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Welcome to the Waves, Pattern Formation and Excitability REU 2025 Wiki!

Team members: Kaan Volkan, Neil Gabrielson, Naiwen Wang

Graduate mentor: Tanishq Bhatia

Faculty mentor: Calina Copos (Office: Mugar 206K; Lab: Mugar 206A)

TODO list:

Week 1:

• Discuss the aims in your group and agree on 3 common aims and decide who will take lead on which aim.
• Read Sections 1-3 in the Hughes et al. paper. What is this mathematical paper concerned with? Who was the first mathematician to think about pattern formation? What equations are proposed? Why are there two sets of equations? What do each of the terms in the equations mean -- who are the unknowns variables? What is done in section 3? Why is this an important section and how could this help your work?
• Read the entire Michaud et al. paper. Where do patterns show up? Why do patterns show up? Are the patterns important for this biological process? What are their governing equations? What do each of the terms in the equations mean -- who are the unknowns variables?
• Numerically solve Michaud et al.'s pattern formation system (Eq. 3) with their boundary conditions, initial conditions, and parameters provided in Table S1. Do you observe the same patterns they shown in Video 11?
• Similarly implement and solve Hughes et al.'s simpler model (Eq. 2.1a-b). What do you expect to see? Then move to their more complex model (Eq. 2.2a-c). What is gained from this complexity?
• Generate sample figures and movies; upload into Overleaf documet
• Discuss how to ensure reproducibility of the generated figures -- I'd like you to think about what piece of code and data you need to make these figures and videos again but for example with a different color scheme? Ensure that steps are taken for reproducibility.
• Document and upload your code to Github
• Complete end-of-the-week reflection
• Complete/alter as needed the todo list for week 2

Week 2:

• Record all equations in further details by Jupyter Markdown cells, and upload to github.
• Update the overleaf with results.
• Read lecture notes on stability analysis for pattern formation
• Read the Mori paper and generate the overleaf glossary document. Maybe, also try to code the Mori paper equations. Reproduce Figure 2 in More using input Stimuli. (Optional) Also, reproduce an animation for figure 3.
  • For refining the mesh, we can do "pretraining", i.e. run the algorithm for a few steps with dx=1.0 and then decrease for later iterations. Need CFL condition instead.
• Check Experimental hypothesis for Michaud: Increase the production for RT. This could be done via modifying constants in equation 2:
  1. Increase $k_0$ and/or $\alpha$
  2. Decrease $k_2$
  3. Decrease $k_3$, $k_4$
• Perform the linear stability analysis for all models: Vary parameters to inspect the number of solutions
• Answer the model analysis questions from last week
• Put all parameters in pandas tables and call from there for documentability

Week 3:

• Gain confidence in the numerical results of Michaud et al.
• Try decreasing $dX$ to remove the checkered pattern and the stading waves. Use the CFL condition, $dt < dX^2/D$, where $D$ is the diffusion coefficient
• Explore CFL conditions and make sure our Michaud simulation parameters align. Add notes to Overleaf document.
• Prepare slides for midterm presentations
  • Start by standing and travelling waves
  • Introduce the equations, maybe a little bit of history
  • Show if Mori can produce travelling wave or if Michaud can produce standing wave

Week 4:

• Sampling parameter space: Sobol (or Latin Hypercube) sampling, classify by pattern, no pattern, type of pattern.
• Explore Hughes et al. numerically to find both traveling and stationary wave patterns (can use AUTO package for bifurcation diagrams)
• $\alpha$ and $k_8$ important for Michaud
• Find numerical metrics to distinguish travelling and standing waves:
  • Compare the amplitude and wave width output from the pipeline on standing and travelling wave
  • Try Vineyards for a michaud video to distinguish standing and travelling waves

Week 5:

Week 6:

Week 7:

Week 8: