Hopfield Networks as Models of Emergent Function in Biology.

Hopfield models, originally developed to study memory retrieval in neural networks, have become versatile tools for modeling diverse biological systems in which function emerges from collective dynamics. In this review, we provide a pedagogical introduction to both classic and modern Hopfield networks from a biophysical perspective. After presenting the underlying mathematics, we build physical intuition through three complementary interpretations of Hopfield dynamics: as noise discrimination, as a geometric construction defining a natural coordinate system in pattern space, and as gradient-like descent on an energy landscape. We then survey applications of Hopfield networks in a variety of biological settings, including cellular differentiation and epigenetic memory, molecular self-assembly, and spatial neural representations.