Saddle-Node on a Limit Cycle

Abstract

In this chapter we study a weakly connected network

$$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right),{\text{ X}}_i \in \mathbb{R}^m {\text{,i = 1,}} \ldots {\text{,n,}} $$

of neurons each having a saddle-node bifurcation on a limit cycle. Such neurons are said to have Class 1 neural excitability. This bifurcation provides an important example of when local analysis at an equilibrium renders global information about the system behavior.