Skip to main content
SpringerLink
Log in

The identification of nonlinear biological systems: LNL cascade models

  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Banyasz CS, Haber R, Keviczky L (1973) Some estimation methods for nonlinear discrete time identification. IFAC Symp Ident Sys Param Est 3:793–802

    Google Scholar 

  • Billings SA, Fakhouri SY (1978) Identification of a class of nonlinear systems using correlation analysis. Proc IEE 7:691–697

    Google Scholar 

  • Billings SA, Fakhouri SY (1982) Identification of systems containing linear dynamic and static nonlinear elements. Automatica 18:15–26

    Google Scholar 

  • de Boer E (1981) Correlation functions as tools for analysis. Adv Physiol Sci 30:79–91

    Google Scholar 

  • Brillinger DR (1977) The identification of a particular nonlinear time series system. Biometrika 64:509–515

    Google Scholar 

  • Bussgang JJ (1952) Crosscorrelation functions of amplitudedistorted Gaussian signals. MIT Res Lab Elec Tech Rep 216:1–14

    Google Scholar 

  • Davenport WB (1953) Signal-to-noise ratios in band-pass limiters. J Appl Phys 24:720–727

    Google Scholar 

  • Douce JL (1976) Identification of a class of non-linear system. IFAC Symp Ident Sys Param Est 4:1655–1668

    Google Scholar 

  • Economakos E (1971) Identification of a group of internal signals of zero-menory non-linear systems. Elect Lett 7:99–100

    Google Scholar 

  • Eykhoff P (1974) System identification: parameter and state estimation. Wiley, London

    Google Scholar 

  • Fakhouri SY, Billings SA, Wormald CN (1981) Analysis of estimation errors in the identification of non-linear systems. Int J Sys Sci 12:205–225

    Google Scholar 

  • Gardiner AB (1966) Elimination of the effect of non-linearities on process cross-correlations. Elect Lett 2:164–165

    Google Scholar 

  • Gardiner AB (1967) Identification of the linear channel of a non-linear process using pseudo-random cross-correlation techniques. IEE Colloq Pseudo-rand Seq Appl Contr Syst, London

  • Gardiner AB (1968) Determination of the linear output signal of a process containing single-valued nonlinearities. Elect Lett 4:224–225

    Google Scholar 

  • Goodwin GC, Payne RL (1977) Dynamic system identification: experimental design and data analysis. Academic Press New York

    Google Scholar 

  • Graupe D (1972) Identification of systems. Van Nostrand Reinhold, New York

    Google Scholar 

  • Graupe D (1984) Time series analysis, identification and adaptive filtering. Krieger, Florida

    Google Scholar 

  • Haddad AH (ed) (1975) Nonlinear systems: processing of random signals-classical analysis. Dowden Hutchinson Ross, Stroudsburg Pennsylvania

    Google Scholar 

  • Hunter IW, Kearney RE (1983a) Generation of randon sequences with jointly specified probability density and autocorrelation functions. Biol Cybern 47:141–146

    Google Scholar 

  • Hunter IW, Kearney RE (1983b) Two-sided linear filter identification. Med Biol Eng Comput 21:203–209

    Google Scholar 

  • Hunter IW, Kearney RE (1984) NEXUS: a computer language for physiological systems and signal analysis. Comput Biol Med 14:385–401

    Google Scholar 

  • Hunter IW, Korenberg MJ (1986) The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol Cybern 55:135–144

    Google Scholar 

  • Isobe E, Sato S (1984) An integro-differential formula on the Wiener kernels and its application to sandwich system identification. IEEE Trans AC-29:595–602

    Google Scholar 

  • Kalaba R, Spingarn K (1982) Control, identification, and input optimization. Plenum Press, New York

    Google Scholar 

  • Korenberg MJ (1973a) Identification of biological cascades of linear and static nonlinear systems. Proc Midwest Symp Circuit Theory 18.2:1–9

    Google Scholar 

  • Korenberg MJ (1973b) Cross-correlation analysis of neural cascades. Proc Ann Rocky Mountain Bioeng Symp 1:47–52

    Google Scholar 

  • Korenberg MJ (1973c) Obtaining differential equation, functional expansion or cascade representation for nonlinear biological systems. Proc New England Bioeng Conf 1:237–245

    Google Scholar 

  • Korenberg MJ (1982) Statistical identification of parallel cascades of linear and nonlinear systems. IFAC Symp Ident Sys Param Est 1:580–585

