The aim of this talk is to suggest a new way of modeling neural
tissue, not as a network, but instead as a three-dimensional
continuum, partitioned by cell membranes into neurons, glia, and
extracellular space.  The fundamental equations of such a system are
the 3D drift-diffusion equations for the various ions,
neurotransmitters, and neuromodulators of the tissue, coupled to the
Poisson equation for the electrical potential, with internal boundary
(jump) conditions at cell membranes.  We do not propose to homogenize
these equations, but instead to solve them on the detailed, intricate
geometry of the actual neuronal and glial cytoarchitecture.  Because
of a small parameter, the ratio of the Debye length to a typical
cellular dimension, we are led to consider the limiting case known as
electroneutrality, in which space charge is confined to thin layers
near membranes, with the rest of the system being electrically
neutral.  An interesting complication here is that the ionic
composition of the space charge layers enters into the boundary
conditions and therefore needs to be determined.  Boundary conditions
across membranes involve the traditional Hodgkin-Huxley type equations
for a membrane patch, taking into account the local densities of the
different channel types.  This talk will focus on the mathematical
formulation of the problem, some preliminary numerical results, and a
discussion of what it would take to do computational neuroscience this
way.