Asymmetrically extremely dilute neural networks with non-trivial dynamics

We study graded response attractor neural networks with asymmetrically extremely dilute interactions and Langevin dynamics. We solve our model in the thermodynamic limit using generating functional analysis, and find (in contrast to the binary neurons case) that even in statics one cannot eliminate the non-persistent order parameters. The macroscopic dynamics is driven by the (non-trivial) joint distribution of neurons and fields, rather than just the (Gaussian) field distribution. We calculate phase transition lines and present simulation results in support of our theory.