Compartmentally?

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ted
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Re: Compartmentally?

Post by ted »

Not a silly question, really.

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.
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