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 is time and  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, and  over time, whilst ,  and  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.