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For a class of vector fields, we show that one can selectively average terms which are of the same order in a small parameter, giving an extension of standard averaging results. Such selective averaging is illustrated for the phase reduction of a system of oscillators with both coupling and external input, for which the coupling can be averaged to give a term which only depends on phase differences, while the external input term is not averaged. For a coupled two-neuron system, we use selectively averaged equations to find the optimal input which takes the in-phase state to the anti-phase state.

Selective averaging with application to phase reduction and neural control