Phase Dynamics on the Modified Oscillators in Bipedal Locomotion

Chapter
Part of the Mathematics for Industry book series (MFI, volume 4)

Abstract

Based on neurophysiological evidence, studies modeling human locomotion system have shown that a bipedal walking is generated by mutual entrainments between the oscillatory activities of a central pattern generator (CPG) and a Body. The walking model could well reproduce human walking. However, it has been also shown that time delay in the sensorimotor loop destabilizes mutual entrainments, which leads a failure to walk. Recently, theoretical studies have discovered a phenomenon in which a CPG can induce the phase of its oscillatory activity to shift forward according to time delay. This self-organized phenomenon overcoming time delay is called “flexible-phase locking”. Then, theoretical studies have hypothesized that one of the essential mechanisms to yield of flexible-phase locking is a stable limit cycle of CPG activity. This study demonstrates the hypothesis in walking models through computer simulation by replacing the CPG model with the one having different oscillation properties.

Keywords

CPG Body Time delay Limit cycle Flexible-phase locking 

References

  1. 1.
    Hammond PH (1956) The influence of prior instruction to the subject on an apparently involuntary neuro-muscular response. J Physiol 132:17–18Google Scholar
  2. 2.
    Chan CWY, Melvill JG, Kearney RE, Watt DG et al (1979) The late electromyographic response to limb displacement in man. I. Evidence for supraspinal contribution. Electroencephalogr Clin Neurophysiol 46:173–181CrossRefGoogle Scholar
  3. 3.
    Chan CWY, Melvill JG, Kearney RE, Watt DG et al (1979) The late electromyographic response to limb displacement in man. ii. sensory origin. Electroencephalogr Clin Neurophysiol 46:182–188CrossRefGoogle Scholar
  4. 4.
    Shinoda Y, Yamaguchi T, Futami T et al (1986) Multiple axon collaterals of single corticospinal axons in the cat spinal cord. J Neorophysiol 55:425–448Google Scholar
  5. 5.
    Taga G, Yamaguchi Y, Shimizu H et al (1991) Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol Cybern 65:147–159CrossRefMATHGoogle Scholar
  6. 6.
    Grillner S (1985) Neurobiological bases of rhythmic motor acts in vertebrates. Science 228:143–149CrossRefGoogle Scholar
  7. 7.
    Taga G (1994) Emergence of bipedal locomotion through entrainment among the neuro-musculo-skeltal system and environment. Phys D 75:190–208CrossRefMATHGoogle Scholar
  8. 8.
    Ohgane K, Ei SI, Kudo K, Ohtsuki T et al (2004) Emergence of adaptability to time delay in bipedal locomotion. Biol Cybern 90:125–132CrossRefMATHGoogle Scholar
  9. 9.
    Matsuoka K (1985) Sustained oscillations generated by mutually inhibiting neurons with adaptation. Biol Cybern 52:367–376CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Matsuoka K (1987) Mechanisms of frequency and pattern control in the neural rhythm generators. Biol Cybern 56:345–353CrossRefGoogle Scholar
  11. 11.
    Fitzhugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys 1:445–466Google Scholar
  12. 12.
    Ohgane K, Ei SI, Mahara H (2009) Neuron phase shift adaptive to time delay in locomotor control. Appl Math Model 33:797–811CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ei SI (2004) A remark on the interpretation of periodic solutions. Bull Jpn Soc Ind Appl Math 14(1):35–47Google Scholar
  14. 14.
    Ei SI, Ohgane K (2011) A new treatment for periodic solutions and coupled oscillators. Kyushu J Math 65:197–217CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Weng WW, Ei SI, Ohgane K (2012) The functional roles of time delay on flexible phase-locking in bipedal locomotion. J Math-for-Industry 4:123–133MathSciNetGoogle Scholar
  16. 16.
    Van der Pol B (1920) A theory of the amplitude of free and forced triode vibrations. Radio Rev 1(701–710):754–762Google Scholar
  17. 17.
    Caruso G, Labianca O, Ferrannini E et al (1973) Effect of ischemia on sensory potentials of normal subjects of different ages. J Neurol Neurosurg Psychiatry 36:455–466CrossRefGoogle Scholar
  18. 18.
    Mankovskij NB, Timko NA et al (1973) Age-related characteristics of the functional condition of the neoromuscular system. Z Alterforsch 27:191–200Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  3. 3.Graduate School of Human SciencesKinjo Gakuin UniversityMoriyama-ku, AichiJapan

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