  The Bayesian view of the brain hypothesizes that the brain constructs a
generative model of the world, and uses it to make inferences via Bayes' rule.
Although many types of approximate inference schemes have been proposed for
hierarchical Bayesian models of the brain, the questions of how these distinct
inference procedures can be realized by hierarchical networks of spiking
neurons remains largely unresolved. Based on a previously proposed
multi-compartment neuron model in which dendrites perform logarithmic
compression, and stochastic spiking winner-take-all (WTA) circuits in which
firing probability of each neuron is normalized by activities of other neurons,
here we construct Spiking Neural Networks that perform \emph{structured}
mean-field variational inference and learning, on hierarchical directed
probabilistic graphical models with discrete random variables. In these models,
we do away with symmetric synaptic weights previously assumed for
\emph{unstructured} mean-field variational inference by learning both the
feedback and feedforward weights separately. The resulting online learning
rules take the form of an error-modulated local Spike-Timing-Dependent
Plasticity rule. Importantly, we consider two types of WTA circuits in which
only one neuron is allowed to fire at a time (hard WTA) or neurons can fire
independently (soft WTA), which makes neurons in these circuits operate in
regimes of temporal and rate coding respectively. We show how the hard WTA
circuits can be used to perform Gibbs sampling whereas the soft WTA circuits
can be used to implement a message passing algorithm that computes the
marginals approximately. Notably, a simple change in the amount of lateral
inhibition realizes switching between the hard and soft WTA spiking regimes.
Hence the proposed network provides a unified view of the two previously
disparate modes of inference and coding by spiking neurons.
