  Networks of strongly-coupled neurons with random connectivity exhibit
chaotic, asynchronous fluctuations. In previous work, we showed that when
endowed with an additional low-rank connectivity consisting of the outer
product of orthogonal vectors, these networks generate large-scale coherent
fluctuations. Although a striking phenomenon, that result depended on a
fine-tuned choice of low-rank structure. Here we extend that work by
generalizing the theory of excitation-inhibition balance to networks with
arbitrary low-rank structure and show that low-dimensional variability emerges
intrinsically through what we call incomplete recurrent alignment. We say that
a low-rank connectivity structure exhibits incomplete alignment if its
row-space is not contained in its column-space. In the setting of incomplete
alignment, recurrent connectivity can be decomposed into a subspace-recurrent
component and an effective-feedforward component. We show that
high-dimensional, microscopic fluctuations are propagated via the
effective-feedforward component to a low-dimensional subspace where they are
dynamically balanced by macroscopic fluctuations. We present biologically
plausible examples from excitation-inhibition networks and networks with
heterogeneous degree distributions. Finally, we define the alignment matrix as
the overlap between left and right-singular vectors of the structured
connectivity, and show that the singular values of the alignment matrix
determine the amplitude of macroscopic variability, while its singular vectors
determine the structure. Our work shows how macroscopic fluctuations can emerge
generically in strongly-coupled networks with low-rank structure. Furthermore,
by generalizing excitation-inhibition balance to arbitrary low-rank structure
our work may find relevance in any setting with strongly interacting units,
whether in biological, social, or technological networks.
