Digraphs vs. dynamics in discrete models of neuronal networks

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July 2012, 17(5): 1365-1381. doi: 10.3934/dcdsb.2012.17.1365

Digraphs vs. dynamics in discrete models of neuronal networks

Sungwoo Ahn ^1, and Winfried Just ^2,

1.	Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, IN 46202, United States
2.	Department of Mathematics, Ohio University, OH 45701, United States

Received January 2011 Revised December 2011 Published March 2012

Abstract
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Under a Creative Commons license

It has recently been shown that discrete-time finite-state models can reliably reproduce the ordinary differential equation (ODE) dynamics of certain neuronal networks. We study which dynamics are possible in these discrete models for certain types of network connectivities. In particular we are interested in the number of different attractors and bounds on the lengths of attractors and transients. We completely characterize these properties for cyclic connectivities and derive additional results on the lengths of attractors in more general classes of networks.

Keywords: directed cycle., neuronal network, transient, Discrete dynamical system, attractor.

Mathematics Subject Classification: 05C20, 05C82, 37F20, 92C20, 92C4.

Citation: Sungwoo Ahn, Winfried Just. Digraphs vs. dynamics in discrete models of neuronal networks. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1365-1381. doi: 10.3934/dcdsb.2012.17.1365

References:

[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. 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), 553-567. 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), 217-235. 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), 525-531. 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), 8697-8702. doi: 10.1073/pnas.0602767103.
[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. 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), 999-1012.
[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. 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), 2856-2857. 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), 1109-1126.
[13]	W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Phys. D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.
[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. doi: 10.1016/0022-5193(69)90015-0.
[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. 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), 661-673. 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), 409-413. 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), 335-350. doi: 10.1016/j.aam.2007.11.003.
[22]	E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3-9. doi: 10.1142/S0218339098000042.
[23]	D. Terman, S. Ahn, X. Wang and W. Just, Reducing neuronal networks to discrete dynamics, Phys. D, 237 (2008), 324-338. 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), 15417-15422. doi: 10.1073/pnas.93.26.15417.
[25]	D. Thieffry, Dynamical roles of biological regulatory circuits, Brief. Bioinform., 8 (2007), 220-225. doi: 10.1093/bib/bbm028.
[26]	R. Thomas and R. D'Ari, "Biological Feedback," CRC Press, 1990.

show all references

References:

[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. 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), 553-567. 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), 217-235. 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), 525-531. 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), 8697-8702. doi: 10.1073/pnas.0602767103.
[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. 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), 999-1012.
[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. 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), 2856-2857. 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), 1109-1126.
[13]	W. Just, S. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Phys. D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.
[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. doi: 10.1016/0022-5193(69)90015-0.
[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. 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), 661-673. 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), 409-413. 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), 335-350. doi: 10.1016/j.aam.2007.11.003.
[22]	E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3-9. doi: 10.1142/S0218339098000042.
[23]	D. Terman, S. Ahn, X. Wang and W. Just, Reducing neuronal networks to discrete dynamics, Phys. D, 237 (2008), 324-338. 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), 15417-15422. doi: 10.1073/pnas.93.26.15417.
[25]	D. Thieffry, Dynamical roles of biological regulatory circuits, Brief. Bioinform., 8 (2007), 220-225. doi: 10.1093/bib/bbm028.
[26]	R. Thomas and R. D'Ari, "Biological Feedback," CRC Press, 1990.

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