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An energy landscape assigns an energy to each possible state and describes its variation between neighbouring states. Its local minima and intervening energy barriers help distinguish favourable states from states that a specified dynamics can reach quickly. In a Hopfield network, the assigned energy is an objective function and need not be a conserved physical quantity.

A recurrent network with symmetric interactions and an energy function. In the binary version, units take values $\pm 1$, self-couplings are zero, and asynchronous threshold updates can retrieve stored patterns from incomplete or corrupted cues. Fixed points are local energy minima, which need not be the intended memories or global minima.

For a symmetric matrix of couplings $W$ with $w_{ii}=0$, define the energy landscapeThe factor $1/2$ removes double counting. The local input is $h_k={\sum }_{j\ne k}{w}_{kj}{x}_j-{\theta }_k$. Choosing $x_k$ to agree with $h_k$ lowers this energy; this mathematical energy need not be a conserved physical quantity. Its pairwise form connects the model to the Ising model; competing couplings also motivate the comparison with a spin glass. Here retrieval seeks a useful nearby local minimum, so reaching the global minimum is not the goal.

John Hopfield's 1982 computer study stored random binary strings in neural networks of 30 and 100 units. Couplings were prepared from the stored patterns; asynchronous updates supplied the computational apparatus. Starting at each nominal memory, the readout counted Hamming distances after settling as storage load varied. Corrupted starting patterns separately tested recovery. Varying the load and cue damage exposed interference between memories. These simulations supported collective retrieval, not a measurement of biological memory capacity.^[ref]