Structural analysis of fluid flow in complex biological systems

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Eisenberg R. Structural analysis of fluid flow in complex biological systems. MAIO [Internet]. 2023 Feb. 7 [cited 2024 May 22];4(1). Available from:

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Copyright (c) 2022 Robert Eisenberg


bidomain model; conservation laws; fluid flow; glymphatic system


Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know the function of the structures. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. The hierarchy of structures allows a handful of atoms (in proteins and nucleic acids) to control macroscopic function. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.


Eisenberg B. Living Transistors: a Physicist’s View of Ion Channels (version 2). 2005. http://arxivorg/qbio/0506016v2

Eisenberg B. Ions in Fluctuating Channels: Transistors Alive. Fluctuation and Noise Letters. 2012;11(01):1240001 available on with Paper ID arXiv:q-bio/0506016v1240003.

Eisenberg B. Engineering channels: Atomic biology. Proceedings of the National Academy of Sciences. 2008;105(17):6211-6212.

Eisenberg RS. Look at biological systems through an engineer’s eyes. Nature. 2007;447(7143):376.

Miedema H, Vrouenraets M, Wierenga J, Meijberg W, Robillard G, Eisenberg B. A Biological Porin Engineered into a Molecular, Nanofluidic Diode. Nano Lett. 2007;7(9):2886-2891.

Shur M. Physics of Semiconductor Devices. New York: Prentice Hall; 1990. 680 p.

Pierret RF. Semiconductor Device Fundamentals. New York: Addison Wesley; 1996.

Sze SM. Semiconductor devices: physics and technology: John Wiley & Sons; 2008.

Laux SE, Hess K. Revisiting the analytic theory of pn junction impedance: Improvements guided by computer simulation leading to a new equivalent circuit. IEEE Transactions on Electron Devices. 1999;46(2):396-412.

Cressler JD. Silicon Heterostructure Handbook: Materials, Fabrication, Devices, Circuits and Applications of SiGe and Si Strained-Layer Epitaxy. Boca Raton, FL: CRC 2005. 1248 p.

Muller RS, Chan M, Kamins TI. Device Electronics For Integrated Circuits, 3rd Ed: Wiley India Pvt. Limited; 2003.

Eisenberg B. Living Devices: The Physiological Point of View. 2012. Available on arXiv as http://arxivorg/abs/12066490

Engl HW, Hanke M, Neubauer A. Regularization of Inverse Problems Dordrecht, The Netherlands: Klouwer; 2000.

Kaipio J, Somersalo E. Statistical and Computational Inverse Problems New York: Springer; 2005. 344 p.

Burger M. Inverse problems in ion channel modelling. Inverse Problems. 2011;27(8):083001.

Burger M, Eisenberg RS, Engl H. Inverse Problems Related to Ion Channel Selectivity. SIAM J Applied Math. 2007;67(4):960-989

Keller JB. Inverse Problems. The American Mathematical Monthly. 1976;83(2):107-118.

Forster J. Mathematical Modeling of Complex Fluids [Master’s]. Wurzburg, Germany: University of Wurzburg; 2013.

Giga M-H, Kirshtein A, Liu C. Variational Modeling and Complex Fluids. In: Giga Y, Novotny A, editors. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Cham: Springer International Publishing; 2017. p. 1-41.

Xu S, Sheng P, Liu C. An energetic variational approach for ion transport. arXiv preprint arXiv:14084114.2014.

Hyon Y, Eisenberg B, Liu C. An energetic variational approach to ion channel dynamics. Mathematical Methods in the Applied Sciences. 2014;37(7):952-961.

Hyon Y, Fonseca JE, Eisenberg B, Liu C. Energy variational approach to study charge inversion (layering) near charged walls. Discrete and Continuous Dynamical Systems Series B (DCDS-B). 2012;17(8):2725 - 2743.

Hyon Y, Kwak DY, Liu C. Energetic Variational Approach in Complex Fluids : Maximum Dissipation Principle. Discrete and Continuous Dynamical Systems (DCDS-A). 2010;26(4: April):1291-1304.

Ryham R, Liu C, Wang ZQ. On electro-kinetic fluids: One dimensional configurations. Discrete and Continuous Dynamical Systems-Series B. 2006;6(2):357-371.

Ryham RJ. An Energetic Variational Approach to Mathematical Moldeling of Charged Fluids, Charge Phases, Simulation and Well Posedness, Ph.D. Thesis [Ph.D.]. State College: The Pennsylvania State University; 2006.

Eisenberg B, Hyon Y, Liu C. Energy Variational Analysis EnVarA of Ions in Water and Channels: Field Theory for Primitive Models of Complex Ionic Fluids. J Chem Phys. 2010;133(10):104104.

Eisenberg B. Life’s Solutions are Complex Fluids. A Mathematical Challenge. 2012. arXiv preprint arXiv:12074737.

Eisenberg RS. Complexities in solution. Trends Biochem Sci. 1990;15(2):51-52.

Ionic Selectivity in Channels: complex biology created by the balance of simple physics [Internet]. Nanohub Purdue University. 2008 [cited Jun]. Available from:

Rice SA, Gray P. The Statistical Mechanics of Simple Fluids. New York: Interscience (Wiley); 1965. 582 p.

Barratt J-L, Hansen J-P. Basic concepts for simple and complex liquids: Cambridge University Press; 2003. 296 p.

Hansen J-P, McDonald IR. Theory of Simple Liquids. First Edition ed. New York: Academic Press; 1986. 395 p.

Hansen J-P, McDonald IR. Theory of Simple Liquids. Third Edition ed. New York: Academic Press; 2006. 428 p.

Hansen JP, Bellissent-Funel MC, Neilson GW. The Physics and Chemistry of Aqueous Ionic Solutions; 1987. 1 p.

Kontogeorgis GM, Folas GK. Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories: John Wiley & Sons; 2009. 721 p.

Kunz W. Specific Ion Effects. Singapore: World Scientific 2009. 348 p.

Kunz W, Neueder R. An Attempt at an Overview. In: Kunz W, editor. Specific Ion Effects. Singapore: World Scientific 2009. p. 11-54.

Liu JL, Eisenberg B. Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Bikerman Model. Entropy 2020;22:550 Preprint available at

Li C-L, Liu J-L. Generalized Debye--Hückel Equation From Poisson--Bikerman Theory. SIAM Journal on Applied Mathematics. 2020;80(5):2003-2023.

Liu J-L, Li C-L. A generalized Debye-Hückel theory of electrolyte solutions. AIP Advances. 2019;9(1):015214.

Vera JH, Wilczek-Vera G. Classical Thermodynamics of Fluid Systems: Principles and Applications: CRC Press; 2016.

Haase W. Crystals that flow. Classic papers from the history of liquid crystals. Compiled with translation and commentary by Tim J. Sluckin, David A. Dunmur and Horst Stegemeyer. Liquid Crystals Book Series, edited by GW Gray, JW Goodby and A. Fukuda. Pp. xxiii+ 738. London: Taylor & Francis, 2004. Price (hardback) GBP 99. ISBN 0-415-25789-1. Journal of Applied Crystallography. 2005;38(3):579.

Gennes P-Gd, Prost J. The Physics of Liquid Crystals. New York: Oxford University Press; 1993. 616 p.

Ericksen J. Conservation laws for liquid crystals. Trans Soc Rheol. 1961;5:22-34.

Lin P, Liu C, Zhang H. An Energy Law Preserving $C^0$ Finite Element Scheme for Simulating the Kinematic Effects in Liquid Crystal Flow Dynamics. J Comput Phys,. 2007;227:1411-1427.

Brannick J, Kirshtein A, Liu C. Dynamics of multi-component flows: diffusive interface methods with energetic variational approaches. In: Hashmi S, Buggy M, editors. Reference Module in Materials Science and Materials Engineering. Reference Module in Materials Science and Materials Engineering: Elsevier; 2016. p. 1-7.

Mestre H, Mori Y, Nedergaard M. The Brain’s Glymphatic System: Current Controversies. Trends in Neurosciences.

Nagelhus EA, Ottersen OP. Physiological Roles of Aquaporin-4 in Brain. Physiological Reviews. 2013;93(4):1543-1562.

Xie L, Kang H, Xu Q, et al. Sleep Drives Metabolite Clearance from the Adult Brain. Science. 2013;342(6156):373-377.

Jessen NA, Munk ASF, Lundgaard I, Nedergaard M. The glymphatic system: a beginner’s guide. Neurochemical research. 2015;40(12):2583-2599.

Jiang Q, Zhang L, Ding G, et al. Impairment of the glymphatic system after diabetes. Journal of Cerebral Blood Flow & Metabolism. 2017;37(4):1326-1337.

