 
Summary: 1
Supplementary Materials
This section is organized into three parts. In the first, we show that when the likelihood
function, p(rs), belongs to the exponential family with linear sufficient statistics, optimal
cue combination can be performed by a simple network in which firing rates from two
population codes are combined linearly. Moreover, we show that the tuning curves of the
two populations don't need to be identical, and that the responses both within and across
populations don't need to be uncorrelated. In the second part, we consider the specific
case of independent Poisson noise, which provides an example of a distribution
belonging to the exponential family with linear sufficient statistics. We also consider a
distribution that does not belong to the exponential family with linear sufficient statistics,
namely, independent Gaussian noise with fixed variance. We show that, for this case,
optimal cue combination requires a nonlinear combination of the population codes. In the
third part, we describe in detail the parameters of the network of conductancebased
integrateandfire neurons.
1. Probabilistic Population Codes for Optimal Cue Combination
1.1 Bayesian inference through linear combinations for the exponential family
Consider two population codes, r1 and r2 (both of which are vectors of firing rates),
which code for the same stimulus, s. As described in the main text, this coding is
probabilistic, so r1 and r2 are related to the stimulus via a likelihood function, p(r1,r2s).
