On the firing maps of a general class of forced integrate and fire neurons

Integrate and fire processes are fundamental mechanisms causing excitable and oscillatory behavior. Van der Pol [Philos. Mag. (7) 2 (11) (1926) 978] studied oscillations caused by these processes, which he called 'relaxation oscillations' and pointed out their relevance, not only to engineering, but also to the understanding of biological phenomena [Acta Med. Scand. Suppl. CVIII (108) (1940) 76], like cardiac rhythms and arrhythmias. The complex behavior of externally stimulated integrate and fire oscillators has motivated the study of simplified models whose dynamics are determined by iterations of 'firing circle maps' that can be studied in terms of Poincaré's rotation theory [Chaos 1 (1991) 20; Chaos 1 (1991) 13; SIAM J. Appl. Math. 41 (3) (1981) 503]. In order to apply this theory to understand the responses and bifurcation patterns of forced systems, it is fundamental to determine the regions in parameter space where the different regularity properties (e.g., continuity and injectivity) of the firing maps are satisfied. Methods for carrying out this regularity analysis for linear systems, have been devised and the response of integrate and fire neurons (with linear accumulation) to a cyclic input has been analyzed [SIAM J. Appl. Math. 41 (3) (1981) 503]. In this paper we are concerned with the most general class of forced integrate and fire systems, modelled by one first-order differential equation. Using qualitative analysis we prove theorems on which we base a new method of regularity analysis of the firing map, that, contrasting with methods previously reported in the literature, does not requires analytic knowledge of the solutions of the differential equation and therefore it is also applicable to non-linear integrate and fire systems. To illustrate this new methodology, we apply it to determine the regularity regions of a non-linear example whose firing maps undergo bifurcations that were unknown for the previously studied linear systems.