Abstract
Neuromuscular reflexes with time-delayed negative feedback, such as the pupil light reflex, have different rates depending on the direction of movement. This asymmetry is modeled by an implicit first-order delay differential equation in which the value of the rate constant depends on the direction of movement. Stability analyses are presented for the cases when the rate is: (1) an increasing and (2) a decreasing function of the direction of movement. It is shown that the stability of equilibria in these dynamical systems depends on whether the rate constant is a decreasing or increasing function. In particular, when the asymmetry has the shape of an increasing step function, it is possible to have stability which is independent of the value of the time delay or the steepness (i.e., gain) of the negative feedback.
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References
an der Heiden, U. and M. C. Mackey (1982). The dynamics of production and destruction: analytic insight into complex behavior. J. Math. Biol. 16, 75–101.
Atay, F. M. (1994). A differential-delay equation arising in the modelling of the pupil light reflex, PhD thesis, Brown University, Providence, RI.
Clynes, M. (1960). Computer dynamic analysis of the pupil light reflex: a unidirectional rate-sensitive sensor, in Proc. 3rd Int. Conf. On Medical Electronics, London: Institution of Electrical Engineers, pp. 356–358.
Clynes, M. (1961). Unidirectional rate sensitivity: a biocybernetic law of reflex and humoral systems as physiologic channels of control and communication. Ann. N. Y. Acad. Sci. 92, 949–969.
Eurich, C. W. and J. G. Milton (1996). Noise-induced transitions in human postural sway. Phys. Rev. E 54, 6681–6684.
Hadeler, K. P. and J. Tomiuk (1977). Periodic solutions of difference-differential equations. Arch. Rat. Mech. Anal. 65, 87–95.
Hale, J. K. (1977). Theory of Functional Differential Equations, Berlin: Springer-Verlag.
Longtin, A. (1988). Nonlinear oscillations, noise and chaos in neural delayed feedback. PhD thesis, McGill University, Montreal, Canada.
Longtin, A. and J. G. Milton (1988). Complex oscillations in the human pupil light reflex with ‘mixed’ and delayed feedback. Math. Biosci. 90, 183–199.
Longtin, A. and J. G. Milton (1989). Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations. Bull. Math. Biol. 51, 605–624.
Longtin, A., J. G. Milton, J. E. Bos, and M. C. Mackey (1990). Noise and critical behavior of the pupil light reflex at oscillation onset. Phys. Rev. A 41, 6992–7005.
Mackey, M. C. and L. Glass (1990). Oscillation and chaos in physiological control systems. Science 197, 287–289.
Mackey, M. C. and U. an der Heiden (1984). The dynamics of recurrent inhibition. J. Math. Biol. 19, 211–225.
Mallet-Paret, J. and R. D. Nussbaum (1986). Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Annali di Mat. Pura ed Appl. 145, 33–128.
Milton, J. G. and A. Longtin (1990). Evaluation of pupil constriction and dilation from cycling measurements. Vision Res. 30, 515–525.
Slotine, J.-J. E. and W. Li (1991). Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall.
Stark, L. (1959). Stability, oscillations, and noise in the human pupil servomechanism. Proc. IRE 47, 1925–1939.
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Atay, F.M., Mallet-Paret, J. Modeling reflex asymmetries with implicit delay differential equations. Bull. Math. Biol. 60, 999–1015 (1998). https://doi.org/10.1006/S0092-8240(98)90000-3
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DOI: https://doi.org/10.1006/S0092-8240(98)90000-3