  Recent experimental advances are producing an avalanche of data on both
neural connectivity and neural activity. To take full advantage of these two
emerging datasets we need a framework that links them, revealing how collective
neural activity arises from the structure of neural connectivity and intrinsic
neural dynamics. This problem of {\it structure-driven activity} has drawn
major interest in computational neuroscience. Existing methods for relating
activity and architecture in spiking networks rely on linearizing activity
around a central operating point and thus fail to capture the nonlinear
responses of individual neurons that are the hallmark of neural information
processing. Here, we overcome this limitation and present a new relationship
between connectivity and activity in networks of nonlinear spiking neurons by
developing a diagrammatic fluctuation expansion based on statistical field
theory. We explicitly show how recurrent network structure produces pairwise
and higher-order correlated activity, and how nonlinearities impact the
networks' spiking activity. Our findings open new avenues to investigating how
single-neuron nonlinearities---including those of different cell
types---combine with connectivity to shape population activity and function.
