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Digraphs vs. dynamics in discrete models of neuronal networks
1. | Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, IN 46202, United States |
2. | Department of Mathematics, Ohio University, OH 45701, United States |
References:
[1] |
S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination,", Ph.D thesis, (2010).
|
[2] |
S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics,, Phys. D, 239 (2010), 515.
doi: 10.1016/j.physd.2009.12.011. |
[3] |
M. Bazhenov, M. Stopfer, M. Rabinovich, R. Huerta, H. D. Abarbanel, T. J. Sejnowski and G. Laurent, Model of transient oscillatory synchronization in the locust antennal lobe,, Neuron, 30 (2001), 553.
doi: 10.1016/S0896-6273(01)00284-7. |
[4] |
J. Best, C. Park, D. Terman and C. J. Wilson, Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks,, J. Comput. Neurosci., 23 (2007), 217.
doi: 10.1007/s10827-007-0029-7. |
[5] |
M. D. Bevan, P. J. Magill, D. Terman, J. P. Bolam and C. J. Wilson, Move to the rhythm: Oscillations in the subthalamic nucleus-external globus pallidus network,, Trends Neurosci., 25 (2002), 525.
doi: 10.1016/S0166-2236(02)02235-X. |
[6] |
G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks,, Proc. Nat. Acad. Sci. U.S.A., 103 (2006), 8697.
doi: 10.1073/pnas.0602767103. |
[7] |
A. Destexhe and T. J. Sejnowski, Synchronized oscillations in thalamic networks: Insights from modeling studies,, in, (1997), 331. Google Scholar |
[8] |
A. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of Necklaces and "Hermite Reciprocity,", J. Algebraic Combin., 10 (1999), 173.
doi: 10.1023/A:1018727630642. |
[9] |
R. Fdez Galán, S. Sachse, C. G. Galizia and A. V. Herz, Odor-driven attractor dynamics in the antennal lobe allow for simple and rapid olfactory pattern classification,, Neural Comput., 16 (2004), 999. Google Scholar |
[10] |
P. C. Fernandez, F. F. Locatelli, N. Person-Rennell, G. Deleo and B. H. Smith, Associative conditioning tunes transient dynamics of early olfactory processing,, J. Neurosci., 29 (2009), 10191.
doi: 10.1523/JNEUROSCI.1874-09.2009. |
[11] |
L. Glass, A topological theorem for nonlinear dynamics in chemical and ecological networks,, Proc. Nat. Acad. Sci. U.S.A., 72 (1975), 2856.
doi: 10.1073/pnas.72.8.2856. |
[12] |
D. Golomb, X. J. Wang and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model,, J. Neurophysiol., 72 (1994), 1109. Google Scholar |
[13] |
W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks,, Phys. D, 237 (2008), 3186.
doi: 10.1016/j.physd.2008.08.011. |
[14] |
W. Just, et al., More phase transitions in digraph systems,, work in progress., (). Google Scholar |
[15] |
S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution,", Oxford University Press, (1993). Google Scholar |
[16] |
S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets,, J. Theor. Biol., 22 (1969), 437.
doi: 10.1016/0022-5193(69)90015-0. |
[17] |
E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen,", Chelsea Publishing Co., (1974). Google Scholar |
[18] |
G. Laurent, Dynamical representation of odors by oscillating and evolving neural assemblies,, Trends Neurosci., 19 (1996), 489.
doi: 10.1016/S0166-2236(96)10054-0. |
[19] |
O. Mazor and G. Laurent, Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons,, Neuron, 48 (2005), 661.
doi: 10.1016/j.neuron.2005.09.032. |
[20] |
E. Plahte, T. Mestl and S. W. Omholt, Feedback loops, stability and multistationarity in dynamical systems,, J. Biol. Syst., 3 (1995), 409.
doi: 10.1142/S0218339095000381. |
[21] |
É. Remy, P. Ruet and D. Thieffry, Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework,, Adv. in Appl. Math., 41 (2008), 335.
doi: 10.1016/j.aam.2007.11.003. |
[22] |
E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity,, J. Biol. Syst., 6 (1998), 3.
doi: 10.1142/S0218339098000042. |
[23] |
D. Terman, S. Ahn, X. Wang and W. Just, Reducing neuronal networks to discrete dynamics,, Phys. D, 237 (2008), 324.
doi: 10.1016/j.physd.2007.09.011. |
[24] |
D. Terman, A. Bose and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms,, Proc. Nat. Acad. Sci. U.S.A., 93 (1996), 15417.
doi: 10.1073/pnas.93.26.15417. |
[25] |
D. Thieffry, Dynamical roles of biological regulatory circuits,, Brief. Bioinform., 8 (2007), 220.
doi: 10.1093/bib/bbm028. |
[26] |
R. Thomas and R. D'Ari, "Biological Feedback,", CRC Press, (1990). Google Scholar |
show all references
References:
[1] |
S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination,", Ph.D thesis, (2010).
