First let's eliminate the notion of "biological compartmentalization" from the discussion.

"Biological compartmentalization" is an inescapable fact: biological processes occur

in restricted spatial and temporal scales, which are set by physical and/or physiological

barriers of greater or lesser permeability. Examples of biological compartmentalization

include membrane-bound receptors, buffers, organelles, the blood-brain barrier,

etc..

So we focus on "compartmentalization" as it is used in computational neuroscience.

It's important to distinguish between conceptual models and computational models.

Conceptual models, i.e. hypotheses about how neurons and networks function,

do not need compartments at all. The literature is full of papers by experimentalists

and theoreticians who propose and analyze conceptual models that make no reference

whatsoever to compartments. This is possible because many conceptual models are

amenable to intuition, logic, and (when necessary) simplifications that allow the

application of powerful mathematical tools (e.g. nonlinear dynamical systems

approaches).

However, there is a large number of conceptual models whose complexity defies

intuition, logic, and mathematical analysis. Many of these are models that are closely

tied to experimental observations, in which specific anatomical and/or biophysical

properties of real cells are important. In such cases, it may be helpful to try to

map the conceptual model onto a computational model. The computational model,

which is used in simulations that generate numerical results, serves as a surrogate

for the original conceptual model.

The question is how to accomplish this mapping. At the scale of interest to most

neuroscientists, space seems continuous: membrane potential and current, ion

concentrations, neurite diameter--all these things appear to be continuous functions

of location. And time also appears to be infinitely divisble.

But in a digital computer, nothing is continuous. In a computational model, implemented

with a digital computer, all of these continuous variables must be represented by

numbers that have finite precision. The partial differential equations that describe the

spread and interaction of electrical and chemical signals over continuous time and

space must be broken into families of ordinary differential equations (ODEs), which

describe the signals at discrtete points in space (the "spatial grid").

The spatially discretized model is not the same as the original, spatially continuous,

conceptual model--it is a

*model of a model*. And, except for very simple cases,

it isn't possible to produce an exact solution of the ODEs that describe the discretized

model. Instead, it is necessary to chop time into bits (temporal discretization) and

generate a numerical approximation to the solution of the ODEs.

So a computer simulation is an approximation to the behavior of a model of a model.

It consists of a sequence of samples over time and space (the "spatiotemporal grid")

of approximations to the variables we are interested in.

For a computational model to be a fair test of a conceptual model, it is necessary to

ensure that the mapping from conceptual to computational model introduces is

faithful. Careful attention to spatial and temporal discretization are an important part

of this task.