Hopfield Model is a recurrent neural network that serves as an associative memory system. The Hopfield model is a collection of binary threshold units with symmetrical connections, and it can be understood using the concepts of statistical mechanics.
Definition
The Hopfield network consists of $N$ neurons, each having a state $s_i$ where $i = 1, 2, \ldots, N$. The state of each neuron can be either +1 or 1, and the neurons are connected through a weight matrix $W_{ij}$.
The energy of the network is given by:
$$E = \frac{1}{2} \sum_{i,j} W_{ij} s_i s_j$$
Dynamics
The dynamics of the Hopfield model can be described by the update rule:
$$s_i \leftarrow \text{sgn}\left(\sum_{j} W_{ij} s_j\right)$$
Here, the sign function ensures that the state remains binary.
Learning
The Hopfield network can learn patterns through the Hebbian learning rule. Given a set of patterns $\xi^{\mu}$, the weights are updated as:
$$W_{ij} = \sum_{\mu} \xi_i^{\mu} \xi_j^{\mu}$$
Associative Memory
The Hopfield model serves as an associative memory, where a partial or noisy pattern can be recalled to its original stored pattern. The network converges to a stable state, representing the stored memory.
Energy Landscape
The energy landscape of the Hopfield model can be visualized as a surface with local minima representing the stored patterns. The dynamics of the network ensures that the energy decreases, leading to convergence to a stable state.
Limitations and Extensions
 Capacity: The Hopfield network has a limited capacity, typically around 0.15$N$ per pattern.
 Spurious States: Besides the stored patterns, the network may converge to undesired local minima.
 Extensions: Various extensions like continuous Hopfield networks and stochastic updates have been proposed to overcome limitations.
Connection to Statistical Mechanics
The Hopfield model's dynamics and energy landscape can be related to the Ising model in statistical mechanics. The stored patterns correspond to the ground states, and the dynamics can be understood through the concepts of temperature, entropy, and free energy.
HopfieldIsing Model Equivalence Theorem. The Hopfield model is the Ising model with a Hamiltonian $\mathcal{H} = \sum_{i,j} J_{ij} \sigma_i \sigma_j$, where $J_{ij} = W_{ij}$ and $\sigma_i = s_i$. 
This connection bridges neural networks and statistical mechanics.

