Bifurcations Dynamics of Single Neurons and Small Networks

What Is a Bifurcation? The word bifurcate is commonly used to denote a split, as in “up ahead, the road bifurcates into two parts.” The use of the term in this context implies that a driver traveling on this road can make a realtime choice of being able to take one or the other branch of the road when the bifurcation point is reached. In mathematics, and in its application to neuroscience, the term bifurcation has an entirely different contextual meaning. While a mathematical bifurcation is similar in that it involves different branches, the decision on which branch is chosen is largely made a priori and depends on the choice of parameters. It is instructive to first gain some insight into situations that are not bifurcations. Different initial conditions converging to different solutions of the same differential equation while all parameters are held constant are not examples of bifurcations. Instead it may be that one initial condition lies in the basin of attraction of a particular solution, while the other one does not. Think, for example, of a double well potential where each well can trap particles starting with different initial conditions. A solution to a differential equation that is perturbed and then suddenly veers off toward another attracting solution is also not an example of a bifurcation. In these examples, just because a change in behavior has occurred or a difference in convergence is observed, a bifurcation has not occurred. To qualify as a bifurcation, the change in behavior must be associated with a change in a parameter value. Bifurcations arise in mathematical equations called dynamical systems. A dynamical system can generically be defined by the equation