Skip to main content

A quantitative dynamical systems approach to differential learning: self-organization principle and order parameter equations

Biological Cybernetics volume 98, pages 19–31 (2008)Cite this article

Abstract

Differential learning is a learning concept that assists subjects to find individual optimal performance patterns for given complex motor skills. To this end, training is provided in terms of noisy training sessions that feature a large variety of between-exercises differences. In several previous experimental studies it has been shown that performance improvement due to differential learning is higher than due to traditional learning and performance improvement due to differential learning occurs even during post-training periods. In this study we develop a quantitative dynamical systems approach to differential learning. Accordingly, differential learning is regarded as a self-organized process that results in the emergence of subject- and context-dependent attractors. These attractors emerge due to noise-induced bifurcations involving order parameters in terms of learning rates. In contrast, traditional learning is regarded as an externally driven process that results in the emergence of environmentally specified attractors. Performance improvement during post-training periods is explained as an hysteresis effect. An order parameter equation for differential learning involving a fourth-order polynomial potential is discussed explicitly. New predictions concerning the relationship between traditional and differential learning are derived.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  • Adams JA (1971) A closed-loop theory of motor learning. J Motor Behav 3: 111–149

    CAS  Google Scholar 

  • Amazeen PG, Amazeen E, Turvey MT (1998) Dynamics of human intersegmental coordination: theory and research. In: Rosenbaum DA, Collyer CE(eds) Timing of behavior. MIT, Cambridge, pp 237–259

    Google Scholar 

  • Beckmann H (2003) MA Thesis: Vergleich von Techniktrainingsansätzen im Kugelstoßen (in German). University of Münster, Münster

  • Beckmann H, Schöllhorn W (2003) Differential learning in shot put. In: Schöllhorn WI, Bohn C, Jäger JM, Schaper H, Alichmann M(eds) European workshop on movement sciences. Sport & Buch Strauß, Cologne, pp 68–68

    Google Scholar 

  • Beek PJ, Peper CE, Stegeman DF (1995) Dynamical models of movement coordination. Hum Movement Sci 14: 573–608

    Article  Google Scholar 

  • Beek PJ, Turvey MT (1992) Temporal patterning in cascade juggling. J Exp Psychol - Hum Percept Perform 18: 934–947

    PubMed  Article  CAS  Google Scholar 

  • Bernstein NA (1967) The coordination and regulation of movements. Pergamon, Oxford

    Google Scholar 

  • Daffertshofer A, van den Berg C, Beek PJ (1999) A dynamical model for mirror movements. Physica D 132: 243–266

    Article  Google Scholar 

  • Frank TD (2005a) Nonlinear Fokker–Planck equations: fundamentals and applications. Springer, Berlin

    Google Scholar 

  • Frank TD (2005b) On the M. Phys Rev E 72: 041703

    Article  CAS  Google Scholar 

  • Haken H (1975) Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev Mod Phys 47: 67–121

    Article  Google Scholar 

  • Haken H (1996) Principles of brain functioning. Springer, Berlin

    Google Scholar 

  • Haken H (2004) Synergetics: introduction and advanced topics. Springer, Berlin

    Google Scholar 

  • Henry RM, Rogers DE (1960) Increased response latency for complicated movements and a “memory drum” theory of neuromotor reactions. Res Quat 31: 448–457

    Google Scholar 

  • Horsthemke W, Lefever R (1984) Noise-induced transitions. Springer, Berlin

    Google Scholar 

  • Humpert V (2004) MA thesis: Vergleichende Analyse von Techniktrainingansätzen zum Tennisaufschlag (in German). University of Münster, Münster

  • Humpert V, Schöllhorn WI (2006) Vergleich von T. In: Ferrouti A, Remmert H(eds) Trainingswissenschaften im F. Czwalina, Hamburg, pp 121–124

    Google Scholar 

  • Huys R, Daffertshofer A, Beek PJ (2004) Multiple time scales and subsystem embedding in the learning of juggling. Hum Mov Sci 23: 315–336

    PubMed  Article  Google Scholar 

  • Jirsa VK, Kelso JAS (2004) Coordination dynamics: issues and trends. Springer, Berlin

    Google Scholar 

  • Kay BA (1988) The dimensionality of movement trajectories and the degrees of freedom problem. Hum Mov Sci 7: 343–364

    Article  Google Scholar 

  • Keele SW (1968) Movement control in skilled performance. Psychol Bull 70: 387–403

    Article  Google Scholar 

  • Kelso JAS (1995) Dynamic patterns—the self-organization of brain and behavior. MIT, Cambridge

    Google Scholar 

  • Neda Z, Ravasz E, Vicsek T, Brechet Y, Barabasi AL (2000) Physics of the rhythmic applause. Phys Rev E 61: 6987–6992

    Article  CAS  Google Scholar 

  • Patanarapeelert K, Frank TD, Beek PJ, Friedrich R, Tang IM (2006) Theoretical analysis of destabilization resonances in time-delayed stochastic second order dynamical systems and some implications for human motor control. Phys Rev E 73: 021901

    Article  CAS  Google Scholar 

  • Peper CE, Beek PJ (1998) Distinguishing between the effects of frequency and amplitude on interlimb coupling in tapping a 2:3 polyrhythm. Exp Brain Res 118: 78–92

