Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph

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August 2015, 9(3): 645-659. doi: 10.3934/ipi.2015.9.645

Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph

Sergei Avdonin ^1, and Jonathan Bell ^2,

1.	Department of Mathematics and Statistics, University of Alaska, Fairbanks, AK 99775-6660
2.	Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250, United States

Received April 2014 Revised November 2014 Published July 2015

Abstract
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In this paper we solve the inverse problem of recovering a single spatially distributed conductance parameter in a cable equation model (one-dimensional diffusion) defined on a metric tree graph that represents a dendritic tree of a neuron. Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. We employ the boundary control method that gives a unique reconstruction and an algorithmic approach.

Keywords: metric graph, parameter recovery., inverse problem, tree graph, boundary control, Cable equation.

Mathematics Subject Classification: 92, 93, 3.

Citation: Sergei Avdonin, Jonathan Bell. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems and Imaging, 2015, 9 (3) : 645-659. doi: 10.3934/ipi.2015.9.645

References:

[1]	S. A. Avdonin, Control problems on quantum graphs, in Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, 77, AMS, Providence, RI, 2008, 507-521. doi: 10.1090/pspum/077/2459889.
[2]	S. A. Avdonin, B. P. Belinskiy and J. V. Matthews, Dynamical inverse problem on a metric tree, Inverse Problems, 27 (2011), 075011, 21pp. doi: 10.1088/0266-5611/27/7/075011.
[3]	S. A. Avdonin, M. I. Belishev and Yu. S. Rozhkov, The BC method in the inverse problem for the heat equation, J. Inverse and Ill-Posed Problems, 5 (1997), 309-322. doi: 10.1515/jiip.1997.5.4.309.
[4]	S. A. Avdonin and J. Bell, Determining a distributed parameter in a neural cable theory model via a boundary control method, J. Mathematical Biology, 67 (2013), 123-141. doi: 10.1007/s00285-012-0537-6.
[5]	S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1.
[6]	S. A. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), 136-150. doi: 10.1002/zamm.200900295.
[7]	S. A. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method, Inverse Problems, 18 (2002), 349-361. doi: 10.1088/0266-5611/18/2/304.
[8]	S. A. Avdonin and V. Mikhailov, Controllability of partial differential equations on graphs, Appl. Math., 35 (2008), 379-393. doi: 10.4064/am35-4-1.
[9]	S. A. Avdonin and V. Mikhailov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19pp. doi: 10.1088/0266-5611/26/4/045009.
[10]	S. A. Avdonin, V. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl m-function, Comm. Math. Phys., 275 (2007), 791-803. doi: 10.1007/s00220-007-0315-2.
[11]	S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory, J. Neurophysiol, 65 (1991), 874-890.
[12]	M. I. Belishev, Canonical model of a dynamical system with boundary control in inverse problem for the heat equation, St. Petersburg Math. Journal, 7 (1996), 869-890.
[13]	M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: 10.1088/0266-5611/20/3/002.
[14]	M. Belishev and A. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46. doi: 10.1515/156939406776237474.
[15]	J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models, Math. Biosciences., 194 (2005), 1-19. doi: 10.1016/j.mbs.2004.07.001.
[16]	J. von Below, Parabolic Network Equations, Habilitation Thesis, Eberhard-Karls-Universitat Tubingen, 1993.
[17]	T. H. Brown, R. A. Friske and D. H. Perkel, Passive electrical constants in three classes of hippocampal neurons, J. Neurophysiol, 46 (1981), 812-827.
[18]	B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243. doi: 10.1098/rspa.2005.1513.
[19]	S. J. Cox, A new method for extracting cable parameters from input impedance data, Math. Biosci., 153 (1998), 1-12. doi: 10.1016/S0025-5564(98)10033-0.
