It has been proposed that populations of neurons process information in terms of probability density functions (PDFs) of analog variables. Such analog variables range, for example, from target luminance and depth on the sensory interface to eye position and joint angles on the motor output side. The requirement that analog variables must be processed leads inevitably to a probabilistic description, while the limited precision and lifetime of the neuronal processing units lead naturally to a population representation of information. We show how a time-dependent probability densityρ(x; t) over variable x, residing in a specified function space of dimension D, may be decoded from the neuronal activities in a population as a linear combination of certain decoding functions φi(x), with coefficients given by the N firing rates ai(t) (generally with DN). We show how the neuronal encoding process may be described by projecting a set of complementary encoding functions i(x) on the probability density ρ(x; t), and passing the result through a rectifying nonlinear activation function. We show how both encoders i (x) and decoders φi(x) may be determined by minimizing cost functions that quantify the inaccuracy of the representation. Expressing a given computation in terms of manipulation and transformation of probabilities, we show how this representation leads to a neural circuit that can carry out the required computation within a consistent Bayesian framework, with the synaptic weights being explicitly generated in terms of encoders, decoders, conditional probabilities, and priors.

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