Abbott NJ, Pizzo ME, Preston JE, Janigro D, Thorne RG. The role of brain barriers in fluid movement in the CNS: is there a ‘glymphatic’system? Acta Neuropathol. 2018;135(3):387-407.

Gakuba C, Gaberel T, Goursaud S, et al. General anesthesia inhibits the activity of the “glymphatic system”. Theranostics. 2018;8(3):710.

Fultz NE, Bonmassar G, Setsompop K, et al. Coupled electrophysiological, hemodynamic, and cerebrospinal fluid oscillations in human sleep. Science. 2019;366(6465):628-631.

Mestre H, Du T, Sweeney AM, et al. Cerebrospinal fluid influx drives acute ischemic tissue swelling. Science. 2020;367(6483):eaax7171.

Nedergaard M, Goldman SA. Glymphatic failure as a final common pathway to dementia. Science. 2020;370(6512):50.

Vilcaes AA, Chanaday NL, Kavalali ET. Interneuronal exchange and functional integration of synaptobrevin via extracellular vesicles. Neuron. 2021.

Falk G, Fatt P. Linear Electrical Properties of Striated Muscle Fibres Observed with Intracellular Electrodes. Proc R Soc Lond B Biol Sci. 1964;160:69-123.

Eisenberg RS. The equivalent circuit of single crab muscle fibers as determined by impedance measurements with intracellular electrodes. J Gen Physiol. 1967;50(6):1785-1806.

Sun HH. Synthesis of RC Networks: Hayden Book Publishers; 1967. 150 p.

Tuttle DF. Network synthesis. Wiley; 1958.

VanValkenburg ME. Introduction to Modern Network Synthesis: CBLS Publishers; 1991.

Weinberg L. Network analysis and synthesis: Krieger Pub. Co.; 1975.

Balabanian N, Bickart TA. Electrical network theory: Wiley; 1969.

Ghausi MS, Kelly JJ. Introduction to distributed-parameter networks: with application to integrated circuits: Holt, Rinehart and Winston; 1968.

Guillemin EA. Communications Networks Vol. 1 The Classical Theory of Lumped Constant Networks: John Wiley; 1931.

LePage WR, Seely S. General Network Analysis: McGraw-Hill; 1952.

Zemanian AH. Infinite electrical networks: Cambridge University Press; 1991.

Bush V, Wiener N. Operational Circuit Analysis: With an Appendix by Norbert Wiener: Chapman & Hall; 1929.

Guillemin EA. Introductory Circuit Theory: Wiley; 1958.

McAllister RE, Noble D, Tsien R. Reconstruction of the electrical activity of cardiac Purkinje fibres. The Journal of physiology. 1975;251(1):1-59.

Fozzard HA, Beeler Jr GW. The voltage clamp and cardiac electrophysiology. Circulation Research. 1975;37(4):403-413.

Bartos DC, Grandi E, Ripplinger CM. Ion Channels in the Heart. Comprehensive Physiology. 2015:1423-1464.

Priest BT, McDermott JS. Cardiac ion channels. Channels. 2015;9(6):352-359.

Schneider MF. Linear electrical properties of the transverse tubules and surface membrane of skeletal muscle fibers. J Gen Physiol. 1970;56(5):640-671.

Valdiosera R, Clausen C, Eisenberg RS. Circuit models of the passive electrical properties of frog skeletal muscle fibers. J Gen Physiol. 1974; 63:432-459.

Valdiosera R, Clausen C, Eisenberg RS. Measurement of the impedance of frog skeletal muscle fibers. Biophys J. 1974;14:295-314.

Valdiosera R, Clausen C, Eisenberg RS. Impedance of frog skeletal muscle fibers in various solutions. J Gen Physiol. 1974;63(4):460-491.

Mathias RT, Eisenberg RS, Valdiosera R. Electrical properties of frog skeletal muscle fibers interpreted with a mesh model of the tubular system. Biophys J. 1977;17(1):57-93.

Eisenberg Robert S. Impedance Measurement of the Electrical Structure of Skeletal Muscle. Comprehensive Physiology, Republished by the American Physiological Society, as Volume 1, Supplement 27 of Handbook of Physiology, part of Comprehensive Physiology, SSN: 20404603 Online ISBN: 9780470650714. 1, Supplement 27, Handbook of Physiology: American Physiological Society and Wiley OnLine; 2011.

Eisenberg B. Electrical Structure of Biological Cells and Tissues: impedance spectroscopy, stereology, and singular perturbation theory. In: Barsoukov E, Macdonald JR, editors. Impedance Spectroscopy: Theory, Experiment, and Applications Third Edition. Third ed. New York: Wiley-Interscience; 2018. p. 472-478 Available on arXiv at

Milton RL, Mathias RT, Eisenberg RS. Electrical properties of the myotendon region of frog twitch muscle fibers measured in the frequency domain. Biophys J. 1985;48(2):253-267.

Eisenberg BR, Milton RL. Muscle fiber termination at the tendon in the frog’s sartorius: a stereological study. Am J Anat. 1984;171(3):273-284.

Eisenberg RS, Mathias RT, Rae JS. Measurement, modeling, and analysis of the linear electrical properties of cells. Ann N Y Acad Sci. 1977;303:342-354.

Mathias RT, Rae JL, Eisenberg RS. Electrical properties of structural components of the crystalline lens. Biophys J. 1979;25(1):181-201.

Rae JL. The electrophysiology of the crystalline lens. Curr Top Eye Res. 1979;1:37-90.

Rae JL, Stacey T. Lanthanum and procion yellow as extracellular markers in the crystalline lens of the rat. Exp Eye Res. 1979;28(1):1-21.

Mathias RT, Rae JL, Eisenberg RS. The lens as a nonuniform spherical syncytium. Biophys J. 1981;34(1):61-83.

Kuszak JR, Rae JL. Scanning electron microscopy of the frog lens. Exp Eye Res. 1982;35(5):499-519.

Rae JL, Mathias RT, Eisenberg RS. Physiological role of the membranes and extracellular space with the ocular lens. Exp Eye Res. 1982;35(5):471-489.

Rae JL, Thomson RD, Eisenberg RS. The effect of 2-4 dinitrophenol on cell to cell communication in the frog lens. Exp Eye Res. 1982;35(6):597-609.

Rae JL, Kuszak JR. The electrical coupling of epithelium and fibers in the frog lens. Exp Eye Res. 1983;36(3):317-326.

Rae JL, Truitt KD, Kuszak JR. The use of procion dyes for light microscopy of the frog lens. Invest Ophthalmol Vis Sci. 1983;24(9):1167-1171.

Mathias RT, Rae JL. Steady state voltages in the frog lens. Curr Eye Res. 1985;4(4):421-430.

Mathias RT, Rae JL, Ebihara L, McCarthy RT. The localization of transport properties in the frog lens. Biophys J. 1985;48(3):423-434.

Baldo GJ, Mathias RT. Spatial variations in membrane properties in the intact rat lens. Biophys J. 1992;63(2):518-529.

Donaldson PJ, Roos M, Evans C, Beyer E, Kistler J. Electrical properties of mammalian lens epithelial gap junction channels. Invest Ophthalmol Vis Sci. 1994;35(9):3422-3428.

Donaldson PJ, Dong Y, Roos M, Green C, Goodenough DA, Kistler J. Changes in lens connexin expression lead to increased gap junctional voltage dependence and conductance. Am J Physiol. 1995;269(3 Pt 1):C590-600.

Rae JL, Bartling C, Rae J, Mathias RT. Dye transfer between cells of the lens. J Membr Biol. 1996;150(1):89-103.

Nicholson C, Hrabětová S. Brain Extracellular Space: The Final Frontier. Biophysical Journal. 2017;113(10):2133-2142.

Sykova E, Nicholson C. Diffusion in brain extracellular space. Physiol Rev. 2008;88(4):1277-1340.

Eisenberg B, Eisenberg RS. Selective disruption of the sarcotubular system in frog sartorius muscle. A quantitative study with exogenous peroxidase as a marker. J Cell Biol. 1968;39(2):451-467.

Eisenberg BR, Mathias RT, Gilai A. Intracellular localization of markers within injected or cut frog muscle fibers. Am J Physiol. 1979;237(1):C50-55.

Baddeley A, Jensen EBV. Stereology for statisticians: Chapman and Hall/CRC; 2004.

Howard V, Reed M. Unbiased Stereology: Three-Dimensional Measurement in Microscopy: CRC Press; 2004.

Mouton PR. Principles and Practices of Unbiased Stereology: An Introduction for Bioscientists: Johns Hopkins University Press; 2002.