[20]	S. J. Cox, An adjoint method for channel localization, Math. Medicine and Biology, 23 (2006), 139-152. doi: 10.1093/imammb/dql004.
[21]	S. J. Cox and B. Griffith, Recovering quasi-active properties of dendrites from dual potential recordings, J. Comput. Neurosci., 11 (2001), 95-110.
[22]	S. J. Cox and L. Ji, Identification of the cable parameters in the somatic shunt model, Biol. Cybern., 83 (2000), 151-159. doi: 10.1007/PL00007972.
[23]	S. J. Cox and L. Ji, Discerning ionic currents and their kinetics from input impedance data, Bull. Math. Biol., 63 (2001), 909-932. doi: 10.1006/bulm.2001.0250.
[24]	S. J. Cox and A. Wagner, Lateral overdetermination of the FitzHugh-Nagumo system, Inverse Problems, 20 (2004), 1639-1647. doi: 10.1088/0266-5611/20/5/019.
[25]	A. Däguanno, B. J. Bardakjian and P. L. Carlen, Passive neuronal membrane parameters: Comparison of optimization and peeling methods, IEEE Trans. Biomed. Eng., 33 (1986), 1188-1196.
[26]	P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, 2001.
[27]	D. M. Durand, P. L. Carlen, N. Gurevich, A. Ho and H. Kunov, Electrotonic parameters of rat dentate granule cells measured using short current pulses and HRP staining, J. Neurophysiol, 50 (1983), 1080-1097.
[28]	G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 86 (2007), 653-667. doi: 10.1080/00036810701303976.
[29]	B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A., 34 (2001), 6061-6068. doi: 10.1088/0305-4470/34/31/301.
[30]	W. R. Holmes and W. Rall, Estimating the electrotonic structure of neurons with compartmental models, J. Neurophysiol, 68 (1992), 1438-1452.
[31]	J. J. B. Jack and S. J. Redman, An electrical description of a motoneurone, and its application to the analysis of synaptic potentials, J. Physiol., 215 (1971), 321-352. doi: 10.1113/jphysiol.1971.sp009473.
[32]	M. Kawato, Cable properties of a neuron model with non-uniform membrane resistivity, J. Theor. Biol., 111 (1984), 149-169. doi: 10.1016/S0022-5193(84)80202-7.
[33]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996. doi: 10.1007/978-1-4612-5338-9.
[34]	C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, New York, 1999.
[35]	T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics, 274 (1999), 76-124. doi: 10.1006/aphy.1999.5904.
[36]	P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A., 38 (2005), 4901-4915. doi: 10.1088/0305-4470/38/22/014.
[37]	A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control and Optimization, 17 (1979), 494-499. doi: 10.1137/0317035.
[38]	W. Rall, Membrane potential transients and membrane time constants of motoneurons, Exp. Neurol., 2 (1960), 503-532. doi: 10.1016/0014-4886(60)90029-7.
[39]	W. Rall, Theory of physiological properties of dendrites, Ann. NY Acad. Sci., 96 (1962), 1071-1092. doi: 10.1111/j.1749-6632.1962.tb54120.x.
[40]	W. Rall, Core conductor theory and cable properties of neurons, in Handbook of Physiology. The Nervous System, American Physiological Society, 1977, 39-97. doi: 10.1002/cphy.cp010103.
[41]	W. Rall, R. E. Burke, W. R. Holmes, J. J. B. Jack, S. J. Redman and I. Segev, Matching dendritic neuron models to experimental data, Physiol. Rev., 172 (1992), S159-S186.
[42]	G. Stuart and N. Spruston, Determinants of voltage attenuation in neocortical pyramidal neuron dendrites, J. Neurosci., 18 (1998), 3501-3510.
[43]	A. K. Schierwagen, Identification problems in distributed parameter neuron models, Automatica, 26 (1990), 739-755. doi: 10.1016/0005-1098(90)90050-R.
[44]	A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1963.
[45]	D. Wang, Partial Differential Equation Constrained Optimization and its Applications to Parameter Estimation in Models of Nerve Dendrites, (unpublished) dissertation, UMBC, 2008.
[46]	J. A. White, P. B. Manis and E. D. Young, The parameter identification problem for the somatic shunt model, Biol. Cybern., 66 (1992), 307-318. doi: 10.1007/BF00203667.
[47]	V. Yurko, Inverse Sturm-Lioville operator on graphs, Inverse Problems, 21 (2005), 1075-1086.

