Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions

We establish two conditions that ensure the nondivergence of additive recur rent networks with unsaturating piecewise linear transfer functions, also c alled linear threshold or semilinear transfer functions. As Hahnloser, Sarp eshkar, Mahowald, Douglas, and Seung (2000) showed, networks of this type c an be efficiently built in silicon and exhibit the coexistence of digital s election and analog amplification in a single circuit. To obtain this behav ior, the network must be multistable and nondivergent, and our conditions a llow determining the regimes where this can be achieved with maximal recurr ent amplification. The first condition can be applied to nonsymmetric netwo rks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron. The second condition is restricted to symmetric networks, but can also tak e into account the stabilizing effect of nonlocal inhibitory interactions. We demonstrate the application of the conditions on a simple example and th e orientation-selectivity mo del of Ben-Yishai, Lev Bar-Or, and Sompolinsky (1995). We show that the conditions can be used to identify in their model regions of maximal orientation-selective amplification and symmetry breaki ng.