Dynamics of Evolving Feed-Forward Neural Networks and Their Topological Invariants

The evolution of a simulated feed-forward neural network with recurrent excitatory connections and inhibitory forward connections is studied within the framework of algebraic topology. The dynamics includes pruning and strengthening of the excitatory connections. The invariants that we define are based on the connectivity structure of the underlying graph and its directed clique complex. The computation of this complex and of its Euler characteristic are related with the dynamical evolution of the network. As the network evolves dynamically, its network topology changes because of the pruning and strengthening of the onnections and algebraic topological invariants can be computed at different time steps providing a description of the process. We observe that the initial values of the topological invariant computed on the network before it evolves can predict the intensity of the activity.