Controlling the bifurcations of attractors in modern Hopfield networks

Hopfield networks model complex systems with attractor states. However, there are many systems where attractors are not static. Attractors may undergo bifurcations under certain conditions; for example, cell fates have been described as attractor states that can be stabilized or destabilized by signalling. In the case of neural networks, retrieving a sequence of memories involves changing attractor states. We provide an extension to the modern Hopfield network that connects network dynamics to the landscape of any potential. With our model, it is possible to control the bifurcations of attractors and simulate the resulting neuron dynamics. By introducing controlled bifurcations, our formulation expands the application of Hopfield models to real-world contexts where attractors do not remain static.