Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Nonlinear analysis with an arbitrary stimulus ensemble

Authors: Jonathan D. Victor and Bruce W. Knight
Journal: Quart. Appl. Math. 37 (1979), 113-136
MSC: Primary 92A09
DOI: https://doi.org/10.1090/qam/542986
MathSciNet review: 542986
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Abstract | References | Similar Articles | Additional Information

Abstract: A family of Wiener-type methods is discussed in a general context. These methods share the concept of expansion of an unknown transducer as an orthogonal series. The terms of the series are drawn from a hierarchy of subspaces of transducers that are orthogonal with respect to a particular stimulus ensemble. Choices of specific stochastic ensembles lead to previously described analytical methods, including the classical one of Wiener.

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  • [1] M. Abramowitz and I. Stegun, eds., Handbook of mathematical functions, Dover, N.Y., 1959
  • [2] J. F. Barrett, The use of functionals in the analysis of nonlinear physical systems, J. Electronics and Control 15, 567-615 (1963)
  • [3] E. Bedrosian and S. O. Rice, The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs, Proc. IEEE 59, 1688-1707 (1971) MR 0396037
  • [4] A. S. Besicovitch, Almost periodic functions, Dover, N.Y., 1954 MR 0068029
  • [5] M. Biederman-Thorson and J. Thorson, Dynamics of excitation and inhibition in the light-adapted Limulus eye in situ, J. Gen. Physiol. 58, 1-19 (1971)
  • [6] H. W. Bode, Network analysis and feedback amplifier design, D. van Nostrand Company, N.Y., 1945
  • [7] A. Borsellino, R. E. Poppele, and C. A. Terzuolo, Transfer functions of the slowly adapting stretch receptor organ of Crustacea, Cold Spring Harbor Symp. Quant. Biol. 30, 581-586 (1965)
  • [8] S. E. Brodie, B. Knight, and F. Ratliff, The response of the Limulus retina to moving stimuli: a prediction by Fourier synthesis, J. Gen. Physiol. 72, 129-166 (1978)
  • [9] S. E. Brodie, B. Knight, and F. Ratliff, The spatiotemporal transfer function of the Limulus lateral eye, J. Gen. Physiol. 72, 167-202 (1978)
  • [10] R H. Cameron and W. T. Martin, The orthogonal development of nonlinear functionals in a series of Fourier-Hermite functionals. Ann. Math. 48, 385-392 (1947) MR 0020230
  • [11] J. Cooley and J. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Computation 19, 297-301 (1965) MR 0178586
  • [12] H. deLange, Attenuation characteristics and phase-shift characteristics of the human fovea-cortex systems in relation to flicker-fusion phenomena, Doctoral dissertation, Technische Hogeschool, Delft, 1957
  • [13] R. DeVoe, Linear superposition of retinal action potentials to predict electrical flicker responses from the eye of the wolf spider, Lycosa baltimoriana (Keyserling), J. Gen. Physiol. 46, 75-96 (1962)
  • [14] F. Dodge, R. Shapley, and B. Knight, Linear systems analysis of the Limulus retina, Behav. Sci. 15, 24-36 (1970)
  • [15] H. Duifhuis, Cochlear nonlinearity and second filter: possible mechanism and implications, J. Acoust. Soc. Am. 59, 408-423 (1976)
  • [16] J. F. Fohlmeister, R. E. Poppele, and R L. Purple, Repetitive firing: a quantitative study of feedback in model encoders. J. Gen. Physiol. 69, 815-848 (1977)
  • [17] J. F. Fohlmeister, R. E. Poppele, and R. L. Purple, Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus, J. Gen. Physiol. 69, 849-877 (1977)
  • [18] A. S. French, Practical nonlinear system analysis by Wiener kernel estimation in the frequency domain, Biol. Cybernetics 24, 111-119 (1976)
  • [19] A. S. French and R. K. S. Wong, Nonlinear analysis of sensory transduction in an insect mechanoreceptor, Biol. Cybernetics 26, 231-240 (1977)
  • [20] K. Fukurotani, K.-I. Hara, and Y. Oomura, Dynamic characteristics of the receptive field of L-cells in the carp retina, Vision Res. 15, 1403-1405 (1975)
  • [21] E. Hille and R. S. Phillips, Functional analysis and semigroups, American Mathematical Society, Providence, R.I., 1957 MR 0089373
  • [22] S. Hochstein and R. Shapley, Linear and nonlinear spatial submits in Y cat retinal ganglion cells, J. Physiol. 262, 265-284 (1976)
  • [23] G. W. Hughes and L. M affei, Retinal ganglion cell response to sinusoidal light stimulation, J. Neurophysiol. 29, 333-352 (1966)
  • [24] H. E. Ives, A theory of intermittent vision, J. Opt. Soc. Am. 6, 343-361 (1922)
  • [25] S. Klein and S. Y asui, Nonlinear systems analysis with non-Gaussian white stimuli: general basis functionals and kernels, IEEE Trans. Inf. Theory, in press (1979)
  • [26] B. W. Knight, J.-I. Toyoda, and F. A. Dodge, A quantitative description of the dynamics of excitation and inhibition in the eye of Limulus, J. Gen. Physiol. 36, 421-437 (1970)
  • [27] H. I. Krausz, Identification of nonlinear systems using random impulse trains, Biol. Cybernetics 19, 217-230 (1975)
  • [28] H. I. Krausz and W. O. Friesen, The analysis of nonlinear synaptic transmission, J. Gen. Physiol. 70, 243-265 (1977)
  • [29] Y. N. Lee and M. Schetzen, Measurement of the kernels of a nonlinear system by cross-correlation, Int. J. Control 2, 237-254 (1965)
