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### Numerical solutions to differential equations

The Hodgkin-Huxley model is a set of four coupled nonlinear differential equations.  How does one solve these equations?  To explain let us abstract to a general nonlinear differential equation of one variable x.

\begin{eqnarray} \frac{dx}{dt}=f(x) \end{eqnarray}

Here is some function of x (like x, x4, or tan(x) for example) and dx/dt is the rate of change of x with respect to time.  Let us assume that at an initial time t0 we know that x=x0  If f is a linear equation, i.e f(x)=ax+b, then we can explicitly write down the solution of x.  Unfortunately, this is not true in general for nonlinear equations, such as the Hodgkin-Huxley model.  To solve nonlinear differential equations it is typical to use an iterative scheme to construct the solution from many sequential calculations.

Choose a discrete set of times t0<t1<t2<…<tN with ti+1 - ti=\Deltat, we call \Deltat the timestep.  We consider a timestep that is much smaller than the timescale over which the system evolves (\Deltat << 1 for a system with timescale 1).  This means that in the small time interval ti+1 – ti we have that x(ti+1)-x(ti)=\Deltaxi is also small.

Since \Deltat and \Deltaxi are both small we can approximate their ratio with the derivative

\begin{eqnarray} \frac{\Delta x_{i}}{\Delta t} \approx \frac{dx}{dt}(t_i)=f(x(t_i)) \end{eqnarray}

We can rearrange the equation, recalling that \Deltaxi=x(ti+1) - x(ti), to give:

\begin{eqnarray} x(t_{i+1})=x(t_i)+\Delta t f(x(t_i)) \end{eqnarray}

We can write x(ti) via the iterative scheme

\begin{eqnarray} x(t_0)&=&x_0 \\ x(t_1)&=&x(t_0)+\Delta t f(x(t_0)) \\ x(t_2)&=&x(t_1)+\Delta t f(x(t_1)) \\ &\vdots& \\ x(t_N)&=&x(t_{N-1})+\Delta t f(x(t_{N-1})) \end{eqnarray}

Given an initial condition x0 we can solve for x(t) with an accuracy that increases as \Deltat decreases.  The price paid for increased accuracy is that the number of iterations needed between t and t0 also increases.  With the computation speed of modern computers a small \Deltat can be easily be chosen without seriously compromising speed.  In the 1950s Hodgkin and Huxley used a crank calculator, so they chose a balance between speed and accuracy.  Here is an example for a linear equation where we compare the Euler approximation to the known solution. PLOT

The simple algorithm presented here explains the basic principals of the more sophisticated numerical techniques used by Hodgkin and Huxley in the 1950s and are still in use today by many researchers in computational science.      Brain visualizations courtesy of Chris Johnson and Nathan Galli, Scientific Computing and Imaging Institute, University of Utah