From a PDE model to an ODE model of dynamics of synaptic depression

Adam Bobrowski; Katarzyna Morawska

doi:10.3934/dcdsb.2012.17.2313

Article Contents

2012, Volume 17, Issue 7: 2313-2327. Doi: 10.3934/dcdsb.2012.17.2313

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From a PDE model to an ODE model of dynamics of synaptic depression

Adam Bobrowski^1, and
Katarzyna Morawska^2,

1.
Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin
2.
Faculty of Mathematics, Physics and Computer Science, Maria Curie-Skŀodowska University in Lublin, Pl. Marii Curie-Skŀodowskiej 1, 20-031 Lublin

Received: January 2012

Revised: February 2012

Published: October 2012

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Abstract

We provide a link between two recent models of dynamics of synaptic depression. To this end, we specify the missing transmission conditions in the PDE model of Bielecki and Kalita, and show that if diffusion is fast and communication between pools is slow, the PDE model is well approximated by the ODE model of Aristizabal and Glavinovič. From the mathematical point of view the ODE model is obtained as a singular perturbation of the PDE model with singularities both in the operator and in the boundary and transmission conditions. The result is put in the context of degenerate convergence of semigroups of operators, where a sequence of strongly continuous semigroups approaches a semigroup that is strongly continuous only on a subspace of the original Banach space. Biologically, our approach allows a new, natural interpretation of the ODE model’s parameters.

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Mathematics Subject Classification: Primary: 47D07; Secondary: 46N60, 60J70, 92B05.

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