The Effect of Changing the Stepsize in Linear Multistep Codes

Shampine, L. F.; Bogacki, P.

doi:10.1137/0910060

Cited by 8 publications

(3 citation statements)

References 3 publications

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“…Another limitation of our approach is the use of the Matlab built-in function ode15s to solve the proposed system of stiff differential equations. Shampine and Bogacki ( 1989 ) advised caution in drastically reducing the stepsize in the discretization implemented in ode15s since this action may in fact increase numerical error and cause instabilities in the solutions.…”

Section: Discussionmentioning

confidence: 99%

An electromechanical model of neuronal dynamics using Hamilton's principle

Drapaca

2015

Front. Cell. Neurosci.

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Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loading from blast, and by chemical imbalances due to neurological diseases and aging that trigger not only neuronal degeneration but also changes in the mechanical properties of brain tissue. An understanding of the interconnected nature of the electro-chemo-mechanical processes that result in brain damage and ultimately loss of functionality is currently lacking. While modern mathematical models that focus on how to link brain mechanics to its biochemistry are essential in enhancing our understanding of brain science, the lack of experimental data required by these models as well as the complexity of the corresponding computations render these models hard to use in clinical applications. In this paper we propose a unified variational framework for the modeling of neuronal electromechanics. We introduce a constrained Lagrangian formulation that takes into account Newton's law of motion of a linear viscoelastic Kelvin–Voigt solid-state neuron as well as the classic Hodgkin–Huxley equations of the electronic neuron. The system of differential equations describing neuronal electromechanics is obtained by applying Hamilton's principle. Numerical simulations of possible damage dynamics in neurons will be presented.

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Section: Discussionmentioning

confidence: 99%

An electromechanical model of neuronal dynamics using Hamilton's principle

Drapaca

2015

Front. Cell. Neurosci.

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“…The only difference between formulas (30) and (4) is an extra force 3 in the expression of 2 which is a work conjugate for the second endfoot (see Figure 4). By adding equations (32) and (33) we obtain an equation only for the unknown ( ):…”

Section: -2-astrocyte With Two Endfeetmentioning

confidence: 99%

“…We solved system (19) - (24) and the system made of the equations (21) - (24), (31), (32), (34) numerically using Matlab's built-in function ode15s with its default values for the relative error tolerance (10 −3 ) and absolute error tolerance (10 −6 ). The algorithm implemented in this solver involves a modified linear multistep backward difference formula of order up to 5 that has good stability, and an adaptive step size that changes according to a numerical scheme that calculates relative and absolute error tolerances [33].…”

Section: -4-parametersmentioning

confidence: 99%

Mathematical Modeling of a Brain-on-a-Chip: A Study of the Neuronal Nitric Oxide Role in Cerebral Microaneurysms

Drapaca

2018

Emerg Sci J

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Brain tissue is a complex material made of interconnected neural, glial, and vascular networks. While the physics and biochemistry of brain’s cell types and their interactions within their networks have been studied extensively, only recently the interactions of and feedback among the networks have started to capture the attention of the research community. Thus, a good understanding of the coupled mechano-electrochemical processes that either provide or diminish brain’s functions is still lacking. One way to increase the knowledge on how the brain yields its functions is by developing a robust controlled feedback engineering system that uses fundamental science concepts to guide and interpret experiments investigating brain’s response to various stimuli, aging, trauma, diseases, treatment and recovery processes. Recently, a mathematical model for an implantable neuro-glial-vascular unit, named brain-on-a-chip, was proposed that can be optimized to perform some fundamental cellular processes that could facilitate monitoring and supporting brain’s functions, and highlight basic brain mechanisms. In this paper we use coupled elastic, viscoelastic and mass elements to model a brain-on-a-chip made of a neuron and its membrane, and astrocyte’s endfeet connected to an arteriole’s wall. We propose two constrained Lagrangian formulations that link the Hodgkin-Huxley model of the neuronal membrane, and the mechanics of the neuron, neuronal membrane, and the glia’s endfeet. The effects of the nitric oxide produced by neurons and endothelial cells on the proposed brain-on-a-chip are investigated through numerical simulations. Our numerical simulations suggest that a non-decaying synthesis of nitric oxide may contribute to the onset of a cerebral microaneurysm.

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