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Sensitivity analysis of the Poisson Nernst-Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model. (English) Zbl 1410.92027
Summary: Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst-Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.
MSC:
92C20 Neural biology
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Software:
ADIC; NEURON
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[1] Adams RA, Fournier JJF (2003) Sobolev spaces, vol 140. Pure and applied mathematics. Academic, New York · Zbl 1098.46001
[2] Appel JR (1997) Sensitivity calculations for conservation laws with application to discontinuous fluid flows. Ph.D. thesis, Virginia Tech., Blacksburg
[3] Ask, M.; Reza, M., Computational models in neuroscience: How real are they? A critical review of status and suggestions, Austin Neurol Neurosci, 1, 1008, (2016)
[4] Banks, HT; Bihari, KL, Modelling and estimating uncertainty in parameter estimation, Inverse Probl, 17, 95-112, (2001) · Zbl 1054.35121
[5] Bathe K (1996) Finite element procedures. Prentice-Hall, Upper Saddle River · Zbl 1326.65002
[6] Belhamadia, Y.; Fortin, A.; Bourgault, Y., On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction-diffusion systems, J Comput Appl Math, 271, 233-246, (2014) · Zbl 1329.65219
[7] Biler, P.; Dolbeault, J., Long time behavior of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems, Ann Henri Poincaré, 1, 461-472, (2000) · Zbl 0976.82046
[8] Biler, P.; Hebisch, W.; Nadzieja, T., The Debye system: existence and large time behavior of solutions, Nonlinear Anal Theory Methods Appl, 23, 1189-1209, (1994) · Zbl 0814.35054
[9] Bischof, CH; Roh, L.; Mauer-Oats, AJ, Adic: an extensible automatic differentiation tool for ANSI-C, Softw Pract Exp, 27, 1427-1456, (1997)
[10] Bolintineanu, DS; Sayyed-Ahmad, A.; Davis, HT; Kaznessis, YN, Poisson-Nernst-Planck models of nonequilibrium ion electrodiffusion through a protegrin transmembrane pore, PLoS Comput Biol, 5, 1-12, (2009)
[11] Borggaard J, Bums J, Cliff E, Schreck S (1997) Computational methods for optimal design and control. In: Proceeding of the AFOSR workshop on optimal design and control, Arlington, Virginia
[12] Brette, R., What is the most realistic single-compartment model of spike initiation?, PLoS Comput Biol, (2015)
[13] Carnevale NT, Hines ML (2004) The NEURON book. Cambridge University Press, Cambridge
[14] Cartailler, J.; Schuss, Z.; Holcman, D., Electrostatics of non-neutral biological microdomains, Sci Rep, 7, 11269, (2017)
[15] Ciarlet P, Luneville E (2009) La méthode des éléments finis: de la théorie à la pratique. Concepts généraux. I. Cours (ENSTA), Les Presses de l’ENSTA
[16] Dione, I.; Deteix, J.; Briffard, T.; Chamberland, E.; Doyon, N., Improved simulation of electrodiffusion in the node of Ranvier by mesh adaptation, PLoS One, (2016)
[17] Eberhard, P.; Bischof, C., Automatic differentiation of numerical integration algorithms, Math Comput Am Math Soc, 68, 717-731, (1999) · Zbl 1017.65062
[18] Einstein, A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann Phys, 322, 549560, (1905) · JFM 36.0975.01
[19] Glykys, J.; Egawa, VDK; Balena, T.; Saponjian, Y.; Kuchibhotla, K.; Bacskai, B.; Kahle, K.; T, TZ; Staley, K., Local impermeant anions establish the neuronal chloride concentration, Science, 343, 670-675, (2014)
[20] Gow, A.; Devaux, J., Model of tight junction function in CNS myelinated axons, Neuron Glia Biol, 4, 307-317, (2008)
[21] Gramse, G.; Dols-Perez, A.; Edwards, MA; Fumagalli, L.; Gomila, G., Nanoscale measurement of the dielectric constant of supported lipid bilayers in aqueous solutions with electrostatic force microscopy, Biophys J, 104, 1257-1262, (2013)
[22] Griewank A, Walther A (2008) Evaluating derivatives, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1159.65026
[23] Haines, J.; Inglese, M.; Casaccia, P., Axonal damage in multiple sclerosis, Mt Sinai J Med, 78, 231-243, (2011)
