July 2018, 23(5): 1917-1930. doi: 10.3934/dcdsb.2018188

Approximating network dynamics: Some open problems

Department of Mathematics, Ohio University, Athens, OH 45701, USA

Received  April 2017 Revised  June 2017 Published  May 2018

Discrete-time finite-state dynamical systems on networks are often conceived as tractable approximations to more detailed ODE-based models of natural systems. Here we review research on a class of such discrete models $N$ that approximate certain ODE models $M$ of mathematical neuroscience. In particular, we outline several open problems on the dynamics of the models $N$ themselves, as well as on structural features of ODE models $M$ that allow for the construction of discrete approximations $N$ whose predictions will be consistent with those of $M$.

Citation: Winfried Just. Approximating network dynamics: Some open problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1917-1930. doi: 10.3934/dcdsb.2018188
References:
[1]

E. Ackermann, T. P. Peixoto and B. Drossel, Reliable dynamics in Boolean and continuous networks, New Journal of Physics, 14 (2012), 123029, 17pp. doi: 10.1088/1367-2630/14/12/123029.

[2]

E. Ackermann, E. M. Weiel and T. Pfaff, Boolean versus continuous dynamics in modules with two feedback loops, European Physical Journal E, 35 (2012), p107. doi: 10.1140/epje/i2012-12107-9.

[3]

S. Ahn, Transient and Attractor Dynamics in Models for Odor Discrimination, Ph. D. Thesis, The Ohio State University, 2010. Available from: https://etd.ohiolink.edu/rws_etd/document/get/osu1280342970/inline.

[4]

S. Ahn and W. Just, Digraphs vs. dynamics in discrete models of neuronal networks, Disc. Contin. Dyn. Syst. Ser. B, 17 (2012), 1365-1381. doi: 10.3934/dcdsb.2012.17.1365.

[5]

S. AhnB. H. SmithA. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics, Physica D, 239 (2010), 515-528. doi: 10.1016/j.physd.2009.12.011.

[6]

M. Aldana, S. Coppersmith and L. P. Kadanoff, Boolean dynamics with random couplings, in: Perspectives and Problems in Nonlinear Science (eds. E. Kaplan, J. E. Marsden and K. R. Sreenivasan), Springer, (2003), 23–89.

[7]

A. D. Barbour, L. Holst and S. Janson, Poisson Approximation, Clarendon Press, Oxford, UK, 1992.

[8]

J. M. Bower and H. Bolouri (eds.), Computational Modeling of Genetic and Biochemical networks, MIT Press, Cambridge, MA, 2001.

[9]

S. Braunewell and S. Bornholdt, Superstability of the yeast cell-cycle dynamics: Ensuring causality in the presence of biochemical stochasticity, J. Theor. Biol., 245 (2007), 638-643. doi: 10.1016/j.jtbi.2006.11.012.

[10]

B. Drossel, Random boolean networks, in: Reviews of Nonlinear Dynamics and Complexity, Volume 1 (ed. H. G. Schuster), Wiley, (2008), 69–110.

[11]

R. Edwards, Analysis of continuous-time switching networks, Physica D, 146 (2000), 165-199. doi: 10.1016/S0167-2789(00)00130-5.

[12]

J. Epperlein and S. Siegmund, Phase-locked trajectories for dynamical systems on graphs, Disc. Contin. Dyn. Syst. Ser. B, 18 (2013), 1827-1844. doi: 10.3934/dcdsb.2013.18.1827.

[13]

M. Franceschetti and R. Meester, Critical node lifetimes in random networks via the Chen-Stein method, IEEE Trans. Inf. Theory, 52 (2006), 2831-2837. doi: 10.1109/TIT.2006.874545.

[14]

N. Friedman, S. Ito, B. A. Brinkman, M. Shimono, R. E. DeVille, K. A. Dahmen, J. M. Beggs and T. C. Butler, Universal critical dynamics in high resolution neuronal avalanche data, Phys. Rev. Lett. , 108 (2012), 208102. doi: 10.1103/PhysRevLett.108.208102.

[15]

E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules, Phys. Rev. E, 82 (2010), 046120, 9pp. doi: 10.1103/PhysRevE.82.046120.

[16]

W. Just and S. Ahn, Lengths of attractors and transients in neuronal networks with random connectivities preprint, (2014) arXiv: 1404.5536. doi: 10.1137/140996045.

[17]

W. Just and S. Ahn, Lengths of attractors and transients in neuronal networks with random connectivities, SIAM J. Disc. Math., 30 (2016), 912-933. doi: 10.1137/140996045.

[18]

W. JustS. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Physica D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.

[19]

W. Just, S. Ahn and D. Terman, Neuronal networks: A discrete model, in Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models (eds. R. Robeva and T. Hodge), Academic Press, (2013), 179–211.

