Integrate-and-fire models with an almost periodic input function

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 a firing map and its displacement, which give information about the spiking behaviour of the system under consideration. We provide conditions under which such maps are well-defined for every t ∈ R and are uniformly continuous. Moreover, we show that the LIF model with a Stepanov almost periodic input has a uniformly almost periodic displacement map. We also show that in the case of a µ-almost periodic drive it may happen that the displacement map corresponding to the LIF model is uniformly continuous, but is not µ-almost periodic (and thus cannot be Stepanov or uniformly almost periodic). By allowing discontinuous inputs, we generalize some results of previous papers, showing, for example, that the firing rate for the LIF model with a Stepanov almost periodic drive exists and is unique. This is a starting point for the investigation of the dynamics of almost-periodically driven integrate-and-fire systems. The work provides also some contributions to the theory of Stepanovand µ-almost periodic functions.2010 Mathematics Subject Classification. 42A75, 37B55, 37E45, 92B20. Key words and phrases. Displacement map, firing map, firing rate, Haar wavelets, leaky integrate-and-fire model, linear differential equation, mean value, µ-almost periodic function, neuron model, Stepanov almost periodic function, uniformly almost periodic function.1 In the special case σ = 0, the LIF model is usually referred to as the perfect integrator model (or the PI model for short).

Further properties of Stepanov--Orlicz almost periodic functions

“…This space is well investigated. The literature is abundant, see for instance [20], [18], [3], [27].…”

“…Since then, there has been a significant attention devoted to these functions. One can mention the works by S. Stoiński [31], L. I. Danilov [12], D. Bugajewski and A. Nawrocki [7] and P. Kasprzak et al [20]. Let µ S be the set function defined on Σ by:…”

“…Definition 3.8 ([30], [7], [20], [31]). We say that a function f ∈ M(R, X) is measurably almost periodic and we write f ∈ meas-AP(R, X), if for any ε, δ > 0, there exists l(ε, δ) > 0 such that any interval of length l(ε, δ) contains at least a real number τ for which…”

“…The concept of Stepanov almost periodic functions introduced by Stepanov in [30] in 1926 is a natural and important generalization of Bohr almost periodicity. In recent decades, this concept has been well developed in connection with the theory of differential equations, because of their significance and applications in different areas such as physics, mathematical biology, control theory, and other related fields, see [7], [17], [20]. Important progresses have been made on this subject, we can cite several works which give a beautiful presentation of the methods and results: S. Zaidman [32], A. S. Rao [29], S. Stoiński [31], J. Andres and D. Pennequin [5], [4], T. Diagana [13], H. S. Ding et al [15], Y. Hu and A.…”

“…Important progresses have been made on this subject, we can cite several works which give a beautiful presentation of the methods and results: S. Zaidman [32], A. S. Rao [29], S. Stoiński [31], J. Andres and D. Pennequin [5], [4], T. Diagana [13], H. S. Ding et al [15], Y. Hu and A. B. Mingarelli [18], F. Bedouhene et al [6], D. Bugajewski and A. Nawrocki [7], P. Kasprzak et al [20].…”