Skip to main content
SpringerLink
Log in

Determining a distributed parameter in a neural cable model via a boundary control method

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Anger G (1990) Inverse problems in differential equations. Plenum, New York

    Google Scholar 

  • Avdonin S (2008) Control problems on quantum graphs. In: Analysis on graphs and its applications. Proceedings of symposia in pure mathematics. AMS, vol 77, pp 507–521

  • Avdonin SA, Ivanov SA (1995) Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press, New York

    MATH  Google Scholar 

  • Avdonin SA, Belishev MI (1996) Boundary control and dynamical inverse problem for nonselfadjoint Sturm–Liouville operator. Control Cybern 25: 429–440

    MathSciNet  MATH  Google Scholar 

  • Avdonin SA, Belishev M (2004) Dynamical inverse problem for the multidimensional Schrödinger equation. In: Proc. St. Petersburg Math. Soc., vol 10, pp 3–18 (Russian). Engl. Transl. in AMS Transl. (2005) Ser. 2, vol 214, pp 1–14, Am. Math. Soc., Providence

  • Avdonin S, Kurasov P (2008) Inverse problems for quantum trees. Inverse Problems and Imaging 2(1): 1–21

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin S, Mikhailov V (2008) Controllability of partial differential equations on graphs. Appl Math 35: 379–393

    MathSciNet  MATH  Google Scholar 

  • Avdonin S, Pandolfi L (2009) Boundary control method and coefficient identification in the presence of boundary dissipation. Appl Math Lett 22(11): 1705–1709

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin S, Mikhailov V (2010) The boundary control approach to inverse spectral theory. Inverse Probl 26(4): 1–19

    Article  Google Scholar 

  • Avdonin SA, Belishev MI, Ivanov SA (1992) Boundary control and matrix inverse problem for the equation u tt − u xx + V (x)u = 0. Math. USSR Sbornik 72: 287–310

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin SA, Belishev MI, Rozhkov YuS (1997) The BC method in the inverse problem for the heat equation. J. Inverse Ill Posed Probl 5: 1–14

    Article  MathSciNet  Google Scholar 

  • Avdonin SA, Belishev MI, Rozhkov YuS (2000a) A dynamic inverse problem for the nonselfadjoint Sturm-Liouville operator. J Math Sci 102(4): 4139–4148

    Article  MathSciNet  Google Scholar 

  • Avdonin SA, Medhin NG, Sheronova TL (2000b) Identification of a piecewise constant coefficient in the beam equation. J Comput Appl Math 114: 11–21

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin S, Lenhart S, Protopopescu V (2002) Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method. Inverse Probl 18: 349–361

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin SA, Lenhart S, Protopopescu V (2005) Determining the potential in the Schrödinger equation from the Dirichlet to Neumann map by the boundary control method. J Inverse Ill Posed Probl 13(5): 317–330

    Article  MathSciNet  MATH  Google Scholar 

  • Avdonin S, Leugering G, Mikhailov V (2010) On an inverse problem for tree-like networks of elastic strings. Zeit Angew Math Mech 90(2): 136–150

    Article  MATH  Google Scholar 

  • Avdonin SA, Belinskiy BP, Matthews JV (2011a) Dynamical inverse problem on a metric tree. Inverse Probl 27: 075011

    Article  MathSciNet  Google Scholar 

  • Avdonin SA, Belinskiy BP, Matthews JV (2011b) Inverse problem on the semi-axis: local approach. Tamkang J Math 42(3): 1–19

    Article  MathSciNet  Google Scholar 

  • Baer SM, Rinzel J (1991) Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J Neurophysiol 65(4): 874

    Google Scholar 

  • Belishev MI (1987) An approach to multidimensional inverse problems for the wave equation. Dokl. AN SSSR 297:524–527 (in Russian); English transl. in Soviet Math Dokl 36:481–484

  • Belishev MI (1996) Canonical model of a dynamical system with boundary control in inverse problem for the heat equation. St Petersburg Math J 7(6): 869–890

