A random matrix theory in eigenspace enables direct control of collective network activity

A common approach to analytically treat neural network models is to assume a random connectivity matrix. But how does our choice of randomness affect the network's behavior? Rather than prescribing the distribution of synaptic strengths, we specify connectivity in the space that directly controls the dynamics – the space of eigenmodes. We develop a thermodynamic theory for a novel ensemble of random matrices, whose eigenvalue distribution can be chosen arbitrarily. As this distribution is varied, we show analytically how the behavior of a stochastic linear rate network changes. 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 (d-1 and d, respectively) slow down ∝ d, which controls the density of the real part of near-critical eigenvalues p( x ) ∼ x^d-1, for x → 0. Also, the network shows a transition from high to low dimensional activity when d < 2, with a minimum at d = 1. We argue that d can be interpreted as the network's spatial dimension in the sense of critical phenomena and the renormalization group. In particular, below a critical dimension d = 1, correlations diverge, hinting at a potential transition to nongaussian criticality in a nonlinear system. Our novel approach uncovers a diverse range of behaviors that only emerges when the collective effect of the subleading-order synaptic strengths' statistics is not neglected.