Integrate-and-fire models with an almost periodic input function |
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Authors: | Piotr Kasprzak Adam Nawrocki Justyna Signerska-Rynkowska |
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Institution: | 1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, ul. Umultowska 87, 61-614 Poznań, Poland;2. Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, ul. G. Narutowicza 11/12, 80-233 Gdańsk, Poland |
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Abstract: | 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. |
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Keywords: | 42A75 37B55 37E45 92B20 Almost periodic function Displacement and firing map Firing rate Leaky integrate-and-fire model Mean value Neuron model |
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