Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T21:30:58.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Volterra representation and Wiener-like identification of nonlinear systems: scope and limitations

Published online by Cambridge University Press:  17 March 2009

Günther Palm  and
Bertram Pöpel
Günther Palm
Affiliation:
Max-Planck-Institute for biological Cybernetics, Spemannstraβe 38, 7400 Tübingen, F.R.G.
Bertram Pöpel
Affiliation:
Department of Physiology, Freie Universität Berlin, Berlin, F.R.G.
Get access Rights & Permissions [Opens in a new window]

Extract

After the work of Marmarelis & Naka (1972, 1973) in the catfish retina, systems analysis using stochastic stimuli has had a boom in the seventies (e.g. McCann & Marmarelis, 1975; Eckert & Bishop, 1975; French & Wong, 1977; Lipson, 1975; McCann, 1974; Naka, Marmarelis & Chan, 1975; Spekreijse & Reits, 1982; Trimble & Phillips, 1978; Terzuolo et al. 1982). White-noise analysis was considered to be a general tool for investigating nonlinear systems gaining a maximum of information with a minimum of assumptions about the system. The modification of the original Wiener theory (Wiener, 1958; Cameron & Martin, 1947; McKean, 1972) by Lee & Schetzen (1965) made the theory fairly easy to implement into widely available computers and thus accessible to a larger number of experimenters.

Type
Research Article
Information
Quarterly Reviews of Biophysics , Volume 18 , Issue 2 , May 1985 , pp. 135 - 164
Copyright
Copyright © Cambridge University Press 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aertsen, A. M. H. J. & Johannesma, P. I. M. (1981 a). The spectra-temporal receptive field. A functional characteristic of auditory neurons. Biol. Cybernet. 42, 133–143.CrossRefGoogle Scholar
Aertsen, A. M. H. J. & Johannesma, P. I. M. (1981 b). A comparison of the spectro-temporal sensitivity of auditory neurons to tonal and natural stimuli. Biol. Cybernet. 42, 145–156.CrossRefGoogle ScholarPubMed
Alper, P. & Poortvliet, D. C. J. (1964). On the use of Volterra series representation and higher order impulse responses for nonlinear systems. Rev. Tijdschrift. 6, 19–33.Google Scholar
Arthur, R. M. (1976). Harmonic analysis of two-tone discharge patterns in cochlear nerve fibers. Biol. Cybernet. 22, 21–31.CrossRefGoogle ScholarPubMed
Baker, C. L. (1978). Nonlinear systems analysis of computer models of repetitive firing. Biol. Cybernet. 29, 115–123.CrossRefGoogle ScholarPubMed
Barrett, J. F. (1963). The use of functionals in the analysis of nonlinear physical systems. J. Electron. Control. 15, 567–615.CrossRefGoogle Scholar
Bedrosian, E. & Rice, S. O. (1971). The output properties of Volterra systems (nonlinear systems with memory). Proc. IEEE 59(12), 1688–1707.CrossRefGoogle Scholar
Billings, S. A. & Fakhouri, S. Y. (1978). Identification of a class of nonlinear systems using correlation analysis. Proc. IEEE. 125, 691–697.Google Scholar
Boer, E. De (1979). Cross correlation function of a bandpass nonlinear network. Proc. IEEE. 64, 1443–1444.CrossRefGoogle Scholar
Boer, E. De (1979). Polynomial correlation. Proc. IEEE. 67, 317–318.CrossRefGoogle Scholar
Brillinger, D. R. (1970). The identification of polynomial systems by means of higher order spectra. J. Sound Vib. 12 (3), 301–313.CrossRefGoogle Scholar
Brillinger, D. R. (1975). The identification of point process systems. Ann. Probability 2 (6), 909–929.Google Scholar
Bussgang, J. J., Ehrman, L. & Graham, J. W. (1974). Analysis of nonlinear systems with multiple inputs. Proc. IEEE 62 (8), 1088–1119.CrossRefGoogle Scholar
