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.

**[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]**Edward Bedrosian and Stephen O. Rice,*The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs*, Proc. IEEE**59**(1971), no. 12, 1688–1707. MR**0396037****[4]**A. S. Besicovitch,*Almost periodic functions*, Dover Publications, Inc., New York, 1955. 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 non-linear functionals in series of Fourier-Hermite functionals*, Ann. of Math. (2)**48**(1947), 385–392. MR**0020230**, https://doi.org/10.2307/1969178**[11]**James W. Cooley and John W. Tukey,*An algorithm for the machine calculation of complex Fourier series*, Math. Comp.**19**(1965), 297–301. MR**0178586**, https://doi.org/10.1090/S0025-5718-1965-0178586-1**[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]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. 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**(1977), no. 2, 195–216. MR**0452959**, https://doi.org/10.1137/0133012**[39]**G. Palm and T. Poggio,*Stochastic identification methods for nonlinear systems: an extension of the Wiener theory*, SIAM J. Appl. Math.**34**(1978), no. 3, 524–534. MR**0476117**, https://doi.org/10.1137/0134041**[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]**Vito Volterra,*Theory of functionals and of integral and integro-differential equations*, With a preface by G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by E. Whittaker, Dover Publications, Inc., New York, 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]**Norbert Wiener,*Nonlinear problems in random theory*, Technology Press Research Monographs, The Technology Press of The Massachusetts Institute of Technology and John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. 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]**Syozo Yasui,*Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis*, IEEE Trans. Automat. Control**24**(1979), no. 2, 230–242. MR**528518**, https://doi.org/10.1109/TAC.1979.1101990**[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