American Institute of Mathematical Sciences

Advanced
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

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 & 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, (2008), 507.  doi: 10.1090/pspum/077/2459889.  Google Scholar

[2]

S. A. Avdonin, B. P. Belinskiy and J. V. Matthews, Dynamical inverse problem on a metric tree,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/7/075011.  Google Scholar

[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.  doi: 10.1515/jiip.1997.5.4.309.  Google Scholar

[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.  doi: 10.1007/s00285-012-0537-6.  Google Scholar

[5]

S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees,, Inverse Problems and Imaging, 2 (2008), 1.  doi: 10.3934/ipi.2008.2.1.  Google Scholar

[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.  doi: 10.1002/zamm.200900295.  Google Scholar

[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.  doi: 10.1088/0266-5611/18/2/304.  Google Scholar

[8]

S. A. Avdonin and V. Mikhailov, Controllability of partial differential equations on graphs,, Appl. Math., 35 (2008), 379.  doi: 10.4064/am35-4-1.  Google Scholar

[9]

S. A. Avdonin and V. Mikhailov, The boundary control approach to inverse spectral theory,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

[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.  doi: 10.1007/s00220-007-0315-2.  Google Scholar

[11]

S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory,, J. Neurophysiol, 65 (1991), 874.   Google Scholar

[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.   Google Scholar

[13]

M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar

[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.  doi: 10.1515/156939406776237474.  Google Scholar

[15]

J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models,, Math. Biosciences., 194 (2005), 1.  doi: 10.1016/j.mbs.2004.07.001.  Google Scholar

[16]

J. von Below, Parabolic Network Equations,, Habilitation Thesis, (1993).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1098/rspa.2005.1513.  Google Scholar

[19]

S. J. Cox, A new method for extracting cable parameters from input impedance data,, Math. Biosci., 153 (1998), 1.  doi: 10.1016/S0025-5564(98)10033-0.  Google Scholar

[20]

S. J. Cox, An adjoint method for channel localization,, Math. Medicine and Biology, 23 (2006), 139.  doi: 10.1093/imammb/dql004.  Google Scholar

[21]

S. J. Cox and B. Griffith, Recovering quasi-active properties of dendrites from dual potential recordings,, J. Comput. Neurosci., 11 (2001), 95.   Google Scholar

[22]

S. J. Cox and L. Ji, Identification of the cable parameters in the somatic shunt model,, Biol. Cybern., 83 (2000), 151.  doi: 10.1007/PL00007972.  Google Scholar

[23]

S. J. Cox and L. Ji, Discerning ionic currents and their kinetics from input impedance data,, Bull. Math. Biol., 63 (2001), 909.  doi: 10.1006/bulm.2001.0250.  Google Scholar

[24]

S. J. Cox and A. Wagner, Lateral overdetermination of the FitzHugh-Nagumo system,, Inverse Problems, 20 (2004), 1639.  doi: 10.1088/0266-5611/20/5/019.  Google Scholar

[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.   Google Scholar

[26]

P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems,, MIT Press, (2001).   Google Scholar

[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.   Google Scholar

[28]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions,, Appl. Anal., 86 (2007), 653.  doi: 10.1080/00036810701303976.  Google Scholar

[29]

B. Gutkin and U. Smilansky, Can one hear the shape of a graph?,, J. Phys. A., 34 (2001), 6061.  doi: 10.1088/0305-4470/34/31/301.  Google Scholar

[30]

W. R. Holmes and W. Rall, Estimating the electrotonic structure of neurons with compartmental models,, J. Neurophysiol, 68 (1992), 1438.   Google Scholar

[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.  doi: 10.1113/jphysiol.1971.sp009473.  Google Scholar

[32]

M. Kawato, Cable properties of a neuron model with non-uniform membrane resistivity,, J. Theor. Biol., 111 (1984), 149.  doi: 10.1016/S0022-5193(84)80202-7.  Google Scholar

[33]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Springer, (1996).  doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[34]

C. Koch, Biophysics of Computation: Information Processing in Single Neurons,, Oxford University Press, (1999).   Google Scholar

[35]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs,, Ann. Physics, 274 (1999), 76.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[36]

