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The fractional-order dynamics of brainstem vestibulo-oculomotor neurons


The vestibulo-ocular reflex (VOR) and other oculomotor subsystems such as pursuit and saccades are ultimately mediated in the brainstem by premotor neurons in the vestibular and prepositus nuclei that relay eye movement commands to extraocular motoneurons. The premotor neurons receive vestibular signals from canal afferents. Canal afferent frequency responses have a component that can be characterized as a fractional-order differentiation (dk x/dt k where k is a nonnegative real number). This article extends the use of fractional calculus to describe the dynamics of motor and premotor neurons. It suggests that the oculomotor integrator, which converts eye velocity into eye position commands, may be of fractional order. This order is less than one, and the velocity commands have order one or greater, so the resulting net output of motor and premotor neurons can be described as fractional differentiation relative to eye position. The fractional derivative dynamics of motor and premotor neurons may serve to compensate fractional integral dynamics of the eye. Fractional differentiation can be used to account for the constant phase shift across frequencies, and the apparent decrease in time constant as VOR and pursuit frequency increases, that are observed for motor and premotor neurons. Fractional integration can reproduce the time course of motor and premotor neuron saccade-related activity, and the complex dynamics of the eye. Insight into the nature of fractional dynamics can be gained through simulations in which fractional-order differentiators and integrators are approximated by sums of integer-order high-pass and low-pass filters, respectively. Fractional dynamics may be applicable not only to the oculomotor system, but to motor control systems in general.

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  1. Anastasio TJ (1991) Neural network models of velocity storage in the horizontal vestibulo-ocular reflex. Biol Cybern 64:187–196

  2. Anastasio TJ (1992) Implications of vestibular nucleus neuron rectification for signal processing in the horizontal vestibuloocular reflex. Ann NY Acad Sci 656:907–909

  3. Anastasio TJ, Correia MJ, Perachio AA (1985) Spontaneous and driven responses of semicircular canal primary afferents in the unanesthetized pigeon. J Neurophysiol 54:335–347

  4. Blanks RHI. Volkind R, Precht W, Baker R (1977) Response of cat prepositus hypoglossi neurons to horizontal angular acceleration. Neuroscience 2:391–403

  5. Boyle R, Goldberg JM, Highstein SM (1992) Inputs from regularly and irregularly discharging vestibular nerve afferents to secondary neurons in squirrel monkey vestibular nuclei. III. Correlation with vestibulospinal and vestibuloocular output pathways. J Neurophysiol 68:471–484

  6. Buettner UW, Büttner U, Henn V (1978) Transfer characteristics of neurons in vestibular nuclei of the alert monkey. J Neurophysiol 41:1614–1628

  7. Correia MJ, Landolt JP, Ni M-D, Eden AR, Rae JL (1981) A species comparison of linear and nonlinear transfer characteristics of primary afferents innervatig the semicircular canal. In: Gualtierotti T (ed) The vestibular system: function and morphology. Springer, Berlin Heidelberg New York, pp 280–316

  8. Delgado-García JM, del Pozo F, Baker R (1986) Behavior of neurons in the abducens nucleus of the alert cat. I. Motoneurons. Neuroscience 17:929–952

  9. Escudero M, de la Cruz RR, Delgado-García JM (1992) A physiological study of vestibular and prepositus hypoglossi neurons projecting to the abducens nucleus in the alert cat. J Physiol 458:539–560

  10. Fernandez C, Goldberg JM (1971) Physiology of peripheral neurons innervating semicircular canals of the squirrel monkey. II. Response to sinusoidal stimulation and dynamics of peripheral vestibular system. J Neurophysiol 34:661–675

  11. Fuchs AF, Kimm J (1975) Unit activity in vestibular nucleus of the alert monkey during horizontal angular acceleration and eye movement. J Neurophysiol 38:1140–1161

  12. Fuchs AF, Scudder CA, Kaneko CRS (1988) Discharge patterns and recruitment order of identified motoneurons and internuclear neurons in the monkey abducens nucleus. J Neurophysiol 60:1874–1895

  13. Gisbergen JAM van, Robinson DA, Gielen S (1981) A quantitative analysis of generation of saccadic eye movements by burst neurons. J Neurophysiol 45:417–442

  14. Goldstein HP, Robinson DA (1986) Hysteresis and slow drift in abducens unit activity. J Neurophysiol 55:1044–1056

  15. Keller EL, Kamath BY (1975) Characteristics of head rotation and eye movement-related neurons in alert monkey vestibular nucleus. Brain Res 100:182–187

  16. Keller EL, Precht W (1979) Adaptive modification of central vestibular neurons in response to visual stimulation through reversing prisms. J Neurophysiol 42:896–911

  17. Lisberger SG, Fuchs AF (1978) Role of primate flocculus during rapid behavioral modification of vestibuloocular reflex. I. Purkinje cell activity during visually guided horizontal smooth-pursuit eye movements and passive head rotation. J Neurophysiol 41:733–763

  18. Lopez-Barneo J, Darlot C, Berthoz A (1979) Functional role of the prepositus hypoglossi nucleus in the control of gaze. Prog Brain Res 50:667–679

  19. McFarland JL, Fuchs AF (1992) Discharge patterns in nucleus prepositus hypoglossi and adjacent medial vestibular nucleus during horizontal eye movement in behaving macaques. J Neurophysiol 68:319–332

  20. Oldham KB, Spanier J (1974) The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. Academic Press, New York

  21. Optican LM, Miles FA (1985) Visually induced adaptive changes in primate saccadic oculomotor control signals. J Neurophysiol 54:940–958

  22. Pastor AM, Torres B, Delgado-García JM, Baker R (1991) Discharge characteristics of medial rectus and abducens motoneurons in the goldfish. J Neurophysiol 66:2125–2140

  23. Robinson DA (1964) The mechanics of human saccadic eye movement. J Physiol 174:245–264

  24. Robinson DA (1981) The use of control systems analysis in the neurophysiology of eye movements. Ann Rev Neurosci 4:462–503

  25. Schneider LW, Anderson DJ (1976) Transfer characteristics of first and second order lateral canal vestibular neurons in gerbil. Brain Res 112:61–76

  26. Scott Blair GW (1950) Measurements of mind and matter. Dennis Dobson, London

  27. Scudder CA, Fuchs AF (1992) Physiological and behavioral identification of vestibular nucleus neurons mediating the horizontal vestibuloocular reflex in trained rhesus monkeys. J Neurophysiol 68:244–264

  28. Shinoda Y, Yoshida K (1974) Dynamic characteristics of responses to horizontal head angular acceleration in vestibuloocular pathway in the cat. J Neurophysiol 37:653–673

  29. Skavenski AA, Robinson DA (1973) Role of abducens neurons in vestibuloocular reflex. J Neurophysiol 36:724–738

  30. Thorson J, Biederman-Thorson M (1974) Distributed relaxation processes in sensory adaptation. Science 183:161–172

  31. Tobolsky AV (1960) Properties and structure of polymers. Wiley, New York

  32. Wilson VJ, Melvill Jones G (1979) Mammalian vestibular physiology. Plenum, New York

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Anastasio, T.J. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72, 69–79 (1994).

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  • Fractional Calculus
  • Fractional Dynamic
  • Fractional Differentiation
  • Vestibular Signal
  • Motor Control System