  The synaptic connectivity of cortical networks features an overrepresentation
of certain wiring motifs compared to simple random-network models. This
structure is shaped, in part, by synaptic plasticity that promotes or
suppresses connections between neurons depending on their spiking activity.
Frequently, theoretical studies focus on how feedforward inputs drive
plasticity to create this network structure. We study the complementary
scenario of self-organized structure in a recurrent network, with spike
timing-dependent plasticity driven by spontaneous dynamics. We develop a
self-consistent theory that describes the evolution of network structure by
combining fast spiking covariance with a fast-slow theory for synaptic weight
dynamics. Through a finite-size expansion of network dynamics, we obtain a
low-dimensional set of nonlinear differential equations for the evolution of
two-synapse connectivity motifs. With this theory in hand, we explore how the
form of the plasticity rule drives the evolution of microcircuits in cortical
networks. When potentiation and depression are in approximate balance, synaptic
dynamics depend on the frequency of weighted divergent, convergent, and chain
motifs. For additive, Hebbian STDP, these motif interactions create
instabilities in synaptic dynamics that either promote or suppress the initial
network structure. Our work provides a consistent theoretical framework for
studying how spiking activity in recurrent networks interacts with synaptic
plasticity to determine network structure.
