In physical systems a particle instead interacts only with a finite number of other particles, hence the density field remains highly fluctuating. The effect of theθ(w x) term is to select one particular half-space over which the integralisdone. To estimate the fluctuations due to a finite number of nodes, we will have to estimate the width of the output distribution for a given set of parameters. Toestimate the error inFigure 1d ofthe main text, we ask what are the values ofxk = xcosθ such that the average output plus or minus a standard deviation, divided by M, would be equal to the threshold. Since the standard deviation involves|x|2, we estimate its average value for points 3 with a givenxk, i.e.