and figural aspects of drawing movements. Acta Psychologica 54: 115–130PubMedCrossRefGoogle Scholar
  44. Lee N, Erickson M, Cherveny P (2002) Measurement of the behavior of a golf club during the golf swing. In: Routledge E (ed) Science and golf IV. Proceedings of the world scientific congress of golfGoogle Scholar
  45. Lewis FL (1992) Applied optimal control and extimation. Prentice Hall, Upper Saddle RiverGoogle Scholar
  46. Liebermann DG, Biess A, Friedman J, Gielen CCAM, Flash T (2006) Intrinsic joint kinematic planning. I: Reassessing the Listing’s law constraint in the control of three-dimensional arm movements, experimental brain research, vol 171, number 2. Springer, Berlin/HeidelbergGoogle Scholar
  47. Macki J, Strauss A (1982) An introduction to optimal control theory. Springer Verlag, New YorkCrossRefGoogle Scholar
  48. Mandelbrot B (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science (New Series) 156(3775): 636–638CrossRefGoogle Scholar
  49. Merryfield KG (1992) A nowhere analytic C function. Missouri J Math Sci 4(3)(Fall 1992):132–138Google Scholar
  50. Meyer DE, Abrams RA, Kornblum S, Wright CE, Smith JEK (1988) Optimality in human motor performance: Ideal control of rapid aimed movements. Psychol Rev 95: 340–370PubMedCrossRefGoogle Scholar
  51. Milner TE (1992) A model for the generation of movements requiring endpoint precision. Neuroscience 49(2): 487–496PubMedCrossRefGoogle Scholar
  52. Morasso P (1981) Spatial control of arm movements. Exp Brain Res 42: 223–227PubMedCrossRefGoogle Scholar
  53. Nakano E, Imamizu H, Osu R, Uno Y, Gomi H, Yoshioka T, Kawato M (1999) Quantitative examinations of internal representations for arm trajectory planning: minimum commanded torque change model. J Neurophysiol 81(5): 2140–2155PubMedGoogle Scholar
  54. Nelson W (1983) Physical principles for economies of skilled movements. Biol Cybern 46: 135–147PubMedCrossRefGoogle Scholar
  55. Novak KE, Miller LE, Houk JC (2000) Kinematic properties of rapid hand movements in a knob turning task. Exp Brain Res 132: 419–433PubMedCrossRefGoogle Scholar
  56. Novak KE, Miller LE, Houk JC (2002) The use of overlapping submovements in the control of rapid hand movements. Exp Brain Res 144: 351–364PubMedCrossRefGoogle Scholar
  57. Novak KE, Miller LE, Houk JC (2003) Features of motor performance that drive adaptation in rapid hand movements. Exp Brain Res 148: 388–400PubMedGoogle Scholar
  58. Patton JL, Mussa-Ivaldi FA (2004) Robot-assisted adaptive training: custom force fields for teaching movement patterns. IEEE Trans Biomed Eng 51: 636–646PubMedCrossRefGoogle Scholar
  59. Paul R (1981) Robot manipulators, mathematics, programming, and control. MIT Press, Cambridge MAGoogle Scholar
  60. Pellionisz A, Llinas R (1980) Tensorial approach to the geometry of brain function: cerebellar coordination via a metric tensor. Neuroscience 5: 1125–1136PubMedCrossRefGoogle Scholar
  61. Prunescu M (2009) Self-similar carpets over finite fields. Eur J Comb 866–878Google Scholar
  62. Richardson MJ, Flash T (2002) Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis. J Neurosci 22(18): 8201–8211PubMedGoogle Scholar
  63. Rizzolatti G, Luppino G (2003) Grasping movements: visuomotor transformations. In: Arbib M. (eds) Handbook of brain theory and neural networks. MIT Press, Cambridge, pp 501–504Google Scholar
  64. Saltzman EL (1979) Levels of sensorimotor representation. J Math Psychol 20: 91–163CrossRefGoogle Scholar
  65. Schaal S, Sternad D (2001) Origins and violations of the 2/3 power law in rhythmic 3D movements. Exp Brain Res 136: 60–72PubMedCrossRefGoogle Scholar
  66. Schaal S (2003) Arm and hand movement control. In: Arbib MA (eds) The handbook of brain theory and neural networks, 2nd edn. MIT Press, Cambridge, MA, pp 110–113Google Scholar
  67. Schubert H (1968) Topology. Macdonald & Co, LondonGoogle Scholar
