On the capacity of continuous-time Hopfield models

The Hopfield model of associative memory stands as a simple example of the use of dynamical systems to answer abstract neuroscientific questions. Notably, it has helped in clarifying phenomena such as memory formation, memory retrieval, and memory capacity. The capacity analysis, a longstanding challenge for physicists and mathematicians, has focused on the discrete-time model with the sign activation function, largely neglecting exploration of continuoustime models and diverse activation functions. In this paper, we provide an initial investigation of the capacity of the continuoustime Hopfield model. First, we extend well-known results from the literature generalizing the sign activation function to functions that saturate at an arbitrary point, which we will call “saturated”. Then, we generalize this result to a wider class of activation functions and we supplement the analysis with numerical results to test our predictions. In particular, the numerical simulation uncovers the potential existence of a phase transition for the continuous model.