  A rigorous bridge between spiking-level and macroscopic quantities is an
on-going and well-developed story for asynchronously firing neurons, but focus
has shifted to include neural populations exhibiting varying synchronous
dynamics. Recent literature has used the Ott--Antonsen ansatz (2008) to great
effect, allowing a rigorous derivation of an order parameter for large
oscillator populations. The ansatz has been successfully applied using several
models including networks of Kuramoto oscillators, theta models, and
integrate-and-fire neurons, along with many types of network topologies. In the
present study, we take a converse approach: given the mean field dynamics of
slow synapses, predict the synchronization properties of finite neural
populations. The slow synapse assumption is amenable to averaging theory and
the method of multiple timescales. Our proposed theory applies to two
heterogeneous populations of N excitatory n-dimensional and N inhibitory
m-dimensional oscillators with homogeneous synaptic weights. We then
demonstrate our theory using two examples. In the first example we take a
network of excitatory and inhibitory theta neurons and consider the case with
and without heterogeneous inputs. In the second example we use Traub models
with calcium for the excitatory neurons and Wang-Buzs{\'a}ki models for the
inhibitory neurons. We accurately predict phase drift and phase locking in each
example even when the slow synapses exhibit non-trivial mean-field dynamics.
