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, experimental data on parallel dynamics of neurons is more readily accessible than their microcircuitry. We therefore propose a paradigm shift, 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. We discover a critical point whose nature is shaped by the distribution of oscillation frequencies of near-critical modes: correlation and response functions can be tuned from an exponential to a power-law decay in time. Their decay exponents slow down proportionally to the density of near-critical eigenvalues. Above a critical density, the network furthermore shows a sharp transition from high to low dimensional activity.
Which synaptic strengths statistics control such diverse range of behaviors? We prove it to be the structure of subleading order statistics. At leading order in the number of neurons, instead, synapses only show simple correlations between reciprocal connections. The richer structures at subleading order are effectively hidden in synaptic space. Yet, their collective effect is macroscopically observable in the space of neuronal dynamics.
Our novel approach provides a largely extended ensemble of connectivity matrices, with which to explore a wider space of possible network dynamics. As an application, we expose novel behaviors whose origin is hardly detectable in synaptic space alone and establish a solid analytical link between observable neuronal dynamics – correlations, responses and dimensionality – and connectivity.