Macroscopic fluctuations emerge in balanced networks with incomplete recurrent alignment

Networks of strongly coupled neurons with random connectivity exhibit chaotic, asynchronous fluctuations. In a previous study, we showed that when endowed with an additional component of low-rank connectivity consisting of the outer product of orthogonal vectors, these networks generate macroscopic coherent fluctuations. Although a striking phenomenon, that result depended on a fine-tuned choice of low-rank structure. Here we extend that result by generalizing the theory of excitation-inhibition balance to networks with arbitrary low-rank structure and show that macroscopic fluctuations emerge 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 generic setting of incomplete alignment, recurrent connectivity can be decomposed into a "subspace-recurrent"component and an "effective-feedforward"component. We show that the balance equations of excitation-inhibition networks generalize naturally to the setting of arbitrary strongly coupled low-rank structure and are determined by the subspace-recurrent component of connectivity. The effective-feedforward component, meanwhile, projects high-dimensional, microscopic fluctuations to a low-dimensional subspace where they are dynamically balanced by emergent 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 matrix of overlaps 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, low-dimensional variability, while its singular vectors determine the structure. Our work shows how low-dimensional 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.