Abstract
Detection, representation, and identification of nonlinearities in biological systems are considered. We begin by briefly but critically examining a well-known test of system nonlinearity, and point out that this test cannot be used to prove that a system is linear. We then concentrate on the representation of nonlinear systems by Wiener's orthogonal functional series, discussing its advantages, limitations, and biological applications. System identification through estimating the kernels in the functional series is considered in detail. An efficient time-domain method of correcting for coloring in inputs is examined and shown to result in significantly improved kernel estimates in a biologically realistic system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barrett, J.F. The use of functionals in the analysis of nonlinear physical systems. J. Elect. Control. 15:567–615; 1963.
Barrett, J.F. Functional series representation of nonlinear systems—some theoretical comments. IFAC Symp. Ident. Sys. Param. Est. 1:251–256; 1982.
Bedrosian, E.; Rice, S.O. The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. Proc. IEEE 59:1688–1707; 1971.
Billings, S.A. An overview of nonlinear systems identification. IFAC Symp. Ident. Sys. Param. Est. 1:725–729; 1985.
Billings, S.A.; Fakhouri, S.Y. Identification of systems containing linear dynamic and static nonlinear elements. Automatica. 18:15–26; 1982.
Billings, S.A.; Voon, W.S.F. Structure detection and model validity tests in the identification of nonlinear systems. IEE Proc. 130, Part D:193–199; 1983.
Billings, S.A.; Voon, W.S.F. Correlation based model validity tests for non-linear models. Int. J. Control. 44:235–244; 1986.
Bose, A.G. A theory of nonlinear systems. MIT Res. Lab. Elec. Tech. Rep. No. 309, 1956.
Emerson, R.C.; Citron, M.C.; Vaughn, W.J.; Klein, S.A. Nonlinear directionally selective subunits in complex cells of cat striate cortex. J. Neurophysiol. 58:33–65; 1987.
Emerson, R.C.; Korenberg, M.J.; Citron, M.C. Measurement of a simple-cell threshold function in cat's striate cortex. Soc. Neurosci. Abstr. 14:899; 1988.
Emerson, R.C.; Korenberg, M.J.; Citron, M.C. Identification of intensive nonlinearities in cascade models of visual cortex and its relation to cell classification. In: Marmarelis, V.Z., ed. Advanced methods of physiological system modeling, Vol. 2. New York: Plenum Press; 1989: pp. 97–111.
Frechet, M. Sur les fonctionnelles continues. Annales Scientifiques de l'Ecole Normal Superieure. 27:193–219; 1910.
French, A.S. Measuring the Wiener kernels of a nonlinear system by use of the fast Fourier transform and Walsh functions. Proc. 1st Symp. Testing Ident. Nonlin. Sys. 1:76–88; 1975.
French, A.S.; Butz, E.G. Measuring the Wiener kernels of a nonlinear system using the fast Fourier transform algorithm. Int. J. Contr. 17:529–539; 1973.
French, A.S.; Butz, E.G. The use of Walsh functions in the Wiener analysis of nonlinear systems. IEEE Trans. Comp. 23:225–232; 1974.
French, A.S.; Korenberg, M.J. A nonlinear cascade model for action potential encoding in an insect sensory neuron. Biophys. J. 55:655–661; 1989.
Goussard, Y. Identification de systemes nonlineaires, representes par developpements fonctionnels et soumis a diverses entree aleatoires, a l'aide d'une methode d'approximation stochastique. These de Docteur-Ingenieur, Universite de Paris-Sud, Centre d'Orsay; 1983.
Goussard, Y. Wiener kernel estimation: A comparison of cross-correlation and stochastic approximation methods. In: Marmarelis, V.Z., ed. Advanced methods of physiological system modeling. Los Angeles: USC Biomedical Simulations Resource, vol. 1; 1987: pp. 289–302.
Goussard, Y.; Krenz, W.C.; Stark, L. An improvement of the Lee and Schetzen cross-correlation method. IEEE Trans. Automat. Contr. 30:895–898; 1985.
