Over the past ten years, several studies based on multielectrode recordings have revealed that cortical pyramidal neurons connect with each other according to patterns that display a non-trivial statistical structure.

One prominent example of such structure is the fact that bidirectional connections between neurons appear more often than one would expect if connections were random. It is still unclear how this prevalence of bidirectional connections affects the dynamics of cortical circuits, and what possible functional role it plays.

To elucidate this question we study the dynamics of large-scale networks of neurons with unclustered, random, and partially symmetric connections. We investigate networks of rate units and show that the nature of the network dynamics is strongly determined by the spectrum of eigenvalues of the connectivity matrix. For weak couplings, the real part of the eigenvalues is small and the network is in an equilibrium state with constant firing rates. For increasing connection strengths, the real part of the eigenvalues gets larger and the network eventually undergoes an instability that leads to a rate-chaotic regime, characterized by heterogeneous and fluctuating firing rates with a characteristic timescale that depends only mildly on the coupling strength.

Using numerical simulations and mathematical analysis, we show that introducing partial symmetry in the couplings slows down the rate fluctuations in the chaotic regime, a result that can be qualitatively explained by the appearance of connection that contribute to activity reverberation. More quantitatively, this slowing down is brought about by the flattening of the eigenspectrum along the real axis and the subsequent concentration of low-frequency modes.

For the non-chaotic regime, we compute how symmetry modulates the timescale of the noise filtered by the network operating at the transition point. In that case symmetry increases the characteristic asymptotic decay time of the autocorrelation function. Furthermore, for sufficiently symmetric connections the system operating in the chaotic regime exhibits aging effects, by which the timescale of the rate fluctuations slowly grows as time evolves, and which is a characteristic of systems out of equilibrium exhibiting a very large relaxation time.