Fig. 4: A recurrent circuit generates samples from the posterior defined by a hierarchical generative model.
From: Sampling-based Bayesian inference in recurrent circuits of stochastic spiking neurons
a Schematic of recurrent circuit dynamics, in which stimulus, s, and stimulus parameter, z, are encoded respectively in the population response, rt, and recurrent inputs, \({{{{{{{{\bf{u}}}}}}}}}_{t}^{{\mathsf{r}}}\). b, c When the feedforward inputs and recurrent inputs share the same tuning profile, summing the two inputs to define the instantaneous firing rate (b) is equivalent to multiplying the conditional distributions encoded by the two inputs to obtain the conditional distribution of the stimulus, \(p(s| {\tilde{z}}_{t},{{{{{{{{\bf{u}}}}}}}}}^{{\mathsf{f}}})\). c The conditional distributions of the stimulus can be explicitly read out from corresponding population responses by a linear decoder (b). d–f) Reading out the joint sampling distribution from the recurrent circuit. The projection of the spiking activity (Eq. (14)) and recurrent inputs (Eq. (29)) onto the stimulus subspace (black curves), can be read out linearly from the population activity and interpreted as a sample of stimulus and stimulus parameter respectively (Eqs. (4b), (4c)). Top right insets: the empirical marginal distributions of samples and marginal posteriors (smooth lines). (f) The joint value (red dots) of instantaneous samples of stimulus (black curve on the surface in (d)), and stimulus parameter (black curve on the surface in (e)) represent samples from the joint posterior of the stimulus and stimulus parameter. The true joint posterior is represented by the blue contour.
