2D ERIC model and beyond

2D ERIC (2DE) model:

The 2D ERIC model is defined by the following dynamical equations,

$$ \frac{\mathrm{d}\theta_{i,j} (t) }{\mathrm{d}t} \Big \vert_{2DE} = \omega_{i,j} + K \Big( \sin(\theta_{i-1,j}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i+1,j}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i,j-1}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i,j+1}(t) -\theta_{i,j}(t) ) $$

$$ + \Lambda \big ( \sin^2(\theta_{i-1,j}(t) -\theta_{i,j} (t)) + \sin^2(\theta_{i+1,j}(t) -\theta_{i,j}(t) ) + \sin^2(\theta_{i,j-1}(t) -\theta_{i,j} (t)) + \sin^2(\theta_{i,j+1}(t) -\theta_{i,j}(t) ) \big) \Big) $$

Here $\theta_{i,j}(t)$ represents the phase of the oscillator at the $(i,j)$-th site in the square lattice. The salient feature of the coupling is that it is asymmetric and contains a genuine higher order harmonic.

2D ERIC model with excitability (2DE + ex):

We introduce excitability into the above model by introducing a simple $b, \sin \theta_{i,j}(t)$ term, such that in the absence of coupling, the individual oscillators can be in a quiescent state ($\omega_{i,j} < b$) or oscillatory state ($\omega_{i,j}>b$). The dynamical equations can be similarly written as,

$$ \frac{\mathrm{d}\theta_{i,j} (t) }{\mathrm{d}t}\Big\vert_{2DE+ex} = \omega_{i,j} - b, \sin \theta_{i,j}(t) + K \Big( \sin(\theta_{i-1,j}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i+1,j}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i,j-1}(t) -\theta_{i,j}(t) ) + \sin(\theta_{i,j+1}(t) -\theta_{i,j}(t) ) $$

$$ + \Lambda \big ( \sin^2(\theta_{i-1,j}(t) -\theta_{i,j} (t)) + \sin^2(\theta_{i+1,j}(t) -\theta_{i,j}(t) ) + \sin^2(\theta_{i,j-1}(t) -\theta_{i,j} (t)) + \sin^2(\theta_{i,j+1}(t) -\theta_{i,j}(t) ) \big) \Big) $$

Details of the repository

This repository contains the Python codes for generating the plots presented in the paper titled "Modeling the spatiotemporal self-organization of embryonic genetic oscillators in the mouse PSM into target phase waves using a phase model with an asymmetric, biharmonic coupling function" by Kaushik Roy and Paul Francois. The code is written in a simple, easy-to-understand manner while invoking some useful capabilities of Python such as: broadcasting in NumPy, multiprocessing modules etc. We have mentioned some salient features of the files in the repository below.

1. Files titled "phasegrids_xxx.py" generate the 10x10 or 5x5 grids of phasemaps at different times or parameter values for the different models discussed in the paper. These include the 2D ERIC model with an asymmetric, biharmonic coupling function which is the primary model that we are interested in. In addition, it contains the 2D ERIC model with excitability and other models that we have mentioned in the supplement such as the 2D Kuramoto model, 2D Rectified KUramoto (ReKU) model and the 2D Kuramoto model for Quadratic-Integrate-and-Fire (QIF) neurons.

In principle, the functions describing the models can all be incorporated in a single function file and invoked in the main code. However, we have included them in separate codes for easy of use and better intelligibility. The code is highly modular meaning that anyone interested in implementing their own 2D model can simply make a few changes in the code and generate the same phasemap grids. The advantage of this visualization is that it allows us to see the phase dynamics for a range of parameter values and at different times simultaneously.

Except for the "phasegrids_2DERIC_excitable_varyabL.py" file, all other "phasegrids_xxx.py" files generate a multipage .pdf file where each page contains a timeshot of the phasemaps for a range of parameter values. For the "phasegrids_2DERIC_excitable_varyabL.py" file, we get a multipage .pdf file but all the phasemaps are obtained at a certain fixed time with each page of the .pdf file containing phasemaps for different values of the excitable parameter $b$ and relative coupling strength $\Lambda$ at a fixed primary coupling strength $K$. This code therefore allows us to see the phasemaps at a certain time in the $(K,b,\Lambda)$ parameter space which is very useful.

2. The "2DERIC+excitable.ipynb" file is a JuPyteR notebook that allows the user to obtain a variety of physical quantities for the 2D ERIC (+excitable) model such as phasemaps at different times for fixed $K$ and $\Lambda$ values, timeshots of local coherence maps, global coherence vs time, timeshots of phase fields, timeshots of instantaneous frequency maps and average frequency maps. The blocks of the code are similar to that used to generate the phasemap grids mentioned above but we do not use multiprocessing module here. Instead, this gives the user an interactive experience by allowing them to study the models discussed in the paper (and beyond) in a highly comprehensive manner.

3. Finally, we have the "Animations" folder which contains videos of the phasemaps generated by the 2D ERIC (+excitable) models for better visualization of the temporal patterns.