Assignment 4: Laplace Temporal Context Model (Laplace TCM)

Assignment 4: Laplace Temporal Context Model (Laplace TCM)#

Overview#

In this assignment, you will implement the Laplace Temporal Context Model (Laplace TCM) as described in Shankar & Howard (2012). The Laplace TCM provides a scale-invariant representation of time by maintaining a Laplace-transformed representation of temporal history, which can be used to predict future stimuli based on past experience.

Unlike previous assignments, instead of fitting the model to an experimental dataset, you will use synthetic examples to explore its behavior. Specifically, you will reproduce Figures 2, 3, and 4 from the paper, analyze how the quality of reconstructions changes with different numbers of nodes, and reflect on the model’s strengths and limitations.

Model Description#

The Laplace TCM constructs a compressed representation of time based on two key steps:

  1. Encoding Past Events as a Laplace Transform

    • A stimulus function \(f(\tau)\) (past stimulus history) is encoded into a set of leaky integrators with different decay rates \(s\).

    • This encoding follows:

      \[ t(\tau, s) = \int_{-\infty}^{\tau} f(\tau') e^{s (\tau' - \tau)} d\tau' \]

      where \(t(\tau, s)\) represents the Laplace transform of the stimulus history up to time \(\tau\).

  2. Reconstructing Time from the Laplace Representation

    • The inverse Laplace transform is approximated using a set of neurons called time cells with different peak latencies \(\tau^*\):

      \[ T(\tau, \tau^*) = (-1)^k \frac{k!}{(s^{k+1})} \frac{d^k}{ds^k} t(\tau, s) \Bigg|_{s=-\frac{k}{\tau^*}} \]

    • The function \(T(\tau, \tau^*)\) reconstructs past stimulus history, allowing temporal predictions.

Implementation Tasks#

Step 1: Implement the Model#

  • Implement the Laplace encoding function to generate \(t(\tau, s)\) from a stimulus function.

  • Implement the approximate inverse Laplace transform using the Post (1930) inversion formula to reconstruct \(T(\tau, \tau^*)\).

  • Implement a simple Hebbian learning rule to associate time cells with past stimuli.

Step 2: Reproduce Key Figures from the Paper#

You will generate synthetic stimuli and reproduce the following figures:

Figure 2: Leaky Integrators Encoding Temporal History#

  • Stimulus: Two pulses at different times.

  • Plot: The responses of leaky integrators with different decay rates \(s\).

  • Insight: Larger \(s\) values decay quickly, while smaller \(s\) retain long-term information.

Figure 3: Time Cell Representation of Stimulus History#

  • Stimulus: Same as Figure 2.

  • Plot: Activity of time cells with different peak latencies \(\tau^*\).

  • Insight: Each time cell stores a different temporal segment of the past.

Figure 4: Time Cells Over Time#

  • Stimulus: Two pulses.

  • Plot: Evolution of two time cells over time.

  • Insight: Cells peak at fixed latencies, but older events are reconstructed with increasing uncertainty.

Step 3: Explore Model Performance#

  • Vary the number of time cells (length of \(T\) nodes) and evaluate how well the past is reconstructed.

  • Hypothesis: With more nodes, reconstructions become more precise.

Step 4: Write Reflections#

  • Discuss the strengths of the Laplace TCM.

  • Consider its limitations (e.g., loss of precision for older memories).

  • Suggest potential improvements.

Grading Criteria#

  • Model Implementation (50%)

    • Correct encoding and reconstruction of time.

    • Accurate implementation of leaky integrators and time cells.

  • Figure Reproduction (30%)

    • 10% each for Figures 2, 3, and 4.

  • Exploration of Node Count Effects (10%)

    • Vary number of nodes and analyze reconstruction quality.

  • Reflections (10%)

    • Thoughtful discussion of model strengths, weaknesses, and improvements.

Submission Instructions#

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

    • Your full implementation of the Laplace TCM.

    • Markdown explanations of your approach.

    • Reproduced figures (2, 3, and 4).

    • Analysis of node count effects.

    • A short written reflection on the model.

Good luck, and happy modeling!

Contents