A change in the definition of spatial frequency bandwidth?

Source

Since the beginning, we have used a definition of bandwidth in the spatial frequency domain which was quite standard (see supp material for instance):

$$ \mathcal{E}(f; sf_0, B_{sf}) \propto \frac {1}{f} \cdot \exp \left( -.5 \frac {\log( \frac{f}{sf_0} ) ^2} {\log( 1 + \frac {B_sf}{sf_0} )^2 } \right) $$

This is implemented in the folowing code which reads:

env = 1./f_radius*np.exp(-.5*(np.log(f_radius/sf_0)**2)/(np.log((sf_0+B_sf)/sf_0)**2))

However the one implemented in the code looks different (thanks to Kiana for spotting this!), so that one can think that the code is using:

$$ \mathcal{E}(f; sf_0, B_{sf}) \propto \frac {1}{f} \cdot \exp \left( -.5 \frac {\log( \frac{f}{sf_0} ) ^2} {\log(( 1 + \frac {B_sf}{sf_0} )^2 ) } \right) $$

The difference is minimal, yet very important for a correct definition of the bandwidth!

In fact, we are really using the correct formula, but to make it clearer, I just changed it to this new code which reads:

env = 1./f_radius*np.exp(-.5*(np.log(f_radius/sf_0)**2)/((np.log((sf_0+B_sf)/sf_0))**2))