Phase reduction beyond the Kuramoto model

Many biological, engineering and natural systems can be modeled as populations of coupled oscillators where each oscillator behaves periodically.
When these units are coupled to each other, emergent phenomena, as synchronization, appears.
However, dealing with those systems is usually difficult due to the large number of degrees of freedom. Conditionality reduction techniques to obtain simple tractable models are usually considered.
The most common method is "phase reduction" that allows to capture the dynamics of each oscillator with just one variable, the phase.
The succeed of the method was clear when the Kuramoto model, derived through phase reduction, gave a simple explanation to collective synchronization.
Despite this success, phase reduction is often limited to the Kuramoto model because of the challenge to obtain analytical expressions.

The porpoise of this talk is to make clear that phase reduction beyond Kuramoto model is possible. On the first part of the talk we introduce phase reduction and its limitations.
Then we show how it is possible to obtain analytical phase reduced model for weakly nonlinear oscillators.
Finally, we talk about second order phase reduction where higher order corrections are included to capture the qualitative dynamics and improve accuracy.