Speaker
Description
The dynamics of large neuronal networks are notoriously difficult to study analytically, because of their high dimensionality. Characterizing network equilibria, instead, is far more tractable. What information about the network's dynamics, then, can we infer from the knowledge of its equilibria? Here, we address this question by focusing on random networks. These networks exhibit a paradigmatic transition from a unique, stable equilibrium to extensive chaos as the synaptic gain increases. Employing the Kac-Rice formalism, we compute the typical number of hyperbolic equilibria, and determine their stability and their geometric organization in phase space. In the chaotic regime, an exponentially large number of equilibria is present; they are all saddles with an extensive, yet fractionally small, number of unstable directions. Surprisingly, despite the network's connectivity being completely random, the equilibria are strongly correlated and, as a result, occupy a small region in phase space. The attractor is inside this region. Because of this geometric organization, the quantitative features of the equilibria provide natural bounds on the network's dynamics. In particular, the fraction of positive Lyapunov exponents is bounded from above by the fraction of unstable directions of typical equilibria. This explains why the dynamics in these models can be described by a fractionally small number of effective degrees of freedom. Our results demonstrate how the spatial and spectral properties of the equilibria place strong, quantitative constraints on the dynamics.