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: Or more generally: 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: 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: 600px|center|thumb|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.