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We investigate the mechanisms responsible for the generation of oscillations in mutually inhibitory cells of non-oscillatory neurons and the transitions from non-relaxation (sinusoidal-like) oscillations to relaxation oscillations. We use a minimal model consisting of a 2D linear resonator, a 1D linear cell and graded synaptic inhibition described by a piecewise linear sigmoidal function. Individually, resonators exhibit a peak in their response to oscillatory inputs at a preferred (resonant) frequency, but they do not show intrinsic (damped) oscillations in response to constant perturbations. We show that network oscillations emerge in this model for appropriate balance of the model parameters, particularly the connectivity strength and the steepness of the connectivity function. For fixed values of the latter, there is a transition from sinusoidal-like to relaxation oscillations as the connectivity strength increases. Similarly, for fixed connectivity strength values, increasing the connectivity steepness also leads to relaxation oscillations. Interestingly, relaxation oscillations are not observed when the 2D linear node is a damped oscillator. We discuss the role of the intrinsic properties of the participating nodes by focusing on the effect that the resonator’s resonant frequency has on the network frequency and amplitude.