Eisenberg BR. Skeletal muscle fibers: stereology applied to anisotropic and periodic structures. In: Weibel E, editor. Stereological Methods: Practical methods for biological morphometry. 11979. p. 274-284.

Eisenberg BR, Kuda AM, Peter JB. Stereological analysis of mammalian skeletal muscle: I. Soleus muscle of the adult guinea pig. The Journal of cell biology. 1974;60(3):732-754.

Eisenberg BR, Kuda AM. Stereological analysis of mammalian skeletal muscle: II. White vastus muscle of the adult guinea pig. Journal of ultrastructure research. 1975;51(2):176-187.

Eisenberg BR, Mobley BA. Size changes in single muscle fibers during fixation and embedding. Tissue and Cell. 1975;7:383-387.

Mobley BA, Eisenberg BR. Sizes of components in frog skeletal muscle measured by methods of stereology. The Journal of general physiology. 1975;66(1):31-45.

Eisenberg BR, Kuda AM. Discrimination between fiber populations in mammalian skeletal muscle by using ultrastructural parameters. J Ultrastruct Res. 1976;54(1):76-88.

Eisenberg BR, Salmons S. The reorganization of subcellular structure in muscle undergoing fast-toslow type transformation. Cell and tissue research. 1981;220(3):449-471.

Eisenberg BR. Quantitative ultrastructure of mammalian skeletal muscle. In: Peachey LD, Adrian RH, editors. Handbook of Physiology. 10: American Physiological Society Bethesda MD; 1983. p. 73-112.

Eisenberg BR, Brown J, Salmons S. Restoration of fast muscle characteristics following cessation of chronic stimulation. Cell and tissue research. 1984;238(2):221-230.

Eisenberg BR. Adaptability of ultrastructure in the mammalian muscle. Journal of experimental biology. 1985;115(1):55-68.

Ling GN. A Physical Theory of the Living State: The Association-induction Hypothesis; with Considerations of the Mechanisms Involved in Ionic Specificity: Blaisdell Publishing Company; 1962.

Pollack GH. The Fourth Phase of Water: Beyond Solid, Liquid, and Vapor: Ebner & Sons; 2013.

Sevostianov I, Mogilevskaya SG, Kushch VI. Maxwell’s methodology of estimating effective properties: Alive and well. International Journal of Engineering Science. 2019;140:35-88.

Maxwell JC. A Treatise on Electricity and Magnetism (reprinted 1954). Third ed. New York: Dover Publications; 1865.

Schanne OF, -Ceretti ERP. Impedance Measurements in Biological Cells: Wiley; 1978.

Mobley BA, Eidt G. Transverse impedance of single frog skeletal muscle fibers. Biophysical Journal. 1982;40(1):51-59.

Eisenberg RS, Johnson EA. Three dimensional electrical field problem in physiology. Prog Biophys Mol Biol 1970;20:1-65

Barcilon V, Cole J, Eisenberg RS. A singular perturbation analysis of induced electric fields in nerve cells. SIAM J Appl Math. 1971;21(2):339-354.

Peskoff A, Eisenberg RS. Interpretation of some microelectrode measurements of electrical properties of cells. Annu Rev Biophys Bioeng. 1973;2:65-79.

Peskoff A, Eisenberg RS. The time-dependent potential in a spherical cell using matched asymptotic expansions. J Math Biol. 1975;2(The time-dependent potential in a spherical cell using matched asymptotic expansions):277-300.

Peskoff A, Ramirez DM. Potential induced in a spherical cell by an intracellular point source and an extracellular point sink. J Math Biol. 1975;2(4):301-316.

Peskoff A, Eisenberg RS, Cole JD. Matched asymptotic expansions of the Green’s function for the electric potential in an infinite cylindrical cell. SIAM J Appl Math 1976;30(2):222-239.

Peskoff A. Electric potential in three-dimensional electrically syncytial tissues. Bulletin of mathematical biology. 1979;41(2):163-181.

Mobley BA, Leung J, Eisenberg RS. Longitudinal impedance of skinned frog muscle fibers. J Gen Physiol. 1974;63(5):625-637.

Mobley BA, Leung J, Eisenberg RS. Longitudinal impedance of single frog muscle fibers. J Gen Physiol. 1975;65(1):97-113.

Hanai T, Haydon DA, Taylor J. The variation of capacitance and conductance of bimolecular lipid membranes with area. J Theor Biol. 1965;9(3):433-443.

Hanai T, Haydon DA, Taylor J. The influence of lipid composition and of some adsorbed proteins on the capacitance of black hydrocarbon membranes. J Theor Biol. 1965;9(3):422-432.

Everitt CT, Haydon DA. Electrical capacitance of a lipid membrane separating two aqueous phases. J Theor Biol. 1968;18(3):371-379.

Haydon DA. Properties of lipid bilayers at a water-water interface. J Am Oil Chem Soc. 1968;45(4):230-240.

Fettiplace R, Andrews DM, Haydon DA. The thickness, composition and structure of some lipid bilayers and natural membranes. J Membr Biol. 1971;5(3):277-296.

Haydon DA, Myers VB. Surface charge, surface dipoles and membrane conductance. Biochim Biophys Acta. 1973;307(3):429-443.

Haydon DA, Kimura J, Requena J, Urban BW. Impedance measurement as an indication of membrane thickness change in the squid giant axon [proceedings]. J Physiol. 1979;287:2P.

Brosseau C, Sabri E. Resistor–capacitor modeling of the cell membrane: A multiphysics analysis. Journal of Applied Physics. 2021;129(1):011101.

Ziebert F, Lacoste D. A planar lipid bilayer in an electric field: membrane instability, flow field, and electrical impedance. Advances in planar lipid bilayers and liposomes. 14: Elsevier; 2011. p. 63-95.

Fernandez J, Bezanilla F, Taylor R. Distribution and kinetics of membrane dielectric polarization. II. Frequency domain studies of gating currents. J Gen Physiol. 1982;79(1):41-67.

Taylor RE, Bezanilla F. Comments on the measurement of gating currents in the frequency domain. Biophys J. 1979;26(2):338-340.

Bezanilla F, Taylor RE, Fernandez JM. Distribution and kinetics of membrane dielectric polarization. 1. Long-term inactivation of gating currents. J Gen Physiol. 1982;79(1):21-40.

Fernandez JM, Taylor RE, Bezanilla F. Induced capacitance in the squid giant axon. Lipophilic ion displacement currents. J Gen Physiol. 1983;82(3):331-346.

Taylor RE. Impedance of the squid axon membrane. J Cell Physiol. 1965;66(S2):21-25.

Clausen C, Lewis SA, Diamond JM, Eisenberg RS. Electrical circuit analysis of tight epithelia by alternating current techniques. Biophys J 1976;16:131a.

Clausen C, Fernandez JM. A low-cost method for rapid transfer function measurements with direct application to biological impedance analysis. Pflügers Archiv. 1981;390(3):290-295.

Clausen C. [32] Impedance analysis in tight epithelia. Methods Enzymol. 171: Elsevier; 1989. p. 628-642.

Clausen C, Machen TE, Diamond JM. Use of AC impedance analysis to study membrane changes related to acid secretion in amphibian gastric mucosa. Biophysical Journal. 1983;41(2):167-178.

Clausen C, Reinach PS, Marcus DC. Membrane transport parameters in frog corneal epithelium measured using impedance analysis techniques. The Journal of membrane biology. 1986;91(3):213-225.

Lewis SA, Clausen C, Wills NK. Impedance analysis of epithelia. Epithelial Transport: Springer; 1996. p. 118-145.

Wills N, Clausen C. Transport-dependent alterations of membrane properties of mammalian colon measured using impedance analysis. The Journal of membrane biology. 1987;95(1):21-35.

Clausen C, Lewis SA, Diamond JM. Impedance analysis of a tight epithelium using a distributed resistance model. Biophysical journal. 1979;26(2):291-317.

Boron W, Boulpaep E. Medical Physiology. New York: Saunders; 2008. 1352 p.

Silverthorn DU, Johnson BR, Ober WC, Ober CE, Impagliazzo A, Silverthorn AC. Human Physiology: An Integrated Approach: Pearson Education, Incorporated; 2019.

Feher JJ. Quantitative human physiology: an introduction: Academic press; 2017.

Keener J, Sneyd J. Mathematical Physiology: I: Cellular Physiology: Springer New York; 2014.

Layton AT, Edwards A. Mathematical Modeling in Renal Physiology: Springer Berlin Heidelberg; 2014.

Wostyn P, Van Dam D, Audenaert K, Killer H, Paul D, Groot V. A new glaucoma hypothesis: A role of glymphatic system dysfunction. Fluids and barriers of the CNS. 2015;12:16.