P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs,, J. Phys. A., 38 (2005), 4901.  doi: 10.1088/0305-4470/38/22/014.  Google Scholar

[37]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem,, SIAM J. Control and Optimization, 17 (1979), 494.  doi: 10.1137/0317035.  Google Scholar

[38]

W. Rall, Membrane potential transients and membrane time constants of motoneurons,, Exp. Neurol., 2 (1960), 503.  doi: 10.1016/0014-4886(60)90029-7.  Google Scholar

[39]

W. Rall, Theory of physiological properties of dendrites,, Ann. NY Acad. Sci., 96 (1962), 1071.  doi: 10.1111/j.1749-6632.1962.tb54120.x.  Google Scholar

[40]

W. Rall, Core conductor theory and cable properties of neurons,, in Handbook of Physiology. The Nervous System, (1977), 39.  doi: 10.1002/cphy.cp010103.  Google Scholar

[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).   Google Scholar

[42]

G. Stuart and N. Spruston, Determinants of voltage attenuation in neocortical pyramidal neuron dendrites,, J. Neurosci., 18 (1998), 3501.   Google Scholar

[43]

A. K. Schierwagen, Identification problems in distributed parameter neuron models,, Automatica, 26 (1990), 739.  doi: 10.1016/0005-1098(90)90050-R.  Google Scholar

[44]

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics,, Dover Publications, (1963).   Google Scholar

[45]

D. Wang, Partial Differential Equation Constrained Optimization and its Applications to Parameter Estimation in Models of Nerve Dendrites,, (unpublished) dissertation, (2008).   Google Scholar

[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.  doi: 10.1007/BF00203667.  Google Scholar

[47]

V. Yurko, Inverse Sturm-Lioville operator on graphs,, Inverse Problems, 21 (2005), 1075.   Google Scholar

show all references

References:
[1]

S. A. Avdonin, Control problems on quantum graphs,, in Analysis on Graphs and Its Applications, (2008), 507.  doi: 10.1090/pspum/077/2459889.  Google Scholar

[2]

S. A. Avdonin, B. P. Belinskiy and J. V. Matthews, Dynamical inverse problem on a metric tree,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/7/075011.  Google Scholar

[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.  doi: 10.1515/jiip.1997.5.4.309.  Google Scholar

[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.  doi: 10.1007/s00285-012-0537-6.  Google Scholar

[5]

S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees,, Inverse Problems and Imaging, 2 (2008), 1.  doi: 10.3934/ipi.2008.2.1.  Google Scholar

[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.  doi: 10.1002/zamm.200900295.  Google Scholar

[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.  doi: 10.1088/0266-5611/18/2/304.  Google Scholar

[8]

S. A. Avdonin and V. Mikhailov, Controllability of partial differential equations on graphs,, Appl. Math., 35 (2008), 379.  doi: 10.4064/am35-4-1.  Google Scholar

[9]

S. A. Avdonin and V. Mikhailov, The boundary control approach to inverse spectral theory,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

[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.  doi: 10.1007/s00220-007-0315-2.  Google Scholar

[11]

S. M. Baer and J. Rinzel, Propagation of dendritic spikes mediated by excitable spines: A continuum theory,, J. Neurophysiol, 65 (1991), 874.   Google Scholar

[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.   Google Scholar

[13]

M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method,, Inverse Problems, 20 (2004), 647.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar

[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.  doi: 10.1515/156939406776237474.  Google Scholar

[15]

J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models,, Math. Biosciences., 194 (2005), 1.  doi: 10.1016/j.mbs.2004.07.001.  Google Scholar

[16]

J. von Below, Parabolic Network Equations,, Habilitation Thesis, (1993).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1098/rspa.2005.1513.  Google Scholar

[19]

S. J. Cox, A new method for extracting cable parameters from input impedance data,, Math. Biosci., 153 (1998), 1.  doi: 10.1016/S0025-5564(98)10033-0.  Google Scholar

[20]

S. J. Cox, An adjoint method for channel localization,, Math. Medicine and Biology, 23 (2006), 139.  doi: 10.1093/imammb/dql004.  Google Scholar

[21]

S. J. Cox and B. Griffith, Recovering quasi-active properties of dendrites from dual potential recordings,, J. Comput. Neurosci., 11 (2001), 95.   Google Scholar