    MathSciNet  Google Scholar 

  • Belishev MI (2007) Recent progress in the boundary control method. Inverse Probl 23(5): R1–R67

    Article  MathSciNet  MATH  Google Scholar 

  • Bell J (2005) Inverse problems for some cable models of dendrites. In: Lindsay KA, Poznanski RR, Rosenberg JR, Sporns O Modeling in the neurosciences, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  • Bell J, Craciun G (2005) A distributed parameter identification problem in neuronal cable theory models. Math Biosci 194(1): 1–19

    Article  MathSciNet  MATH  Google Scholar 

  • Brown TH, Friske RA, Perkel DH (1981) Passive electrical constants in three classes of hippocampal neurons. J Neurophysiol 46: 812

    Google Scholar 

  • Cox SJ (1998) A new method for extracting cable parameters from input impedance data. Math Biosci 153: 1

    Article  MathSciNet  MATH  Google Scholar 

  • Cox SJ (2004) Estimating the location and time course of synaptic input from multi-site potential recordings. J Comput Neurosci 17(20): 225–243

    Article  MATH  Google Scholar 

  • Cox SJ (2006) An adjoint method for channel localization. Math Med Biol 23: 139–152

    Article  MATH  Google Scholar 

  • Cox SJ, Griffith B (2001) Recovering quasi-active properties of dendrites from dual potential recordings. J Comput Neurosci 11(2): 95

    Article  Google Scholar 

  • Cox SJ, Ji L (2000) Identification of the cable parameters in the somatic shunt model. Biol Cybern 83: 151–159

    Article  MATH  Google Scholar 

  • Cox SJ, Ji L (2001) Discerning ionic currents and their kinetics from input impedance data. Bull Math Biol 63: 909

    Article  Google Scholar 

  • Cox SJ, Wagner A (2004) Lateral overdetermination of the FitzHugh-Nagumo system. Inverse Probl 20: 1639–1647

    Article  MathSciNet  MATH  Google Scholar 

  • D’Aguanno A, Bardakjian BJ, Carlen PL (1986) Passive neuronal membrane parameters: comparison of optimization and peeling methods. IEEE Trans Biomed Eng 33: 1188

    Article  Google Scholar 

  • Dayan P, Abbott LF (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT Press, Cambridge

    Google Scholar 

  • Durand DM, Carlen PL, Gurevich N, Ho A, Kunov H (1983) Electrotonic parameters of rat dentate granule cells measured using short current pulses and HRP staining. J Neurophysiol 50: 1080

    Google Scholar 

  • Engl HW, Rundell W (eds) (1995) Inverse problems in diffusion processes. SIAM, Philadelphia

  • Gel’fand I, Levitan B (1951) On the determination of a differential equation from its spectral function (Russian). Izvestiya Akad Nauk SSSR Ser Mat 15:309–360; translation in AMS Transl (1955) 2(1):253–304

    Google Scholar 

  • Gesztesy F, Simon B (2000) A new approach to inverse spectral theory, II. General real potential and the connection to the spectral measure. Ann Math (2) 152(2): 593–643

    Article  MathSciNet  MATH  Google Scholar 

  • Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its applications to conduction and excitation in nerve. J Physiol 117: 500

    Google Scholar 

  • Holmes WR, Rall W (1992) Estimating the electrotonic structure of neurons with compartmental models. J Neurophysiol 68(4): 1438

    Google Scholar 

  • Isakov V (1998) Inverse problems for partial differential equations. Springer, New York

    Book  MATH  Google Scholar 

  • Jack JJB, Redman SJ (1971) An electrical description of a motoneurone, and its application to the analysis of synaptic potentials. J Physiol 215: 321

    Google Scholar 

  • Johnston D, Magee JC, Colbert CM, Christie BR (1996) Active properties of neuronal dendrites. Annu Rev Neurosci 19: 165

    Article  Google Scholar 

  • Katchalov A, Kurylev Y, Lassas M (2001) Inverse boundary spectral problems. Chapman & Hall/CRC, Boca Raton

    Book  MATH  Google Scholar 

  • Kawato M (1984) Cable properties of a neuron model with non-uniform membrane resistivity. J Theor Biol 111: 149

