Neuronal Networks: Fast/Slow Analysis

Abstract

In this chapter, we consider a very different approach to studying networks of neurons from that presented in Chap. 8. In Chap. 8, we assumed each cell is an intrinsic oscillator, the coupling is weak, and details of the spikes are not important. By assuming weak coupling, we were able to exploit powerful analytic techniques such as the phase response curve and the method of averaging. In this chapter, we do not assume, in general, weak coupling or the cells are intrinsic oscillators. The main mathematical tool used in this chapter is geometric singular perturbation theory. Here, we assume the model has multiple timescales so we can dissect the full system of equations into fast and slow subsystems. This will allow us to reduce the complexity of the full model to a lower-dimensional system of equations. We have, in fact, introduced this approach in earlier chapters when we discussed bursting oscillations and certain aspects of the Morris–Lecar model.