Specification of Rallpack 2 :
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1. Objective
Tests the ability of a simulator to handle branching in passive compartmental
models .
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2. Simulation
This simulation models a binary branched structure obeying Rall's
3/2 power law. It consists of 1023 compartments corresponding to 10 levels
of branching. Each branch has an electrotonic length of 0.008 . At each
branch away from the root the diameter decreases by a factor of 2^(2/3),
in accordance with the power law, and the length by a factor of 2^(1/3).
The detailed geometry of the compartments is
Depth Number Length (microns) Diameter (microns)
0 1 32.0 16.0
1 2 25.4 10.08
2 4 20.16 6.35
3 8 16.0 4.0
4 16 12.7 2.52
5 32 10.08 1.587
6 64 8.0 1.0
7 128 6.35 0.63
8 256 5.04 0.397
9 512 4.0 0.25
The membrane properties are :
RA = 1.0 ohms meter = 100 ohms cm
RM = 4.0 ohms meter^2 = 40000 ohms cm^2
CM = 0.01 Farads/meter^2 = 1.0 uf/cm^2
EM = -0.065 Volts = -65 mV
A current of 0.1 nA is injected into the root compartment. Membrane voltages
are recorded for the root and a terminal branch compartment.
This model is run for a simulated time of 0.25 second.
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3. Theoretical solution
By Rall's power law this branched structure is equivalent to a single
cylinder of diameter 16 microns and length 320 microns, (ie., 0.08 length
constants) for stimuli presented at the root. The electrotonic length
lambda is 0.004 metres. We use the same 'correct' solution as for the simple
cable test, rallpack1, with the following arguments :
currinj -l 320e-6 -L 0.004 -d 16e-6 -n 200
This calculation takes a considerable length of time, and the precomputed
solution is therefore provided. It is accurate to at least 6 figures.
When using asymmetric compartments the solution is slightly different
from the 'correct' solution because of the poor spatial discretization. This
is an intrinsic property of models based on asymmetric compartments and should
be borne in mind when building neuronal models.
The correct solution is based on the Laplace-transform solution
of the finite cable with constant current injection, with a factor of 2.0
scaling using the method of 'reflections' since our system has current
injection at one end. See "Electrical current flow in excitable cells",
OUP, 1983; Jack, Noble and Tsien, pp 79,80, also pp 68. This solution is
a sum of a series, and therefore its accuracy depends on the number of
terms used in the sum. The shorter length constant of this cable means that
we need to use a large number of terms in the series (the given files are
the results of summing 200 terms), for sufficient accuracy.
The correct solution is implemented in the directory rallpack/theor. The
error function it uses is from "Numerical recipes in C", CUP 1988; Press,
Flannery, Teukolsky and Vetterling.
The correct waveforms (provided in this directory) are :
ref_branch.0 Laplace transform solution for current injection site
ref_branch.x Laplace transform solution for terminal branch
The program rallpack/comparison/rms.c will evaluate rms differences in
voltage traces.
NOTE : An additional, sometimes useful test for numerical stability is to
perform the same current injection into a terminal branch and record at the
root and the terminal branch. The recording at the root should be identical
to the theoretical curve for the branch (ref_branch.x) as computed for
the previous case (Rall, Ann NY Acad Sci 96, 1071-1092 (1962)).
The recording at the terminal branch (which differs from these solutions)
should reveal if the integration method displays instability due to the
high voltage change rates at this site. The Crank Nicolson method is prone
to this problem.
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4. Performance measures. (See ../README for definitions)
General information
Rallpack name, Simulator name and version
Peak speed and model size at which the speed is attained
Asymptotic accuracy (error %)
Semi-accurate timestep (Timestep for 2x asymptotic error)
Hardware information : model and MIPS rating.
Simulation setup time for 1000 compartment model
Integration method
Compartment equivalents : Description, value
Detailed report :
1 Accuracy vs. Timestep
2 Accuracy vs. Simulation speed
A set of simulations of the same model size should be run at
a range of timesteps, and the accuracy and simulation speed
should be calculated for each case.
The timesteps should cover the 'useful' range for the model,
within which the accuracy goes from close to its asymptotic
(best) value to a few % error. Typical timesteps are
10,20,50,100,200,500 and 1000 usec. The recommended model size
1023 compartments (10 levels of branching). The
model size should be quoted.
These results can be tabulated and/or graphed.
If the raw speed is independent of timestep (i.e. the
simulation speed is directly proportional to timestep, which is
usually true) the two graphs can be merged, and the respective
x axis scales should be displayed.
The x axis may be displayed on a log scale.
3 Raw speed vs. Model size
4 Model memory per compartment vs. Model size
A set of simulations with a range of model sizes should
be carried out, calculating the raw speed and model memory
for each case. All simulations should use the same timestep,
which should be quoted. The suggested timestep is 50 usec.
Recommended model sizes are 1, 8, 128, 1023 and 8191
compartments, corresponding to 1, 3, 7, 10 and 13 levels
of branching.
These results can be tabulated and/or graphed.
If graphed, they may be displayed on the same graph with
the two y axis scales displayed.
The x axis may be displayed on a log scale.
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