Fixed Points and Periodic Orbits

To understand how bursting behavior in neurons occurs, we first have to be familiar with fixed points and periodic orbits of dynamical systems. Consider the system of ordinary differential equations

$$\dot{\bf x} = \frac{d {\bf x}}{dt} = {\bf f}({\bf x},\lambda),$$ (1)

where ${\bf x} = {\bf x}(t)$ is a vector of variables which are functions of time and $\lambda$ is a vector of parameters. For example, in the Hodgkin/Huxley equations, ${\bf x} = (V,n,m,h)$ and $\lambda = (I,\bar{g}_{Na},\cdots)$. There are several important classes of solutions of such systems of ODEs, including:

Fixed Points - A fixed point ${\bf x_f}$ satisfies ${\bf f(x_f};\lambda)=0$. Thus, if the system starts at ${\bf x_f}$, it will remain at ${\bf x_f}$ forever.

Periodic Orbits - A periodic orbit is a solution for (1) for which there exists a $T$ with $0<T<\infty$ such that ${\bf x}(t) = {\bf x}(t+T)$ for all $t$. An example of a periodic orbit is the periodically firing action potentials which you found for the Hodgkin/Huxley equations in Homework #1.

A fixed point or periodic orbit is said to be stable if solutions starting close to it tends to it under the evolution of the flow. Not surprisingly, if a solution is not stable, it is called unstable.