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:

\<center> \<math> x(1) = 1.03 * x(0)\ \</math> \</center>

Or more generally:

\<center> \<math> x(t) = 1.03 * x(t-1)\ \</math> \</center>

Where {{#tag:math|t}} is time and {{#tag:math|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:

\<center>

\<math>

\begin{array}{lcl} \dfrac{dx}{dt} & = & \sigma (y - x) \ \ \dfrac{dy}{dt} & = & x (\rho - z) - y \ \ \dfrac{dz}{dt} & = & xy - \beta z \end{array}\ \</math> \</center>

These equations give you the rate of change of variables {{#tag:math|x}}, {{#tag:math|y}} and {{#tag:math|z}} over time, whilst {{#tag:math|\sigma}}, {{#tag:math|\rho}} and {{#tag:math|\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.

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.