Statistical-mechanical formulation of the Willshaw model with local inhibition

The Willshaw model is a neural network with binary synaptic strengths. In the Willshaw model with local inhibition, only one neuron in each block of q neurons is active. We formulate the model using the Potts spin variables and extend the dynamics to a stochastic one. The equilibrium states of the system are characterized by a set of variables representing the overlap between a group of patterns and the network state. Stable Mattis-like solutions exist at zero temperature, and the phase transition is found to be of the first order. A related model whose Hamiltonian is linear in interaction is also studied. The linear model does not function as an associative memory, suggesting that functioning of the Willshaw net stems from the binary truncation of the synaptic strength.