Norman RE, Flanagan JG, Sigal IA, Rausch SM, Tertinegg I, Ethier CR. Finite element modeling of the human sclera: influence on optic nerve head biomechanics and connections with glaucoma. Experimental Eye Research. 2011;93(1):4-12.

Ethier C, Norman R, Sigal I, et al. Finite Element Modeling of the Human Sclera: Influence on ONH Biomechanics and Connections With Glaucoma. Invest Ophthalmol Vis Sci. 2009;50(13):4889-4889.

Gardiner BS, Smith DW, Coote M, Crowston JG. Computational modeling of fluid flow and intra-ocular pressure following glaucoma surgery. PLoS One. 2010;5(10):e13178.

Stefan C, Cojocaru I, Pop A. [Glaucoma and ocular surface]. Oftalmologia. 2011;55(1):30-33.

Wang N. Intraocular and Intracranial Pressure Gradient in Glaucoma: Springer Singapore; 2019.

Jin B-J, Smith AJ, Verkman AS. Spatial model of convective solute transport in brain extracellular space does not support a “glymphatic” mechanism. Journal of General Physiology. 2016;148(6):489-501.

Kaur J, Davoodi-Bojd E, Fahmy LM, et al. Magnetic Resonance Imaging and Modeling of the Glymphatic System. Diagnostics. 2020;10(6):344.

Zhu Y, Xu S, Eisenberg RS, Huang H. A Tridomain Model for Potassium Clearance in Optic Nerve. 2020.

Stephenson JL. 23 - The Mathematical Theory of Renal Function. In: Brown JHU, Gann DS, editors. Engineering Principles in Physiology: Academic Press; 1973. p. 283-320.

Kuo IY, Ehrlich BE. Ion Channels in Renal Disease. Chemical Reviews. 2012;112(12):6353-6372.

Thomas SR, Layton AT, Layton HE, Moore LC. Kidney Modeling: Status and Perspectives. Proceedings of the IEEE. 2006;94(4):740-752.

Layton AT, Layton HE. Countercurrent multiplication may not explain the axial osmolality gradient in the outer medulla of the rat kidney. American Journal of Physiology-Renal Physiology. 2011;301(5):F1047-F1056.

Eisenberg RS, Rae JL. Current-voltage relationships in the crystalline lens. J Physiol. 1976;262(2):285-300.

Eisenberg RS, Engel E. The spatial variation of membrane potential near a small source of current in a spherical cell. J Gen Physiol. 1970;55(6):736-757.

Engel E, Barcilon V, Eisenberg RS. The interpretation of current-voltage relations recorded from a spherical cell with a single microelectrode. Biophys J. 1972;12(4):384-403.

Kevorkian J, Cole JD. Multiple Scale and Singular Perturbation Methods. New York: Springer-Verlag; 1996. pp. 1-632. p.

Eisenberg RS, Barcilon V, Mathias RT. Electrical properties of spherical syncytia. Biophys J. 1979;25(1):151-180.

Peskoff A. Electric potential in cylindrical syncytia and muscle fibers. Bulletin of mathematical biology. 1979;41(2):183-192.

Forrester JV, Dick AD, McMenamin PG, Roberts F, Pearlman E. The Eye: Basic Sciences in Practice: Elsevier; 2021.

Donaldson P, Kistler J, Mathias RT. Molecular solutions to mammalian lens transparency. News Physiol Sci. 2001;16:118-123.

Mathias RT, Kistler J, Donaldson P. The lens circulation. Journal of Membrane Biology. 2007;216(1):1-

Rae JL. The potential difference of the frog lens. Exp Eye Res. 1973;15(4):485-494.

Mathias RT, Rae JL. The lens: local transport and global transparency. Exp Eye Res. 2004;78(3):689-698.

Donaldson PJ, Musil LS, Mathias RT. Point: A critical appraisal of the lens circulation model--an experimental paradigm for understanding the maintenance of lens transparency? Invest Ophthalmol Vis Sci. 2010;51(5):2303-2306.

Donaldson PJ, Grey AC, Maceo Heilman B, Lim JC, Vaghefi E. The physiological optics of the lens. Prog Retin Eye Res. 2017;56:e1-e24.

Mathias RT, Rae JL. Transport properties of the lens. Am J Physiol. 1985;249(3 Pt 1):C181-190.

Mathias RT, Riquelme G, Rae JL. Cell to cell communication and pH in the frog lens. J Gen Physiol. 1991;98(6):1085-1103.

Mathias RT, Rae JL, Baldo GJ. Physiological properties of the normal lens. Physiol Rev. 1997;77(1):21-50.

Gong X, Baldo GJ, Kumar NM, Gilula NB, Mathias RT. Gap junctional coupling in lenses lacking alpha3 connexin. Proc Natl Acad Sci U S A. 1998;95(26):15303-15308.

Kushmerick C, Varadaraj K, Mathias RT. Effects of lens major intrinsic protein on glycerol permeability and metabolism. J Membr Biol. 1998;161(1):9-19.

Baldo GJ, Gong X, Martinez-Wittinghan FJ, Kumar NM, Gilula NB, Mathias RT. Gap junctional coupling in lenses from alpha(8) connexin knockout mice. J Gen Physiol. 2001;118(5):447-456.

Gao J, Sun X, Martinez-Wittinghan FJ, Gong X, White TW, Mathias RT. Connections between connexins, calcium, and cataracts in the lens. J Gen Physiol. 2004;124(4):289-300.

Martinez-Wittinghan FJ, Sellitto C, White TW, Mathias RT, Paul D, Goodenough DA. Lens gap junctional coupling is modulated by connexin identity and the locus of gene expression. Invest Ophthalmol Vis Sci. 2004;45(10):3629-3637.

McNulty R, Wang H, Mathias RT, Ortwerth BJ, Truscott RJ, Bassnett S. Regulation of tissue oxygen levels in the mammalian lens. J Physiol. 2004;559(Pt 3):883-898.

Varadaraj K, Kumari S, Shiels A, Mathias RT. Regulation of aquaporin water permeability in the lens. Invest Ophthalmol Vis Sci. 2005;46(4):1393-1402.

Wang H, Gao J, Sun X, et al. The effects of GPX-1 knockout on membrane transport and intracellular homeostasis in the lens. J Membr Biol. 2009;227(1):25-37.

Gao J, Sun X, Moore LC, White TW, Brink PR, Mathias RT. Lens intracellular hydrostatic pressure is generated by the circulation of sodium and modulated by gap junction coupling. J Gen Physiol. 2011;137(6):507-520.

Gao J, Wang H, Sun X, et al. The effects of age on lens transport. Invest Ophthalmol Vis Sci. 2013;54(12):7174-7187.

Hall JE, Mathias RT. The aquaporin zero puzzle. Biophys J. 2014;107(1):10-15.

Sindhu Kumari S, Gupta N, Shiels A, et al. Role of Aquaporin 0 in lens biomechanics. Biochem Biophys Res Commun. 2015;462(4):339-345.

Gao J, Minogue PJ, Beyer EC, Mathias RT, Berthoud VM. Disruption of the lens circulation causes calcium accumulation and precipitates in connexin mutant mice. Am J Physiol Cell Physiol. 2018;314(4):C492-C503.

Berthoud VM, Gao J, Minogue PJ, Jara O, Mathias RT, Beyer EC. Connexin Mutants Compromise the Lens Circulation and Cause Cataracts through Biomineralization. Int J Mol Sci. 2020;21(16).

Delamere NA, Shahidullah M, Mathias RT, et al. Signaling Between TRPV1/TRPV4 and Intracellular Hydrostatic Pressure in the Mouse Lens. Invest Ophthalmol Vis Sci. 2020;61(6):58.

Weinstein A, Stephenson J. Coupled water transport in standing gradient models of the lateral intercellular space. Biophysical journal. 1981;35(1):167-191.

Vereshchaga Y, Arnold N, Baumgartner W. Physiological relevance of epithelial geometry: New insights into the standing gradient model and the role of LI cadherin. PLOS ONE. 2018;13:e0208791.

O’Brien S. Lin & Segel’s Standing Gradient Problem Revisited: A Lesson in Mathematical Modeling and Asymptotics. SIAM Review. 2011;53:775-796.

McLaughlin S, Mathias RT. Electro-osmosis and the reabsorption of fluid in renal proximal tubules. J Gen Physiol. 1985;85(5):699-728.

Mathias RT. Epithelial water transport in a balanced gradient system. Biophys J. 1985;47(6):823-836.

Mathias RT, Wang H. Local osmosis and isotonic transport. J Membr Biol. 2005;208(1):39-53.

