Find synaptic topology from spike trains

Can you retrieve the underlying neuronal network topology which generates an ensemble of desired spiking trains? This is one of the key questions if one wants to implement learning in spiking neuronal network. We propose an approach to solve the question. Our approach ensures that the retrieved spiking neuronal network not only generates the desired spike timing pattern but also has a sparse topology. We analyze the solvability and robustness of our algorithm in details based on the linear programming theory. Two numerical examples are included to illustrate the approach. One example is artificial and the spike trains are generated by a leaky integrate-and-fire neuronal network. The other is from experimental data of the neuronal spikes recorded in hippocampal CA3 area. Our results demonstrate that the approach can provide us with a framework to deal with the learning problem in spiking neuronal networks.