References
- J. Arino. Fundamental Theory of Ordinary Differential Equations. Lecture Notes. University of Manitoba, 2006.
- V.I. Arnold. Cardiac arrhythmias and circle maps. Chaos, 1(1):20–24, 1991. http://dx.doi.org/10.1063/1.165812. doi: 10.1063/1.165812
- R. Bartoszyn´ski. A characterization of the weak convergence of measures. Ann. Math. Statist., 32(2):561–576, 1961. http://dx.doi.org/10.1214/aoms/1177705061. doi: 10.1214/aoms/1177705061
- R. Brette. Dynamics of one-dimensional spiking neuron model. J. Math. Biol., 48(1):38–56, 2004. http://dx.doi.org/10.1007/s00285-003-0223-9. doi: 10.1007/s00285-003-0223-9
- H. Carrillo and F. Hoppensteadt. Unfolding an electronic integrate-and-fire circuit. Biol. Cybern., 102(1):1–8, 2010. http://dx.doi.org/10.1007/s00422-009-0358-x. doi: 10.1007/s00422-009-0358-x
- H. Carrillo and F.A. Ongay. On the firing maps of a general class of forced integrate-and-fire neurons. Math. Biosci., 172(1):33–53, 2001. http://dx.doi.org/10.1016/S0025-5564(01)00060-8. doi: 10.1016/S0025-5564(01)00060-8
- S. Coombes and P. Bressloff. Mode locking and arnold tongues in integrate-and- fire oscillators. Phys. Rev. E, 60(2):2086–2096, 1999. http://dx.doi.org/10.1103/PhysRevE.60.2086. doi: 10.1103/PhysRevE.60.2086
- A. Denjoy. Sur les courbes d´efinies par les ´equations diff´erentielles a` la surface du tore. J. Math. Pures Appl., 11(9):333–375, 1932.
- M. di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk. Piecewise Smooth Dynamical Systems. Taylor & Francis, 2008.
- T. Gedeon and M. Holzer. Phase locking in integrate-and-fire models with re-fractory periods and modulation. J. Math. Biol., 49(6):577–603, 2004. http://dx.doi.org/10.1007/s00285-004-0268-4. doi: 10.1007/s00285-004-0268-4
- D. George and F.T. Sommer. Computing with inter-spike interval codes in network of integrate and fire neurons. Neurocomputing, 65–66:415–420, 2005. http://dx.doi.org/10.1016/j.neucom.2004.10.038. doi: 10.1016/j.neucom.2004.10.038
- A.L. Gibbs and F.E. Su. On choosing and bounding probability metrics. Int. Stat. Rev., 70(30):419–435, 2002. http://dx.doi.org/10.1111/j.1751-5823.2002.tb00178.x. doi: 10.1111/j.1751-5823.2002.tb00178.x
- W.H. Gottschalk. Minimal sets: An introduction to topological dynamics. Bull. Amer. Math. Soc., 64(6):336–351, 1958. http://dx.doi.org/10.1090/S0002-9904-1958-10223-2. doi: 10.1090/S0002-9904-1958-10223-2
- P. Hartman. OrdinaryDifferential Equations. Classics Appl. Math. SIAM, (2nd edition), 2002.
- A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia Math. Appl. Taylor & Francis, 1995.
- J.P. Keener, F.C. Hoppensteadt and J. Rinzel. Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math, 41(3):503–517, 1981. http://dx.doi.org/10.1137/0141042. doi: 10.1137/0141042
- J. Kwapisz. Poincar´e rotation number for maps of the real line with almost periodic displacement. Nonlinearity, 13(5):1841–1854, 2000. http://dx.doi.org/10.1088/0951-7715/13/5/320. doi: 10.1088/0951-7715/13/5/320
- W. Marzantowicz and J. Signerska. Firing map of an almost periodic input function. DCDS Suppl., 2011(2):1032–1041, 2011.
- W. Marzantowicz and J. Signerska. Distribution of the displacement sequence of an orientation preserving circle homeomorphism. Dynam. Syst., 29(2):153–166, 2014. http://dx.doi.org/10.1080/14689367.2013.868867. doi: 10.1080/14689367.2013.868867
- W. Marzantowicz and J. Signerska. On the interspike-intervals of periodically-driven integrate-and-fire models. J. Math. Anal. Appl., 423(1):456–479, 2015. http://dx.doi.org/10.1016/j.jmaa.2014.10.013. doi: 10.1016/j.jmaa.2014.10.013
- Y. Ono, H. Suzuki and K. Aihara. Grazing bifurcation and mode locking in reconstructing chaotic dynamics with a leaky integrate and fire model. Artif. Life Robotics, 7(1):55–62, 2003. http://dx.doi.org/10.1007/BF02480886. doi: 10.1007/BF02480886
- R.R. Rao. Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist., 33(2):659–680, 1962. http://dx.doi.org/10.1214/aoms/1177704588. doi: 10.1214/aoms/1177704588
- D.S. Reich, F. Mechler, K.P. Purpura and J.D. Victor. Interspike intervals, receptive fields, and information encoding in primary visual cortex. J. Neurosci., 20(5):1964–1974, 2000.
- J. Signerska. Dynamical properties of maps arising in some models of neuron activity and electrical circuits. PhD Thesis, Institute of Mathematics of the Polish Academy of Sciences, Warsaw, 2013.
- P.H.E. Tiesinga. Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons. Phys. Rev. E, 65(4):041913, 2002. http://dx.doi.org/10.1103/PhysRevE.65.041913. doi: 10.1103/PhysRevE.65.041913
- J. Touboul and R. Brette. Spiking dynamics of bidimensional integrate-and-fire neurons. SIAM J. Appl. Dyn. Syst., 8(4):1462–1506, 2009. http://dx.doi.org/10.1137/080742762. doi: 10.1137/080742762