  • [30] E. D. Lipson, White noise analysis of Phycomyces light growth response system I., II., III, Biophys. J. 15, 989-1045 (1975)
  • [31] G. Marchesini and G. Picci, Sull' identificazione funzionale di sistemi nonlinearri in regime periodico, Rendicotti dell' A.E.I. (1969)
  • [32] P. Z. Marmarelis and K.-I. Naka, White noise analysis of a neuron chain: An application of the Wiener theory. Science 175, 1276-1278 (1972)
  • [33] P. Z. Marmarelis and K.-I. Naka, Nonlinear analysis of receptive field responses in the catfish retina, II. One-input white noise analysis, J. Neurophysiol. 36, 619-633 (1973)
  • [34] V. Z. Marmarelis, A family of quasi-white random signals and its optimum use in biological system identification I. Theory, Biol. Cybernetics 27, 49-56 (1977)
  • [35] G. D. McCann, Nonlinear identification theory models for successive stages of visual nervous systems of flies, J. Neurophysiol. 37, 869-895 (1974)
  • [36] A. R. Møller, Statistical evaluation of the dynamic properties of cochlear nucleus units using stimuli modulated with pseudorandom noise, Brain Res. 57, 443-456 (1973)
  • [37] K.-I. Naka, P. Z. Marmarelis, and R. Y. Chan, Morphological and functional identification of catfish retinal neurons, III. Functional identification, J. Neurophysiol. 38, 92-131 (1975)
  • [38] G. Palm and T. Poggio, The Volterra representation and the Wiener expansion: validity and pitfalls, SIAM J. Appl. Math., 33, 195-216 (1977) MR 0452959
  • [39] G. Palm and T. Poggio, Stochastic identification methods for nonlinear systems: An extension of the Wiener theory, personal communication (1977) MR 0476117
  • [40] R. B. Pinter, Sinusoidal and delta function responses of visual cells of the Limulus eye, J. Gen. Physiol. 49, 565-593 (1966)
  • [41] R. A. Price, A useful theorem for nonlinear devices having Gaussian inputs, IRE Trans. Information Theory IT-4, 69-72 (1958)
  • [42] J. W. S. Pringle and V. J. Wilson, The response of a sense organ to a harmonic stimulus, J. Exp. Biol. 29, 220-234 (1952)
  • [43] R. L. Purple and F. A. Dodge, Interaction of excitation and inhibition in the eccentric cell in the eye of Limulus, Cold Spring Harbor Symp. Quant. Biol. 30, 529-537 (1965)
  • [44] F. Ratliff, B. Knight, F. Dodge, and H. K. Hartline, Fourier analysis of dynamics of excitation and inhibition in the eye of Limulus: amplitude, phase, and distance, Vision Res. 14, 1155-1168 (1974)
  • [45] A. Sandberg and L. Stark, Wiener G-functional analysis as an approach to nonlinear characteristics of human pupil light reflex, Brain Res. 11, 194-211 (1968)
  • [46] R. Shapley and J. Victor, The effect of contrast on the transfer properties of cat retinal ganglion cells, J. Physiol. 285, 275-298 (1978)
  • [47] L. Sirovich and I. Abramov, Photopigments and pseudopigments. Vision Res. 17, 5-16 (1977)
  • [48] H. Spekreijse, Rectification in the goldfish retina: analysis by sinusoidal and auxiliary stimulation, Vision Res. 9, 1461-1472 (1969)
  • [49] H. Spekreijse, O. Estevez, and D. Reits, Visual evoked potentials and the physiological analysis of visual processes in man, in J. E. Desmedt (ed.), Visual evoked potentials in man. Clarendon Press, Oxford, 1977
  • [50] G. J. St.-Cyr and D. H. Fender, Nonlinearities of the human oculomotor system: gain. Vision Res. 9, 1235-1246 (1969)
  • [51] G. Szego, Orthogonal polynomials, American Mathematical Society, Providence, R. I., 1939
  • [52] J. Thorson and M. Biederman-Thorson, Distributed relaxation processes in sensory adaptation, Science 183, 161-172 (1974)
  • [53] J. Victor, Nonlinear systems analysis: comparison of white noise and sum of sinusoids in a biological system, Proc. Nat. Acad. Sci. USA 76, 996-998 (1979)
  • [54] J. Victor and R. Shapley, A method of nonlinear analysis in the frequency domain, Biophys. J., submitted (1979)
  • [55] J. Victor, R. Shapley, and B. Knight, Nonlinear analysis of cat retinal ganglion cells in the frequency domain, Proc. Nat. Acad. Sci. USA 74, 3068-3072 (1977)
  • [56] V. Volterra, Theory of functionals and of integral and integro-differential equations, London and Glasgow: Blackie and Sons, N.Y.: Dover, N.Y., 1959 MR 0100765
  • [57] W. von Seelen and K. P. Hoffman, Analysis of neuronal networks in the visual system of the cat using statistical signals, Biol. Cybernetics 22, 7-20 (1976)
  • [58] N. Wiener, Nonlinear problems in random theory, M.I.T. Press and Wiley, N.Y., 1965 MR 0100912
  • [59] V. J. Wilson, B. Peterson, K. Fukushima, N. Hirai, and Y. Uchino, Analysis of vestibulocollic reflexes by sinusoidal polarization of vestibular afferent fibers, J. Neurophysiol. 42, 331-346 (1979)
  • [60] S. Yasui, Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis, IEEE Trans. Auto. Control, in press (1979) MR 528518
  • [61] L. R. Young and L. Stark, Variable feedback experiments testing a sampled data model for eye tracking movements, IEEE Trans. HFE-4, 38-51 (1963)

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DOI: https://doi.org/10.1090/qam/542986
Article copyright: © Copyright 1979 American Mathematical Society

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