Hidden synaptic structures control collective network dynamics

A common approach to model local neural circuits is to assume random connectivity. But how is our choice of randomness informed by known network properties? And how does it affect the network's behavior? Previous approaches have focused on prescribing increasingly sophisticated statistics of synaptic strengths and motifs. However, at the same time experimental data on parallel dynamics of neurons is readily accessible. We therefore propose a complementary approach, specifying connectivity in the space that directly controls the dynamics - the space of eigenmodes. We develop a theory for a novel ensemble of large random matrices, whose eigenvalue distribution can be chosen arbitrarily. We show analytically how varying such distribution induces a diverse range of collective network behaviors, including power laws that characterize the dimensionality, principal components spectrum, autocorrelation, and autoresponse of neuronal activity. The power-law exponents are controlled by the density of nearly critical eigenvalues, and provide a minimal and robust measure to directly link observable dynamics and connectivity. The density of nearly critical modes also characterizes a transition from high to low dimensional dynamics, while their maximum oscillation frequency determines a transition from an exponential to power-law decay in time of the correlation and response functions. We prove that the wide range of dynamical behaviors resulting from the proposed connectivity ensemble is caused by structures that are invisible to a motif analysis. Their presence is captured by motifs appearing with vanishingly small probability in the number of neurons. Only reciprocal motifs occur with finite probability. In other words, a motif analysis can be blind to synaptic structures controlling the dynamics, which instead become apparent in the space of eigenmode statistics.