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Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos
1. | Department of Mathematics, University of Pittsburgh, Pittsburgh, USA |
2. | The Mathematical Neuroscience Team, CIRB-Collége de France, (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL), Paris, France, Inria Paris, Mycenae Team, Paris, France |
3. | Faculty of Appl. Phys. and Math., Gdańsk University of Technology, Gdańsk, Poland |
4. | Department of Mathematics, Brandeis University, Waltham MA 02454, USA |
5. | Laboratoire de Mathématiques et Modélisation d'Évry (LaMME), CNRS UMR 8071, Université d'Évry-Val-d'Essonne, France |
In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.
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show all references
References:
[1] |
B. Aulbach and B. Kieninger,
On three definitions of chaos, Nonlinear Dyn. Syst. Theory, 1 (2001), 23-37.
|
[2] |
J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey,
On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332-334.
doi: 10.2307/2324899. |
[3] |
I. Belykh, E. de Lange and M. Hasler, Synchronization of bursting neurons: What matters in the network topology,
Phys. Rev. Lett. , 94 (2005), 188101.
doi: 10.1103/PhysRevLett.94.188101. |
[4] |
L. S. Block and W. A. Coppel,
Dynamics in One Dimension, Springer-Verlag, 1992.
doi: 10.1007/BFb0084762. |
[5] |
A. M. Blokh and M. Yu. Lyubich,
Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. École Norm. Sup., 24 (1991), 545-573.
doi: 10.24033/asens.1636. |
[6] |
R. Brette and W. Gerstner,
Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642.
doi: 10.1152/jn.00686.2005. |
[7] |
H. Bruin, G. Keller, T. Nowicki and S. van Strien,
Wild Cantor attractors exist, Ann. of Math., 143 (1996), 97-130.
doi: 10.2307/2118654. |
[8] |
N. Brunel and M. Van Rossum,
Lapicque's 1907 paper: From frogs to integrate-and-fire, Biol. Cybernet., 97 (2007), 337-339.
doi: 10.1007/s00422-007-0190-0. |
[9] |
P. Collet and J-P. Eckmann,
Concepts and Results in Chaotic Dynamics: A Short Course, Theoretical and Mathematical Physics, Springer-Verlag, Berlin, 2006. |
[10] |
S. Coombes and C. Bressloff,
Bursting: The Genesis of Rhythm in the Nervous System, World Scientific, 2005.
doi: 10.1142/5944. |
[11] |
P. de Maesschalck and F. Dumortier,
Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.
doi: 10.1016/j.jde.2005.01.004. |
[12] |
W. de Melo and S. van Strien,
One-dimensional dynamics: The Schwarzian derivative and beyond, Bull. Amer. Math. Soc. (N.S.), 18 (1988), 159-162.
doi: 10.1090/S0273-0979-1988-15633-9. |
[13] |
——,
One-Dimensional Dynamics, Results in Mathematics and Related Areas (3), Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
[14] |
M. Desroche, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster,
Chaos, 23 (2013), 046106, 13 pp.
doi: 10.1063/1.4827026. |
[15] |
A. Destexhe, D. Contreras and M. Steriade,
Mechanisms underlying the synchronizing action of corticothalamic feedback through inhibition of thalamic relay cells, J. Neurophysiol., 79 (1998), 999-1016.
|
[16] |
R. L. Devaney,
An Introduction to Chaotic Dynamical Systems, Westview Press, 2003. |
[17] |
F. Dumortier and R. H. Roussarie, Canard cycles and center manifolds Memoirs of the American Mathematical Soc. , 121 (1996), x+100 pp.
doi: 10.1090/memo/0577. |
[18] |
E. Foxall, R. Edwards, S. Ibrahim and P. van den Driessche,
A contraction argument for two-dimensional spiking neuron models, SIAM J. Appl. Dyn. Syst., 11 (2012), 540-566.
doi: 10.1137/10081811X. |
[19] |
A. Granados, F. Clément and M. Krupa,
Border collision bifurcations of stroboscopic maps in periodically driven spiking models, SIAM J. Appl. Dyn. Syst., 13 (2014), 1387-1416.
doi: 10.1137/13094637X. |
[20] |
J. Guckenheimer,
Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160.
doi: 10.1007/BF01982351. |
[21] |
P. Hartman,
Ordinary Differential Equations, Classics in Applied Mathematics, 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
doi: 10.1137/1.9780898719222. |
[22] |
L. D. Iasemidis and J. C. Sackellares,
Review: Chaos theory and epilepsy, The Neuroscientist, 2 (1996), 118-126.
