  Dynamics and function of neuronal networks are determined by their synaptic
connectivity. Current experimental methods to analyze synaptic network
structure on the cellular level, however, cover only small fractions of
functional neuronal circuits, typically without a simultaneous record of
neuronal spiking activity. Here we present a method for the reconstruction of
large recurrent neuronal networks from thousands of parallel spike train
recordings. We employ maximum likelihood estimation of a generalized linear
model of the spiking activity in continuous time. For this model the point
process likelihood is concave, such that a global optimum of the parameters can
be obtained by gradient ascent. Previous methods, including those of the same
class, did not allow recurrent networks of that order of magnitude to be
reconstructed due to prohibitive computational cost and numerical
instabilities. We describe a minimal model that is optimized for large networks
and an efficient scheme for its parallelized numerical optimization on generic
computing clusters. For a simulated balanced random network of 1000 neurons,
synaptic connectivity is recovered with a misclassification error rate of less
than 1% under ideal conditions. We show that the error rate remains low in a
series of example cases under progressively less ideal conditions. Finally, we
successfully reconstruct the connectivity of a hidden synfire chain that is
embedded in a random network, which requires clustering of the network
connectivity to reveal the synfire groups. Our results demonstrate how synaptic
connectivity could potentially be inferred from large-scale parallel spike
train recordings.