Ebihara L, Mathias RT. Linear impedance studies of voltage-dependent conductances in tissue cultured chick heart cells. Biophys J. 1985;48(3):449-460.

Mathias RT. Steady-state voltages, ion fluxes, and volume regulation in syncytial tissues. 1985;48(3):435-448.

Kelvin L. On the theory of the electric telegraph. Proceedings of the Royal Society (London). 1855;7:382-399.

Kelvin L. On the theory of the electric telegraph. Philosophical Magazine. 1856;11:146-160.

Gordon JS. A Thread Across the Ocean: The Heroic Story of the Transatlantic Cable: Paw Prints; 2008.

Hodgkin AL, Rushton WAH. The electrical constants of a crustacean nerve fiber. Proc Roy Soc (London) Ser B. 1946;133:444-479.

Davis LD, Jr., de No RL. Contribution to the Mathematical Theory of the electrotonus. Studies from the Rockefeller Institute for Medical Research. 1947;131:442-496.

Jack JJB, Noble D, Tsien RW. Electric Current Flow in Excitable Cells. New York: Oxford, Clarendon Press.; 1975.

Weinberg AM. Nerve Conduction with Distributed Capacitance. Journal of Applied Physics. 1939;10(2):128-134.

Weinberg A. On the formal theory of nerve conduction. The bulletin of mathematical biophysics. 1940;2(3):127-133.

Weinberg AM. Weber’s theory of the Kernleiter. The bulletin of mathematical biophysics. 1941;3(2):39-55.

Weinberg AM. Green’s functions in biological potential problems. The bulletin of mathematical biophysics. 1942;4(3):107-115.

Weinstein AM. A mathematical model of the rat proximal tubule. American Journal of Physiology-Renal Physiology. 1986;250(5):F860-F873.

Weinstein AM. Analysis of volume regulation in an epithelial cell model. Bull Math Biol. 1992;54(4):537-561.

Weinstein AM. Mathematical models of tubular transport. Annu Rev Physiol. 1994;56:691-709.

Weinstein AM. A mathematical model of rat proximal tubule and loop of Henle. Am J Physiol Renal Physiol. 2015;308(10):F1076-1097.

Weinstein AM. A mathematical model of the rat kidney: K(+)-induced natriuresis. Am J Physiol Renal Physiol. 2017;312(6):F925-F950.

Starling EH. On the Absorption of Fluids from the Connective Tissue Spaces. The Journal of Physiology. 1896;19(4):312-326.

Michel CC, Woodcock TE, Curry FRE. Understanding and extending the Starling principle. Acta Anaesthesiologica Scandinavica. 2020;64(8):1032-1037.

Nikolaev P. A new method to obtain the Carnahan–Starling equation and its generalization. Moscow University Physics Bulletin. 2017;72(1):23-28.

Dohrn R, Prausnitz JM. A simple perturbation term for the Carnahan-Starling equation of state. Fluid phase equilibria. 1990;61(1-2):53-69.

Hansen-Goos H, Roth R. A new generalization of the Carnahan-Starling equation of state to additive mixtures of hard spheres. The Journal of chemical physics. 2006;124(15):154506.

Erstad BL. The revised Starling equation: the debate of albumin versus crystalloids continues. Annals of Pharmacotherapy. 2020;54(9):921-927.

Song Y, Mason E, Stratt RM. Why does the Carnahan-Starling equation work so well? The Journal of Physical Chemistry. 1989;93(19):6916-6919.

Civetta JM. A new look at the Starling equation. Crit Care Med. 1979;7(3):84-91.

Rowlinson JS. The Perfect Gas. New York: Macmillan; 1963. 136 p.

Eisenberg RS. Computing the field in proteins and channels. Journal of Membrane Biology. 1996;150:1–25. Preprint available on physics arXiv as document 1009.2857

Eisenberg B. Self-organized model of selectivity. Institute of Mathematics and its Applications. 2009;IMA University of Minnesota (on-line): and also

Eisenberg B. Ion Channels, Natural Nanovalves. In: Savinell RF, Ota K-i, Kreysa G, editors. Encyclopedia of Applied Electrochemistry (in the press) Also available at http://arxivorg/abs/12061253 as arXiv:12061253v1. New York: Springer; 2013.

Eisenberg B. Electrostatic effects in living cells. Physics Today. 2013;66(7):10-11.

Eisenberg B. Shouldn’t we make biochemistry an exact science? ASBMB Today. 2014;13(9, October):36-38, Available on arXiv as

Eisenberg B. Rate Constant Models cannot Describe Movement of Charged Atoms or Molecules. Biophysical Journal. 2015;108(2):577a.

Eisenberg R. PNP what is in a name july 25-1 2019. pdf 10.31224/ engrXiv August 3. 2019.

Eisenberg RS. Kirchhoff’s Law can be Exact. 2019. arXiv preprint available at https://arxivorg/abs/190513574

Eisenberg RS. Updating Maxwell with Electrons, Charge, and More Realistic Polarization. 2019. arXiv preprint available at https://arxivorg/abs/190409695

Eisenberg RS. Electrodynamics Correlates Knock-on and Knock-off: Current is Spatially Uniform in Ion Channels. 2020. Preprint on arXiv at https://arxivorg/abs/200209012

Eisenberg RS. Core Maxwell Equations are Exact, Universal, and Scary, for that reason.

Eisenberg R. A Necessary Addition to Kirchhoff’s Current Law of Circuits, Version 2. Engineering Archive EngArXiv. 2022;

Shockley W. Electrons and Holes in Semiconductors to applications in transistor electronics. New York: van Nostrand; 1950. 558 p.

Van Roosbroeck W. Theory of flow of electrons and holes in germanium and other semiconductors. Bell System Technical Journal. 1950;29:560-607.

Blotekjaer K. Transport equations for electrons in two-valley semiconductors. Electron Devices, IEEE Transactions on. 1970;17(1):38-47.

Streetman BG. Solid State Electronic Devices. 4th ed. Englewood Cliffs, NJ: Prentice Hall; 1972. 462 p.

Sze SM. Physics of Semiconductor Devices. New York: John Wiley & Sons; 1981. 838. p.

Selberherr S. Analysis and Simulation of Semiconductor Devices. New York: Springer-Verlag; 1984. pp. 1-293. p.

Ferry DK. Semiconductor Transport. New York: Taylor and Francis; 2000. 384 p.

Hess K. Advanced Theory of Semiconductor Devices. New York: IEEE Press; 2000. 350 p.

Vasileska D, Goodnick SM, Klimeck G. Computational Electronics: Semiclassical and Quantum Device Modeling and Simulation. New York: CRC Press; 2010. 764 p.

Eisenberg B. The value of Einstein’s mistakes. Letter to the Editor: “Einstein should be allowed his mistakes …” Physics Today. 2006;59(4):12.

Mott NF. The theory of crystal rectifiers. Proc Roy Soc A. 1939;171:27-38.

Grimley TB, Mott NF. I. General and theoretical. The contact between a solid and a liquid electrolyte. Discuss Faraday Soc. 1947;1:3.

Mott NF, Gurney RW. Electronic processes in ionic crystals. 1948.

Hille B. Ion Channels of Excitable Membranes. 3rd ed. Sunderland: Sinauer Associates Inc.; 2001. 1-814. p.

Goldman DE. Potential, impedance and rectification in membranes. J Gen Physiol. 1943;27:37–60.

Hodgkin AL, Katz B. The effect of sodium ions on the electrical activity of the giant axon of the squid. J Physiol. 1949; 108:37–77.

Eisenberg B. Proteins, Channels, and Crowded Ions. Biophysical Chemistry. 2003;100:507 - 517.

Eisenberg B. Crowded Charges in Ion Channels. In: Rice SA, editor. Advances in Chemical Physics. New York: John Wiley & Sons, Inc.; 2011. p. 77-223 also on the arXiv at

Boda D, Nonner W, Henderson D, Eisenberg B, Gillespie D. Volume Exclusion in Calcium Selective Channels. Biophys J. 2008;94(9):3486-3496.

Boda D, Nonner W, Valisko M, Henderson D, Eisenberg B, Gillespie D. Steric selectivity in Na channels arising from protein polarization and mobile side chains. Biophys J. 2007;93(6):1960-1980.

Boda D, Valisko M, Henderson D, Eisenberg B, Gillespie D, Nonner W. Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion. Journal of General Physiology. 2009;133(5):497-509.

Chen R-C, Li C-L, Chen J-H, Eisenberg B, Liu J-L. Differential Capacitance of Electric Double Layers: A Poisson-Bikerman Formula. 2020. arxiv:2012.13141. Availabe at https://arxivorg/pdf/201213141pdf.

