Numerical Integration of Ordinary Differential Equations

Abstract

This chapter presents an overview of numerical integration techniques for solving ODE systems, as implemented in Matlab and COMSOL. These techniques are broadly classified into one-step and multistep methods. One-step methods include forward and backward Euler, Runge–Kutta as well as generalized-$\alpha$ methods, whilst multistep algorithms include the predictor-corrector schemes as well as the backward differentiation and numerical differentiation formula (BDF/NDF) methods. The chapter ends with problems with fully-worked answers in the Solutions section of the text.

Problems

3.1

The Beeler–Reuter [3] model of cardiac ventricular myocyte electrical activity is given by the following system of 8 ODEs:

$$\begin{aligned} \frac{\mathrm{d}V_m}{\mathrm{d}t}= & {} -\frac{1}{C_m}\left[ i_{K_1}+i_{x_1}+i_{Na}+i_s -i_{stim}\right] \nonumber \\ \frac{\mathrm{d}\mathrm {[Ca]}_i}{\mathrm{d}t}= & {} -r_{Ca}i_s + k_{up}\left( \mathrm {[Ca]}_{SR}-\mathrm {[Ca]}_i\right) \nonumber \\ \frac{\mathrm{d}x_1}{\mathrm{d}t}= & {} \alpha _{x_1}\left( 1-x_1\right) - \beta _{x_1}x_1 \nonumber \\ \frac{\mathrm{d}m}{\mathrm{d}t}= & {} \alpha _m\left( 1-m\right) - \beta _mm \nonumber \\ \frac{\mathrm{d}h}{\mathrm{d}t}= & {} \alpha _h\left( 1-h\right) - \beta _hh \nonumber \\ \frac{\mathrm{d}j}{\mathrm{d}t}= & {} \alpha _j\left( 1-j\right) - \beta _jj \nonumber \\ \frac{\mathrm{d}d}{\mathrm{d}t}= & {} \alpha _d\left( 1-d\right) - \beta _dd \nonumber \\ \frac{\mathrm{d}f}{\mathrm{d}t}= & {} \alpha _f\left( 1-f\right) - \beta _ff \nonumber \end{aligned}$$

with four membrane ionic currents: a time-independent outward K$^{+}$ current ($i_{K_1}$), a voltage-dependent outward K$^{+}$ current ($i_{x_1}$), an inward Na$^{+}$ current ($i_{Na}$), and a slow inward Ca$^{2{+}}$ current ($i_s$), given by the expressions:

$$\begin{aligned} i_{K_1}= & {} A_{K_1}\left\{ \frac{4\left( \mathrm{e}^{0.04(V_m+85)}-1\right) }{\mathrm{e}^{0.08(V_m+53)}+\mathrm{e}^{0.04(V_m+53)}} + \frac{0.2(V_m+23)}{1-\mathrm{e}^{-0.04(V_m+23)}} \right\} \nonumber \\ i_{x_1}= & {} A_{x_1}x_1\frac{\mathrm{e}^{0.04(V_m+77)}-1}{\mathrm{e}^{0.04(V_m+35)}} \nonumber \\ i_{Na}= & {} \left( g_{Na}m^3hj+g_{NaC}\right) (V_m-V_{Na}) \nonumber \\ i_s= & {} g_sdf(V_m-V_{Ca}) \nonumber \end{aligned}$$

with

$$\begin{aligned} V_{Ca} = -82.3 - 13.0287\ln \mathrm {[Ca]}_i \nonumber \end{aligned}$$

where $V_m$ is the transmembrane potential (in mV) and [Ca]$_i$ denotes the free intracellular Ca$^{2{+}}$ ion concentration (in M). The remaining variables $x_1$, m, h, j, d, f are dimensionless kinetic gating terms whose voltage-dependent forward ($\alpha$) and reverse ($\beta$) rates (in ms$^{-1}$) are given by:

