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Biological Bases
Published in Bahrain Nabet, Robert B. Pinter, Sensory Neural Networks: Lateral Inhibition, 2017
Bahrain Nabet, Robert B. Pinter
There are several methodologies to elicit nonlinear behavior of these receptive fields. The most general is to apply a band-limited white noise spatio-temporal stimulus and apply the Lee-Schetzen cross-correlation algorithm to this input and the output to obtain the linear convolution kernel (first order) and higher order kernels of the Wiener series describing the system (Marmarelis and Marmarelis 1978). The second order kernel often contains the major part of the nonlinear operators’ effects, and can be visualized as two linear operators each feeding a single multiplier or a set of such three component blocks in parallel (Schetzen 1980). Other methods are to apply combinations of dots or bars along the receptive field two at a time (Emerson et al. 1987) or M-sequences (Sutter 1987). Often the result of these experimental analyses contains a model whose significant nonlinearity is a multiplier, where the response of the interneuron depends on the product of activities in two portions of the receptive field. On the other hand, from a purely mathematical point of view, a nonlinear receptive field function can also be approximated to second order by multiplicative terms, that is, product terms of two variables, the second term in a Taylor series expansion of the function.
Nonlinear Systems Analysis in Vision: Overview of Kernel Methods
Published in Robert B. Pinter, Bahram Nabet, Nonlinear Vision: Determination of Neural Receptive Fields, Function, and Networks, 1992
These idealized examples point out that a Wiener series for a nonlinearity with a singularity such as a sharp corner cannot be converted into a Volterra series. The distinction between Volterra and Wiener approximations is of practical importance, even though the typical biological nonlinearity, when regarded on a sufficiently small scale, is likely to be smooth rather than abrupt. This is because for some nonlinearities, a signal size small enough to be sensitive to the “smoothness” of the nonlinearity may result in a response which is lost in the noise. Stated another way, a sequence of Wiener approximations built on inputs of successively smaller variances will approach a Volterra limit, provided that the contributions from higher-order components become sufficiently small as the variance decreases. This is the behavior which characterizes an “analytic” nonlinearity (such as y = ex). For a “nonanalytic” nonlinearity (such as y = max{0,x}), higher-order contributions do not become negligible as the variance decreases. An example of a nonlinearity that appears nonanalytic for signals of limitingly small size is that of the Y cell of the cat retina (Victor and Shapley 1979), whose response consists of second- and fourth-order components down to a contrast on the order of 1%.
Applications in Systems Control
Published in Phil Mars, J.R. Chen, Raghu Nambiar, Learning Algorithms, 1996
Phil Mars, J.R. Chen, Raghu Nambiar
This series is called a Volterra series, and the functions hn(τ1,…, τn) are called the Volterra kernels of the system. The analysis assumes that the system is time invariant. However there are two basic difficulties associated with the practical application of the Volterra series. The first difficulty concerns the measurement of the Volterra kernels of a given system and the second concerns the convergence of the series. Other functional series expansion methods for nonlinear system representation include Wiener series [Sch80] and the Uryson operator [Gal75]. In spite of the theoretical promise, all these models have some practical difficulties for general applicability. While the input-output finite order differential or difference equation model achieves wide acceptance in representation and identification of linear systems, it is natural to try to extend the input-output model to nonlinear systems. The input-output difference equation model for discrete nonlinear systems was proposed by Leontaritis and Billings in Reference [LB85]. Narendra and Parthasarathy proposed a nonlinear system identification scheme based on an finite order input-output difference equation model and MLP network [NP89]. There are many open questions concerning the theoretical and practical issues of the identification of nonlinear systems with neural networks. Examples are the excitation condition and the convergence of the weights. In this chapter, we discuss some of these fundamental problems and provide some computer simulations. Because of the theoretical difficulties of nonlinear systems, computer simulation is still an indispensable approach for the study of nonlinear systems.
Volterra and Wiener Model Based Temporally and Spatio-Temporally Coupled Nonlinear System Identification: A Synthesized Review
Published in IETE Technical Review, 2021
Saurav Gupta, Ajit Kumar Sahoo, Upendra Kumar Sahoo
Many times accurate mathematical description of numerous practical systems are not known. Hence, data-based system identification techniques got importance and became a prominent stream of control theory. Applications of these include economic data and financial systems, industrial processes, control systems, biology and the life sciences, and many more [3–7]. Data-based system modeling is a key issue for a wide range of real-time applications. Such an approach first selects the appropriate model structure and then estimate the parameters of interest using some identification methodology [8]. A variety of mathematical theories and methods have been developed for modeling the nonlinear systems. Popular methods are Taylor series, Volterra series, Wiener Series, nonlinear state–space model, nonlinear autoregressive with exogenous (NARX) and nonlinear autoregressive moving average with exogenous (NARMAX) models, and block-structured models. Volterra series is one among them which is widely used and has well-established methodology. Also, there exist a close relationship between Volterra series and many other models as described in [9].
Influence of harmonic and pulse excitations on response of foundations
Published in International Journal of Geotechnical Engineering, 2023
Kirtika Samanta, Priti Maheshwari
In view of the above discussion, this paper tries to address some of the above issues. The obstacles one foresees are listed below: Solution methodology: There are various solution methodologies available for such nonlinear systems including, Volterra series, Taylor series, Wiener series, NARMAX model, Hammerstein model, Wiener model, Wiener-Hammerstein model, harmonic balance method, perturbation method, and Adomian decomposition. The details have been presented by Cheng et al. (2017). Few other studies dealing with solution of nonlinear systems include Worden and Manson (2005) and Carassale and Kareem (2010). Numerical analysis using finite element method can be used to obtain the dynamic response of structures to blast loadings (Børvik et al. 2009; Zakrisson, Wikman, and Haggblad 2011; Wang et al. 2013). Although results from finite element analysis are usually in good agreement with experimental studies. However, modelling the structure using an SDOF system is often recommended as it is easy to apply, and has fewer input parameters. Several past researchers like Gantes and Pnevmatikos (2004), Li, Rong, and Pan (2007), and Rigby and Bennett (2014) have found the application of the SDOF method suitable for the analysis of structures to blast loads. For SDOF systems under specific loading conditions, it is possible to derive the differential equations and therefore obtain the solutions analytically. In the context of machine foundations, derivation of closed form solutions of soil-foundation system (as SDOF system) under simultaneous harmonic loading and pulse loading will lead to nonlinearity and the phase difference in two different types of loadings will come into picture influencing the vibration of the foundation system.Transition from elastic to elastic-plastic response: Foundation vibrating only under the influence of harmonic loading is analysed as an elastic system due to stringent requirement of amplitude of vibrations. Since the interaction of harmonic loading and pulse loading is highly nonlinear, it is expected that such interaction can result in the onset of large deformations in the system. Consequent to that, one can expect that an elastic response for harmonic loading condition can show elastic-plastic response when the interaction effect (harmonic and pulse) comes into play.System response after pulse loading: It is expected that maximum response of the system will happen after the occurrence of pulse loading and therefore it needs to be investigated.