Resumen
- Rationale: The aim of the present work is to understand what the propagation of epileptogenic activity in the cortex can tell us about the underlying connectivity between different cortical area. We have constructed a simple geometrical model of the cerebral hemisphere using a projection of the cortex to a tangential plane defined by the Delaunay triangulation of a sphere. These triangular elements can be chosen of an arbitrary size and can be connected to any other according to specific rules.Methods: The model is governed by a stochastic process that follows the equation U(x,t,T)=E(exp(- (x(s))ds) where the function is related to the probability of a process being absorbed at a point x and the average is taken over all the paths. We take this function U to represent the propagation of an electrical wave and the function to represent the connectivity of the Delaunay elements. We show that U satisfies an equation of the form tU+b xU+1/2a xxU- U=0 . We solve this equation on the Delaunay sphere and show how the propagation of the electrical waves depends on the connectivity pattern of the underlying model cortex . The coefficients b and a are related to the mean and the variance of the underlying stochastic process.Results: It is clear that epileptic areas have been identified where there is an abnormal pattern of connection between neurons such as in dysplasias, or other types of epilepsies, such as those induced by kindling or injuries, where there is abnormal dendritic growth. We reason that in these pathological areas of the cortex there would be lower connectivity to passing axonal fibers in the first case(dysplasias), and excessive connectivity in the second(injury or kindling). We model the lower connectivity region by changing the probability of absoption, and we find that in lower connectivity areas there is a border effect that gives rise to abnormal electrical propagation, whereas increasing the connectivity of an area beyond a certain critical level leads to an abnormal locus of electrical activity. We want to understand what is the effect of long range connections between cortical areas and between inter-hemispherical connectivity and how these type of connections affect the electrical waves in the cortex. Conclusions: We have developed a geometrical model to model epileptogenic activity in the cortex. The mode of propagation of the resulting wave is sensitive to the intrinsic properties of connectivity of the Delaunay elements such that border effects and foci of electrical activity result when there are areas of low or high connectivity. These effects could explain why certain types of abnormal cortical structures such as the ones found in dysplasias and injuries could generate foci of abnormal electrical activity that lead to unprovoked seizures and epilepsy.Translational Research