Jimenez-Morales D, Liang J, Eisenberg B. Ionizable side chains at catalytic active sites of enzymes. European Biophysics Journal. 2012;41(5):449-460.

Edsall JT, Wyman J. Biophysical Chemistry: Thermodynamics, Electrostatics, and the Biological Significance of the Properties of Matter: Elsevier Science; 2014.

Tanford C. Physical Chemistry of Macromolecuiles. New York: Wiley; 1961. 710 p.

Kubo R, Toda M, Hashitsume N. Statistical Physics II: Nonequilibrium Statistical Mechanics Second Edition ed. New York: Springer-Verlag; 1995. 295 p.

Eisenberg B. Vignette Applications of Physical Chemistry, a Biological Example. In: Berry S, Rice SA, Ross J, editors. Physical Chemistry: Oxford; 2005.

Hille E, Schwartz W. Potassium channels as multi-ion single-file pores. J Gen Physiol. 1978;72:409-442.

Chen D, Xu L, Tripathy A, Meissner G, Eisenberg R. Rate Constants in Channology. Biophys J. 1997;73(3):1349-1354.

Eisenberg RS. Atomic Biology, Electrostatics and Ionic Channels. In: Elber R, editor. New Developments and Theoretical Studies of Proteins. 7. Philadelphia: World Scientific; 1996. p. 269-357. Published in the Physics ArXiv as arXiv:0807.0715.

Barcilon V, Chen D, Eisenberg RS, Ratner M. Barrier crossing with concentration boundary conditions in biological channels and chemical reactions. J Chem Phys. 1993;98:1193–1211.

Eisenberg RS, Kłosek MM, Schuss Z. Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations. J Chem Phys. 1995;102:1767-1780.

Schuss Z, Nadler B, Eisenberg RS. Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model. Phys Rev E Stat Nonlin Soft Matter Phys. 2001;64(3 Pt 2):036116.

Schuss Z, Nadler B, Singer A, Eisenberg R. A PDE formulation of non-equilibrium statistical mechanics for ionic permeation. AIP Conference Proceedings, 3-6 September 2002: Unsolved Problems Of Noise And Fluctuations, UPoN 2002, 3rd International Conference on Unsolved Problems of Noise and Fluctuations in Physics, Biology, and High Technology 2002;665.

Nadler B, Schuss Z, Singer A, Eisenberg B. Diffusion through protein channels: from molecular description to continuum equations. Nanotechnology. 2003;3:439.

Schuss Z, Nadler B, Singer A, Eisenberg RS, editors. A PDE Formulation of Non-Equilibrium Statistical Mechanics for Ionic Permeation. AIP Conference Proceedings , 3-6 September 2002: Unsolved Problems Of Noise And Fluctuations, UPoN 2002, 3rd International Conference on Unsolved Problems of Noise and Fluctuations in Physics, Biology, and High Technology 2003: AIP.

Nadler B, Schuss Z, Singer A, Eisenberg R. Ionic diffusion through confined geometries: from Langevin equations to partial differential equations Journal of Physics: Condensed Matter. 2004;16:S2153-S2165.

Singer A, Schuss Z, Nadler B, Eisenberg RS. Models of boundary behavior of particles diffusing between two concentrations. In: Abbot D, Bezrukov SM, Der A, Sanchez A, editors. Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems II: series Vol 5467. 5467. New York: SPIE Proc.; 2004. p. 345-358.

Cooper K, Jakobsson E, Wolynes P. The theory of ion transport through membrane channels. Prog Biophys Molec Biol. 1985;46:51–96.

Cooper KE, Gates PY, Eisenberg RS. Surmounting barriers in ionic channels. Quarterly Review of Biophysics. 1988;21: 331–364.

Cooper KE, Gates PY, Eisenberg RS. Diffusion theory and discrete rate constants in ion permeation. J Membr Biol. 1988;109:95–105.

Gates P, Cooper K, Rae J, Eisenberg R. Predictions of diffusion models for one-ion membrane channels. Prog Biophys Mol Biol. 1989;53(3):153-196.

Xu X, Liu C, Qian T. Hydrodynamic boundary conditions for one-component liquid-gas flows on non-isothermal solid substrates. Communications in Mathematical Sciences. 2012;10(4 (December 2012)):1027-1053.

Eisenberg B. Multiple Scales in the Simulation of Ion Channels and Proteins. The Journal of Physical Chemistry C. 2010;114(48):20719-20733.

Eisenberg B. Mass Action in Ionic Solutions. Chemical Physics Letters. 2011;511:1-6.

Eisenberg B. Life’s Solutions are Not Ideal. Posted on arXivorg with Paper ID arXiv:11050184v1. 2011.

Eisenberg B. A Leading Role for Mathematics in the Study of Ionic Solutions. SIAM News. 2012;45(9):11-12.

Eisenberg B. Ionic Interactions Are Everywhere. Physiology. 2013;28(1):28-38.

Eisenberg B. Interacting ions in Biophysics: Real is not ideal. . Biophysical Journal. 2013;104:1849-1866.

Martell AE, Smith RM. Critical Stability Constants: First Supplement: Springer US; 2013.

Martell AE, Smith RM. Critical Stability Constants: Second Supplement: Springer US; 2013.

Martell AE, Motekaitis R. Determination and Use of Stability Constants. New York: VCH; 1988. 216 p.

Martell AE, Hancock RD. Metal Complexes in Aqueous Solutions: Springer US; 2013.

Martell AE, Hancock RD. Metal Complexes in Aqueous Solutions. New York: Plenum; 1996. 253 p.

Zhu Y, Xu S, Eisenberg RS, Huang H. A Bidomain Model for Lens Microcirculation. Biophysical Journal. 2019;116(6):1171-1184 Preprint available at

Zhu Y. Mathematical Modeling of Coupled Ion and Water Transport in Biological Tissues. 2021.

Mathias RT. An analysis of the electrical properties of a skeletal muscle fiber containing a helicoidal T system. Biophys J. 1978;23(2):277-284.

Eisenberg RS, Mathias RT. Structural analysis of electrical properties of cells and tissues. CRC Crit Rev Bioeng. 1980;4(3):203-232.

Mathias RT. Analysis of Membrane Properties Using Extrinsic Noise. In: Eisenberg R, Frank M, Stevens C, editors. Membranes, Channels, and Noise: Springer US; 1984. p. 49-116.

Xu S, Eisenberg B, Song Z, Huang H. Osmosis through a Semi-permeable Membrane: a Consistent Approach to Interactions. 2018. arXiv preprint arXiv:180600646.

Chanson H. Applied Hydrodynamics: An Introduction: CRC Press; 2013.

Varadaraj K, Gao J, Mathias RT, Kumari S. C-Terminal End of Aquaporin 0 Regulates Lens Gap Junction Channel Function. Invest Ophthalmol Vis Sci. 2019;60(7):2525-2531.

Berthoud VM, Gao J, Minogue PJ, Jara O, Mathias RT, Beyer EC. The Connexin50D47A Mutant Causes Cataracts by Calcium Precipitation. Invest Ophthalmol Vis Sci. 2019;60(6):2336-2346.

Vaghefi E, Donaldson PJ. The lens internal microcirculation system delivers solutes to the lens core faster than would be predicted by passive diffusion. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology. 2018.

Petrova RS, Webb KF, Vaghefi E, Walker K, Schey KL, Donaldson PJ. Dynamic functional contribution of the water channel AQP5 to the water permeability of peripheral lens fiber cells. Am J Physiol Cell Physiol. 2018;314(2):C191-C201.

Schey KL, Petrova RS, Gletten RB, Donaldson PJ. The Role of Aquaporins in Ocular Lens Homeostasis. Int J Mol Sci. 2017;18(12).

Kumari S, Gao J, Mathias RT, et al. Aquaporin 0 Modulates Lens Gap Junctions in the Presence of Lens-Specific Beaded Filament Proteins. Invest Ophthalmol Vis Sci. 2017;58(14):6006-6019.

Vaghefi E, Kim A, Donaldson PJ. Active Maintenance of the Gradient of Refractive Index Is Required to Sustain the Optical Properties of the Lens. Invest Ophthalmol Vis Sci. 2015;56(12):7195-7208.

Young JZ. Fused neurons and synaptic contacts in the giant nerve fibres of cephalopods. Philosophical Transactions of the Royal Society of London Series B, Biological sciences. 1939;229(564):465-503.

Young J, Keynes R. The Functioning of the Giant Nerve Fibres of the Squid. 1938-JZ and the discovery of squid giant nerve fibres. The Journal of experimental biology. 2005;208(Pt 2):179-180.

