Glassy phase in dynamically balanced networks

We study the dynamics of (inhibitory) balanced networks at varying (i) the level of symmetry in the synaptic connectivity; and (ii) the ariance of the synaptic efficacies (synaptic gain). We find three regimes of activity. For suitably low synaptic gain, regardless of the level of symmetry, there exists a unique stable fixed point. Using a cavity-like approach, we develop a quantitative theory that describes the statistics of the activity in this unique fixed point, and the conditions for its stability. Increasing the synaptic gain, the unique fixed point destabilizes, and the network exhibits chaotic activity for zero or negative levels of symmetry (i.e., random or antisymmetric). Instead, for positive levels of symmetry, there is multi-stability among a large number of marginally stable fixed points. In this regime, ergodicity is broken and the network exhibits non-exponential relaxational dynamics. We discuss the potential relevance of such a “glassy” phase to explain some features of cortical activity.