Cameron, R. H. & Martin, W. T. (1947). The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. of Math. 48 (2), 385–392.CrossRefGoogle Scholar
Davis, S., Shapley, R., Kaplan, E. & Tranchina, D. (1984). The receptive field organization of X cells in the cat: Spatiotemporal coupling and asymmetry. Vision Res. 24, 549–564.Google Scholar
Eckert, H. & Bishop, L. G. (1975). Nonlinear dynamic transfer characteristics of cells in the peripheral visual pathway of flies. Biol. Cybernet. 17, 1–6.CrossRefGoogle ScholarPubMed
Eckhorn, R. & Pöpel, B. (1979). Generation of Gaussian noise with improved quasi-white properties. Biol. Cybernet. 32, 243–248.CrossRefGoogle ScholarPubMed
Eggermont, J. J., Johannesma, P. I. M. & Aertsen, A. M. H. J. (1983).Reverse correlation methods in auditory research. Q. Rev. Biophys 16, 341–414.CrossRefGoogle ScholarPubMed
Enroth-Cugell, C., Robson, J. G., Schweitzer-Tong, D. E. & Watson, A. B. (1983). Spatio-temporal interactions in cat retinal ganglion cells showing linear spatial summation. J. Physiol. 341, 279–307.CrossRefGoogle ScholarPubMed
Fernald, R. D. & Gerstein, G. L. (1972). Response of cat cochlear nucleus neurones to frequency and amplitude modulated tones. Brain Res. 45, 417–435.CrossRefGoogle ScholarPubMed
French, A. S. & Butz, E. G. (1973). Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm. Internal. J. Control. 17, 529–539.CrossRefGoogle Scholar
French, A. S. & Holden, A. V. (1971). Alias-free sampling of neuronal spike trains. Kybernetik. 8, 165–171.CrossRefGoogle ScholarPubMed
French, A. S. & Wong, R. K. S. (1977). Nonlinear analysis of sensory transduction in an insect mechanoreceptor. Biol. Cybernet. 26, 231–240.CrossRefGoogle Scholar
Goblick, T. J. & Pfeiffer, R. R. (1969). Time-domain measurements of cochlear nonlinearities using combination click stimuli. J. acoust. Soc. Am. 46, 924–938.CrossRefGoogle ScholarPubMed
Halme, A. & Orava, I. (1972). Generalized polynomial operators for nonlinear system analysis. IEEE Trans. on Aut. Contr. AC. 17, 226–228.CrossRefGoogle Scholar
Hung, G., Stark, L. & Eykhoff, P. (1977). On the interpretation of kernels. I. Computer simulation of responses to impulse pairs. Ann. Biomed. Engin. 5, 130–143.CrossRefGoogle Scholar
Isobe, E. & Sato, S. (1984). An integro-difFerential formula on the Wiener kernels and its application to sandwich system identification. IEEE Trans. on Aut. Contr. AC. 29, 595–602.CrossRefGoogle Scholar
Johannesma, P. I. M. (1971). Dynamical aspects of the transmission of stochastic neural signals. In Proc. First European Biophysics Congress (ed. Broda, E., Locker, A. and Springer-Lederer, H.), pp. 329–333. Vienna: Verlag der Wiener Medizinischen Akademie.Google Scholar
Johannesma, P. I. M. (1972). The pre-response stimulus ensemble of neurons in the cochlear nucleus. In Proc. of the IPO Symp. on Hearing Theory (ed. Cardozo, B. L.), pp. 58–69. Eindhoven: IPO.Google Scholar
Johannesma, P. I. M. & Aertsen, A. M. H. J. (1982). Statistical and dimensional analysis of the neural representation of the acoustic biotope of the frog. Journal of Medical Systems. 6, 399–421.CrossRefGoogle ScholarPubMed
Johannesma, P. I. M., Aertsen, A. M. H. J., Van Den Boogard, H., Eggermont, J. & Epping, W. (1985). From synchrony to harmony: ideas on the function of neural assemblies and on the interpretation of neural synchrony. In Brain Theory (ed. Palm, G. and Aertsen, A.). Springer (in the Press).Google Scholar
Johnson, D. H. (1980). Applicability of white-noise nonlinear system analysis to the peripheral auditory system. J. acoust. Soc. Am. 68, 876–884.CrossRefGoogle Scholar
