# American Institute of Mathematical Sciences

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

## 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

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.
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.
 [1] Nataša Djurdjevac Conrad, Ralf Banisch, Christof Schütte. Modularity of directed networks: Cycle decomposition approach. Journal of Computational Dynamics, 2015, 2 (1) : 1-24. doi: 10.3934/jcd.2015.2.1 [2] Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091 [3] Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899 [4] Mustapha Yebdri. Existence of $\mathcal{D}-$pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036 [5] Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 [6] T. Jäger. Neuronal coding of pacemaker neurons -- A random dynamical systems approach. Communications on Pure and Applied Analysis, 2011, 10 (3) : 995-1009. doi: 10.3934/cpaa.2011.10.995 [7] S. Mohamad, K. Gopalsamy. Neuronal dynamics in time varying enviroments: Continuous and discrete time models. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 841-860. doi: 10.3934/dcds.2000.6.841 [8] Gheorghe Craciun, Baltazar Aguda, Avner Friedman. Mathematical Analysis Of A Modular Network Coordinating The Cell Cycle And Apoptosis. Mathematical Biosciences & Engineering, 2005, 2 (3) : 473-485. doi: 10.3934/mbe.2005.2.473 [9] Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91 [10] Chiun-Chuan Chen, Ting-Yang Hsiao, Li-Chang Hung. Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 153-187. doi: 10.3934/dcds.2020007 [11] Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699 [12] Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 [13] Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893 [14] Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264 [15] P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692 [16] Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 [17] Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229 [18] Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757 [19] A. Azhagappan, T. Deepa. Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2843-2856. doi: 10.3934/jimo.2019083 [20] Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665

2021 Impact Factor: 1.497