Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Digraphs vs. dynamics in discrete models of neuronal networks
Pages: 1365 - 1381, Issue 5, July 2012

doi:10.3934/dcdsb.2012.17.1365      Abstract        References        Full text (394.5K)                  Related Articles

Sungwoo Ahn - Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, IN 46202, United States (email)
Winfried Just - Department of Mathematics, Ohio University, OH 45701, United States (email)

1 S. Ahn, "Transient and Attractor Dynamics in Models for Odor Discrimination," Ph.D thesis, The Ohio State University, 2010.       
2 S. Ahn, B. H. Smith, A. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics, Phys. D, 239 (2010), 515-528.       
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), 553-567.
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), 217-235.       
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), 525-531.
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), 8697-8702.
7 A. Destexhe and T. J. Sejnowski, Synchronized oscillations in thalamic networks: Insights from modeling studies, in "Thalamus" (ed. M. Steriade, E. G. Jones and D. A. McCormick), Elsevier, (1997), 331-371.
8 A. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of Necklaces and "Hermite Reciprocity," J. Algebraic Combin., 10 (1999), 173-188.       
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), 999-1012.
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), 10191-10202.
11 L. Glass, A topological theorem for nonlinear dynamics in chemical and ecological networks, Proc. Nat. Acad. Sci. U.S.A., 72 (1975), 2856-2857.       
12 D. Golomb, X. J. Wang and J. Rinzel, Synchronization properties of spindle oscillations in a thalamic reticular nucleus model, J. Neurophysiol., 72 (1994), 1109-1126.
13 W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Phys. D, 237 (2008), 3186-3196.       
14 W. Just, et al., More phase transitions in digraph systems, work in progress.
15 S. A. Kauffman, "The Origins of Order: Self-Organization and Selection in Evolution," Oxford University Press, Oxford, UK, 1993.
16 S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol., 22 (1969), 437-467.
17 E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen," Chelsea Publishing Co., 1974.
18 G. Laurent, Dynamical representation of odors by oscillating and evolving neural assemblies, Trends Neurosci., 19 (1996), 489-496.
19 O. Mazor and G. Laurent, Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons, Neuron, 48 (2005), 661-673.
20 E. Plahte, T. Mestl and S. W. Omholt, Feedback loops, stability and multistationarity in dynamical systems, J. Biol. Syst., 3 (1995), 409-413.
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), 335-350.       
22 E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3-9.
23 D. Terman, S. Ahn, X. Wang and W. Just, Reducing neuronal networks to discrete dynamics, Phys. D, 237 (2008), 324-338.       
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), 15417-15422.
25 D. Thieffry, Dynamical roles of biological regulatory circuits, Brief. Bioinform., 8 (2007), 220-225.
26 R. Thomas and R. D'Ari, "Biological Feedback," CRC Press, 1990.

Go to top