Saddle point and Hamiltonian structure in excitatory-inhibitory networks

Because the dynamics of a neural network with symmetric interactions is similar to a gradient descent dynamics, convergence to a fixed point is the general behavior. In this paper, we analyze the global behavior of networks with distinct excitatory and inhibitory populations of neurons, under the assumption that the interactions between the populations are antisymmetric. Our analysis exploits the similarity of such a dynamics to a saddle point dynamics. This analogy gives some intuition as to why such a dynamics can either converge to a fixed point or a limit cycle, depending on parameters. We also show that the network dynamics can be written in a dissipative Hamiltonian form. Dynamic neural networks with symmetric interactions provably converge to fixed points under quite general hypotheses [1, 2]. The convergence theory helped to establish the paradigm of dynamic neural computation with attractors[3, 4]. Little is known about the dynamics of networks whose connections are not symmet..