Table 2.

Closed form solution for optimal neural populations with sigmoidal tuning curves, for objective functions specified by Eq. (25).

		Infomax	Discrimax	General
Optimized:		f(x) = log x	f(x) = −x−1	$f(x)=-x^{\alpha },\alpha <\frac{1}{3}$
Density	d(s)	Np(s)	$\propto Np{(s)}^{\frac{1}{3}}{[1-P(s)]}^{\frac{1}{3}}$	$\propto Np{(s)}^{\frac{1}{1-2\alpha }}{[1-P(s)]}^{\frac{\alpha }{2\alpha -1}}$
Gain	g(s)	RN−1 [1 − P(s)]−1	RN−1 [1 − P (s)]−1	RN−1 [1 − P (s)]−1
Fisher	If (s)	∝ RNp2(s) [1 − P(s)]−1	$\propto R{Np}^2(s){[1-P(s)]}^{-1}$	$\propto R{Np}^{\frac{2}{1-2\alpha }}(s){[1-P(s)]}^{\frac{1}{2\alpha -1}}$
Discrim.	δmin(s)	$\propto p^{-1}(s){[1-P(s)]}^{\frac{1}{2}}$	$\propto p^{-\frac{1}{3}}(s){[1-P(s)]}^{\frac{1}{6}}$	$\propto p^{\frac{1}{2\alpha -1}}(s){[1-P(s)]}^{\frac{1}{2-4\alpha }}$