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Mathematical models for the swimming pattern of a lamprey

I. Analysis of collective oscillators with time-delayed interaction and multiple coupling

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Abstract

A significant characteristic in a swimming pattern of a lamprey is the generation of a constant phase lag along its body in spite of the wide range of undulation frequencies. In this paper, we discuss a mathematical treatment for coupled oscillators with time-delayed interaction and propose a model for the central pattern generator (CPG) of a lamprey to account for the generation of a constant phase relation, with consideration of the signal conduction time. From this model, it is suggested that the desired phase relation can be produced by long ascending connections from the tail to the neck region of the CPG.

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References

  • Buchanan JT (1982) Identification of interneurons with contralateral, caudal axons in the lamprey spinal cord: synaptic interaction and morphology. J Neurophysiol 47:961–975

    PubMed  Google Scholar 

  • Bachanan JT, Grillner S, Cullheim S, Risling M (1989) Identification of excitatory interneurons contributing to generation of locomotion in lamprey: structure, pharmacology and function. J Neurophysiol 62:59–69

    PubMed  Google Scholar 

  • Cohen AH (1987) Intersegmental coordinating system of the lamprey control pattern generator for locomotion. J Comp Physiol [A]160:181–193

    Article  Google Scholar 

  • Cohen AH, Holmes PJ, Rand RH (1982) The nature of the coupling between segmental oscillators of the lamprey spinal gene rator for locomotion: a mathematical model. J Math Biol 13:345–369

    Article  PubMed  Google Scholar 

  • Ermentrout GB, Kopell N (1984) Frequency plateaus in a chain of weakly coupled oscillators. 1. SIAM J Appl Math 15:215–237

    Article  Google Scholar 

  • Ermentrout GB, Kopell N (1990) Oscillator death in systems of coupled neural oscillators. SIAM J Appl Math 50:125–146

    Article  Google Scholar 

  • Grillner S, Matsushima T (1991) The neural network underlying locomotion in lamprey-synaptic and cellular mechanisms. Neuron7:1–15

    Article  PubMed  Google Scholar 

  • Grillner S, Buchanan JT, Wallen P, Brodin L (1988) Neural control of locomotion in lower vertebrates. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 1–40

    Google Scholar 

  • Grillner S, Wallen P, Brodin L (1991) Neuronal network generating locomotor behavior in lamprey: circuitry, transmitters, membrane, properties and simulation. Annu Rev Neurosci 14:169–199

    Article  PubMed  Google Scholar 

  • Hassard B, Wan YH (1978) Bifurcation formulae derived from center manifold theory. J Math Anal Appl 63:297–312

    Article  Google Scholar 

  • Kasicki S, Grillner S, Ohta Y, Dubuc R, Brodin L (1989) Phasic modulation of reticulospinal neurones during locomotion and other types of spinal motor activity in lamprey. Brain Res484:203–216

    Article  PubMed  Google Scholar 

  • Kopell N, Ermentrout GB (1990) Phase transitions and other phenomena in chains of coupled oscillators. SIAM J Appl Math50:1014–1052

    Article  Google Scholar 

  • Pavlidis T (1973) Biological oscillators: their mathematical analysis. Academic Press, New York

    Google Scholar 

  • Rand RH, Cohen AH, Holmes PJ (1988) Systems of coupled oscillators as models of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 333–367

    Google Scholar 

  • Wallen P, Williams TL (1984) Fictive locomotion in the lamprey spinal cord in vitro compared with swimming in the intact and spinal animals. J Physiol 347:225–239

    PubMed  Google Scholar 

  • Williams TL, Sigvardt KA, Kopell N, Ermentrout GB, Remler MP (1990) Forcing of coupled nonlinear oscillators: studies of intersegmental coordination in the lamprey locomotor central pattern generator. J Neurophysiol 64:862–871

    PubMed  Google Scholar 

  • Winfree AT (1980) Geometry of biological time. Springer, Berlin Heidelberg New York

    Google Scholar 

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Nishii, J., Uno, Y. & Suzuki, R. Mathematical models for the swimming pattern of a lamprey. Biol. Cybern. 72, 1–9 (1994). https://doi.org/10.1007/BF00206233

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  • DOI: https://doi.org/10.1007/BF00206233

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