UPC_Minimal_Calcium_Model

This repository contains files which implement a fourth order Runge-Kutta method to solve the system introduced in the paper Minimal model for calcium alternans due to SR release refractoriness

$$ \dot{c}_j = I_{s}(t)+g_{rel}c_{rel}\frac{k_ac_j^2}{k_{om}+k_ac_j^2}(c_{SR}-c_j) - (c_j-c_o)/\tau_{diff}$$ $$\dot{c}_{rel} = k_{im}(1-c_{rel}) - k_ic_jc_{rel} $$

The programs have some dependencies:

1. Python: requires numba, os, sys and numpy
2. Julia: requires CSV and DataFrames, These two packages are only used to save the results. Of course other methods can be used.
3. C: uses cmath and iostream.

To compile the C version I used g++ -o calcium calcium.cc.

The file cark.cc is the file exposing the functions compiled in WASM. This file was compiled using:

emcc cark.cc -o cark.js -sEXPORTED_RUNTIME_METHODS=ccall,cwrap -sEXPORT_ES6 -sMODULARIZE -sMALLOC=dlmalloc -sALLOW_MEMORY_GROWTH=1 -sEXPORTED_FUNCTIONS=_malloc,_free,_I_s,_c_i,_c_r,_rk

Usage.

The three versions take three arguments: $T_s$, $T_r$ and $N$.

$T_s$ and $T_r$ are the values in miliseconds of the stimulus current signal $I_s(t)$ and the recovery time $k_{im}=T_r^{-1}$. And $N$ is the number of beats to run. This means that the integration time is the interval $(0.0,NT_s)$

Thus, to use the programs:

C:

$./calcium 400  500 20 > Ca.dat

This redirects the output to Ca.dat, which contains as columns: $t$, $c_j$, $c_{rel}$ and $I_s(t)$.

Python:

$./caKi.py 400 500 20

This produces two files KCa.npy and Kical.npy which contain arrays with the solution and the stimulus. These can be accessed later using something like:

Sn = np.load('KCa.npy');
In = np.load('KIcal.npy');
t= Sn[:,0];
C=Sn[:,1];
q=Sn[:,2];

Julia:

$./ca_rk.jl 400 500 20

This will produce a csv file Ca_Results.csv with colums corresponding to $t$, $c_j$, $c_{rel}$ and $I_s(t)$.

As described in the paper, depending of the relationshp between $T_s$ and $T_r$ the calcium concentration can exhibit different behaviours. As shown below for the cases $T_s = 450 ms$ and $T_r = 500$ and $200 ms$.

Alternans

Periodic release.

The former case show a 2:1 alternating regime. The latter, a regular periodic calcium release regime.

A live demo can be found here