$$\begin{aligned} \begin{array}{ll} \alpha _{x_1} = \frac{0.0005\mathrm{e}^{0.083(V_m+50)}}{\mathrm{e}^{0.057(V_m+50)}+1}&{}\quad \beta _{x_1} = \frac{0.0013\mathrm{e}^{-0.06(V_m+20)}}{\mathrm{e}^{-0.04(V_m+20)}+1} \nonumber \\ \alpha _m = \frac{-(V_m+47)}{\mathrm{e}^{-0.1(V_m+47)}-1} &{}\quad \beta _{m} = 40\mathrm{e}^{-0.056(V_m+72)} \nonumber \\ \alpha _h = 0.126\mathrm{e}^{-0.25(V_m+77)} &{}\quad \beta _h = \frac{1.7}{\mathrm{e}^{-0.082(V_m+22.5)}+1} \nonumber \\ \alpha _j = \frac{0.055\mathrm{e}^{-0.25(V_m+78)}}{\mathrm{e}^{-0.2(V_m+78)}+1} &{}\quad \beta _j = \frac{0.3}{\mathrm{e}^{-0.1(V_m+32)}+1} \nonumber \\ \alpha _d = \frac{0.095\mathrm{e}^{-0.01(V_m-5)}}{\mathrm{e}^{-0.072(V_m-5)}+1} &{}\quad \beta _d = \frac{0.07\mathrm{e}^{-0.017(V_m+44)}}{\mathrm{e}^{0.05(V_m+44)}+1} \nonumber \\ \alpha _f = \frac{0.012\mathrm{e}^{-0.008(V_m+28)}}{\mathrm{e}^{0.15(V_m+28)}+1} &{}\quad \beta _f = \frac{0.0065\mathrm{e}^{-0.02(V_m+30)}}{\mathrm{e}^{-0.2(V_m+30)}+1} \nonumber \end{array} \end{aligned}$$

Finally, $i_{stim}$ is the applied stimulus current given by

$$\begin{aligned} i_{stim} = \left\{ \begin{array}{ll} A_s \qquad &{} t_{on} \le t < t_{on} + t_{dur} \\ 0 &{} \mathrm {otherwise} \end{array} \right. \ \nonumber \end{aligned}$$

where $A_s$, $t_{on}$ and $t_{dur}$ represent the stimulus current amplitude, onset time and duration respectively.

Initial variable values at $t=0$ are $V_m=-83.3\,\mathrm {mV}$, $\mathrm {[Ca]}_i=1.87\times 10^{-7}\,\mathrm {M}$, $x_1=0.1644$, $m=0.01$, $h = 0.9814$, $j=0.9673$, $d=0.0033$ and $f=0.9884$, with remaining model parameters given below:

Parameter	Value	Parameter	Value
$C_m$	1 $\upmu$F cm$^{-2}$	$g_{Na}$	4 ms cm${-2}$
$r_{Ca}$	$1\times 10^{-7}$ M cm$^{2}$ nC$^{-1}$	$g_{NaC}$	0.003 ms cm$^{-2}$
$\mathrm {[Ca]}_{SR}$	$1\times 10^{-7}$ M	$g_s$	0.09 ms cm$^{-2}$
$k_{up}$	0.07 ms$^{-1}$	$A_s$	40 $\upmu$A cm$^{-2}$
$A_{K_1}$	0.35 $\upmu$A cm$^{-2}$	$t_{on}$	50 ms
$A_{x_1}$	0.8 $\upmu$A cm$^{-2}$	$t_{dur}$	1 ms
$V_{Na}$	50 mV

(a) Use Matlab’s ode15s solver to solve and plot for membrane potential $V_m$ against time from $t=0$ to 500 ms.

(b) Write your own Matlab code to solve the same model using the forward-Euler method with a fixed step-size of 0.01 ms, plotting on the same graph $V_m$ obtained with both the forward-Euler and ode15s methods.

(c) Verify that when the forward-Euler step-size is increased to 0.03 ms, the method becomes unstable.