Young JZ. The functioning of the giant nerve fibres of the squid. J Exp Biol. 1938;15(2):170-185.

Hodgkin AL. Chance and Design. New York: Cambridge University Press; 1992. 401 p.

Huxley AF. From overshoot to voltage clamp. Trends in Neurosciences 2002;25 (11):553-558.

Huxley A. Hodgkin Obituary. The Independent (newspaper). 1999;January 4, 1999

Huxley A. Sir Alan Lloyd Hodgkin, O. M., K. B. E. 5 February 1914-20 December 1998. Biographical Memoirs of Fellows of the Royal Society. 2000;46:221-241.

Hodgkin AL. Evidence for electrical transmission in nerve: Part II. J Physiol. 1937;90(2):211-232.

Hodgkin AL. Evidence for electrical transmission in nerve: Part I. J Physiol. 1937;90(2):183-210.

Hill AV. Chemical Wave Transmission in Nerve: Cambridge University Press; 1932. 74 p.

Huxley AF. Kenneth Stewart Cole 1900-1984. A biographical memoir by Sir Andrew Huxley. Washington DC: National Academies Press; 1996.

Rubinstein I, Zaltzman B. Convective diffusive mixing in concentration polarization: from Taylor dispersion to surface convection. Journal of Fluid Mechanics. 2013;728:239-278.

Abu-Rjal R, Chinaryan V, Bazant MZ, Rubinstein I, Zaltzman B. Effect of concentration polarization on permselectivity. Phys Rev E Stat Nonlin Soft Matter Phys. 2014;89(1):012302.

Tanaka Y. 6 - Concentration Polarization. Ion Exchange Membranes (Second Edition). Amsterdam: Elsevier; 2015. p. 101-121.

Wang Y, Liu C, Eisenberg B. On variational principles for polarization in electromechanical systems. arXiv preprint arXiv:210811512. 2021.

Frankenhaeuser B, Hodgkin AL. The after-effects of impulses in the giant nerve fibres of Loligo. Journal of Physiology (London). 1956;131:341-376.

Taylor RE, Bezanilla F, Rojas E. Diffusion models for the squid axon Schwann cell layer. Biophys J. 1980;29(1):95-117.

Salzberg BM, Bezanilla F. An optical determination of the series resistance in Loligo. J Gen Physiol. 1983;82(6):807-817.

Kuffler SW, Nicholls JG, Orkand RK. Physiological properties of glial cells in the central nervous system of amphibia. J Neurophysiol. 1966;29(4):768-787.

Orkand RK, Nicholls JG, Kuffler SW. Effect of nerve impulses on the membrane potential of glial cells in the central nervous system of amphibia. J Neurophysiol. 1966;29(4):788-806.

Kuffler SW, Nicholls JG. The physiology of neuroglial cells. Ergeb Physiol. 1966;57:1-90.

Kuffler SW. Neuroglial cells: physiological properties and a potassium mediated effect of neuronal activity on the glial membrane potential. Proc R Soc Lond B Biol Sci. 1967;168(1010):1-21.

Cohen MW, Gerschenfeld HM, Kuffler SW. Ionic environment of neurones and glial cells in the brain of an amphibian. J Physiol. 1968;197(2):363-380.

Zhu Y, Xu S, Eisenberg RS, Huang H. Optic nerve microcirculation: Fluid flow and electrodiffusion. Physics of Fluids. 2021;33(4):041906.

Zhu Y, Xu S, Eisenberg RS, Huang H. Membranes in Optic Nerve Models. arXiv preprint arXiv:210514411. 2021.

Gabbiani F, Cox SJ. Mathematics for Neuroscientists. New York: Academic Press; 2010.

Huxley AF. The quantitative analysis of excitation and conduction in nerve1963. 242-260 p.

Gao J, Sun X, Yatsula V, Wymore R, Mathias R. Isoform-specific function and distribution of Na/K pumps in the frog lens epithelium. The Journal of membrane biology. 2000;178(2):89-101.

Cheng X, Mwaura BW, Chang Stauffer SR, Bezanilla M. A Fully Functional ROP Fluorescent Fusion Protein Reveals Roles for This GTPase in Subcellular and Tissue-Level Patterning. Plant Cell. 2020;32(11):3436-3451.

Moss J, Williams A. Opening the floodgates to the brain. Science. 2020;367(6483):1195.

Ray L, Iliff JJ, Heys JJ. Analysis of convective and diffusive transport in the brain interstitium. Fluids and Barriers of the CNS. 2019;16(1):6.

Iliff JJ, Wang M, Liao Y, et al. A paravascular pathway facilitates CSF flow through the brain parenchyma and the clearance of interstitial solutes, including amyloid β. Sci Transl Med. 2012;4(147):147ra111-147ra111.

Witthoft A, Filosa JA, Karniadakis GE. Potassium buffering in the neurovascular unit: models and sensitivity analysis. Biophysical journal. 2013;105(9):2046-2054.

Killer H, Jaggi G, Flammer J, Miller NR, Huber A, Mironov A. Cerebrospinal fluid dynamics between the intracranial and the subarachnoid space of the optic nerve. Is it always bidirectional? Brain. 2007;130(2):514-520.

Kofuji P, Newman EA. Potassium homeostasis in glia. Encyclopedia of Neuroscience: Elsevier Ltd; 2009. p. 867-872.

Rouach N, Duc KD, Sibille J, Holcman D. Dynamics of ion fluxes between neurons, astrocytes and the extracellular space during neurotransmission. Opera Medica et Physiologica. 2018;4(1).

Murakami S, Kurachi Y. Mechanisms of astrocytic K+ clearance and swelling under high extracellular K+ concentrations. The Journal of Physiological Sciences. 2016;66(2):127-142.

Lewcock JW, Schlepckow K, Di Paolo G, Tahirovic S, Monroe KM, Haass C. Emerging Microglia Biology Defines Novel Therapeutic Approaches for Alzheimer’s Disease. Neuron. 2020.

Pajevic S. Exploring the Dynamics of Brain Extracellular Space. Biophys J. 2019;117(10):1781-1782.

O’Connell R, Mori Y. Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression. Bull Math Biol. 2016;78(10):1943-1967.

Filippidis AS, Zarogiannis SG, Ioannou M, Gourgoulianis K, Molyvdas P-A, Hatzoglou C. Permeability of the arachnoid and pia mater. The role of ion channels in the leptomeningeal physiology. Child’s Nervous System. 2012;28(4):533-540.

Qiao Q, Zhang W, Vencent C, Ren Q. Electric Stimulation of Optic Nerve Fiber: A Simulation Study. Advances in Cognitive Neurodynamics ICCN 2007: Springer; 2008. p. 609-615.

Wei Y, Ullah G, Schiff SJ. Unification of Neuronal Spikes, Seizures, and Spreading Depression. The Journal of Neuroscience. 2014;34(35):11733-11743.

Ullah G, Wei Y, Dahlem MA, Wechselberger M, Schiff SJ. The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization. PLoS Comput Biol. 2015;11(8):e1004414.

Ayata C, Lauritzen M. Spreading Depression, Spreading Depolarizations, and the Cerebral Vasculature. 2015; 953-993.

Nye BL, Thadani VM. Migraine and Epilepsy: Review of the Literature. Headache: The Journal of Head and Face Pain. 2015;55(3):359-380.

Somjen G. Ions in the Brain: Normal Function, Seizures, and Stroke. New York: Oxford; 2004. 504 p.

Miura RM, Huang H, Wylie JJ. Cortical spreading depression: An enigma. The European Physical Journal. 2007;147:287-302.

Filippidis A, Zarogiannis S, Ioannou M, Gourgoulianis K, Molyvdas PA, Hatzoglou C. Transmembrane resistance and histology of isolated sheep leptomeninges. Neurol Res. 2010;32.

Verikouki CH, Hatzoglou CH, Gourgoulianis KI, Molyvdas PA, Kallitsaris A, Messinis IE. Rapid effect of progesterone on transepithelial resistance of human fetal membranes: evidence for non-genomic action. Clin Exp Pharmacol Physiol. 2008;35.

Adams EA, Choi HM, Cheung CY, Brace RA. Comparison of amniotic and intramembranous unidirectional permeabilities in late-gestation sheep. Am J Obstet Gynecol. 2005;193.

Vogiatzidis K, Hatzoglou C, Zarogiannis S, Matafia G, Gourgoulianis K, Molyvdas PA. mu-opioid influence on transmesothelial resistance of isolated sheep pleura and parietal pericardium. Eur J Pharmacol. 2006;530.

