Weakly Connected Oscillators

In this chapter we study weakly connected networks (9.1)$$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n,}} $$ of oscillatory neurons. Our basic assumption is that there is a value of λ ∈ Λ such that every equation in the uncoupled system (ε = 0) (9.2)$$ \dot X_i = F_i \left( {X_i ,\lambda } \right),{\text{ X}}_i \in \mathbb{R}^m , $$ has a hyperbolic stable limit cycle attractor γ ⊂ ℝm 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 Ω_i(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: $$ \dot \theta _i = \Omega _i \left( \lambda \right), + \varepsilon g_i \left( {\theta ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n}}{\text{.}} $$