Paper:
Inference by Reparameterization in Neural Population Codes
Behavioral experiments on humans and animals suggest that the brain performs probabilistic inference to interpret its environment. Here we present a new generalpurpose, biologically-plausible neural implementation of approximate inference. The neural network represents uncertainty using Probabilistic Population Codes (PPCs), which are distributed neural representations that naturally encode probability distributions, and support marginalization and evidence integration in a biologically-plausible manner. By connecting multiple PPCs together as a probabilistic graphical model, we represent multivariate probability distributions. Approximate inference in graphical models can be accomplished by message-passing algorithms that disseminate local information throughout the graph. An attractive and often accurate example of such an algorithm is Loopy Belief Propagation (LBP), which uses local marginalization and evidence integration operations to perform approximate inference efficiently even for complex models. Unfortunately, a subtle feature of LBP renders it neurally implausible. However, LBP can be elegantly reformulated as a sequence of Tree-based Reparameterizations (TRP) of the graphical model. We re-express the TRP updates as a nonlinear dynamical system with both fast and slow timescales, and show that this produces a neurally plausible solution. By combining all of these ideas, we show that a network of PPCs can represent multivariate probability distributions and implement the TRP updates to perform probabilistic inference. Simulations with Gaussian graphical models demonstrate that the neural network inference quality is comparable to the direct evaluation of LBP and robust to noise, and thus provides a promising mechanism for general probabilistic inference in the population codes of the brain.

Review:
A fundamental challenge for probabilistic interpretations of brain computation is to figure out how can neural networks represent complex probability distributions and use them to guide actions. Two main classes of solutions have been proposed - PPCs which work well for univariate distributions and very restricted parametric families of multivariate distributions and sampling based representations which have more flexibility but at the cost of potentially long sampling times. This paper considers a tempting alternative within the class of variational algs, using a continuous time approximation of TRP. They show that the resulting network dynamics are biologically plausible (and similar to traditional PPCs) and does numerically not much worse than the original. Overall, a neat idea which is worth exploring further in particular in terms of neural implications.  A few comments: - the main advantage of such a representations seem to me to be twofold a) easy access to posterior marginals throughout the computation (which is what probably drives individual decisions) and b) convergence speed.  It would be nice to have a better illustration of this in the results, In particular how long does it take to reach convergence in neural terms (some way to tie dynamics to membrane time constant? how slow are the slow dynamics relative to this)? How much faster is this compared to a standard sampling based alternative implementation? - the idea of state space expansion when mapping random variable is something which has been put forward in the context of sampling as well (Savin& Deneve, Nips 2013). It would be interesting to make these parallels more explicit ...
Would be nice to have a clearer understanding of what the different scales of the dynamics mean in neural terms, beyond the fact that the network activity looks complicated. Fig.4,6: what are the units for the time axis (fast time scale = membrane time const? slow timescale = synaptic facilitation/depression or NMDA/GABAb currents or sth even slower ?)? Still not 100% sure how the graphical model structure translates into the network connectivity structure   Minor typo: last page "biologically plauble interpretable"