Analytic solution of neural network with disordered lateral inhibition.

The replica method has played a key role in analyzing systems with disorder, e.g., the Sherrington-Kirkpatrick (SK) model, and associative neural networks. Here we study the influence of disorder in the lateral inhibition type interactions on the cooperative and uncooperative behavior of recurrent neural networks by using the replica method. Although the interaction between neurons has a dependency on distance, our model can be solved analytically. Bifurcation analysis identifies the boundaries between paramagnetic, ferromagnetic, spin-glass, and localized phases. In the localized phase, the network shows a bump like activity, which is often used as a model of spatial working memory or columnar activity in the visual cortex. Simulation results show that disordered interactions can stabilize the drift the of bump position, which is commonly observed in conventional lateral inhibition type neural networks.