Wednesday, June 24, 2015

Using Recursion to Define a Periodic Function

I'm working my way through Heikki Ruskeepaa's book, Mathematica Navigator and found this elegant one-liner that shows how to define a periodic function with recursion. Note that the base case required to stop the recursion is not a single value but the condition 0 <= t <= 2.

Clear@sawtooth;sawtooth[time_,scale_:1]:=If[0<=time<=2,scale time,sawtooth[scale (time-2)]]
Plot[sawtooth@t,{t,0,10}]



How does the recursion work? If t => 2, sawtooth keeps subtracting 2 from t recursively until t is back in the range 0 <= t <= 2 and computes that value for y. We can see sample values of a function plotted with DiscretePlot.

DiscretePlot[sawtooth@t,{t,0,6,0.2},PlotTheme->"Web"]



Of course a faster way of defining the domain is to use Mod, but there is a lesson here. We don't need the speed of Mod, so either implementation is fine - using Mod for the most compact and fast function, or using an elegant new recursive technique to learn it. This function omits the values at multiples of 2, but is effectively the same function:

Clear@sawtooth2;sawtooth2[time_,scale_:1]:=scale Mod[time,2]

Mathematica has built-in functions to produce various canonical waveforms  such as SawtoothWave. We are working with sawtooth and triangle waves in the lab since, interestingly, there is a lot of nonlinearity at the electrode-tissue interface and in the kHz+ frequency range, a rectangular wave is transformed by what is called partial half-wave rectification into something like a triangle offset in the negative (cathodic) direction from V = 0. We believe this is a key to understanding how biphasic neuromodulation works in the kHz+ frequency range.

Plot[SquareWave[{-50,50},t],{t,0,10},ExclusionsStyle->Dotted,AxesLabel->{"time (mS)","mV"},PlotLabel->"Electric potential at the electrode"]




Plot[TriangleWave[{-40,10},t],{t,0,10},AxesLabel->{"time (mS)","mV"},PlotLabel->"Electric potential seen by the axon"]




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