Zarogiannis S, Vogiatzidis K, Hatzoglou C, et al. mu-opioid stimulation of isolated parietal sheep peritoneum decreases peritoneal permeability in vitro. Adv Perit Dial. 2007;23.

Zarogiannis S, Liakopoulos V, Hatzoglou C, et al. Effect of sodium-potassium pump inhibition by ouabain on the permeability of isolated visceral sheep peritoneum. Adv Perit Dial. 2007;23.

Zarogiannis S, Kourti P, Hatzoglou C, et al. Influence of the sodium transport inhibition by amiloride on the transmesothelial resistance of isolated visceral sheep peritoneum. Adv Perit Dial. 2005;21.

Stefanidis I, Liakopoulos V, Kourti P, et al. Amiloride-sensitive sodium channels on the parietal human peritoneum: evidence by Ussing-type chamber experiments. ASAIO J. 2007;53.

Simon M. Peritoneal mesothelium in vitro: an electrophysiologic study. Perit Dial Int. 1996;16.

Li FK, To CH, Leung JK, Chan TM, Lai KN. Electrophysiology and glucose transport of human peritoneal mesothelial cells: implications for peritoneal dialysis. Perit Dial Int. 2001;21.

Zarogiannis S, Deligiorgi T, Stefanidis I, et al. Dexamethasone decreases the transmesothelial electrical resistance of the parietal and visceral pleura. J Physiol Sci. 2009;59.

Zarogiannis S, Hatzoglou C, Stefanidis I, Liakopoulos V, Gourgoulianis K, Molyvdas PA. Adrenergic influence on the permeability of sheep diaphragmatic parietal pleura. Respiration. 2007;74.

Zarogiannis S, Hatzoglou C, Stefanidis I, et al. Comparison of the electrophysiological properties of the sheep isolated costal and diaphragmatic parietal pleura. Clin Exp Pharmacol Physiol. 2007;34.

Sarkos S, Hatzoglou C, Dahabre J, Gourgoulianis KI, Molyvdas PA. Effect of amiloride in human and sheep parietal pleura. Respir Physiol Neurobiol. 2002;132.

Payne DK, Kinasewitz GT, Gonzalez E. Comparative permeability of canine visceral and parietal pleura. J Appl Physiol. 1988;65.

Hatzoglou CH, Gourgoulianis KI, Molyvdas PA. Effects of SNP, ouabain, and amiloride on electrical potential profile of isolated sheep pleura. J Appl Physiol. 2001;90.

Weller RO. Microscopic morphology and histology of the human meninges. Morphol. 2005;89.

McLone DG. The subarachnoid space: a review. Childs Brain. 1980;6.

Jayatilaka AD. An electron microscopic study of sheep arachnoid granulations. J Anat. 1965;99.

Barshes N, Demopoulos A, Engelhard HH. Anatomy and physiology of the leptomeninges and CSF space. Cancer Treat Res. 2005;125.

Alcolado R, Weller RO, Parrish EP, Garrod D. The cranial arachnoid and pia mater in man: anatomical and ultrastructural observations. Neuropathol Appl Neurobiol. 1988;14.

Redzic ZB, Segal MB. The structure of the choroid plexus and the physiology of the choroid plexus epithelium. Adv Drug Deliv Rev. 2004;56.

Johanson CE, Duncan JA, Klinge PM, Brinker T, Stopa EG, Silverberg GD. Multiplicity of cerebrospinal fluid functions: new challenges in health and disease. Cereb Fluid Res. 2008;5.

Amin MS, Reza E, Wang H, Leenen FH. Sodium transport in the choroid plexus and salt-sensitive hypertension. Hypertension. 2009;54.

Pollay M, Hisey B, Reynolds E, Tomkins P, Stevens FA, Smith R. Choroid plexus Na+/K+-activated adenosine triphosphatase and cerebrospinal fluid formation. Neurosurgery. 1985;17.

Filippidis AS, Kalani MY, Rekate HL. Hydrocephalus and aquaporins: lessons learned from the bench. Child Nerv Syst ChNS Off J Int Soc Pediatr Neurosurg. 2011;27.

Damkier HH, Brown PD, Praetorius J. Epithelial pathways in choroid plexus electrolyte transport. Physiol Bethesda. 2010;25.

Segal MB. Extracellular and cerebrospinal fluids. J Inherit Metab Dis. 1993;16.

Segal MB, Pollay M. The secretion of cerebrospinal fluid. Exp Eye Res. 1977;25.

McComb J. Recent research into the nature of cerebrospinal fluid formation and absorption. J Neurosurg. 1983;59.

Cserr HF. Physiology of the choroid plexus. Physiol Rev. 1971;51.

Mori Y. A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression. Physica D: Nonlinear Phenomena. (0).

Mori Y, Jerome JW, Peskin CS. A three-dimensional model of cellular electrical activity. Bulletin of the Institute of Mathematics, Academia Sinica (New Series). 2007;2(2):367-390.

Mori Y, Fishman GI, Peskin CS. Ephaptic conduction in a cardiac strand model with 3D electrodiffusion. Proc Natl Acad Sci U S A. 2008;105(17):6463-6468.

Mori Y, Liu C, Eisenberg RS. A multidomain model for electrodiffusion and water flow. Poster 511. Biophyiscal Journal. 2010;98:96a.

Mori Y, Liu C, Eisenberg RS. A model of electrodiffusion and osmotic water flow and its energetic structure. Physica D: Nonlinear Phenomena. 2011;240(22):1835-1852.

Mori Y. Mathematical properties of pump-leak models of cell volume control and electrolyte balance. J Math Biol. 2012;65(5):875-918.

Mori Y. A multidomain model for ionic electrodiffusion and osmosis with an application to cortical spreading depression. Physica D: Nonlinear Phenomena. 2015;308:94-108.

Stinchcombe AR, Mori Y, Peskin CS. Well-Posed Treatment of Space-Charge Layers in the Electroneutral Limit of Electrodiffusion. Communications on Pure and Applied Mathematics. 2015:n/a-n/a.

Wei N, Mori Y, Tolkacheva EG. The dual effect of ephaptic coupling on cardiac conduction with heterogeneous expression of connexin 43. J Theor Biol. 2016;397:103-114.

Mori Y, Young Y-N. Electrohydrodynamics of Leaky Dielectrics as the Weak Electrolyte Limit of an Electrodiffusion Model. 2017. arXiv preprint arXiv:170902321.

Tuttle A, Riera Diaz J, Mori Y. A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model. PLoS Comput Biol. 2019;15(12):e1007455.

Ellingsrud AJ, Solbrå A, Einevoll GT, Halnes G, Rognes ME. Finite element simulation of ionic electrodiffusion in cellular geometries. Frontiers in Neuroinformatics. 2020;14:11.

Genocchi B, Cunha A, Jain S, Hyttinen J, Lenk K, Ellingsrud AJ, editors. Parametric exploration of cellular swelling in a computational model of cortical spreading depression. 2020 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC); 2020: IEEE.

Daversin-Catty C, Richardson CN, Ellingsrud AJ, Rognes ME. Abstractions and automated algorithms for mixed domain finite element methods. ACM Transactions on Mathematical Software (TOMS). 2021;47(4):1-36.

Ellingsrud AJ, Boullé N, Farrell PE, Rognes ME. Accurate numerical simulation of electrodiffusion and water movement in brain tissue. Mathematical Medicine and Biology: A Journal of the IMA. 2021;38(4):516-551.

Ellingsrud AJ, Daversin-Catty C, Rognes ME. A cell-based model for ionic electrodiffusion in excitable tissue. Modeling Excitable Tissue: Springer, Cham; 2021. p. 14-27.

Ellingsrud AJ, Enger R, Dukefoss DB, Halnes G, Pettersen KH, Rognes ME. Validating a computational framework for ionic electrodiffusion with cortical spreading depression as a case study. bioRxiv. 2021.

Sacco R, Airoldi P, Mauri AG, Jerome j. Three-dimensional simulation of biological ion channels under mechanical, thermal, and fluid forces. available on arXiv at http://arxivorg/pdf/150907301pdf

Zhu Y, Xu S, Eisenberg RS, Huang H. A tridomain model for potassium clearance in optic nerve of Necturus. Biophysical journal. 2021;120(15):3008-3027.

Hill A. Solute-solvent coupling in epithelia: a critical examination of the standing-gradient osmotic flow theory. Proceedings of the Royal Society of London Series B Biological Sciences. 1975;190(1098):99-114.

Diamond JM, Bossert WH. Standing-gradient osmotic flow: A mechanism for coupling of water and solute transport in epithelia. The Journal of general physiology. 1967;50(8):2061-2083.

Curran PF, Macintosh JR. A model system for biological water transport. Nature. 1962;193:347-348.