doi: 10.1177/107385849600200213. |
[23] |
B. Ibarz, J. M. Casado and M. A. F. Sanjuán,
Map-based models in neuronal dynamics, Phys. Rep., 501 (2011), 1-74.
doi: 10.1016/j.physrep.2010.12.003. |
[24] |
E.M. Izhikevich,
Neural excitability, spiking, and bursting, Internat, J. Bifur. Chaos Appl., 10 (2000), 1171-1266.
doi: 10.1142/S0218127400000840. |
[25] |
——, Simple model of spiking neurons, IEEE Trans. Neural Netw. , 14 (2003), 1569-1572. |
[26] |
——, Which model to use for cortical spiking neurons?, IEEE Trans. Neural Netw. , 15
(2004), 1063-1070. |
[27] |
——,
Dynamical Systems in Neuroscience: The Geometry of Excitability And Bursting, MIT Press, 2007. |
[28] | |
[29] |
E. M. Izhikevich and G. M. Edelman,
Large-scale model of mammalian thalamocortical systems, Proc. Natl. Acad. Sci. USA, 105 (2007), 3593-3598.
doi: 10.1073/pnas.0712231105. |
[30] |
E. M. Izhikevich, N. S. Desai, E. C. Walcott and F. C. Hoppensteadt,
Bursts as a unit of neural information: Selective communication via resonance, Trends in neurosciences, 26 (2003), 161-167.
doi: 10.1016/S0166-2236(03)00034-1. |
[31] |
B. Jia, H. Gu, L. Li and X. Zhao,
Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns, Cognitive Neurodynamics, 6 (2012), 89-106.
doi: 10.1007/s11571-011-9184-7. |
[32] |
N. D Jimenez, S. Mihalas, R. Brown, E. Niebur and J. Rubin,
Locally contractive dynamics in generalized integrate-and-fire neurons, SIAM J. Appl. Dyn. Syst., 12 (2013), 1474-1514.
doi: 10.1137/120900435. |
[33] |
R. Jolivet, R. Kobayashi, A. Rauch, R. Naud, S. Shinomoto and W. Gerstner,
A benchmark test for a quantitative assessment of simple neuron models, J. Neurosci. Meth, 169 (2008), 417-424.
doi: 10.1016/j.jneumeth.2007.11.006. |
[34] |
M. Juan, L. Yu-Ye, W. Chun-Ling, Y. Ming-Hao, G. Hua-Guang, Q. Shi-Xian and R. Wei, Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable Chin. Phys. B, 19 (2010), 080513.
doi: 10.1088/1674-1056/19/8/080513. |
[35] |
A. Kepecs and J. Lisman,
Information encoding and computation with spikes and bursts, Network: Comp. Neural Syst., 14 (2003), 103-118.
doi: 10.1080/net.14.1.103.118. |
[36] |
A. Kepecs, X.-J. Wang and J. Lisman,
Bursting neurons signal input slope, J. Neurosci., 22 (2002), 9053-9062.
|
[37] |
L. Lapicque,
Recherches quantitatifs sur l'excitation des nerfs traitee comme une polarisation, J. Physiol. Paris, 9 (1907), 620-635.
|
[38] |
E. Lee and D. Terman,
Uniqueness and stability of periodic bursting solutions, J. Differential Equations, 158 (1999), 48-78.
doi: 10.1016/S0022-0396(99)80018-7. |
[39] |
M. Levi,
A period-adding phenomenon, SIAM J. Appl. Math., 50 (1990), 943-955.
doi: 10.1137/0150058. |
[40] |
T.-Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke,
No division implies chaos, Trans. Amer. Math. Soc., 273 (1982), 191-199.
doi: 10.1090/S0002-9947-1982-0664037-3. |
[41] |
B. G. Lindsey, I. A. Rybak and J. C. Smith, Computational models and emergent properties of respiratory neural networks,
Compr. Physiol. (2012).
doi: 10.1002/cphy.c110016. |
[42] |
E. Manica, G. Medvedev and J. E. Rubin,
First return maps for the dynamics of synaptically coupled conditional bursters, Biol. Cybern., 103 (2010), 87-104.
doi: 10.1007/s00422-010-0399-1. |
[43] |
E. Marder,
Motor pattern generation, Curr. Opin. Neurobiol., 10 (2000), 691-698.
doi: 10.1016/S0959-4388(00)00157-4. |
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