Biol Cybern (2008) 99:443–458 DOI 10.1007/s00422-008-0250-0 ORIGINAL PAPER Compartment model of neuropeptide synaptic transport with impulse control Andrzej Bielecki · Piotr Kalita · Marian Lewandowski · Marek Skomorowski Received: 29 October 2007 / Accepted: 25 July 2008 / Published online: 20 September 2008 © Springer-Verlag 2008 Abstract In this paper a mathematical description of a pre- synaptic episode of slow synaptic neuropeptide transport is proposed. Two interrelated mathematical models, one based on a system of reaction diffusion partial differential equa- tions and another one, a compartment type, based on a sys- tem of ordinary differential equations (ODE) are formulated. Processes of inflow, calcium triggered activation, diffusion and release of neuropeptide from large dense core vesicles (LDCV) as well as inflow and diffusion of ionic calcium are represented. The models assume the space constraints on the motion of inactive LDCVs and free diffusion of activa- ted ones and ions of calcium. Numerical simulations for the ODE model are presented as well. Additionally, an electro- nic circuit, reflecting the functional properties of the mathe- matically modelled presynaptic slow transport processes, is introduced. 1 Introduction Synaptic transmission in neural systems has usually a che- mical character which means that chemical compounds are released to the synaptic cleft from presynaptic boutons. A. Bielecki · P. Kalita (B ) · M. Skomorowski Institute of Computer Science, Jagiellonian University, Nawojki 11, 30-072 Kraków, Poland e-mail: kalita@softlab.ii.uj.edu.pl A. Bielecki e-mail: bielecki@softlab.ii.uj.edu.pl M. Skomorowski e-mail: skomorowski@softlab.ii.uj.edu.pl M. Lewandowski Institute of Zoology, Jagiellonian University, Ingardena 6, 30-060 Kraków, Poland e-mail: lew@zuk.iz.uj.edu.pl Neurotransmitter and neuropeptide transport plays a key role in this process. An episode of fast transport, namely the release of neu- rotransmitter from presynaptic bouton to the synaptic cleft, has been intensively studied, described well on a biological level and modelled using mathematical, cybernetic and elec- tronic models—papers by Aristizabal and Glavinovic (2004) and Kruckenberg and Sandweg (1968) are examples. Mathe- matical models are usually based on differential equations, both ordinary and partial (Aristizabal and Glavinovic 2004; Friedman and Craciun 2005; Magleby and Stevens 1972) and on difference schemes (Scott and Rusakov 2006). Pro- babilistic models have been stated as well—see, for ins- tance, Boyd and Martin (1956); del Castillo and Katz (1954); Kalkstein and Magleby 2003 and Keener and Sneyd (1998, Sect. 7.1.1). In recent years the process of slow transport, involving neuropeptides, has been being studied intensively. Fluores- cence microscopy allows for both in vitro and in vivo imaging (Han et al. 1999; Levitan et al. 2007; Shakiryanova et al. 2005, 2006) of LDCVs, constituting a large step towards the understanding of underlying mechanisms. It is known that this phenomenon is strictly related to transport of cal- cium ions and those two processes mutually interact (Green- gard 2001): neuropeptide exocytosis and fusion of LDCVs with the membrane are allowed after their activation by ionic calcium. Results concerning a slow excitatory postsynaptic potential (EPSP) and mathematical models of this pheno- menon were obtained already in the 1980s, see Adams et al. (1986); Bertrand et al. (2000); however it seems that the kine- tics of neuropeptide in a presynaptic bouton has not been so far modelled intensively. This paper is intended to fill this gap. It should be stressed, that the knowledge concerning the slow synaptic transport is, generally, nowadays far incom- 123