    Google Scholar 

  • Korenberg MJ (1984) Statistical identification of Volterra kernels of high order systems. IEEE Int Symp Circ Sys 2:570–575

    Google Scholar 

  • Korenberg MJ (1985a) Identifying noisy cascades of linear and static nonlinear systems. IFAC Symp Ident Sys Param Est 1:421–426

    Google Scholar 

  • Korenberg MJ (1985b) Orthogonal identification of nonlinear difference equation models. Midwest Symp Sirc Sys 1:90–95

    Google Scholar 

  • Lee YW, Schetzen M (1965) Measurement of the Wiener kernels of a nonlinear system by cross-correlation. Int J Contr 2:237–254

    Google Scholar 

  • Ljung L, Soderstrom T (1983) Theory and practice of recursive identification. MIT Press, Cambridge, MA

    Google Scholar 

  • Marmarelis PZ, Marmarelis VZ (1978) Analysis of physiological systems. Plenum Press, New York

    Google Scholar 

  • Middleton D (1948) Some general results in the theory of noise through non-linear devices. Q J Appl Math 5:445–498

    Google Scholar 

  • Powell DR, Macdonald JR (1972) A rapidly convergent iterative method for the solution of the generalized nonlinear least squares problem. Comput J 15:148–155

    Google Scholar 

  • Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 23:282–332

    Google Scholar 

  • Robinson EA (1978) Multichannel time series analysis with digital computer programs (Rev ed only). Holden-Day, San Francisco

    Google Scholar 

  • Rugh WJ (1981) Nonlinear system theory: the Volterra/Wiener approach. Johns Hopkins Univ Press, Baltimore

    Google Scholar 

  • Schetzen M (1980) The Volterra and Wiener theories of nonlinear systems. Wiley, New York

    Google Scholar 

  • Segal BN, Outerbridge JS (1982) A nonlinear model of semicircular canal primary afferents in bullfrog. J Neurophys 47:563–578

    Google Scholar 

  • Sinha NK, Kuszta B (1983) Modeling and identification of dynamic systems. Van Nostrand Reinhold, New York

    Google Scholar 

  • Southwell WH (1975) Fitting data to nonlinear functions with uncertainties in all measurement variables. Comput J 19:69–73

    Google Scholar 

  • Spekreijse H (1969) Rectification in the goldfish retina: analysis by sinsoidal and auxiliary stimulation. Vision Res 9:1461–1472

    Google Scholar 

  • Spekreijse H, Oosting J (1970) Linearizing: a method for analysing and synthesizing nonlinear systems. Kybernetik 7:23–31

    Google Scholar 

  • Spekreijse H, Reits D (1982) Sequential analysis of the visual evoked potential system in man: nonlinear analysis of a sandwich system. Ann New York Acad Sci 388:72–97

    Google Scholar 

  • Stoica P (1981) On the convergence of an iterative algorithm used for Hammerstein system identification. IEEE Trans AC- 26:967–969

    Google Scholar 

  • Swerup C (1978) On the choice of noise for the analysis of the peripheral auditory system. Biol Cybern 29:97–104

    Google Scholar 

  • Victor J, Shapley R (1980) A method of nonlinear analysis in the frequency domain. Biophys J 29:459–484

    Google Scholar 

  • Victor J, Shapley R (1981) A comment on nonlinear analysis. Biophys J 35:807–808

    Google Scholar 

  • Weiss TF (1966) A model of the peripheral auditory system. Kybernetik 3:153–175

    Google Scholar 

  • Yasui S (1982) Wiener-like Fourier kernels for nonlinear system identification and synthesis (nonanalytic cascade, bilinear, and feedback cases). IEEE Trans AC-27:677–685

    Google Scholar 

  • Zohar S (1979) Fortran subroutines for the solution of Toeplitz sets of linear equations. IEEE Trans ASSP-27:656–658

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Queens University, K7L 3N6, Kingston, Ontario, Canada

    M. J. Korenberg

  2. Biomedical Engineering Unit, McGill University, 3775 University Street, H3A 2B4, Montreal, PQ, Canada

    I. W. Hunter

Authors
  1. M. J. Korenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. W. Hunter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korenberg, M.J., Hunter, I.W. The identification of nonlinear biological systems: LNL cascade models. Biol. Cybern. 55, 125–134 (1986). https://doi.org/10.1007/BF00341928

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00341928

Keywords

Search

Navigation