Integrate-and-fire Models with an Almost Periodic Input Function

We investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and μ-almost periodic functions. Special attention is paid to the properties of the firing map and its displacement, which give information about the spiking behavior of the considered system. We provide conditions under which such maps are well-defined and are uniformly continuous. We show that the LIF models with Stepanov almost periodic inputs have uniformly almost periodic displacements. We also show that in the case of μ-almost periodic drives it may happen that the displacement map is uniformly continuous, but is not μ-almost periodic (and thus cannot be Stepanov or uniformly almost periodic). By allowing discontinuous inputs, we extend some previous results, showing, for example, that the firing rate for the LIF models with Stepanov almost periodic input exists and is unique. This is a starting point for the investigation of the dynamics of almost-periodically driven integrate-and-fire systems.