On the methodological side, the points of interest are the following:
First, deriving the properties of the membrane potential fluctuations as a function of the synaptic input properties for Rall’s dendritic model was not so easy (although not conceptually elaborated, basic distribution theory and complex analysis, see Supp. Material). This was a relatively long and nasty calculus, so prone to mistakes… Hence I performed the derivation with sympy, the python library for symbolic mathematics. You can then directly export your results to numpy (for the numerical implementation) and to LateX (for the paper, the supplementary material equations have been automatically generated) without any manual manipulation (the mistakes). See the notebook for how to do this. This is, to my opinion, a great workflow for those analytically demanding problems. Note that, of course, you need to carefully set up the problem conceptually before, sympy would never have been able to give me the solution “out of the box”.
Second, the numerical simulations of the dendritic model (just to check the relative accuracy with the analytical description), I implemented it in NEURON (with its python layer, well…). I provide the code for this implementation. Note that the simulations are rather long to obtain the statistical properties of the membrane potential fluctuations (this is what motivates the analytical derivation, see paper).
The rest of the paper is quite trivial in terms of implementation once you have the analytical estimate.
The Ipython notebook associated to this paper can be found on the following link:
[[ in progress ..]]