If you just want to do a polynomial fit, polyfit would be enough
p=polyfit(x,y,2);
Because the spikes will interfere with the fit (by any algorithm), you might want to do a second pass, removing the outliers,
keep=(y-polyval(p,x))<2e-4;
p=polyfit(x(keep), y(keep), 2); %refinement
Or you could do a pre-pass over the data with a median filter.