    PubMed  Article  CAS  Google Scholar 

  • Peper CE, Beek PJ, van Wieringen PCW (1995) Multifrequency coordination in bimanual tapping: a symmetrical coupling and signs of supercriticality. J Exp Psychol - Hum Percept Perform 21: 1117–1138

    Article  Google Scholar 

  • Plischke M, Bergersen B (1994) Equilibrium statistical physics. World Scientific, Singapor

    Google Scholar 

  • Römer J, Schöllhorn WI, Jaitner T (2003) Differentielles lernen bei der Aufschlagannahme im Volleyball. In: Krug J, Müller T (eds) Messplätze, Messtraining, Motorisches Lernen (in German). Academia Verlag, Sankt Augustin, pp 129–133

  • Schmidt RA, Lee TD (1999) Motor control and learning: a behavioral emphasis. Human Kinetics, Champaign

    Google Scholar 

  • Schöllhorn WI (1999a) Individualität - ein vernachlässigter Parameter? (in German) Leistungssport 2:4–11

  • Schöllhorn WI (1999b) Practical consequences of biomechanically determined individuality and fluctuations on motor learning. In: Herzog W, Jinha A (eds) International Society of Biomechanics XVIIth Congress. Calgary, p 147

  • Schöllhorn WI (2000) Applications of systems dynamic principles to technique and strength training. Acta Acad Estonia 8: 25–37

    Google Scholar 

  • Schöllhorn WI, Beckmann H, Michelbrink M, Trockel M, Sechelmann M, Davids K (2006) Does noise provide a basis for unifying different motor learning theories?. Int J Sport Psychol 2: 34–42

    Google Scholar 

  • Schöllhorn WI, Sechelmann M, Trockel M, Westers R (2004) Nie das Richtige trainieren, um richtig zu spielen (in German). Leistungsport 5:13–17

    Google Scholar 

  • Scholz JP, Kelso JAS, Schöner GS (1987) Non-equilibrium phase transitions in coordinated biological motion: critical slowing down and switching time. Phys Lett A 123: 390–394

    Article  Google Scholar 

  • Schöner GS (1989) Learning and recall in a dynamic theory of coordination patterns. Biol Cybern 62: 39–54

    PubMed  Article  Google Scholar 

  • Schöner GS, Haken H, Kelso JAS (1986) A stochastic theory of phase transitions in human hand movement. Biol Cybern 53: 247–257

    PubMed  Article  Google Scholar 

  • Schöner GS, Kelso JAS (1988a) A synergetic theory of environmentally-specified and learned patterns of movement coordination. I. R. Biol Cybern 58: 71–80

    PubMed  Article  Google Scholar 

  • Schöner GS, Kelso JAS (1988b) A synergetic theory of environmentally-specified and learned patterns of movement coordination. II. Component oscillator dynamics. Biol Cybern 58: 81–89

    PubMed  Article  Google Scholar 

  • Schöner GS, Zanone PG, Kelso JAS (1992) Learning as change in coordination dynamics. J Motor Behav 64: 455–462

    Google Scholar 

  • Shea CH, Shebilske WL, Worchel S (1993) Motor learning and control. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Shea CH, Wulf G (2005) Schema theory: a critical appraisal and reevaluation. J Motor Behav 37: 85–101

    Article  Google Scholar 

  • Sternad D (2000) Debates in dynamics: a dynamical systems perspective on action and perception. Hum Mov Sci 19: 407–423

    Article  Google Scholar 

  • Sternad D, Duarte M, Katsumata H, Schaal S (2001) Dynamics of a bouncing ball in human performance. Phys Rev E 63: 011902

    Article  CAS  Google Scholar 

  • Turvey MT (1990) Coordination. Am Psychol 45: 938–953

    PubMed  Article  CAS  Google Scholar 

  • van Rossum JHA (1990) Schmidt’s schema theory: the empirical base of the variability of practice hypothesis. Hum Mov Sci 9: 387–435

    Article  Google Scholar 

  • Wallenstein GV, Kelso JAS, Bressler SL (1995) Phase transitions in spatiotemporal patterns of brain activity and behavior. Physica D 84: 626–634

    Article  Google Scholar 

  • Welminski D (2005) MA thesis: Vergleich von Techniktrainingsansätzen im leichtathletischen Hochsprung (in German). University of Münster, Münster

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149, Münster, Germany

    T. D. Frank

  2. Center for the Ecological Study of Perception and Action, University of Connecticut, 406 Babbidge Road, Storrs, CT, 06269, USA

    T. D. Frank

  3. Institute for Sport Science, University of Mainz, Albert-Schweitzer Str. 22, 55099, Mainz, Germany

    M. Michelbrink & W. I. Schöllhorn

  4. Institute of Sport Science, University of Münster, Horstmarer Landweg 62b, 48149, Münster, Germany

    H. Beckmann

Authors
  1. T. D. Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Michelbrink
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Beckmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. W. I. Schöllhorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. D. Frank.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Frank, T.D., Michelbrink, M., Beckmann, H. et al. A quantitative dynamical systems approach to differential learning: self-organization principle and order parameter equations. Biol Cybern 98, 19–31 (2008). https://doi.org/10.1007/s00422-007-0193-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-007-0193-x

Keywords

  • Learning Rate
  • Dynamical System Approach
  • Motor Control System
  • Differential Learning
  • Dynamical System Perspective