Katznelson, J. & Gould, L. A. (1964). The design of nonlinear filters and control systems. II. Information and Control. 7, 117–145.CrossRefGoogle Scholar
Krausz, H. (1975). Characterization of nonlinear systems using random impulse train inputs. Biol. Cybernet. 19, 217.CrossRefGoogle Scholar
Krausz, H. & Friesen, W. O. (1975). Identification of discrete input nonlinear systems using Poisson impulse trains. Proc. 1st Symp. on Testing and Identification of Nonlinear Systems, (ed. G. D. McCann and P. Z. Marmarelis), 124–146. Calif. Inst. of Tech., Pasadena.Google Scholar
Kroeker, J. P. (1980). Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals. Biol. Cybernet. 36, 243–248.CrossRefGoogle Scholar
Lammers, H. C. & Boer, E. De. (1979). Regression function of a bandpass nonlinear (BPNL) network. Proc. IEEE. 67, 432–434.CrossRefGoogle Scholar
Lee, Y. W. & Schetzen, M. (1965). Measurement of the Wiener kernels of a non-linear system by crosscorrelation. Internat. J. Control. 2, 237–254.CrossRefGoogle Scholar
Lipson, E. D. (1975). White noise analysis of phycomyces light growth response system. I, II, III. Biophys. J. 15, 989–1045.CrossRefGoogle Scholar
Marmarelis, P. Z. (1971). Nonlinear dynamic transfer function for certain retinal neuronal systems. Thesis, Calif. Inst. of Tech., Pasadena.Google Scholar
Marmarelis, P. Z. & Marmarelis, V. Z. (1978). Analysis of physiological systems. In The White Noise Approach. New York: Plenum.Google Scholar
Marmarelis, P. Z. & Naka, K. I. (1972). White-noise analysis of a neuron chain: An application of the Wiener theory. Science. 175, 1276–1278.CrossRefGoogle ScholarPubMed
Marmarelis, P. Z. & Naka, K. I. (1973). Nonlinear analysis of synthesis of receptive field responses in the catfish retina. I, II, III. J. Neurophysiol. 36, 605.CrossRefGoogle Scholar
Marmarelis, P. Z. & Naka, K. I. (1974). Identification of multi-input biological systems. IEEE Transactions on Biomed. Engrg., BME. 21, 88–101.CrossRefGoogle ScholarPubMed
Marmarelis, V. Z. (1977). A family of quasi-white random signals and its optimal use in biological system identification. I. Theory. Biol. Cybernetics. 27, 49–56.CrossRefGoogle Scholar
Marmarelis, V. Z. & McCann, G. D. (1977). A family of quasi-white random signals and its optimal use in biological system identification. II. Application to the photoreceptor of Calliphora erythrocephala. Biol. Cybernet. 27, 57–62.CrossRefGoogle Scholar
McCann, G. D. (1974). Nonlinear identification theory models for successive stages of visual neurons systems of flies. J. Neurophysiol. 37, 869–875.CrossRefGoogle ScholarPubMed
McCann, G. D. & Marmarelis, P. Z.(eds.) (1975). Proceedings of the first symposium on testing and identification of nonlinear systems, Calif. Inst. of Tech., Pasadena.Google Scholar
McKean, H. P. (1972). Wieners theory of nonlinear noise. In Stochastic Differential Equations (ed. Keller, J. B. and McKean, H. P.). New York: Courant Institute of Math. Sciences.Google Scholar
Møller, A. R. (1973). Statistical evaluation of the dynamic properties of cochlear nucleus units using stimuli modulated with pseudorandom noise. Brain Res. 57, 443–456.CrossRefGoogle ScholarPubMed
Møller, A. R. (1976 a). Dynamic properties of the response of single neurones in the cochlear nucleus of the rat. J. Physiol. 259, 63–82.CrossRefGoogle ScholarPubMed
Møller, A. R. (1976 b). Dynamic properties of primary auditory fibers compared with cells in the cochlear nucleus. Acta physiol. Scand. 98, 157–167.CrossRefGoogle ScholarPubMed
Naka, K. I., Marmarelis, P. Z. & Chan, R. Y. (1975). Morphological and functional identifications of catfish retinal neurons. III. Functional identification. J. Neurophysiol. 38, 92–131.CrossRefGoogle ScholarPubMed
