Wavelet Analysis of Impulses in Axon Physiology Carlo Cattani 1 and Massimo Scalia 2 1 DiFarma University of Salerno, Via Ponte Don Melillo I-84084 Fisciano (SA) 2 Dept. of Mathematics, “G. Castelnuovo”, University of Rome, “La Sapienza”, P.le A. Moro 2, I-00185 Roma ccattani@unisa.it, massimo.scalia@uniroma1.it Abstract. The nonlinear dynamical system which models the axon im- pulse activity is studied through the analysis of the wavelet coefficients. A system with a pulse source is compared with the corresponding source- less, through the wavelet coefficients. Keywords: Haar wavelets, Short Haar Wavelet Transform, Competi- tion model, multidimensional analysis. 1 Introduction In this paper we consider the nonlinear Fitzhugh-Nagumo system [1,2] ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dx dt = 1 ε [x(1 − x 2 ) − y], dy dt = x − β, (1) where ε is a small parameter 0 <ε ≪ 1 and β is a crucial parameter. This system was proposed in the early 60ties in order to describe the neural activity. Stimulated axons shows their activity by a suddenly change in their electrical potential. These pulse in a short time were called spikes or axons firing. However, the normal activity of neurons is usually describes by a continuous axons firing, even in absence of external stimulations. Therefore one of the main problems is to recognize among all generated spikes those which are caused by some external stimulations. The same system, with an additional wave structure, has been also used to describe solitary wave propagation (spikes or pulses) in a spatial regions in order to modelling neural communication or calcium waves. There follows that small variations in the parameter for signal which have a very short duration gives rise to different physical phenomena. From mathematical point of view system (1) is a nonlinear system which can be derived from the van der Pol by Lienard’s change of variables. This dynamical system is strongly depending [1,2,6,7] on the crucial parameter β in the sense that the evolution would be completely different, starting from a M. Elloumi et al. (Eds.): BIRD 2008, CCIS 13, pp. 538–545, 2008. c  Springer-Verlag Berlin Heidelberg 2008