Haber, R. Nonlinearity tests for dynamic processes. IFAC Symp. Ident. Sys. Param. Est. 1:409–413; 1985.
Haber, R. Structure identification of block-oriented models based on the Volterra kernels. IFAC Symp. Ident. Sys. Param. Est. (Appendix); 1985.
Hunter, I.W. Experimental comparison of Wiener and Hammerstein cascade models of frog muscle fiber mechanics. Biophys. J. 49:81a; 1986.
Hunter, I.W.; Kearney, R.E. Generation of random sequences with jointly specified probability density and autocorrelation functions. Biol. Cybern. 47:141–146; 1983.
Hunter, I.W.; Kearney, R.E. Two-sided linear filter identification. Med. Biol. Eng. Comput. 21:203–209; 1983.
Hunter, I.W.; Kearney, R.E. NEXUS: A computer language for physiological systems and signal analysis. Comp. Biol. Med. 14:385–401; 1984.
Hunter, I.W.; Korenberg, M.J. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 55:135–144; 1986.
Klein, S.; Yasui, S. Nonlinear systems analysis with non-Gaussian white stimuli: General basis functionals and kernels. IEEE Trans. Inform. Theory. 25:495–500; 1979.
Korenberg, M.J. Cross-correlation analysis of neural cascades. Proc. 10th Annual Rocky Mountain Bioeng. Symp., 47–52; 1973.
Korenberg, M.J. Identification of biological cascades of linear and static nonlinear systems. Proc. 16th Midwest Symp. Circuit Theory. 18.2:1–9, 1973.
Korenberg, M.J. Statistical identification of parallel cascades of linear and nonlinear systems. IFAC Symp. Ident. Sys. Param. Est. 1:580–585; 1982.
Korenberg, M.J. Orthogonal identification of nonlinear difference equation models. Midwest Symp. Circuit Sys. 1:90–95; 1985.
Korenberg, M.J. Identifying noisy cascades of linear and static nonlinear systems. IFAC Symp. Ident. Sys. Param. Est. 1:421–426; 1985.
Korenberg, M.J. Functional expansions, parallel cascades and nonlinear difference equations. In: Marmarelis, V.Z. ed. Advanced methods of physiological system modeling. Los Angeles: USC Biomedical Simulations Resource, vol 1; 1987:pp. 221–240.
Korenberg, M.J. Fast orthogonal identification of nonlinear difference equation and functional expansion models. Midwest Symp. Circuit Sys. 1:270–276; 1987.
Korenberg, M.J. Identifying nonlinear difference equation and functional expansion representations: The fast orthogonal algorithm. Ann. Biomed. Eng. 16:123–142; 1988.
Korenberg, M.J. A robust orthogonal algorithm for system identification and time-series analysis. Biol. Cybern. 60:267–276; 1989.
Korenberg, M.J. A rapid and accurate method for estimating the kernels of a nonlinear system with lengthy memory. 15th Biennial Symp. Commun. June, 1990; Queen's University, Kingston, Canada, pp. 57–60.
Korenberg, M.J.; Bruder, S.B.; McIlroy, P.J. Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems. Ann. Biomed. Eng. 16:201–214; 1988.
Korenberg, M.J.; French, A.S.; Voo, S.K.L. White-noise analysis of nonlinear behavior in an insect sensory neuron: Kernel and cascade approaches. Biol. Cybern. 58:313–320; 1988.
Koenberg, M.J.; Hunter, I.W. The identification nonlinear biological systems: LNL cascade models. Biol. Cybern. 55:125–134; 1986.
Korenberg, M.J.; Sakai, H.M.; Naka, K-I. Dissection of the neuron network in the catfish inner retina. III. Interpretation of spike kernels. J. Neurophysiol. 61:1110–1120; 1989.
