# American Institute of Mathematical Sciences

October  2012, 17(7): 2313-2327. doi: 10.3934/dcdsb.2012.17.2313

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

 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, Poland

Received  January 2012 Revised  February 2012 Published  July 2012

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.
Citation: Adam Bobrowski, Katarzyna Morawska. From a PDE model to an ODE model of dynamics of synaptic depression. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2313-2327. doi: 10.3934/dcdsb.2012.17.2313
##### References:
 [1] F. Aristizabal and M. I. Glavinovič, Simulation and parameter estimation of dynamics of synaptic depression, Biol. Cybern., 90 (2004), 3-18. doi: 10.1007/s00422-003-0432-8. [2] A. Bielecki and P. Kalita, Model of neurotransmitter fast transport in axon terminal of presynaptic neuron, J. Math. Biol., 56 (2008), 559-576. doi: 10.1007/s00285-007-0131-5. [3] A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327. doi: 10.1007/BF02573493. [4] A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. [5] A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [6] A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in 77 (2008), 520-521 (MR2457335). doi: 10.1007/s00233-006-0633-2. [7] A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4. [8] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications," With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. doi: 10.1002/zamm.19920720316. [9] T. Eisner, "Stability of Operators and Operator Semigroups," Operator Theory: Advances and Applications, 209, Birkahäuser Verlag, Basel, 2010. [10] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [11] K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Universitext, Springer, New York, 2006. [12] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [13] W. J. Ewens, "Mathematical Population Genetics," Biomathematics, 9, Springer-Verlag, Berlin-New York, 1979, Second edition, 2004. [14] W. Feller, Diffusion processes in genetics, in "Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950," University of California Press, Berkeley and Los Angeles, 1951. [15] W. Feller, Two singular diffusion problems, Ann. of Math. (2), 54 (1951), 173-182. doi: 10.2307/1969318. [16] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468-519. doi: 10.2307/1969644. [17] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31. doi: 10.1090/S0002-9947-1954-0063607-6. [18] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. of Mathematics, 13 (1987), 213-229. [19] K. Itô and H. P. McKean, Jr., "Diffusion Processes and their Sample Paths," reprint of the 1974 edition, Classics in Mathematics, Springer, 1996. doi: 10.1214/aoms/1177699390. [20] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991. doi: 10.1002/zamm.19890691124. [21] P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, Springer-Verlag New York, Inc., New York, 1968. [22] E. Neher and R. S. Zucker, Multiple calcium-dependant process related to secretion in bovine chromaffin cells, Neuron, 10 (1993), 2-30. doi: 10.1016/0896-6273(93)90238-M. [23] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3$^rd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999. doi: 10.1214/aop/1176989417. [24] K. Taira, "Semigroups, Boundary Value Problems and Markov Processes," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. [25] A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, (in Russian), Teoriya Veroyat. i Primen., 4 (1959), 172-185, English translation: Theory Prob. and its Appl., 4 (1959), 164-177.

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##### References:
 [1] F. Aristizabal and M. I. Glavinovič, Simulation and parameter estimation of dynamics of synaptic depression, Biol. Cybern., 90 (2004), 3-18. doi: 10.1007/s00422-003-0432-8. [2] A. Bielecki and P. Kalita, Model of neurotransmitter fast transport in axon terminal of presynaptic neuron, J. Math. Biol., 56 (2008), 559-576. doi: 10.1007/s00285-007-0131-5. [3] A. Bobrowski, Degenerate convergence of semigroups, Semigroup Forum, 49 (1994), 303-327. doi: 10.1007/BF02573493. [4] A. Bobrowski, A note on convergence of semigroups, Ann. Polon. Math., 69 (1998), 107-127. [5] A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction," Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511614583. [6] A. Bobrowski, Degenerate convergence of semigroups related to a model of stochastic gene expression, Semigroup Forum, 73 (2006), 345-366, with correction in 77 (2008), 520-521 (MR2457335). doi: 10.1007/s00233-006-0633-2. [7] A. Bobrowski, On limitations and insufficiency of the Trotter-Kato theorem, Semigroup Forum, 75 (2007), 317-336. doi: 10.1007/s00233-006-0676-4. [8] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3. Spectral Theory and Applications," With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. doi: 10.1002/zamm.19920720316. [9] T. Eisner, "Stability of Operators and Operator Semigroups," Operator Theory: Advances and Applications, 209, Birkahäuser Verlag, Basel, 2010. [10] K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. [11] K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Universitext, Springer, New York, 2006. [12] S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. [13] W. J. Ewens, "Mathematical Population Genetics," Biomathematics, 9, Springer-Verlag, Berlin-New York, 1979, Second edition, 2004. [14] W. Feller, Diffusion processes in genetics, in "Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950," University of California Press, Berkeley and Los Angeles, 1951. [15] W. Feller, Two singular diffusion problems, Ann. of Math. (2), 54 (1951), 173-182. doi: 10.2307/1969318. [16] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468-519. doi: 10.2307/1969644. [17] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31. doi: 10.1090/S0002-9947-1954-0063607-6. [18] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. of Mathematics, 13 (1987), 213-229. [19] K. Itô and H. P. McKean, Jr., "Diffusion Processes and their Sample Paths," reprint of the 1974 edition, Classics in Mathematics, Springer, 1996. doi: 10.1214/aoms/1177699390. [20] I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991. doi: 10.1002/zamm.19890691124. [21] P. Mandl, "Analytical Treatment of One-Dimensional Markov Processes," Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia Publishing House of the Czechoslovak Academy of Sciences, Prague, Springer-Verlag New York, Inc., New York, 1968. [22] E. Neher and R. S. Zucker, Multiple calcium-dependant process related to secretion in bovine chromaffin cells, Neuron, 10 (1993), 2-30. doi: 10.1016/0896-6273(93)90238-M. [23] D. Revuz and M. Yor, "Continuous Martingales and Brownian Motion," 3$^rd$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999. doi: 10.1214/aop/1176989417. [24] K. Taira, "Semigroups, Boundary Value Problems and Markov Processes," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. [25] A. D. Ventcel', On boundary conditions for multi-dimensional diffusion processes, (in Russian), Teoriya Veroyat. i Primen., 4 (1959), 172-185, English translation: Theory Prob. and its Appl., 4 (1959), 164-177.
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