[20]

W. JustM. KorbB. Elbert and T. R. Young, Two classes of ODE models with switch-like behavior, Physica D, 264 (2013), 35-48. doi: 10.1016/j.physd.2013.08.008.

[21]

R. K. C., Random Connectivities in Neuronal and Other Biologically Relevant Networks, Ph. D. Dissertation Proposal, Ohio University, May 2016.

[22]

S. A. Kauffman, Origins of Order: Self-Organization and Selection in Evolution, Oxford U Press, Oxford, UK, 1993.

[23]

G. Laurent, Olfactory network dynamics and the coding of multidimensional signals, Nat. Rev. Neurosci., 3 (2002), 884-895. doi: 10.1038/nrn964.

[24]

F. LiT. LongY. LuQ. Ouyang and Ch. Tang, The yeast cell-cycle network is robustly designed, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 4781-4786. doi: 10.1073/pnas.0305937101.

[25]

A. PolynikisS. J. Hogan and M. di Bernardo, Comparing different ODE modelling approaches for gene regulatory networks, J. Theoret. Biol., 261 (2009), 511-530. doi: 10.1016/j.jtbi.2009.07.040.

[26]

J. RinzelD. TermanX.-J. Wang and B. Ermentrout, Propagating activity patterns in large-scale inhibitory neuronal networks, Science, 279 (1998), 1351-1355. doi: 10.1126/science.279.5355.1351.

[27]

A. Saadatpour and R. Albert, A comparative study of qualitative and quantitative dynamic models of biological regulatory networks, EPJ Nonlinear Biomedical Physics, 4 (2016), 5pp.

[28]

S. Scarpetta and A. de Candia, Neural avalanches at the critical point between replay and non-replay of spatiotemporal patterns, PLoS One, 8 (2013), e64162. doi: 10.1371/journal.pone.0064162.

[29]

T. U. SinghK. ManchandaR. Ramaswamy and A. Bose, Excitable nodes on random graphs: Relating dynamics to network structure, SIAM J. Appl. Dyn. Syst., 10 (2011), 987-1012. doi: 10.1137/100802864.

[30]

D. TermanS. AhnX. Wang and W. Just, Reducing neuronal networks to discrete dynamics, Physica D, 237 (2008), 324-338. doi: 10.1016/j.physd.2007.09.011.

[31]

D. TermanA. Bose and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), 15417-15422. doi: 10.1073/pnas.93.26.15417.

[32]

D. M. Wittmann, J. Krumsiek, J. Saez-Rodriguez, D. A. Lauffenburger, S. Klamt and F. J. Theis, Transforming Boolean models to continuous models: Methodology and application to T -cell receptor signaling, BMC Syst. Biol., 3 (2009), p98. doi: 10.1186/1752-0509-3-98.

show all references

References:
[1]

E. Ackermann, T. P. Peixoto and B. Drossel, Reliable dynamics in Boolean and continuous networks, New Journal of Physics, 14 (2012), 123029, 17pp. doi: 10.1088/1367-2630/14/12/123029.

[2]

E. Ackermann, E. M. Weiel and T. Pfaff, Boolean versus continuous dynamics in modules with two feedback loops, European Physical Journal E, 35 (2012), p107. doi: 10.1140/epje/i2012-12107-9.

[3]

S. Ahn, Transient and Attractor Dynamics in Models for Odor Discrimination, Ph. D. Thesis, The Ohio State University, 2010. Available from: https://etd.ohiolink.edu/rws_etd/document/get/osu1280342970/inline.

[4]

S. Ahn and W. Just, Digraphs vs. dynamics in discrete models of neuronal networks, Disc. Contin. Dyn. Syst. Ser. B, 17 (2012), 1365-1381. doi: 10.3934/dcdsb.2012.17.1365.

[5]

S. AhnB. H. SmithA. Borisyuk and D. Terman, Analyzing neuronal networks using discrete-time dynamics, Physica D, 239 (2010), 515-528. doi: 10.1016/j.physd.2009.12.011.

[6]

M. Aldana, S. Coppersmith and L. P. Kadanoff, Boolean dynamics with random couplings, in: Perspectives and Problems in Nonlinear Science (eds. E. Kaplan, J. E. Marsden and K. R. Sreenivasan), Springer, (2003), 23–89.

[7]

A. D. Barbour, L. Holst and S. Janson, Poisson Approximation, Clarendon Press, Oxford, UK, 1992.

[8]

J. M. Bower and H. Bolouri (eds.), Computational Modeling of Genetic and Biochemical networks, MIT Press, Cambridge, MA, 2001.

[9]

S. Braunewell and S. Bornholdt, Superstability of the yeast cell-cycle dynamics: Ensuring causality in the presence of biochemical stochasticity, J. Theor. Biol., 245 (2007), 638-643. doi: 10.1016/j.jtbi.2006.11.012.

