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.
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