Volterra series

About: Volterra series is a(n) research topic. Over the lifetime, 2731 publication(s) have been published within this topic receiving 46199 citation(s).

The Volterra and Wiener Theories of Nonlinear Systems

Abstract: This text presents a complete and detailed development of the analysis, design and characterization of non-linear systems using the Volterra and Wiener theories, as well as gate functions, thus yielding new insights and a better comprehension of the subject. The Volterra and Wiener theories are useful in the study of systems in biological, mechanical, and electrical fields.

Nonlinear responses in fMRI: The balloon model, volterra kernels, and other hemodynamics

Abstract: There is a growing appreciation of the importance of nonlinearities in evoked responses in fMRI, particularly with the advent of event-related fMRI. These nonlinearities are commonly expressed as interactions among stimuli that can lead to the suppression and increased latency of responses to a stimulus that are incurred by a preceding stimulus. We have presented previously a model-free characterization of these effects using generic techniques from nonlinear system identification, namely a Volterra series formulation. At the same time Buxton et al. (1998) described a plausible and compelling dynamical model of hemodynamic signal transduction in fMRI. Subsequent work by Mandeville et al. (1999) provided important theoretical and empirical constraints on the form of the dynamic relationship between blood flow and volume that underpins the evolution of the fMRI signal. In this paper we combine these system identification and model-based approaches and ask whether the Balloon model is sufficient to account for the nonlinear behaviors observed in real time series. We conclude that it can, and furthermore the model parameters that ensue are biologically plausible. This conclusion is based on the observation that the Balloon model can produce Volterra kernels that emulate empirical kernels. To enable this evaluation we had to embed the Balloon model in a hemodynamic input-state-output model that included the dynamics of perfusion changes that are contingent on underlying synaptic activation. This paper presents (i) the full hemodynamic model (ii), how its associated Volterra kernels can be derived, and (iii) addresses the model's validity in relation to empirical nonlinear characterizations of evoked responses in fMRI and other neurophysiological constraints.

Fading memory and the problem of approximating nonlinear operators with Volterra series

Abstract: Using the notion of fading memory we prove very strong versions of two folk theorems. The first is that any time-invariant (TI) continuous nonlinear operator can be approximated by a Volterra series operator, and the second is that the approximating operator can be realized as a finite-dimensional linear dynamical system with a nonlinear readout map. While previous approximation results are valid over finite time intervals and for signals in compact sets, the approximations presented here hold for all time and for signals in useful (noncompact) sets. The discretetime analog of the second theorem asserts that any TI operator with fading memory can be approximated (in our strong sense) by a nonlinear moving- average operator. Some further discussion of the notion of fading memory is given.

Reduced-order modeling: new approaches for computational physics

Abstract: In this paper, we review the development of new reduced-order modeling techniques and discuss their applicability to various problems in computational physics. Emphasis is given to methods ba'sed on Volterra series representations and the proper orthogonal decomposition. Results are reported for different nonlinear systems to provide clear examples of the construction and use of reduced-order models, particularly in the multi-disciplinary field of computational aeroelasticity. Unsteady aerodynamic and aeroelastic behaviors of two- dimensional and three-dimensional geometries are described. Large increases in computational efficiency are obtained through the use of reduced-order models, thereby justifying the initial computational expense of constructing these models and inotivatim,- their use for multi-disciplinary design analysis.

Nonlinear event-related responses in fMRI

Abstract: This paper presents an approach to characterizing evoked hemodynamic responses in fMRI based on nonlinear system identification, in particular the use of Volterra series. The approach employed enables one to estimate Volterra kernels that describe the relationship between stimulus presentation and the hemodynamic responses that ensue. Volterra series are essentially high-order extensions of linear convolution or "smoothing." These kernels, therefore, represent a nonlinear characterization of the hemodynamic response function that can model the responses to stimuli in different contexts (in this work, different rates of word presentation) and interactions among stimuli. The nonlinear components of the responses were shown to be statistically significant, and the kernel estimates were validated using an independent event-related fMRI experiment. One important manifestation of these nonlinear effects is a modulation of stimulus-specific responses by preceding stimuli that are proximate in time. This means that responses at high-stimulus presentation rates saturate and, in some instances, show an inverted U behavior. This behavior appears to be specific to BOLD effects (as distinct from evoked changes in cerebral blood flow) and may represent a hemodynamic "refractoriness." The aim of this paper is to describe the theory and techniques upon which these conclusions were based and to discuss the implications for experimental design and analysis.