SPM/The DCM Equation. 2. Dynamical Systems

What is a dynamic equation?

To understand DCM, you'll need to a know what a dynamic equation is. It's extremely simple. A dynamic equation describes how a process (a system) changes in time or space.

Here's a couple of examples - one from the real world, and one from the world of maths. (The latter is considerably more exciting.)

Example 1: Let's say the bank gives you 3% interest on your savings. We're now at the end of year zero, and your extremely successful business has made you £50. How much will you have next year? We can work out the answer with a dynamic equation:

$x(1)=1.03*x(0)\,$

Or more generally:

$x(t)=1.03*x(t-1)\,$

Where $t$ is time and $x$ is your bank balance. You can apply this equation over and over again to see how your bank balance will develop. In reality, you probably know that there's a one-off equation to calculate compound interest for any number of years, but the point of this example is that the state equation is a simple rule describing how the system (your bank account) changes over time.

Example 2: A dynamic equation may represent how a system changes in space, rather than time. Take these three equations, which describe the rates of change of three numbers:

${\begin{array}{lcl}{\dfrac {dx}{dt}}&=&\sigma (y-x)\\\\{\dfrac {dy}{dt}}&=&x(\rho -z)-y\\\\{\dfrac {dz}{dt}}&=&xy-\beta z\end{array}}\,$

These equations give you the rate of change of variables $x$, $y$ and $z$ over time, whilst $\sigma$, $\rho$ and $\beta$ are numbers selected in advance - they are the parameters of the system, which fine tune it. Don't worry about what they mean.

Together these equations form the Lorenz Attractor, and if you plot them on a graph, you get something which is not only crucial to chaos theory, but something quite pretty:

The Lorenz Attractor, generated by a simple set of equations

So we've seen that repeatedly applying a short dynamic equation to its own output can describe the change of a system over time or space. As we'll explore next, such an equation forms the basis of Dynamic Causal Modelling.