Potassium Waves and Neural Excitation: Propagation in Spatially Extended Reaction Diffusionsystems

Propagation of functional or pathological ionicdisturbances in biological systems plays an importantrole in normal regulatory mechanisms and in disease. For example, potassium diffusion in brain tissue isinvolved in spreading excitation. Models of thistype of phenomenon often take the form of areaction-diffusion system in one spatial dimensionwith continuous dynamic variables. This text examinespropagation in three spatial dimensions through anetwork of discrete dynamic elements coupled bydiffusion. Conditions permissive of pulse origination andpropagation can be determined analytically forsystems in one spatial dimension. However, in threespatial dimensions or in dynamic systems containingdiscontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable systemat a point in space are analyzed. The effectivediffusive flux or current at a point is interpretedas a slowly varying parameter. The bifurcationstructure of the dynamics with respect to thisparameter and the effect of waveform on the timecourse of the parameter are examined. Propagationresults when an excursion at a point produces adiffusion current sufficient to move its restingneighbor above some threshold value. The formation ofa pulse back depends on the stability of equilibriaof the local dynamics. Propagation in some cases mayalso depend on the geometry of the wavefront. Predictions are verified by numerical simulationusing a software package developed by the author. Athree dimensional lattice allows for description ofthe local dynamics at nodal elements and diffusionbetween elements and throughout the lattice.