Abstract
Overparametrization of models in natural sciences, including neuroscience, is a problem that is widely recognized but often not addressed in experimental studies. The systematic reduction of complex models to simpler ones for which the parameters may be reliably estimated is based on asymptotic model reduction procedures taking into account the presence of vastly different time scales in the natural phenomena being studied. The steps of the reduction process, which are reviewed here, include basic model formulation (e.g., using the law of mass action applied routinely for problems in neuroscience, biological and chemical kinetics, and other fields), model non-dimensionalization using characteristic scales (of times, species concentrations, etc.), application of an asymptotic algorithm to produce a reduced model, and analysis of the reduced model (including suggestions for experimental design and fitting the reduced model to experimental data). In addition to the review of some classical results and basic examples, we illustrate how the approach can be used in a more complex realistic case to produce several reduced kinetic models for N-methyl-D-aspartate receptors, a subtype of glutamate receptor expressed on neurons in the brain, with models applied to different experimental protocols. Simultaneous application of the reduced models to fitting the data obtained in a series of specially designed experiments allows for a stepwise estimation of parameters of the original conventional model which is otherwise overparameterized with respect to the existing data.
This is a preview of subscription content, log in via an institution.
References
Benveniste M, Clements J, Vyklicý L, Mayer ML (1990) A kinetic analysis of the modulation of N–methyl–D–aspartic acid receptors by glycine in mouse cultured hippocampal neurones. J Physiol 428:333–357
Clements JD, Westbrook GL (1991) Activation kinetics reveal the number of glutamate and glycine binding sites on the N-methyl-D-aspartate receptor. Neuron 7(4):605–613
Collingridge GL, Kehl SJ, McLennan H (1983) Excitatory amino acids in synaptic transmission in the Schaffer collateral-commissural pathway of the rat hippocampus. J Physiol 334:33–46
FitzHugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 17(4):257–278
Henneberger C, Papouin T, Oliet SH, Rusakov DA (2010) Long-term potentiation depends on release of D-serine from astrocytes. Nature 463(7278):232–236
Henri V (1903) Lois générales de l’Action des diastases. Hermann, Paris
Hodgkin A, Huxley A (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544
Iacobucci GJ, Popescu GK (2018) Kinetic models for activation and modulation of NMDA receptor subtypes. Curr Opin Physiol 2:114–122
Kalachev L (2006) Reduced model of neurotransmitter transport in the presence of generic receptors and transporters. J Phys Conf Ser 55:114–129
Krupp JJ, Vissel B, Heinemann SF, Westbrook GL (1998) N–terminal domains in the NR2 subunit control desensitization of NMDA receptors. Neuron 20:317–327
Le Bail M, Martineau M, Sacchi S, Yatsenko N, Radzishevsky I, Conrod S, Ait Ouares K, Wolosker H, Pollegioni L, Billard JM, Mothet JP (2014) Identity of the NMDA receptor coagonist is synapse specific and developmentally regulated in the hippocampus. Proc Natl Acad Sci USA 112(2):204–313
Lester RA, Jahr CE (1992) NMDA channel behavior depends on agonist affinity. J Neurosci 12(2):635–643
Lester RA, Tong G, Jahr CE (1993) Interactions between the glycine and glutamate binding sites of the NMDA receptor. J Neurosci 13(3):1088–1096
Mayer ML, Vyklicky L Jr, Clements J (1989) Regulation of NMDA receptor desensitization in mouse hippocampal neurons by glycine. Nature 338(6214):425–427
Michaelis L, Menten M (1913) Die Kinetik der Invertinwirkung. Biochem Zeitsch 49:333–369
Mothet JP, Le Bail M, Billard JM (2015) Time and space profiling of NMDA receptor coagonist functions. J Neurochem 135:210–225
Murray J (1993) Mathematical biology. Springer, Berlin/Heidelberg
Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070
Nahum-Levy R, Lipinski D, Shavit S, Benveniste M (2001) Desensitization of NMDA receptor channels is modulated by glutamate agonists. Biophys J 80:2152–2166
Nicoll RA (2017) A brief history of long-term potentiation. Neuron 93(2):281–290
Radzishevsky I, Sason H, Wolosker H (2013) D-serine: physiology and pathology. Curr Opin Clin Nutr Metab Care 16(1):72–75
Rusakov D, Kullmann D (1998) Geometric and viscous components of the tortuosity of the extracellular space in the brain. Proc Natl Acad Sci USA 95(15):8975–8980
Sather W, Johnson JW, Henderson G, Ascher P (1990) Glycine–insensitive desensitization of NMDA responses in cultured mouse embryonic neurons. Neuron 4:725–731
Savtchenko L, Rusakov D (2007) The optimal height of the synaptic cleft. Proc Natl Acad Sci USA 104(6):1823–1828
Segel L, Slemrod M (1989) The quasi-steady state assumption: a case study in perturbation. SIAM Rev 31:446–477
Schorge S, Elenes S, Colquhoun D (2005) Maximum likelihood fitting of single channel NMDA activity with a mechanism composed of independent dimers of subunits. J Physiol 569(Pt 2):395–418
Stiefenhofer M (1998) Quasi-steady-state approximation for chemical reaction networks. J Math Biol 36:593–609
Tikhonov A (1948) On the dependence of the solutions of differential equations on a small parameter. Mat Sb (NS) 22(64)2:193–204
Tikhonov A (1950) On systems of differential equations containing parameters. Mat Sb (NS) 27(69)1:147–156
Tong G, Jahr CE (1994) Regulation of glycine–insensitive desensitization of the NMDA receptor in outside–out patches. J Neurophysiol 72(2):754–761
Traynelis SF, Wollmuth LP, McBain CJ, Menniti FS, Vance KM, Ogden KK, Hansen KB, Yuan H, Myers SJ, Dingledine R (2010) Glutamate receptor ion channels: structure, regulation, and function. Pharmacol Rev 62(3):405–496
Vasil’eva A, Butuzov V, Kalachev L (1995) The boundary function method for singular perturbation problems. SIAM studies in applied mathematics. SIAM, Philadelphia
Voit E, Martens H, Omholt S (2015) 150 years of the mass action law. PLoS Comp Biol 11(1):e1004012
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this entry
Cite this entry
Shchepakin, D., Kalachev, L., Kavanaugh, M. (2020). Mathematical Models in Neuroscience: Approaches to Experimental Design and Reliable Parameter Determination. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_134-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-70658-0_134-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70658-0
Online ISBN: 978-3-319-70658-0
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering