J Comput Neurosci DOI 10.1007/s10827-013-0448-6 Lyapunov exponents computation for hybrid neurons Federico Bizzarri · Angelo Brambilla · Giancarlo Storti Gajani Received: 16 December 2012 / Revised: 5 February 2013 / Accepted: 11 February 2013 © Springer Science+Business Media New York 2013 Abstract Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for con- tinuous time dynamical systems is straightforward when- ever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a varia- tional model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a stan- dard numerical approach even in the case of neurons firing synchronously. Keywords Lyapunov exponent · Hybrid model · Integrate and fire neuron · Neuron networks · Variational model · Saltation Matrix This research work was supported by Regione Lombardia and Politecnico di Milano thorough the financing of the “SPUMA” project. Action Editor: N. Koppell F. Bizzarri () · A. Brambilla · G. Storti Gajani Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, p.za. Leonardo da Vinci, n. 32, 20133 Milano, Italy e-mail: federico.bizzarri@polimi.it A. Brambilla e-mail: brambill@elet.polimi.it G. Storti Gajani e-mail: storti@elet.polimi.it 1 Introduction Hybrid dynamical systems consist of piecewise defined con- tinuous time evolution processes interfaced with some log- ical or decision making process (Peters and Parlitz 2003). This definition can be applied in several frameworks, from mechanics, where there are components that impact with each other (such as gear assemblies) or have “free play”, or problems with friction, sliding or squealing, to many con- trol systems and models in the social and financial sciences where continuous changes can trigger discrete actions. A significant amount of literature can be found dealing with these kinds of systems (see for instance Di Bernardo et al. 2008; Hiskens and Pai 2000; Hristu-Varsakelis et al. 2005 and references therein). Recently, this modeling approach has been successfully proposed also for the analysis and simulation of analog/ digital mixed signal circuits (Bizzarri et al. 2011b, 2012a, b), thus allowing to extend algorithms and simulation meth- ods, whose applicability was originally limited to smooth systems, to non-smooth ones. It is straightforward to realize that, when dealing with neuron models characterized by an I&F behavior, the aforementioned definition of hybrid system sounds appro- priate. To be convinced one can simply focus on (i) the switching between sub-threshold dynamics and refractori- ness (continuous processes) caused by both firing events and the end of refractory periods (decisional processes) and (ii) the simple post-firing reset (impact event) of the state variable mimicking the membrane potential. The use of these models (Izhikevich 2003, 2007; Dayan and Abbott 2005) is now widely accepted for the study of oscillations, synchrony, and other collective, nonlinear and statistical behaviors in neural systems (Gerstner and Kistler 2002) and this has been the driver in the development of tools