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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 |
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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