[10]

B. Drossel, Random boolean networks, in: Reviews of Nonlinear Dynamics and Complexity, Volume 1 (ed. H. G. Schuster), Wiley, (2008), 69–110.

[11]

R. Edwards, Analysis of continuous-time switching networks, Physica D, 146 (2000), 165-199. doi: 10.1016/S0167-2789(00)00130-5.

[12]

J. Epperlein and S. Siegmund, Phase-locked trajectories for dynamical systems on graphs, Disc. Contin. Dyn. Syst. Ser. B, 18 (2013), 1827-1844. doi: 10.3934/dcdsb.2013.18.1827.

[13]

M. Franceschetti and R. Meester, Critical node lifetimes in random networks via the Chen-Stein method, IEEE Trans. Inf. Theory, 52 (2006), 2831-2837. doi: 10.1109/TIT.2006.874545.

[14]

N. Friedman, S. Ito, B. A. Brinkman, M. Shimono, R. E. DeVille, K. A. Dahmen, J. M. Beggs and T. C. Butler, Universal critical dynamics in high resolution neuronal avalanche data, Phys. Rev. Lett. , 108 (2012), 208102. doi: 10.1103/PhysRevLett.108.208102.

[15]

E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules, Phys. Rev. E, 82 (2010), 046120, 9pp. doi: 10.1103/PhysRevE.82.046120.

[16]

W. Just and S. Ahn, Lengths of attractors and transients in neuronal networks with random connectivities preprint, (2014) arXiv: 1404.5536. doi: 10.1137/140996045.

[17]

W. Just and S. Ahn, Lengths of attractors and transients in neuronal networks with random connectivities, SIAM J. Disc. Math., 30 (2016), 912-933. doi: 10.1137/140996045.

[18]

W. JustS. Ahn and D. Terman, Minimal attractors in digraph system models of neuronal networks, Physica D, 237 (2008), 3186-3196. doi: 10.1016/j.physd.2008.08.011.

[19]

W. Just, S. Ahn and D. Terman, Neuronal networks: A discrete model, in Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models (eds. R. Robeva and T. Hodge), Academic Press, (2013), 179–211.

[20]

W. JustM. KorbB. Elbert and T. R. Young, Two classes of ODE models with switch-like behavior, Physica D, 264 (2013), 35-48. doi: 10.1016/j.physd.2013.08.008.

[21]

R. K. C., Random Connectivities in Neuronal and Other Biologically Relevant Networks, Ph. D. Dissertation Proposal, Ohio University, May 2016.

[22]

S. A. Kauffman, Origins of Order: Self-Organization and Selection in Evolution, Oxford U Press, Oxford, UK, 1993.

[23]

G. Laurent, Olfactory network dynamics and the coding of multidimensional signals, Nat. Rev. Neurosci., 3 (2002), 884-895. doi: 10.1038/nrn964.

[24]

F. LiT. LongY. LuQ. Ouyang and Ch. Tang, The yeast cell-cycle network is robustly designed, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 4781-4786. doi: 10.1073/pnas.0305937101.

[25]

A. PolynikisS. J. Hogan and M. di Bernardo, Comparing different ODE modelling approaches for gene regulatory networks, J. Theoret. Biol., 261 (2009), 511-530. doi: 10.1016/j.jtbi.2009.07.040.

[26]

J. RinzelD. TermanX.-J. Wang and B. Ermentrout, Propagating activity patterns in large-scale inhibitory neuronal networks, Science, 279 (1998), 1351-1355. doi: 10.1126/science.279.5355.1351.

[27]

A. Saadatpour and R. Albert, A comparative study of qualitative and quantitative dynamic models of biological regulatory networks, EPJ Nonlinear Biomedical Physics, 4 (2016), 5pp.

[28]

S. Scarpetta and A. de Candia, Neural avalanches at the critical point between replay and non-replay of spatiotemporal patterns, PLoS One, 8 (2013), e64162. doi: 10.1371/journal.pone.0064162.

[29]

T. U. SinghK. ManchandaR. Ramaswamy and A. Bose, Excitable nodes on random graphs: Relating dynamics to network structure, SIAM J. Appl. Dyn. Syst., 10 (2011), 987-1012. doi: 10.1137/100802864.

[30]

D. TermanS. AhnX. Wang and W. Just, Reducing neuronal networks to discrete dynamics, Physica D, 237 (2008), 324-338. doi: 10.1016/j.physd.2007.09.011.

[31]

D. TermanA. Bose and N. Kopell, Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), 15417-15422. doi: 10.1073/pnas.93.26.15417.

[32]

D. M. Wittmann, J. Krumsiek, J. Saez-Rodriguez, D. A. Lauffenburger, S. Klamt and F. J. Theis, Transforming Boolean models to continuous models: Methodology and application to T -cell receptor signaling, BMC Syst. Biol., 3 (2009), p98. doi: 10.1186/1752-0509-3-98.

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