American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. 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), 10191-10202. doi: 10.1523/JNEUROSCI.1874-09.2009.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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, Oxford, UK, 1993. Google Scholar

[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.  Google Scholar

[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), 489-496. doi: 10.1016/S0166-2236(96)10054-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3-9. doi: 10.1142/S0218339098000042.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

D. Thieffry, Dynamical roles of biological regulatory circuits, Brief. Bioinform., 8 (2007), 220-225. doi: 10.1093/bib/bbm028.  Google Scholar

[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, The Ohio State University, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. 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), 10191-10202. doi: 10.1523/JNEUROSCI.1874-09.2009.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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, Oxford, UK, 1993. Google Scholar

[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.  Google Scholar

[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), 489-496. doi: 10.1016/S0166-2236(96)10054-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3-9. doi: 10.1142/S0218339098000042.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

D. Thieffry, Dynamical roles of biological regulatory circuits, Brief. Bioinform., 8 (2007), 220-225. doi: 10.1093/bib/bbm028.  Google Scholar

[26]

R. Thomas and R. D'Ari, "Biological Feedback," CRC Press, 1990. Google Scholar

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