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

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

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.
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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