Assignment 1: Hopfield Networks

Assignment 1: Hopfield Networks#

Overview#

In this assignment, you will explore computational memory models by implementing a Hopfield network. Here is a suggested template to help get you started.

In the original article (Hopfield, 1982), neuronal activations were set to either 0 (“not firing”) or 1 (“firing”). Modern Hopfield networks nearly always follow an updated implementation, first proposed by Amit et al. (1985). In their framing, neurons take on activation values of either –1 (“down state”) or +1 (“up state”). This has three important benefits:

  • It provides a cleaner way to implement the Hebbian learning rule (i.e., without subtracting means or shifting values).

  • It avoids a bias toward 0 (i.e., +1 and –1 are equally “attractive,” whereas 0-valued neurons have a stronger “pull”).

  • The energy function (which describes the attractor dynamics of the network) can be directly mapped onto the Ising model from statistical physics.

You should start by reading Amit et al. (1985) closely. Then implement the model in a Google Colaboratory notebook. Unless otherwise noted, all references to “the paper” refer to Amit et al. (1985).

Tasks#

1. Implement Memory Storage and Retrieval#

Objective#

Write functions that implement the core operations of a Hopfield network.

Memory Storage#

Implement the Hebbian learning rule to compute the weight matrix, given a set of network configurations (memories). This is described in Equation 1.5 of the paper:

Let \(p\) be the number of patterns and \(\xi_i^\mu \in \{-1, +1\}\) the value of neuron \(i\) in pattern \(\mu\). The synaptic coupling between neurons \(i\) and \(j\) is:

\[ J_{ij} = \sum_{\mu=1}^p \xi_i^\mu \xi_j^\mu \]

Note that the matrix is symmetric (\(J_{ij} = J_{ji}\)), and there are no self-connections by definition (\(J_{ii} = 0\)).

Memory Retrieval#

Implement the retrieval rule using Equation 1.3 and surrounding discussion. At each time step, each neuron updates according to its local field:

\[ h_i = \sum_{j=1}^N J_{ij} S_j \]

Each neuron updates its state by aligning with the sign of the field:

\[ S_i(t+1) = \text{sign}(h_i(t)) = \text{sign} \left( \sum_{j} J_{ij} S_j(t) \right) \]

Here, \(S_i \in \{-1, +1\}\) is the current state of neuron \(i\). To “retrieve” a memory:

  • Start by setting all of the neural activations to the cue.

  • Loop through all neurons (in a random order), updating one at a time according to the above equation, given the weight matrix (\(J\)) and the current activities of each of the other neurons.

  • Continue looping until either (a) you have “updated” every neuron in the latest loop, but no activities have changed, or (b) a maximum number of iterations is reached.

  • Return the current state of the network as the retrieved memory.

Optional: instead of always following the update rule, add “noise” to the network by (in a very small proportion of updates) randomly flipping the neuron’s activity. This can sometimes improve a network’s performance by “kicking” the network out of local minima! It can be instructive to explore a network’s performance with vs. without noise, and/or as a function of how much noise is being added.

2. Test with a Small Network#

Encode the following test memories in a Hopfield network with \(N = 5\) neurons:

\[\begin{split} \xi^1 = [+1, -1, +1, -1, +1] \\ \xi^2 = [-1, +1, -1, +1, -1] \end{split}\]

  • Store these memories using the Hebbian rule.

  • Test retrieval by presenting the network with noisy versions (e.g., flipping a sign, or setting some entries to 0).

  • Briefly discuss your observations.

Questions to consider:

  • Can you tell how and why the network stores memories?

  • Why do some memories interfere while others don’t?

  • Can you construct memory sets that do or don’t work in a small network?

  • What factors do you think affect the capacity of the network?

3. Evaluate Storage Capacity#

Objective#

Determine how memory recovery degrades as you vary:

  • Network size (number of neurons)

  • Number of stored memories

To generate \(m\) memories \(\xi_1, \dots, \xi_m\) for a network of size \(N\), use:

import numpy as np
xi = 2 * (np.random.rand(m, N) > 0.5) - 1

Method#

  • For each configuration, run multiple trials.

  • For each trial, measure whether at least 99% of the memory is recovered.