show all references

References:

[1]	S. A. Avdonin, Control problems on quantum graphs, in Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, 77, AMS, Providence, RI, 2008, 507-521. doi: 10.1090/pspum/077/2459889.
[2]	S. A. Avdonin, B. P. Belinskiy and J. V. Matthews, Dynamical inverse problem on a metric tree, Inverse Problems, 27 (2011), 075011, 21pp. doi: 10.1088/0266-5611/27/7/075011.
[3]	S. A. Avdonin, M. I. Belishev and Yu. S. Rozhkov, The BC method in the inverse problem for the heat equation, J. Inverse and Ill-Posed Problems, 5 (1997), 309-322. doi: 10.1515/jiip.1997.5.4.309.
[4]	S. A. Avdonin and J. Bell, Determining a distributed parameter in a neural cable theory model via a boundary control method, J. Mathematical Biology, 67 (2013), 123-141. doi: 10.1007/s00285-012-0537-6.
[5]	S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Problems and Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1.
[6]	S. A. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010), 136-150. doi: 10.1002/zamm.200900295.
[7]	S. A. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method, Inverse Problems, 18 (2002), 349-361. doi: 10.1088/0266-5611/18/2/304.
[8]	S. A. Avdonin and V. Mikhailov, Controllability of partial differential equations on graphs, Appl. Math., 35 (2008), 379-393. doi: 10.4064/am35-4-1.
[9]	S. A. Avdonin and V. Mikhailov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19pp. doi: 10.1088/0266-5611/26/4/045009.
[10]	S. A. Avdonin, V. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl m-function, Comm. Math. Phys., 275 (2007), 791-803. doi: 10.1007/s00220-007-0315-2.
[11]	S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory, J. Neurophysiol, 65 (1991), 874-890.
[12]	M. I. Belishev, Canonical model of a dynamical system with boundary control in inverse problem for the heat equation, St. Petersburg Math. Journal, 7 (1996), 869-890.
[13]	M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: 10.1088/0266-5611/20/3/002.
[14]	M. Belishev and A. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46. doi: 10.1515/156939406776237474.
[15]	J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models, Math. Biosciences., 194 (2005), 1-19. doi: 10.1016/j.mbs.2004.07.001.
[16]	J. von Below, Parabolic Network Equations, Habilitation Thesis, Eberhard-Karls-Universitat Tubingen, 1993.
[17]	T. H. Brown, R. A. Friske and D. H. Perkel, Passive electrical constants in three classes of hippocampal neurons, J. Neurophysiol, 46 (1981), 812-827.
[18]	B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243. doi: 10.1098/rspa.2005.1513.
[19]	S. J. Cox, A new method for extracting cable parameters from input impedance data, Math. Biosci., 153 (1998), 1-12. doi: 10.1016/S0025-5564(98)10033-0.
[20]	S. J. Cox, An adjoint method for channel localization, Math. Medicine and Biology, 23 (2006), 139-152. doi: 10.1093/imammb/dql004.
[21]	S. J. Cox and B. Griffith, Recovering quasi-active properties of dendrites from dual potential recordings, J. Comput. Neurosci., 11 (2001), 95-110.
[22]	S. J. Cox and L. Ji, Identification of the cable parameters in the somatic shunt model, Biol. Cybern., 83 (2000), 151-159. doi: 10.1007/PL00007972.
[23]	S. J. Cox and L. Ji, Discerning ionic currents and their kinetics from input impedance data, Bull. Math. Biol., 63 (2001), 909-932. doi: 10.1006/bulm.2001.0250.
