Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions PDF Download
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Author: Philip James Hahn
Publisher:
ISBN:
Category :
Languages : en
Pages : 100
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Book Description
Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. Potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. Examined here is propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author for this dissertation. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice. Three models are studied using the method developed. First, the Fitzhugh-Nagumo equation is used to illustrate the method. Second, the continuous Nelkin-Yaari model, describing spreading excitation in brain tissue, is examined. Third, a novel model of non-synaptic pulse propagation in hippocampal slices is developed and analyzed. Investigation of this new model shows that potassium wave behavior in the CA1 region can be explained using descriptions of only two phenomena: action potential spike dynamics in response to elevated potassium and simple sink functions that allow for the formation of a wave backside and refractory time.
Author: Philip James Hahn
Publisher:
ISBN:
Category :
Languages : en
Pages : 100
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Book Description
Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. Potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. Examined here is propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author for this dissertation. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice. Three models are studied using the method developed. First, the Fitzhugh-Nagumo equation is used to illustrate the method. Second, the continuous Nelkin-Yaari model, describing spreading excitation in brain tissue, is examined. Third, a novel model of non-synaptic pulse propagation in hippocampal slices is developed and analyzed. Investigation of this new model shows that potassium wave behavior in the CA1 region can be explained using descriptions of only two phenomena: action potential spike dynamics in response to elevated potassium and simple sink functions that allow for the formation of a wave backside and refractory time.
Author: Philip Hahn
Publisher:
ISBN: 9783836493277
Category : Technology & Engineering
Languages : de
Pages : 120
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Book Description
Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. For example, potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. This text examines propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice.
Author: Arnaud Ducrot
Publisher: Editions Publibook
ISBN: 2748346319
Category : Differential operators
Languages : en
Pages : 119
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Book Description
Author: Dongyong Wang
Publisher:
ISBN: 9781321094428
Category :
Languages : en
Pages : 99
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Book Description
Reaction diffusion equations are widely used to model biological phenomena and in some situation, the spatial dimension can be much larger. Numerical efficiently solving high-dimensional reaction-diffusion equations is a huge challenge. To solve the high-dimensional equation, the ``curse of dimensions" has to be dealt with. Also, an efficiently time integration method is needed to solve the afterwords time dependent problem. The sparse grid technique can deal with the problem, and the semi-implicit integration factor method can handles the second one. The combination of two methods will be an efficient method to solve high-dimensional reaction-diffusion equations.
Author:
Publisher:
ISBN:
Category : Dissertations, Academic
Languages : en
Pages : 804
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Book Description
Author: M. R. Duffy
Publisher:
ISBN:
Category : Perturbation (Mathematics)
Languages : en
Pages : 380
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Author: Randy J. Apsel
Publisher:
ISBN:
Category :
Languages : en
Pages : 77
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Book Description
The Boundary Integral Equation (BIE) approach for simulating wave propagation in three-dimensional, irregular multilayered, viscoelastic media is formulated. The BIE formulation takes advantage of known wave propagation properties within an individual layer, leaving only the interactions at the layer boundaries to be treated numerically. This essentially reduces the problem by one spatial dimension and represents a concise treatment of the pertinent physics involved. The resulting system of singular boundary integral equations is much smaller than the corresponding system of equations using the Finite Difference or Finite Element approach, but the block diagonal matrices are much more dense. Two methods are presented for dealing with these dense matrix equations. First an approximate Kirchhoff technique is derived in which only local values of the wave field are allowed to interact with the layer boundaries and the propagation through multilayered structures is accomplished by cascading up and down through the stack to get higher order reflections. Since the Kirchhoff approximation is not valid for critical reflections and some diffraction effects, a second and more complete BIE solution technique was developed which iteratively deals with the singular matrix equation from a perturbation point of view with respect to known flat layer solutions. While the Kirchoff algorithm is a fully three dimensional code for any number of layers, the current iterative BIE algorithm solves a more specialized class of problems and planned extensions to the general case are outlined.
Author: J. H. Merkin
Publisher:
ISBN:
Category :
Languages : en
Pages : 84
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Author: National Aeronautics and Space Administration (NASA)
Publisher: Createspace Independent Publishing Platform
ISBN: 9781722769239
Category :
Languages : en
Pages : 32
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Book Description
Quantitative error analysis for simulation of wave propagation in three dimensional random media assuming narrow angular scattering are presented for the plane wave and spherical wave geometry. This includes the errors resulting from finite grid size, finite simulation dimensions, and the separation of the two-dimensional screens along the propagation direction. Simple error scalings are determined for power-law spectra of the random refractive index of the media. The effects of a finite inner scale are also considered. The spatial spectra of the intensity errors are calculated and compared to the spatial spectra of intensity. The numerical requirements for a simulation of given accuracy are determined for realizations of the field. The numerical requirements for accurate estimation of higher moments of the field are less stringent. Coles, William A. and Filice, J. P. and Frehlich, R. G. and Yadlowsky, M. Unspecified Center NAG8-253...
Author: Anne-Charline Coulon Chalmin
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
This thesis focuses on the long time behaviour, and more precisely on fast propagation, in Fisher-KPP reaction diffusion equations involving fractional diffusion. This type of equation arises, for example, in spreading of biological species. Under some specific assumptions, the population invades the medium and we want to understand at which speed this invasion takes place when fractional diffusion is at stake. To answer this question, we set up a new method and apply it on different models. In a first part, we study two different problems, both including fractional diffusion : Fisher-KPP models in periodic media and cooperative systems. In both cases, we prove, under additional assumptions, that the solution spreads exponentially fast in time and we find the precise exponent of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. In a second part, we deal with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as "the field" and the line to "the road", as a reference to the biological situations we have in mind. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. The expansion shape of the level sets in the field is investigated through numerical simulations.
