Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part II

Application of comparison theorem is used to examine the validitiy of the "lumped parameter assumption" in describing the behavior of solutions of the continuous cable equation Ut = DUxx + f (U) with the discrete cable equation dV(n)/d(t) = d*(Vn+1 - 2V(n) + Vn+1) + f(V-n), where f is a nonlinear functional describing the internal diffusion of electrical potential in single neurons. While the discrete cable equation looks like a finite difference approximation of the continuous cable equation, solutions of the two reveal significantly different behavior which imply that the compartmental models (spiking neurons) are poor quantifiers of neurons, contrary to what is commonly accepted in computational neuroscience.