    Article  Google Scholar 

  • Kirsch A (1996) An introduction to the mathematical theory of inverse problems. Springer, New York

    Book  MATH  Google Scholar 

  • Krein M (1953) A transmission function of a second order one-dimensional boundary value problem. Dokl Akad Nauk SSSR 88(30): 405–408

    MathSciNet  Google Scholar 

  • Krein M (1954) On the one method of effective solving the inverse boundary value problem. Dokl Akad Nauk SSSR 94(6): 987–990

    MathSciNet  Google Scholar 

  • Kurylev Ya, Lassas M (1997) The multidimensional Gel’fand inverse problem for non-self-adjoint operators. Inverse Probl 13(6): 1495–1501

    Article  MathSciNet  MATH  Google Scholar 

  • Magee JC (1998) Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal Ca1 pyramidal neurons. J Neurosci 18(19): 7613

    Google Scholar 

  • Pierce A (1979) Unique identification of eigenvalues and coefficients in a parabolic problem. SIAM J Control Optim 17(4): 494–499

    Article  MathSciNet  MATH  Google Scholar 

  • Rall W (1960) Membrane potential transients and membrane time constants of motoneurons. Exp Neurol 2: 503

    Article  Google Scholar 

  • Rall W (1962) Theory of physiological properties of dendrites. Ann NY Acad Sci 96: 1071

    Article  Google Scholar 

  • Rall W (1977) Core conductor theory and cable properties of neurons. In: Handbook of physiology. The nervous system, vol 1, p 39

  • Rall W, Burke RE, Holmes WR, Jack JJB, Redman SJ, Segev I (1992) Matching dendritic neuron models to experimental data. Physiol Rev 172: S159

    Google Scholar 

  • Remling C (2002) Schrodinger operators and de Branges spaces. J Funct Anal 196(2): 323–394

    Article  MathSciNet  MATH  Google Scholar 

  • Remling C (2003) Inverse spectral theory for one-dimensional Schrodinger operators: the A-function. Math Z 245(3): 597–617

    Article  MathSciNet  MATH  Google Scholar 

  • Safronov BV (1999) Spatial distribution of Na and K channels in spinal dorsal horn neurones: role of soma, axon and dendrites in spike generation. Prog Neurobiol 59: 217

    Article  Google Scholar 

  • Stuart G, Spruston N (1998) Determinants of voltage attenuation in neocortical pyramidal neuron dendrites. J Neurosci 18(10): 3501–3510

    Google Scholar 

  • Schierwagen AK (1990) Identification problems in distributed parameter neuron models. Automatica 26: 739

    Article  MathSciNet  MATH  Google Scholar 

  • Simon B (1999) A new approach to inverse spectral theory, I. Fundamental formalism. Ann Math 150: 1029–1057

    Article  MATH  Google Scholar 

  • Tikhonov AN, Samarskii AA (1963) Equations of mathematical physics. Dover Publications, Inc., New York

    MATH  Google Scholar 

  • Vinokurov VA, Sadovnichii VA (2000) Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary value problem on a segment with a summable potential. Izvestia Math 64: 47–108

    MathSciNet  Google Scholar 

  • von Below J (1993) Parabolic network equations. Habilitation Thesis, Eberhard-Karls-Universitat, Tubingen

  • Wang D (2008) Partial differential equation constrained optimization and its applications to parameter estimation in models of nerve dendrites, PhD dissertation, UMBC

  • White JA, Manis PB, Young ED (1992) The parameter identification problem for the somatic shunt model. Biol Cybern 66: 307

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775, USA

    Sergei Avdonin

  2. Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD, 21250, USA

    Jonathan Bell

Authors
  1. Sergei Avdonin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan Bell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Bell.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Avdonin, S., Bell, J. Determining a distributed parameter in a neural cable model via a boundary control method. J. Math. Biol. 67, 123–141 (2013). https://doi.org/10.1007/s00285-012-0537-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-012-0537-6

Keywords

Mathematics Subject Classification

Search

Navigation