The more detail that is included in a model, the more difficulty there is in selecting model parameters and getting desired results. That's why so many computational neuroscience models don't bother with ion accumulation--it is just assumed that concentrations are constant.
Many modelers have used what might be called "a physicist's first order approximation to representing the dynamics of calcium concentration." The conceptual model is very simple. Calcium accumulates in a shell adjacent to the inner surface of the cell membrane, where the concentration is cai. From there it can exchange with calcium in the core of the cell, or be pumped out of the cell. The effect of these processes on cai is described by a first order ordinary differential equation. The very simplest implementation of this model ignores buffering, so, ignoring scale factors needed for dimensional consistency, the ODE is
cai' = -ica/depth/F/2 + (cai0 - cai)/tau
where depth is the shell thickness, F is Faraday's constant, cai0 is the initial value of cai, and tau is the time constant of cai equilibration. Such a mechanism is implemented in the file cacum.mod, which you will find in the nrn/share/examples/nrniv/nmodl subdirectory of the directory tree that results from cloning NEURON's latest development code, or by downloading and expanding the gzipped tar file of NEURON source code from http://www.neuron.yale.edu/neuron/download/getstd
If you want your calcium accumulation mechanism to include instantaneous buffering, change the ODE to
cai' = -ica/depth/F/2/B + (cai0 - cai)/tau
where B is a dimensionless parameter that specifies the ratio of bound calcium to free calcium. To do this, make a copy of cacum.mod that you call cacumb.mod, then edit cacumb.mod and make these changes:
1. Just before the ENDCOMMENT line, insert these lines:
Based on cacum.mod
Includes instantaneous calcium buffering.
The dimensionless parameter B specifies
the ratio of bound calcium to free calcium.
: SUFFIX cacum
3. In the PARAMETER block insert these lines just before the line that has the closing curly bracket }.
Code: Select all
B (1) : a dimensionless parameter that specifies
: the ratio of bound to free calcium
The advantage of cacum and cacumb is that it's simple and you have complete control over the time course of cai equilibration.
If you prefer the more complex model from chapter 9, you'll have more parameters to deal with, such as the pump's rate constants, and the pump's density parameter. And for each section you'll have to tinker with at least the pump's density parameter to make sure you get the desired cai dynamics, because surface to volume ratio, which varies with segment diameter, affects the cai dynamics. And you'll have people asking whether it makes sense to have compartment thickness vary with segment diameter.