Amplitude models are constructed to describe the resonance structure of ${D^{0}to K^{-}pi^{+}pi^{+}pi^{-}}$ and ${D^{0} to K^{+}pi^{-}pi^{-}pi^{+}}$ decays using $pp$ collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of $3.0mathrm{fb}^{-1}$. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $D^{0} to a_1(1260)^{+} K^{-}$ and $D^{0} to K_1(1270/1400)^{+} pi^{-}$ being prominent in ${D^{0}to K^{-}pi^{+}pi^{+}pi^{-}}$ and $D^{0}to K^{+}pi^{-}pi^{-}pi^{+}$, respectively. Precise measurements of the lineshape parameters and couplings of the $a_1(1260)^{+}$, $K_1(1270)^{-}$ and $K(1460)^{-}$ resonances are made, and a quasi model-independent study of the $K(1460)^{-}$ resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be $R_{K3pi} = 0.459pm 0.010,(mathrm{stat}) pm 0.012,(mathrm{syst}) pm 0.020,(mathrm{model})$, which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle $gamma$ and studies of charm mixing and $C!P$ violation.
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