Specification of Rallpack 2 : ============================================================================= 1. Objective Tests the ability of a simulator to handle branching in passive compartmental models . ============================================================================= 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. ============================================================================= 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. ============================================================================= 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. =============================================================================