[22]

S. J. Cox and L. Ji, Identification of the cable parameters in the somatic shunt model,, Biol. Cybern., 83 (2000), 151.  doi: 10.1007/PL00007972.  Google Scholar

[23]

S. J. Cox and L. Ji, Discerning ionic currents and their kinetics from input impedance data,, Bull. Math. Biol., 63 (2001), 909.  doi: 10.1006/bulm.2001.0250.  Google Scholar

[24]

S. J. Cox and A. Wagner, Lateral overdetermination of the FitzHugh-Nagumo system,, Inverse Problems, 20 (2004), 1639.  doi: 10.1088/0266-5611/20/5/019.  Google Scholar

[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.   Google Scholar

[26]

P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems,, MIT Press, (2001).   Google Scholar

[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.   Google Scholar

[28]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions,, Appl. Anal., 86 (2007), 653.  doi: 10.1080/00036810701303976.  Google Scholar

[29]

B. Gutkin and U. Smilansky, Can one hear the shape of a graph?,, J. Phys. A., 34 (2001), 6061.  doi: 10.1088/0305-4470/34/31/301.  Google Scholar

[30]

W. R. Holmes and W. Rall, Estimating the electrotonic structure of neurons with compartmental models,, J. Neurophysiol, 68 (1992), 1438.   Google Scholar

[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.  doi: 10.1113/jphysiol.1971.sp009473.  Google Scholar

[32]

M. Kawato, Cable properties of a neuron model with non-uniform membrane resistivity,, J. Theor. Biol., 111 (1984), 149.  doi: 10.1016/S0022-5193(84)80202-7.  Google Scholar

[33]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Springer, (1996).  doi: 10.1007/978-1-4612-5338-9.  Google Scholar

[34]

C. Koch, Biophysics of Computation: Information Processing in Single Neurons,, Oxford University Press, (1999).   Google Scholar

[35]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs,, Ann. Physics, 274 (1999), 76.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[36]

P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs,, J. Phys. A., 38 (2005), 4901.  doi: 10.1088/0305-4470/38/22/014.  Google Scholar

[37]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem,, SIAM J. Control and Optimization, 17 (1979), 494.  doi: 10.1137/0317035.  Google Scholar

[38]

W. Rall, Membrane potential transients and membrane time constants of motoneurons,, Exp. Neurol., 2 (1960), 503.  doi: 10.1016/0014-4886(60)90029-7.  Google Scholar

[39]

W. Rall, Theory of physiological properties of dendrites,, Ann. NY Acad. Sci., 96 (1962), 1071.  doi: 10.1111/j.1749-6632.1962.tb54120.x.  Google Scholar

[40]

W. Rall, Core conductor theory and cable properties of neurons,, in Handbook of Physiology. The Nervous System, (1977), 39.  doi: 10.1002/cphy.cp010103.  Google Scholar

[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).   Google Scholar

[42]

G. Stuart and N. Spruston, Determinants of voltage attenuation in neocortical pyramidal neuron dendrites,, J. Neurosci., 18 (1998), 3501.   Google Scholar

[43]

A. K. Schierwagen, Identification problems in distributed parameter neuron models,, Automatica, 26 (1990), 739.  doi: 10.1016/0005-1098(90)90050-R.  Google Scholar

[44]

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics,, Dover Publications, (1963).   Google Scholar

[45]

D. Wang, Partial Differential Equation Constrained Optimization and its Applications to Parameter Estimation in Models of Nerve Dendrites,, (unpublished) dissertation, (2008).   Google Scholar

[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.  doi: 10.1007/BF00203667.  Google Scholar

[47]

V. Yurko, Inverse Sturm-Lioville operator on graphs,, Inverse Problems, 21 (2005), 1075.   Google Scholar

[1]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[2]

Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093
[3]

Jie Chen, Maarten de Hoop. The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter. Inverse Problems & Imaging, 2016, 10 (3) : 641-658. doi: 10.3934/ipi.2016015
[4]

Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
[5]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[6]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[7]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17
[8]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
[9]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[10]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[11]

Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404
[12]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
[13]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036
[14]

Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
[15]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
[16]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[17]

Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015
[18]

Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139
[19]

Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001
[20]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences