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Using \emph{ab initio} linear response techniques we calculate spin wavespectra in K$_{2}$Fe$_{4+x}$Se$_{5}$, and find it to be in excellent agreementwith a recent experiment. The spectrum can be alternatively described ratherwell by localized spin Hamiltonian restricted to first and second nearestneighbor couplings. We confirm that exchange coupling between nearest neighborFe magnetic moments is strongly anisotropic, and show directly that in theideal system this anisotropy has itinerant nature which can be imitated byintroducing higher order terms in effective localized spin Hamiltonian(biquadratic coupling). In the real system, structural relaxation provides anadditional source of the exchange anisotropy of approximately the samemagnitude. The dependence of spin wave spectra on filling of Fe vacancy sitesis also discussed.

Spin excitations in K$_{2}$Fe$_{4+x}$Se$_{5}$: linear response approach