Low-dimensional descriptions of neural networks
Mathematics, Southern Methodist University
(April 2, 2013 3:00 PM - 3:50 PM)
Biological neural circuits display both spontaneous asynchronous activity, and complex, yet ordered activity while actively responding to input. When can model neural networks demonstrate both regimes? Recently, researchers have demonstrated this capability in large, recurrently connected neural networks, or “liquid state machines", with chaotic activity. We study the transition to chaos in a family of such networks, and use principal orthogonal decomposition (POD) techniques to provide a lower-dimensional description of network activity.
We find that key characteristics of this transition depend critically on whether a fundamental neurobiological constraint — that most neurons are either excitatory or inhibitory — is satisfied. Specifically, we find that constrained networks exhibit the transition to chaos at much higher coupling strengths than unconstrained networks. This property is the consequence of the fact that the constrained system may be described as a perturbation from a system with non-trivial symmetries. These symmetries imply the presence of both fixed points and periodic orbits that continue to act as an organizing center for solutions, even for large perturbations. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.