Palm, G. (1978). On representation and approximation of nonlinear systems. Biol. Cybernet. 31, 119–124.CrossRefGoogle Scholar
Palm, G. (1979). On representation and approximation of nonlinear systems. II. Discrete time. Biol. Cybernet. 34, 49–52.CrossRefGoogle Scholar
Palm, G. & Poggio, T. (1977 a). The Volterra representation and the Wiener expansion: validity and pitfalls. SIAM J. Appl. Math. 33, 195–216.CrossRefGoogle Scholar
Palm, G. & Poggio, T. (1977 b). Wiener-like system identification in physiology. J. Math. Biol. 4, 375–381.CrossRefGoogle ScholarPubMed
Palm, G. & Poggio, T. (1978). Stochastic identification methods for nonlinear systems: an extension of the Wiener theory. SIAM J. Appi. Math. 34, 524–534.CrossRefGoogle Scholar
Poggio, T. & Reichardt, W. (1976). Visual control of orientation behaviour in the fly. II. Q. Rev. Biophys. 9, 377–438.CrossRefGoogle ScholarPubMed
Pöpel, B. & Eckhorn, R. (1981). Dynamic aspects of cat retinal ganglion cell's centre and surround mechanisms. A white noise analysis. Vision Res. 21, 1693–1696.CrossRefGoogle ScholarPubMed
Pöpel, B. & Querfurth, H. (1984). The transducer and encoder of frog muscle spindles are essentially nonlinear. Physiological conclusions from a white-noise analysis. Biol. Cybernet. 51, 21–32.CrossRefGoogle ScholarPubMed
Ratliff, F., Knight, B. W., Dodge, F. Jr. & Hartline, H. K. (1974). Fourier analysis of dynamics of excitation and inhibition in the eye of Limulus: amplitude, phase and distance. Vision Res. 14, 1155–1168.CrossRefGoogle ScholarPubMed
Sandberg, A. & Stark, L. (1968). Wiener G-function analysis as an approach to non-linear characteristics of human pupil light reflex. Brain Res. 11, 194–211.CrossRefGoogle ScholarPubMed
Smolders, J. W. T., Aertsen, A. M. H. J. & Johannesma, P. I. M. (1979). Neural representation of the acoustic biotope, a comparison of the response of auditory neurons to tonal and natural stimuli in the cat. Biol. Cybernet. 35, 11–20.CrossRefGoogle ScholarPubMed
Spekreijse, H. & Reits, D. (1982). Sequential analysis of the visual evoked potential system in man: nonlinear analysis in a sandwich system. Ann. N. Y. Acad. Sci. 388, 72–97.CrossRefGoogle Scholar
Swerup, C. (1978). On the choice of noise for the analysis of the peripheral auditory system. Biol. Cybernet. 29, 97–104.CrossRefGoogle ScholarPubMed
Terzuolo, C., Fohlmeister, J. F., Maffei, L., Poppele, R. E., Soechting, J. F. & Young, L. (1982). On the application of systems analysis to neurophysiological problems. Arch. ital. Biol. 120, 18–71.Google ScholarPubMed
Trimble, J. & Phillips, G. (1978). Non-linear analysis of the human visual evoked response. Biol. Cybernet. 30, 55–61.CrossRefGoogle Scholar
Victor, J. D. & Knight, B. W. (1979). Nonlinear analysis with an arbitrary stimulus ensemble. Q. appl. Math. 37, 112–136.CrossRefGoogle Scholar
Victor, J. D., Shapley, R. & Knight, B. W. (1977). Nonlinear analysis of cat retinal ganglion cells in the frequency domain. Proc. natn Acad. Sci. U.S.A. 74, 3068–3072.CrossRefGoogle ScholarPubMed
Volterra, V. (1930). Theory of functionals and of integral and integro-differential equations. New York: Dover.Google Scholar
Wickesberg, R. E. & Geisler, C. D. (1984). Artifacts in Wiener kernels estimated using Gaussian white noise. IEEE Trans. Biomed. Ens. 31, 454–461.CrossRefGoogle ScholarPubMed
Wiener, N. (1958). Nonlinear problems in random theory. New York: John Wiley.Google Scholar
Yasui, S. (1979). Stochastic functional Fourier series, Volterra series and nonlinear systems analysis. IEEE Trans. Autom. Control AC. 24, 230–242.CrossRefGoogle Scholar