Krausz, H.I. Identification of non-linear systems using random impulse train inputs. Biol. Cybern. 19:217–230; 1975.
Lee, Y.W. Contribution of Norbert Wiener to linear theory and nonlinear theory in engineering. In Selected Papers of Norbert Wiener, SIAM, MIT Press, 17–33; 1964.
Lee, Y.W.; Schetzen, M. Measurement of the Wiener kernels of a non-linear system by cross-correlation. Int. J. Control. 2:237–254; 1965.
Leontaritis, I.J.; Billings, S.A. Input-output parametric models for nonlinear systems; Part I—Deterministic nonlinear systems; Part II—Stochastic nonlinear systems. Int. J. Control. 41:303–359; 1985.
Marmarelis, P.Z.; Marmarelis, V.Z. Analysis of physiological systems. New York: Plenum Press; 1978.
Marmarelis, P.Z.; Naka, K.I. White-noise analysis of a neuron chain: An application of the Wiener theory. Science. 175:1276–1278; 1972.
Marmarelis, P.Z.; Naka, K.I. Nonlinear analysis and synthesis of receptive-field responses in the catfish retina. J. Neurophys. 36:605–648; 1973.
Marmarelis, V.Z. Identification of nonlinear systems through quasi-white test signals. Pasadena California: Calif. Inst. Tech.; 1976. Ph.D. Thesis.
Marmarelis, V.Z.; McCann, G.D. A family of quasi-white random signals and its optimal use in biological system identification. Part II: Application to the photoreceptor of Calliphora Erythrocephala. Biol. Cybern. 27:57–62; 1977.
McIlroy, P.J.H. Applications of nonlinear systems identification. Kingston, Ontario, Canada: Queen's University; 1986. MSc Thesis.
Melton, R.B.; Kroeker, J.P. Wiener functional for an N-level uniformly distributed discrete random process. IFAC Symp. Ident. Sys. Param. Est. 2:1169–1174; 1982. Cybern. 34:49–52; 1979.
Palm, G. On the representation and approximation of nonlinear systems. Part II: Discrete time. Biol. Cybern. 34:49–52; 1979.
Palm, G.; Poggio, T. The Volterra representation and the Wiener expansion: Validity and pitfalls. SIAM J. Appl. Math. 33:195–216; 1977.
Palm, G.; Poggio, T. Stochastic identification methods for nonlinear systems: an extension of the Wiener theory, SIAM J. Appl. Math. 34:524–534; 1978.
Rugh, W.J. Nonlinear system theory: The Volterra/Wiener approach. Johns Hopkins Univ Press, Baltimore; 1981.
Sakai, H.M.; Naka, K-I. Signal transmission in the catfish retina. IV transmission to ganglion cells. J. Neurophysiol. 58:1307–1328; 1987.
Sakai, H.M.; Naka, K-I. Signal transmission in the catfish retina. V. Sensitivity and circuit. J. Neurophysiol. 58:1329–1350; 1987.
Sakai, H.M.; Naka, K-I. Dissection of the neuron network in the catfish inner retina. I. Transmission to ganglion cells. J. Neurophysiol. 60:1549–1567; 1988.
Sakai, H.M.; Naka, K-I. Dissection of the neuron network in the catfish inner retina. II. Interactions between ganglion cells. J. Neurophysiol. 60:1568–1583; 1988.
Sakai, H.M.; Naka, K-I. Dissection of the neuron network in the catfish inner retina. IV. Bidirectional interactions between amacrine and ganglion cells. J. Neurophysiol. 63:105–119; 1990.
Sakai, H.M.; Naka, K-I. Dissection of the neuron network in the catfish inner retina. V. Interactions between NA and NB amacrine cells. J. Neurophysiol. 63:120–130; 1990.
Sakai, H.M.; Naka, K-I.; Korenberg, M.J. White noise analysis in visual neuroscience. Vis. Neurosci. 1:287–296; 1988.
