Wave Propagation In A Coupled Reaction Diffusion System Microform

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.

Consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical/biochemical systems. These equations are of the FitzHugh-Nagumo type and contain three parameters. For certain ranges of the parameters the system exhibits two stable singular points. A singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves. A complete, rigorous, description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity. Numerical results of solutions to these equations are also presented. These calculations illustrate how waves are generated from initial data, how they interact during collisions, and how they are influenced by local disturbances and boundary conditions.

"Partial Differential Equations and Solitary Waves Theory" is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirota’s bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons. While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II will be most useful for graduate students and researchers in mathematics, engineering, and other related fields. Dr. Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University, Chicago, Illinois, USA.