  Networks of randomly connected neurons are among the most popular models in
theoretical neuroscience. The connectivity between neurons in the cortex is
however not fully random, the simplest and most prominent deviation from
randomness found in experimental data being the overrepresentation of
bidirectional connections among pyramidal cells. Using numerical and analytical
methods, we investigated the effects of partially symmetric connectivity on
dynamics in networks of rate units. We considered the two dynamical regimes
exhibited by random neural networks: the weak-coupling regime, where the firing
activity decays to a single fixed point unless the network is stimulated, and
the strong-coupling or chaotic regime, characterized by internally generated
fluctuating firing rates. In the weak-coupling regime, we computed analytically
for an arbitrary degree of symmetry the auto-correlation of network activity in
presence of external noise. In the chaotic regime, we performed simulations to
determine the timescale of the intrinsic fluctuations. In both cases, symmetry
increases the characteristic asymptotic decay time of the autocorrelation
function and therefore slows down the dynamics in the network.
