Criticality of a stochastic modern Hopfield network model with exponential interaction function

The Hopfield network (HN) is a classical model of associative memory whose dynamics are closely related to the Ising spin system with 2-body interactions. Stored patterns are encoded as minima of an energy function shaped by a Hebbian learning rule, and retrieval corresponds to convergence towards these minima. Modern Hopfield Networks (MHNs) introduce p-body interactions among neurons with p greater than 2 and, more recently, also exponential interaction functions, which significantly improve network's storing and retrieval capacity. While the criticality of HNs and p-body MHNs were extensively studied since the 1980s, the investigation of critical behavior in exponential MHNs is still in its early stages. Here, we study a stochastic exponential MHN (SMHN) with a multiplicative salt-and-pepper noise. While taking the noise probability p as control parameter, the average overlap parameter Q and a diffusion scaling H are taken as order parameters. In particular, H is related to the time correlation features of the network, with H greater than 0.5 signaling the emergence of persistent time memory. We found the emergence of a critical transition in both Q and H, with the critical noise level weakly decreasing as the load N increases. Notably, for each load N, the diffusion scaling H highlights a transition between a sub- and a super-critical regime, both with short-range correlated dynamics. Conversely, the critical regime, which is found in the range of p around 0.23-0.3, displays a long-range correlated dynamics with highly persistent temporal memory marked by the high value H around 1.3.