|
[2] |
S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics,, Phys. D, 239 (2010), 515.
doi: 10.1016/j.physd.2009.12.011. |
[3] |
M. Bazhenov, M. Stopfer, M. Rabinovich, R. Huerta, H. D. Abarbanel, T. J. Sejnowski and G. Laurent, Model of transient oscillatory synchronization in the locust antennal lobe,, Neuron, 30 (2001), 553.
doi: 10.1016/S0896-6273(01)00284-7. |
[4] |
J. Best, C. Park, D. Terman and C. J. Wilson, Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks,, J. Comput. Neurosci., 23 (2007), 217.
doi: 10.1007/s10827-007-0029-7. |
[5] |
M. D. Bevan, P. J. Magill, D. Terman, J. P. Bolam and C. J. Wilson, Move to the rhythm: Oscillations in the subthalamic nucleus-external globus pallidus network,, Trends Neurosci., 25 (2002), 525.
doi: 10.1016/S0166-2236(02)02235-X. |
[6] |
G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks,, Proc. Nat. Acad. Sci. U.S.A., 103 (2006), 8697.
doi: 10.1073/pnas.0602767103. |
[7] |
A. Destexhe and T. J. Sejnowski, Synchronized oscillations in thalamic networks: Insights from modeling studies,, in, (1997), 331. Google Scholar |
[8] |
A. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of Necklaces and "Hermite Reciprocity,", J. Algebraic Combin., 10 (1999), 173.
doi: 10.1023/A:1018727630642. |
[9] |
R. Fdez Galán, S. Sachse, C. G. Galizia and A. V. Herz, Odor-driven attractor dynamics in the antennal lobe allow for simple and rapid olfactory pattern classification,, Neural Comput., 16 (2004), 999. Google Scholar |
[10] |
P. C. Fernandez, F. F. Locatelli, N. Person-Rennell, G. Deleo and B. H. Smith, Associative conditioning tunes transient dynamics of early olfactory processing,, J. Neurosci., 29 (2009), 10191.
doi: 10.1523/JNEUROSCI.1874-09.2009. |
[11] |
L. Glass, A topological theorem for nonlinear dynamics in chemical and ecological networks,, Proc. Nat. Acad. Sci. U.S.A., 72 (1975), 2856.
doi: 10.1073/pnas.72.8.2856. |
[12] |
D. Golomb, X. J. Wang and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model,, J. Neurophysiol., 72 (1994), 1109. Google Scholar |
[13] |
W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks,, Phys. D, 237 (2008), 3186.
doi: 10.1016/j.physd.2008.08.011. |
[14] |
W. Just, et al., More phase transitions in digraph systems,, work in progress., (). Google Scholar |
[15] |
S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution,", Oxford University Press, (1993). Google Scholar |
[16] |
S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets,, J. Theor. Biol., 22 (1969), 437.
doi: 10.1016/0022-5193(69)90015-0. |
[17] |
E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen,", Chelsea Publishing Co., (1974). Google Scholar |
[18] |
G. Laurent, Dynamical representation of odors by oscillating and evolving neural assemblies,, Trends Neurosci., 19 (1996), 489.
doi: 10.1016/S0166-2236(96)10054-0. |
[19] |
O. Mazor and G. Laurent, Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons,, Neuron, 48 (2005), 661.
doi: 10.1016/j.neuron.2005.09.032. |
[20] |
E. Plahte, T. Mestl and S. W. Omholt, Feedback loops, stability and multistationarity in dynamical systems,, J. Biol. Syst., 3 (1995), 409.
doi: 10.1142/S0218339095000381. |
[21] |
É. Remy, P. Ruet and D. Thieffry, Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework,, Adv. in Appl. Math., 41 (2008), 335.
doi: 10.1016/j.aam.2007.11.003. |
[22] |
E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity,, J. Biol. Syst., 6 (1998), 3.
doi: 10.1142/S0218339098000042. |
[23] |
D. Terman, S. Ahn, X. Wang and W. Just, Reducing neuronal networks to discrete dynamics,, Phys. D, 237 (2008), 324.
doi: 10.1016/j.physd.2007.09.011. |
[24] |
D. Terman, A. Bose and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms,, Proc. Nat. Acad. Sci. U.S.A., 93 (1996), 15417.
doi: 10.1073/pnas.93.26.15417. |
[25] |
D. Thieffry, Dynamical roles of biological regulatory circuits,, Brief. Bioinform., 8 (2007), 220.
doi: 10.1093/bib/bbm028. |
[26] |
R. Thomas and R. D'Ari, "Biological Feedback,", CRC Press, (1990). Google Scholar |
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