A higher order local linearization method for solving ordinary differential equations

Cruz, H. de la; Biscay, Rolando J.; Carbonell, Félix; Ozaki, Tohru; Jiménez, Juan Carlos

doi:10.1016/j.amc.2006.06.096

Cited by 25 publications

(17 citation statements)

References 40 publications

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“…The LL method is accurate, stable and conserves the dynamical properties of the original continuous time system. It was first introduced by [Ozaki, 1989[Ozaki, , 1992a, later elaborated by [Biscay et al, 1996] and extensively developed in recent years [Carbonell et al, 2002De la Cruz et al, 2007;De la Cruz et al, 2006;Jimenez et al, 1998Jimenez et al, , 1999Jimenez et al, , 2005Jimenez, 2002;. Though originally restricted in its applications to neuroimaging [Valdes et al, 1999a,b;Riera et al, 2004Riera et al, , 2006Riera et al, , 2007, the LL method has gained popularity in recent years.…”

Section: Simulation Of State Space Modelsmentioning

confidence: 98%

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Model driven EEG/fMRI fusion of brain oscillations

Valdés‐Sosa

Sánchez-Bornot

Sotero

et al. 2008

Human Brain Mapping

187

153

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This article reviews progress and challenges in model driven EEG/fMRI fusion with a focus on brain oscillations. Fusion is the combination of both imaging modalities based on a cascade of forward models from ensemble of post-synaptic potentials (ePSP) to net primary current densities (nPCD) to EEG; and from ePSP to vasomotor feed forward signal (VFFSS) to BOLD. In absence of a model, data driven fusion creates maps of correlations between EEG and BOLD or between estimates of nPCD and VFFS. A consistent finding has been that of positive correlations between EEG alpha power and BOLD in both frontal cortices and thalamus and of negative ones for the occipital region. For model driven fusion we formulate a neural mass EEG/fMRI model coupled to a metabolic hemodynamic model. For exploratory simulations we show that the Local Linearization (LL) method for integrating stochastic differential equations is appropriate for highly nonlinear dynamics. It has been successfully applied to small and medium sized networks, reproducing the described EEG/BOLD correlations. A new LL-algebraic method allows simulations with hundreds of thousands of neural populations, with connectivities and conduction delays estimated from diffusion weighted MRI. For parameter and state estimation, Kalman filtering combined with the LL method estimates the innovations or prediction errors. From these the likelihood of models given data are obtained. The LL-innovation estimation method has been already applied to small and medium scale models. With improved Bayesian computations the practical estimation of very large scale EEG/fMRI models shall soon be possible.

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Section: Simulation Of State Space Modelsmentioning

confidence: 98%

“…These numerical variants are known as ''LL-schemes'' De la Cruz et al, 2007] and care should be taken because some have proven to be more accurate than others. The major computational burden in these schemes is the evaluation of matrix exponentials such as e F x (x t ,Y)Dt .…”

Section: Simulation Of State Space Modelsmentioning

confidence: 99%

Model driven EEG/fMRI fusion of brain oscillations

Valdés‐Sosa

Sánchez-Bornot

Sotero

et al. 2008

Human Brain Mapping

187

153

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“…Para solucionarlo se construyen los métodos de Linealización Local de Orden Superior [6,7,9], los cuales retienen las propiedades mencionadas, pero con orden de convergencia mayor. Este método se basa en agregar a la aproximación (6) un término adicional r de orden superior conformado por la solución numérica de una ecuación diferencial.…”

Section: Aproximación Lineal Local De Orden Superiorunclassified

“…Los esquemas de Linealización Local (LL) [15,16] y de Linealización Local de Orden Superior (LLOS) [6,7] forman parte de los llamados integradores exponenciales [5,12,13] para la solución de Ecuaciones Diferenciales Ordinarias (EDO), los cuales tienen en común el cálculo de una exponencial matricial en cada paso. En general, el desarrollo de esta clase de métodos ha sido estimulado por su capacidad para preservar numerosas propiedades dinámicas de las EDO con tamaños de pasos mucho mayores que los integradores explícitos y con menor costo computacional que los implícitos [9].…”

Section: Introductionunclassified

“…Resultados teóricos y de simulaciones muestran que los métodos LL poseen un gran número de propiedades deseables entre las que se encuentran la A-estabilidad, ausencia de puntos de equilibrio espurios bajo condiciones bastante generales y la preservación del comportamiento dinámico de la solución alrededor de puntos de equilibrio hiperbólicos y órbitas periódicas [6,7,9,15]. Adicionalmente, permite integrar ciertos tipos de ecuaciones "stiff" con menor costo computacional y mejor precisión que los integradores explícitos convencionales.…”

Section: Introductionunclassified

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Construcción y estudio de códigos adaptativos de linealización local para ecuaciones diferenciales ordinarias

Aguiar

Sobrino

2014

RMTA

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Probabilistic analysis of a class of compartmental models formulated by random differential equations

Bevia,

Carlos Cortés,

Luisovna Pérez

et al. 2024

Math Methods in App Sciences

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This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references.

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A higher order local linearization method for solving ordinary differential equations

Cited by 25 publications

References 40 publications

Model driven EEG/fMRI fusion of brain oscillations

Model driven EEG/fMRI fusion of brain oscillations

Construcción y estudio de códigos adaptativos de linealización local para ecuaciones diferenciales ordinarias

Probabilistic analysis of a class of compartmental models formulated by random differential equations

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