Is a Hopfield network with a continuous activation variable and a discrete time variable possible?
After doing some additional research, I think the answer is yes. It just means using a fixed timestep for the continuous-time activation equation (as described here). Since this is a differential equation, implementing it in software requires implementing a numerical integration method. I recommend the Exponential Euler Method as a starting point, because it's relatively simple and it's designed for differential equations of the sort that's used for the continuous Hopfield network.