Dynamical regimes of partially symmetric, dynamically balanced networks

The statistics of the connectivity in the cortex is significantly different from that of a random network. In particular, there is strong evidence that reciprocal connections are over-represented as compared to random networks. Theory describing collective neuronal dynamics in model networks only exists for symmetric and random connectivity. Thus, the impact of this partial symmetry in the synaptic connectivity on the neuronal dynamics is presently unknown. Using numerical simulations, we found that partially symmetric, dynamically balanced networks exhibit two regimes of activity, depending on the level of symmetry and the variance of the synaptic efficacies (synaptic gain). For suitably low synaptic gains, there exists a unique fixed point. Using a “cavity-like” approach, we developed a quantitative theory that describes (i) the statistics of the activity in this unique fixed point; and (ii) the conditions for its stability. As the unique fixed point destabilizes, two regimes are possible depending on the level of symmetry. For low levels, the network exhibits chaotic activity similarly to what is observed in random networks. When the network is fully symmetric, the network exhibits multi-stability among a large number of fixed points. This raises the question of the regime in which the network is operating at intermediate levels of symmetry.