[24]	S. J. Cox and A. Wagner, Lateral overdetermination of the FitzHugh-Nagumo system, Inverse Problems, 20 (2004), 1639-1647. doi: 10.1088/0266-5611/20/5/019.
[25]	A. Däguanno, B. J. Bardakjian and P. L. Carlen, Passive neuronal membrane parameters: Comparison of optimization and peeling methods, IEEE Trans. Biomed. Eng., 33 (1986), 1188-1196.
[26]	P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT Press, 2001.
[27]	D. M. Durand, P. L. Carlen, N. Gurevich, A. Ho and H. Kunov, Electrotonic parameters of rat dentate granule cells measured using short current pulses and HRP staining, J. Neurophysiol, 50 (1983), 1080-1097.
[28]	G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 86 (2007), 653-667. doi: 10.1080/00036810701303976.
[29]	B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A., 34 (2001), 6061-6068. doi: 10.1088/0305-4470/34/31/301.
[30]	W. R. Holmes and W. Rall, Estimating the electrotonic structure of neurons with compartmental models, J. Neurophysiol, 68 (1992), 1438-1452.
[31]	J. J. B. Jack and S. J. Redman, An electrical description of a motoneurone, and its application to the analysis of synaptic potentials, J. Physiol., 215 (1971), 321-352. doi: 10.1113/jphysiol.1971.sp009473.
[32]	M. Kawato, Cable properties of a neuron model with non-uniform membrane resistivity, J. Theor. Biol., 111 (1984), 149-169. doi: 10.1016/S0022-5193(84)80202-7.
[33]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1996. doi: 10.1007/978-1-4612-5338-9.
[34]	C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, New York, 1999.
[35]	T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics, 274 (1999), 76-124. doi: 10.1006/aphy.1999.5904.
[36]	P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A., 38 (2005), 4901-4915. doi: 10.1088/0305-4470/38/22/014.
[37]	A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control and Optimization, 17 (1979), 494-499. doi: 10.1137/0317035.
[38]	W. Rall, Membrane potential transients and membrane time constants of motoneurons, Exp. Neurol., 2 (1960), 503-532. doi: 10.1016/0014-4886(60)90029-7.
[39]	W. Rall, Theory of physiological properties of dendrites, Ann. NY Acad. Sci., 96 (1962), 1071-1092. doi: 10.1111/j.1749-6632.1962.tb54120.x.
[40]	W. Rall, Core conductor theory and cable properties of neurons, in Handbook of Physiology. The Nervous System, American Physiological Society, 1977, 39-97. doi: 10.1002/cphy.cp010103.
[41]	W. Rall, R. E. Burke, W. R. Holmes, J. J. B. Jack, S. J. Redman and I. Segev, Matching dendritic neuron models to experimental data, Physiol. Rev., 172 (1992), S159-S186.
[42]	G. Stuart and N. Spruston, Determinants of voltage attenuation in neocortical pyramidal neuron dendrites, J. Neurosci., 18 (1998), 3501-3510.
[43]	A. K. Schierwagen, Identification problems in distributed parameter neuron models, Automatica, 26 (1990), 739-755. doi: 10.1016/0005-1098(90)90050-R.
[44]	A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1963.
[45]	D. Wang, Partial Differential Equation Constrained Optimization and its Applications to Parameter Estimation in Models of Nerve Dendrites, (unpublished) dissertation, UMBC, 2008.
[46]	J. A. White, P. B. Manis and E. D. Young, The parameter identification problem for the somatic shunt model, Biol. Cybern., 66 (1992), 307-318. doi: 10.1007/BF00203667.
[47]	V. Yurko, Inverse Sturm-Lioville operator on graphs, Inverse Problems, 21 (2005), 1075-1086.

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