Amorphous Solid Model of Vectorial Hopfield Neural Networks

We present a vectorial extension of the Hopfield associative memory model inspired by the theory of amorphous solids, where binary neural states are replaced by unit vectors $\mathbf{s}_i \in \mathbb{R}^3$ on the sphere $S^2$. The generalized Hebbian learning rule creates a block-structured weight matrix through outer products of stored pattern vectors, analogous to the Hessian matrix structure in amorphous solids. We demonstrate that this model exhibits quantifiable structural properties characteristic of disordered materials: energy landscapes with deep minima for stored patterns versus random configurations (energy gaps $\sim 7$ units), strongly anisotropic correlations encoded in the weight matrix (anisotropy ratios $\sim 10^2$), and order-disorder transitions controlled by the pattern density $\gamma = P/(N \cdot d)$. The enhanced memory capacity ($\gamma_c \approx 0.55$ for a fully-connected network) compared to binary networks ($\gamma_c \approx 0.138$) and the emergence of orientational correlations establish connections between associative memory mechanisms and amorphous solid physics, particularly in systems with continuous orientational degrees of freedom. We also unveil the scaling with the coordination number $Z$ of the memory capacity: $\gamma_c \sim (Z-6)$ from the isostatic point $Z_c =6$ of the 3D elastic network, which closely mirrors the scaling of the shear modulus $G \sim (Z-6)$ in 3D central-force spring networks.