(d) Now write Matlab code to solve this model using the backward-Euler method with fixed step-sizes of 0.01 and 0.1 ms, plotting these solutions with the $V_m$ obtained using ode15s. Note that because the backward-Euler is implicit, you will need to implement Newton’s method (Eq. 3.12) to iteratively obtain the solution at each step.

(e) Finally, solve the model using Matlab’s in-built ODE solvers ode15s , ode23s , ode23t , ode23tb , ode45 , ode23 and ode113 , plotting the $V_m$ obtained for each method on the same plot. Based on the time taken for each of these to solve the model, which solvers would you recommend for this system? HINT: use Matlab’s tic and toc timing commands to determine the computational time taken for each solver.

3.2

A minimal model of neural spiking, known as the $I_{Na,p}+I_K$ model [13] (pronounced persistent sodium plus potassium), is defined by the ODE pair

$$\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t}= & {} -\frac{1}{C}\left[ g_{Na}m_{\infty }(V-E_{Na}) + g_Kn(V-E_K) +g_L(V-E_L)-I\right] \nonumber \\ \frac{\mathrm{d}n}{\mathrm{d}t}= & {} \left( n_{\infty } - n\right) /{\tau _n} \nonumber \end{aligned}$$

with

$$\begin{aligned} m_{\infty } = \frac{1}{1+\exp {\left[ -\frac{V+20}{15}\right] }}, \quad n_{\infty } = \frac{1}{1+\exp {\left[ -\frac{V+25}{5}\right] }} \nonumber \end{aligned}$$

where V represents the neuronal membrane potential, n governs the kinetics of the outward K$^{+}$ current, and I is the applied stimulus. Initial values for V and n are −72.9 mV and 0.36 respectively, with all remaining model parameters given below:

Parameter	Value	Parameter	Value
C	1 $\upmu$F cm$^{-2}$	$\tau _n$	1 ms
$g_{Na}$	20 ms cm$^{-2}$	$g_L$	8 ms cm$^{-2}$
$E_{Na}$	60 mV	$E_L$	−80 mV
$g_K$	10 ms cm$^{-2}$	I	40 $\upmu$A cm$^{-2}$
$E_K$	−90 mV

Write custom Matlab code to solve the model and plot V from $t=0$ to 30 ms for the following two fixed-step methods:

(a) A third-order Runge–Kutta algorithm with coefficients

$$\begin{aligned} \begin{array}{l|rrr} 0 &{} &{} &{} \\ \frac{1}{2} &{} \frac{1}{2} &{} &{} \\ \frac{3}{4} &{} 0 &{} \frac{3}{4} &{} \\ \hline &{} \frac{2}{9} &{} \frac{1}{3} &{} \frac{4}{9} \\ \end{array} \nonumber \end{aligned}$$

using step-sizes of 0.01 and 0.1 ms. Plot both solutions for V on the same plot, along with the solution obtained from ode15s.

(b) The generalized-$\alpha$ method with step size of 0.1 ms and high-frequency damping factors of 1 (no-damping), 0.5 and 0 (maximum damping). As above, plot your solutions on the same plot with the solution obtained from ode15s.

3.3

For the BDF family of implicit solvers , Newton’s method is used with an initial estimate for the solution at the next step $y_n$ determined from the predictor of Eq. 3.60:

$$\begin{aligned} y_n^{p} = \sum _{i=0}^k \nabla ^i\,y_{n-1} \nonumber \end{aligned}$$

Verify that this predictor is equivalent to extrapolation of a polynomial of degree k passing through the previous $k+1$ step solutions $y_{n-1}$, $y_{n-2}, \ldots , y_{n-k-1}$.

3.4

Determine the formula for the 7-step BDF method. For the test ODE

$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}t} = \lambda y \nonumber \end{aligned}$$

with $\lambda$ real and negative, numerically estimate the absolute stability constraint on the step-size, namely step-sizes ensuring $|y_n| < |y_{n-1}|$. Note that for this test ODE, BDF methods of orders 1-6 are absolutely stable for all step-sizes, but orders 7 and above have only limited stability and are therefore not used in practice.