Neuronal excitability manifests itself through a number of key markers of the dynamics and it allows to classify neurons into different groups with identifiable voltage responses to input currents. In particular, two main types of excitability can be defined based on experimental observations, and their underlying mathematical models can be distinguished through separate bifurcation scenarios. Related to these two main types of excitable neural membranes, and associated models, is the distinction between integrator and resonator neurons. One important difference between integrator and resonator neurons, and their associated model representations, is the presence in resonators, as opposed to integrators, of subthreshold oscillations following spikes. Switches between one neural category and the other can be observed and/or created experimentally, and reproduced in models mostly through changes of the bifurcation structure. In the present work, we propose a new scenario of switch between integrator and resonator neurons based upon multiple-timescale dynamics and the possibility to force an integrator neuron with a specific time-dependent slowly-varying current. The key dynamical object organising this switch is a so-called folded-saddle singularity. We also showcase the reverse switch via a folded-node singularity and propose an experimental protocol to test our theoretical predictions.