[24] Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary differential equations I: nonstiff problems, 2, Revised edn. Springer, New York · Zbl 0789.65048
[25] Hille B (1992) Ionic channels of excitable membranes. Sinauer Associates, Sunderland
[26] Hobbie RK, Roth BJ (2007) Intermediate physics for medicine and biology. Biological and medical physics, biomedical engineering. Springer, New York · Zbl 1320.92004
[27] Hodgkin, AL; Huxley, AF, A quantitative description of membrane current and its application to conduction and excitation in nerve, J Physiol, 117, 500-544, (1952)
[28] Holcman, D.; Yuste, R., The new nanophysiology: regulation of ionic flow in neuronal subcompartments, Nat Rev Neurosci, 16, 685-692, (2015)
[29] Laing, CR, Numerical bifurcation theory for high-dimensional neural models, J Math Neurosci, 4, 13, (2014) · Zbl 1333.92018
[30] Li, S.; Petzold, L., Adjoint sensitivity analysis for time-dependent partial differential equations with adaptive mesh refinement, J Comput Phys, 198, 310-325, (2004) · Zbl 1052.65089
[31] Lopreore, CL; Bartol, TM; Coggan, JS; Keller, DX; Sosinsky, GE; Ellisman, MH; Sejnowski, TJ, Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier, Biophys J, 95, 2624-2635, (2008)
[32] Lu, B.; Holst, MJ; McCammon, JA; Zhou, Y., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: finite element solutions, J Comput Phys, 229, 6979-6994, (2010) · Zbl 1195.92004
[33] Malmberg, CG; Maryott, AA, Dielectric constant of water from 0 to 100 C, J Res Natl Bur Stand, 56, 1-8, (1956)
[34] Nymeyer, H.; Zhou, HX, A method to determine dielectric constants in nonhomogeneous systems: application to biological membranes, Biophys J, 94, 1185-1193, (2008)
[35] Pods J (2014) Electrodiffusion Models of axon and extracellular space using the Poisson-Nernst-Planck equations. Ph.D. thesis, Heidelberg University Library · Zbl 1304.92027
[36] Pods, J.; Schonke, J.; Bastian, P., Electrodiffusion models of neurons and extracellular space using the Poisson-Nernst-Planck equations—numerical simulation of the intra- and extracellular potential for an axon model, Biophys J, 105, 242-254, (2013)
[37] Qian, N.; Sejnowski, T., An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons, Biol Cybern, 62, 1-15, (1989) · Zbl 0683.92004
[38] Quarteroni AM, Valli A (2008) Numerical approximation of partial differential equations. Springer, Berlin · Zbl 1151.65339
[39] Schutter ED (2000) Computational neuroscience: realistic modeling for experimentalists. CRC Press, Boca Raton
[40] Sun NZ, Sun A (2015) Model calibration and parameter estimation, for environmental and water resource systems. Springer, Berlin · Zbl 1319.00002
[41] Sun, Y.; Sun, P.; Zheng, B.; Lin, G., Error analysis of finite element method for Poisson-Nernst-Planck equations, J Comput Appl Math, 301, 28-43, (2016) · Zbl 1382.65326
[42] Sylantyev, S.; Savtchenko, L.; Ermolyuk, Y.; Michaluk, P.; Rusakov, D., Spike-driven glutamate electrodiffusion triggers synaptic potentiation via a homer-dependent mGLUR-NMDAR link, Neuron, 77, 528-541, (2013)
[43] Tora, A.; Rovira, X.; Dione, I.; Bertrand, H.; Brabet, I.; Koninck, YD; Doyon, N.; Pin, J.; Acher, F.; Goudet, C., Allosteric modulation of metabotropic glutamate receptors by chloride ions, FASEB J, 29, 4174-4188, (2015)
[44] Troparevsky, MI; Rubio, D.; Saintier, N., Sensitivity analysis for the EEG forward problem, Front Comput Neurosci, 4, 138, (2010)
[45] Yuste, R., Electrical compartmentalization in dendritic spines, Annu Rev Neurosci, 36, 429-449, (2013)
[46] Zhao, J.; Cui, S., Remarks on the local existence of solutions to the Debye system, J Math Anal Appl, 383, 337-343, (2011) · Zbl 1222.35004
[47] Zheng, K.; Jensen, TP; Savtchenko, LP; Levitt, JA; Suhling, K.; Rusakov, DA, Nanoscale diffusion in the synaptic cleft and beyond measured with time-resolved fluorescence anisotropy imaging, Sci Rep, 7, 42022, (2017)
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