  Monte Carlo approaches have recently been proposed to quantify connectivity
in neuronal networks. The key problem is to sample from the conditional
distribution of a single neuronal spike train, given the activity of the other
neurons in the network. Dependencies between neurons are usually relatively
weak; however, temporal dependencies within the spike train of a single neuron
are typically strong. In this paper we develop several specialized
Metropolis--Hastings samplers which take advantage of this dependency
structure. These samplers are based on two ideas: (1) an adaptation of fast
forward--backward algorithms from the theory of hidden Markov models to take
advantage of the local dependencies inherent in spike trains, and (2) a
first-order expansion of the conditional likelihood which allows for efficient
exact sampling in the limit of weak coupling between neurons. We also
demonstrate that these samplers can effectively incorporate side information,
in particular, noisy fluorescence observations in the context of
calcium-sensitive imaging experiments. We quantify the efficiency of these
samplers in a variety of simulated experiments in which the network parameters
are closely matched to data measured in real cortical networks, and also
demonstrate the sampler applied to real calcium imaging data.