Sakuranaga, M.; Ando, Y-J.; Naka, K-I. Dynamics of ganglion cell response in the catfish and frog retinas. J. Gen. Physiol. 90:229–259; 1987.
Sakuranaga, M.; Sato, S.; Hida, E.; Naka, K-I. Nonlinear analysis: Mathematical theory and biological applications. CRC Crit. Rev. Biomed. Eng. 14:127–184; 1986.
Sandberg, A.; Stark, L. Wiener G-function analysis as an approach to nonlinear characteristics of human pupil light reflex. Brain Res. 11:194–211; 1968.
Schetzen, M. Determination of optimum nonlinear systems for generalized error criteria based on the use of gate functions. IEEE Trans. Inform. Theory. 11:117–125; 1965.
Schetzen, M. A theory of nonlinear system identification. Int. J. Control. 20:557–592; 1974.
Schetzen, M. The Volterra and Wiener theories of nonlinear systems. New York: Wiley; 1980.
Schetzen, M. Nonlinear system modeling based on the Wiener theory, Proc. IEEE. 69:1557–1573; 1981.
Stark, L.W. The pupillary control system: its nonlinear adaptive and stochastic engineering design characteristics. Automatica. 5:655–676; 1969.
Stark, L.W. The pupil as a paradigm for neurological control systems. IEEE Trans. Biomed. Eng. 31:919–924; 1984.
Strang, G. Linear algebra and its applications. (2nd Ed.) New York: Academic Press; 1980.
Sutter, E. A practical nonstochastic approach to nonlinear time-domain analysis. In: Marmarelis, V.Z., ed. Advanced methods of physiological system modeling. Los Angeles; USC Biomedical Simulations Resource, vol. 1. 1987:pp.303–315.
Varlaki, P.; Terdik, G.; Lototsky, V.A. Tests for linearity and bilinearity of dynamic systems. IFAC Symp. Ident. Sys. Param. Est. 1:427–432; 1985.
Victor, J.D. The fractal dimension of a test signal: implications for system identification procedures. Biol. Cybern. 57:421–426; 1987.
Victor, J.D.; Johannesma, P. Maximum-entropy approximations of stochastic nonlinear transducers: An extension of the Wiener theory. Biol. Cybern. 54:289–300; 1986.
Victor, J.D.; Knight, B. Nonlinear analysis with an arbitrary stimulus emsemble. Quart. Appl. Math. 37:113–136; 1979.
Victor, J.D.; Shapley, R. The nonlinear pathway of Y ganglion cells in the cat retina. J. Gen. Physiol. 74:671–687; 1979.
Volterra, V. Sur les fonctions qui dependent d'autres fonction. Comptes Rendus Ac. des Sciences. 142:691–695; 1906.
Volterra, V. Theory of functionals and of integral and integro-differential equations. New York: Dover; 1959.
Watanabe, A.; Stark, L. Kernel method for nonlinear analysis: Identification of a biological control system. Math. Biosci. 27:99–108; 1975.
Wiener, N. Nonlinear problems in random theory. Massachusetts: MIT Press; 1958.
Yasui, S. Stochastic functional Fourier series, Volterra series, and nonlinear systems analysis. IEEE Trans. Automat. Contr. 24:230–242; 1979.
Yasui, S. Wiener-like Fourier kernels for nonlinear system identification and synthesis (nonanalytic cascade, bilinear, and feedback cases). IEEE Trans. Automat. Contr. 27:677–685; 1982.
Zadeh, L. On the representation of nonlinear operators. IRE Wescon Conv. Rec. 2:105–113; 1957.
Zohar, S. Fortran subroutines for the solution of Toeplitz setz of linear equations. IEEE Trans. Acoust. Speech. Sig. Proc. 27:656–658; 1979.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Korenberg, M.J., Hunter, I.W. The identification of nonlinear biological systems: Wiener kernel approaches. Ann Biomed Eng 18, 629–654 (1990). https://doi.org/10.1007/BF02368452
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02368452