Visualization 1#

Create a heatmap:

  • \(x\)-axis: network size

  • \(y\)-axis: number of stored memories

  • Color: proportion of memories retrieved with ≥99% accuracy

Visualization 2#

Plot the expected number of accurately retrieved memories vs. network size.

Let:

  • \(P[m, N] \in [0, 1]\): proportion of \(m\) memories accurately retrieved in a network of size \(N\)

  • \(\mathbb{E}[R_N]\): expected number of successfully retrieved memories

Then:

\[ \mathbb{E}[R_N] = \sum_{m=1}^{M} m \cdot P[m, N] \]

Where \(M\) is the maximum number of memories tested.

Follow-Up#

  • What relationship (if any) emerges between network size and capacity?

  • Can you develop rules or intuitions that help predict a network’s capacity?

4. Simulate Cued Recall#

Objective#

Evaluate how the network performs associative recall when only a cue is presented.

Setup: A–B Pair Structure#

  • Each memory consists of two parts:

    • First half: Cue (\(A\))

    • Second half: Response (\(B\))

If \(N\) is odd:

  • Let cue length = \(\lfloor N/2 \rfloor\)

  • Let response length = \(\lceil N/2 \rceil\)

Each full memory:

\[\begin{split} \xi^\mu = \begin{bmatrix} A^\mu \\ B^\mu \end{bmatrix} \end{split}\]

Simulation Procedure#

  1. Choose a memory \(\xi^\mu\)

  2. Construct initial state \(x\):

    • Cue half: set to \(A^\mu\)

    • Response half: set to 0

  3. Evolve the network using the update rule:

    \[ x_i \leftarrow \text{sign} \left( \sum_j J_{ij} x_j \right) \]

    • Optionally: clamp the cue (i.e., hold cue values fixed)

  4. Evaluate success:

    • Compare recovered response to \(B^\mu\)

    • Mark as successful if ≥99% of bits match:

      \[ \frac{1}{|B|} \sum_{i \in \text{response}} \mathbb{1}[x^*_i = B^\mu_i] \geq 0.99 \]

Analysis#

  • Repeat across many \(A\)–\(B\) pairs

  • For each network size \(N\), compute the expected number of correctly retrieved responses

  • Plot this value as a function of \(N\)

Optional Extensions#

  • Compare performance with and without clamping the cue

  • Try cueing with noisy or partial versions of \(A\)

5. Simulate Contextual Drift#

Objective#

Investigate how gradual changes in context influence which memories are recalled.

Setup: Item–Context Representation#

  • Use a Hopfield network with 100 neurons.

  • Each memory:

    • First 50 neurons: Item

    • Last 50 neurons: Context

Create a sequence of 10 memories:

\[\begin{split} \xi^t = \begin{bmatrix} \text{item}^t \\ \text{context}^t \end{bmatrix} \end{split}\]

Context drift:

  • Set \(\text{context}^1\) randomly

  • For each subsequent \(\text{context}^{t+1}\), copy \(\text{context}^t\) and flip ~5% of the bits

Simulation Procedure#

  1. Store all 10 memories in the network.

  2. For each memory \(i = 1, \dots, 10\):

    • Cue the network with \(\text{context}^i\)

    • Set item neurons to 0

    • Run until convergence

    • For each stored memory \(j\), compare recovered item to \(\text{item}^j\)

    • If ≥99% of bits match, record \(j\) as retrieved

    • Record \(\Delta = j - i\) (relative offset)

Analysis#

  • Repeat the procedure (e.g., 100 trials)

  • For each \(\Delta \in [-9, +9]\), compute:

    • Probability of retrieval

    • 95% confidence interval

Visualization#

Create a line plot:

  • \(x\)-axis: Relative position \(\Delta\)

  • \(y\)-axis: Retrieval probability

  • Error bars: 95% confidence intervals

Write a brief interpretation of the observed pattern.

Optional Extensions#

  • Vary the drift rate and observe the effect

  • Try random (non-gradual) context changes

  • Explore links to recency effects or memory generalization

Submission Instructions#

  • Submit a single standalone Google Colaboratory notebook (or similar) that includes:

    • Your full model implementation

    • Markdown cells explaining your methods, assumptions, and findings

    • Plots and results for each section

  • Your notebook should run without errors in Google Colaboratory.

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