Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part i - Brzychczy, S.L. (Imperial College Press, 2010)
Methods of comparing discrete and continuous cable models of single neurons and dynamical phenomena observed in neurobiology can be described with infinite-coupled systems of semilinear parabolic differential-functional equations of the reaction-diffusion-convection type or infinite systems of ordinary integro-differential equations. It is known that numerous problems in computational neuroscience use finite systems of equations based on the so-called compartmental model. It seems a natural idea to extend the results obtained in the theory of finite systems onto infinite systems. However, this requires stringent assumptions to be adopted to achieve compatibility. In most instances the dynamics of infinite systems behave differently to their finite-dimensional projections. The truncation method applied to infinite systems of equations and presented herein yields a truncated system consisting of the first N equations of the infinite system in N unknown functions. A solution of infinite system is defined as the limit when N -> infinity of the sequence of approximations {z(N)} (N=1,2,...,) where z(N) = (z(N)(1), z(N)(2),..., z(N)(N)) are defined as solutions of suitable finite truncated systems with corresponding initial-boundary conditions. Geometrically, it may be described as the projection of an infinite system of differential equations considered in a function abstract space of infinite dimension (such as Banach or Hilbert space) onto its finite-dimensional subspaces.