Optimal representation in biological systems

An optimal representation constitutes an efficient set. It is known that for an aggregating system if the cost of representation increases linearly with the number of bases, ternary coding is superior to binary, and coding in e is optimal. This paper investigates the relative efficiency of bases for the cases when the cost complexity is affine (slope-intercept linear), exponential, and logistic and presents new results. It is shown that for representation of structure in logistic maps, which applies often to biological systems and is true for input-output maps of neurons, the optimal base value is near 1.7632, which is consistent with the unary and space coding of information in songbirds. It is shown that the mathematical basis of this result is the solution to the equation $b^b=e.$