Extended Data Fig. 1: Parallels and differences between neural network models and self-assembly models as exemplars of collective behaviour.

In this rough metaphor, a neuron corresponds to a tile. While Hopfield networks allow full connectivity, multifarious self-assembly (like place cell networks) restricts connectivity to a superposition of grids with different unit permutations. The state of a Hopfield network consists of the set of active neurons, while the state of an assembly consists of the set of tiles present and their arrangement, which is restricted to be connected. We use x_i ∈ {−1, +1} for the activity of neuron i, and \({x}_{p}^{i}\in \{0,1\}\) for the occupancy of tile i in position p. The energy of a state is a quadratic function governed by synaptic weights w_i,j and biases b_i for neural activities, or for assemblies, by directional binding energies \({J}_{i,j}^{\delta }\) for tiles i and j at positions p and \(p{\prime} \) that are neighbors in direction δ, along with (inverted) tile chemical potentials Θ_i. An environment presents a sequence of outside influences driving system state, either stimulating neural activity or spatially organizing tiles. Learning in Hopfield networks occurs any time neurons are simultaneously active. For self-assembly, learning an interaction requires tiles i,j to be located next to each other; we envision a hypothetical proximity-based ligation process^65,66 that creates interaction mediating glues ij for molecules i,j that spend time together in spatial proximity. Qualitative system behaviors depend on the number of memories being stored and the operating temperature, including phases where system state randomizes (paramagnetic/disoriented/dissolved), gets locked in a spurious local minimum (spin glass/random aggregation), or successfully retrieves learned memories. Due in large part to the restrictions on connectivity, the capacity of place cell networks and multifarious self-assembly is less than for the Hopfield model. See Supplementary Information section 1 for details and discussion.