  68. Schwartz AB (1993) Primate motor cortex and free arm motor cortical activity during drawing movements: population response during sinusoid tracing. J Neurophysiol 70: 28–36PubMedGoogle Scholar
  69. Schwartz AB (1994) Direct cortical representation of drawing. Science 265: 540–542PubMedCrossRefGoogle Scholar
  70. Shadmehr R, Mussa-Ivaldi FA (1994) Adaptive representation of dynamics during learning of a motor task. J Neurosci 14: 3208–3224PubMedGoogle Scholar
  71. Shadmehr (2003) Equilibrium point hypothesis. In: Arbib M (eds) Handbook of brain theory and neural networks.. MIT Press, Cambridge, pp 409–412Google Scholar
  72. Soechting JF, Lacquaniti F (1981) Invariant characteristics of a pointing movement in man. J Neurosci 1: 710–720PubMedGoogle Scholar
  73. Sternad D, Schaal S (1999) Segmentation of endpoint trajectories does not imply segmented control. Exp Brain Res 124(1): 118–136PubMedCrossRefGoogle Scholar
  74. Todorov E, Jordan MI (2002) Optimal feedback control as a theory of motor coordination. Nat Neurosci 5: 1226–1235PubMedCrossRefGoogle Scholar
  75. Todorov E (2004) Optimality principles in sensorimotor control. Nat Neurosci 7: 907–915PubMedCrossRefGoogle Scholar
  76. Uno Y, Kawato M, Suzuki R (1989) Formation and control of optimal trajectory in human multi-joint arm movement. Biol Cybern 61: 89–101PubMedCrossRefGoogle Scholar
  77. van Hemmen JL, Schwartz AB (2008) Population vector code: a geometric universal as actuator. Biol Cybern 98: 509–518PubMedCrossRefGoogle Scholar
  78. Van Horn JD (2003) Imaging the motor brain. In: Arbib M (eds) Handbook of brain theory and neural networks. MIT Press, Cambridge, pp 556–562Google Scholar
  79. Viviani P, Terzuolo CA (1982) Trajectory determines movement dynamics. Neuroscience 7: 431–437PubMedCrossRefGoogle Scholar
  80. Von Koch H (1904) On a continuous curve without tangents, constructible from elementary geometry. In: Classics on fractals. (Westview Press, 2004), Boulder, pp 25–45Google Scholar
  81. Wale AP, Weir MK (2002) Measurement and design of goal-directed behavior. In: Proceedings of the seventh neural computation and psychology workshop. World Scientific, Singapore, pp 78–89Google Scholar
  82. Wale AP (2006) Non-analytic shifts in smooth goal-directed human behavior. PhD Thesis, St Andrews UniversityGoogle Scholar
  83. Weir DJ, Stein JF, Miall RC (1989) Cues and control strategies in visually guided tracking. J Mot Behav 21(3): 185–204PubMedGoogle Scholar
  84. Weir MK (1984) Goal-directed behavior, studies in cybernetics. Gordon and Breach, New YorkGoogle Scholar
  85. Weir MK, Wale AP (2003) Smooth shifts in animate goal-directed action. Invited talk to the annual meeting of the computing section of the British Psychological SocietyGoogle Scholar
  86. Weir MK, Wale AP (2005) Finding cognitive action strategies through smooth kinematic shifts. Invited talk to the annual meeting of the computing section of the British Psychological SocietyGoogle Scholar
  87. Wilson VJ, Melvill Jones G (1979) Mammalian vestibular physiology. Plenum Press, New YorkGoogle Scholar
  88. Woodworth RS (1899) The accuracy of voluntary movement. Psychol Rev Monogr 3(13): 1–106Google Scholar
  89. Wu C-H, Houk JC, Young KY, Miller LE (1990) Nonlinear damping of limb motion. In: Winters JM, Woo SL-Y (eds) Multiple muscle systems: biomechanics and movement organization. Springer, Heidelberg, Berlin, New York, pp 214–235Google Scholar
  90. Zelaznik HN, Hawkins B, Kisselburgh L (1983) Rapid visual feedback processing in single-aiming movements. J Mot Behav 23: 75–85Google Scholar
  91. Zelaznik HN (ed) (1996) Advances in motor learning and control. Human Kinetics, Champaign, USAGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Computer ScienceSt. Andrews UniversitySt. AndrewsUK

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