Weakly Connected Oscillators

In this chapter we study weakly connected networks $$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n,}} $$ (9.1) of oscillatory neurons. Our basic assumption is that there is a value of λ ∈ Λ such that every equation in the uncoupled system (e = 0) $$ \dot X_i = F_i \left( {X_i ,\lambda } \right),{\text{ X}}_i \in \mathbb{R}^m , $$ (9.2) has a hyperbolic stable limit cycle attractor γ ⊂ ℝ m The activity on the limit cycle can be described in terms of its phase \( \theta \in \mathbb{S}^1 \) of oscillation $$ \dot \theta _i = \Omega _i \left( \lambda \right), $$ where Ωi(λ) is the natural frequency of oscillations. The dynamics of the oscillatory weakly connected system (9.1) can also be described in terms of phase variables: $$ \dot \theta _i = \Omega _i \left( \lambda \right), + \varepsilon g_i \left( {\theta ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n}}{\text{.}} $$