I have seen a presentation in a lab recently, from a group working precisely on the simulation of lightning. I was surprised to see, at a small scale, how crooked and twisted on itself, was the path of the lighting, due to retro-action of the magnetic field, created by the current, and the varying electric field, on the path itself. The path ends up rolled many times on itself because of that.
About calculus, indeed one would need PIC (particle in cell) as stated above, to model the dynamics of electons and species at low pressures, together with finite elements, to model the electric and magnetic field, and the concentrations at higher pressures. These algorithms are very intensive, and as stated above again, it is difficult to reproduce the correct branching of lighting. Branching in plasma simulations is a difficult thing, subject to research. It is not impossible that in the case of lighting, a lot of "luck" - or chaos, small perturbations changing the path a lot - takes place.