Networks of well-known dynamical units but unknown interaction topology arise across various fields of biology, including genetics, ecology, and neuroscience. The collective dynamics of such networks is often sensitive to the presence (or absence) of individual interactions, but there is usually no direct way to probe for their existence. Here we present an explicit method for reconstructing interaction networks of leaky integrate-and-fire neurons from the spike patterns they exhibit in response to external driving. Given the dynamical parameters are known, the approach works well for networks in simple collective states but is also applicable to networks exhibiting complex spatio-temporal spike patterns. In particular, stationarity of spiking time series is not required.