Fast sampling for Bayesian inference in neural circuits
  Time is at a premium for recurrent network dynamics, and particularly so when they are stochastic and correlated: the quality of inference from such dynamics fundamentally depends on how fast the neural circuit generates new samples from its stationary distribution. Indeed, behavioral decisions can occur on fast time scales (~100 ms), but it is unclear what neural circuit dynamics afford sampling at such high rates.   We analyzed a stochastic form of rate-based linear neuronal network dynamics with synaptic weight matrix $W$, and the dependence on $W$ of the covariance of the stationary distribution of joint firing rates. This covariance $\Sigma$ can be actively used to represent posterior uncertainty via sampling under a linear-Gaussian latent variable model. The key insight is that the mapping between $W$ and $\Sigma$ is degenerate: there are infinitely many $W$'s that lead to sampling from the same $\Sigma$ but differ greatly in the speed at which they sample. We were able to explicitly separate these extra degrees of freedom in a parametric form and thus study their effects on sampling speed.   We show that previous proposals for probabilistic sampling in neural circuits correspond to using a symmetric $W$ which violates Dale's law and results in critically slow sampling, even for moderate stationary correlations. In contrast, optimizing network dynamics for speed consistently yielded asymmetric $W$'s and dynamics characterized by fast transients, such that samples of network activity became fully decorrelated over ~10 ms. Importantly, networks with separate excitatory/inhibitory populations proved to be particularly efficient samplers, and were in the balanced regime. Thus, plausible neural circuit dynamics can perform fast sampling for efficient decoding and inference. 
Guillaume Hennequin, Laurence Aitchison, M\'at\'e Lengyel
q-bio.NC