December  2017, 22(10): 4003-4039. doi: 10.3934/dcdsb.2017205

Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations

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. 

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

* Corresponding author: justyna.signerska@pg.edu.pl

Received  November 2016 Revised  June 2017 Published  August 2017

Fund Project: J. E. Rubin was partly supported by US National Science Foundation awards DMS 1312508 and 1612913. J. Signerska-Rynkowska was supported by Polish National Science Centre grant 2014/15/B/ST1/01710

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left-and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

Citation: Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205
References:
[1]

A. Alonso and R. Klink, Differential electroresponsiveness of stellate and pyramidal-like cells of medial entorhinal cortex layer Ⅱ, J. Neurophysiol., 70 (1993), 128-143.

[2]

A. Alonso and R. Llinás, Subthreshold Na+-dependent theta-like rhythmicity in stellate cells of entorhinal cortex layer Ⅱ, Nature, 342 (1989), 175-177. doi: 10.1038/342175a0.

[3]

L. AlsedáJ. LlibreM. Misiurewicz and Ch. Tresser, Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier (Grenoble), 39 (1989), 929-952. doi: 10.5802/aif.1195.

[4]

L. Alsedá, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One (second ed. ), Advanced Series in Nonlinear Dynamics, 5. World Scientific Publishing Co. , Inc. , River Edge, NJ, 1993. doi: 10.1142/4205.

[5]

R. AmirM. Michaelis and M. Devor, Membrane potential oscillations in dorsal root ganglion neurons: Role in normal electrogenesis and neuropathic pain, J. Neurosci., 19 (1999), 8589-8596.

[6]

L. S. Bernardo and R. E. Foster, Oscillatory behavior in inferior olive neurons: Mechanism, modulation, cell agregates, Brain Research Bulletin, 17 (1986), 773-784.

[7]

R. Brette, Rotation numbers of discontinuous orientation-preserving circle maps, Set-Valued Anal., 11 (2003), 359-371. doi: 10.1023/A:1025644532200.

[8]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642.

[9]

N. Brunel and P. Latham, Firing rate of noisy quadratic integrate-and-fire neurons, Neural Comput., 15 (2006), 2281-2306. doi: 10.1162/089976603322362365.

[10]

S. Coombes and P. Bressloff, Mode locking and Arnold tongues in integrate-and-fire oscillators, Phys. Rev. E., 60 (1999), 2086-2096. doi: 10.1103/PhysRevE.60.2086.

[11]

N. Fourcaud-Trocme, D. Hansel, C. van Vreeswijk and N. Brunel, How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci. , 23 (2003), 11628.

[12]

E. FoxallR. EdwardsS. 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.

[13]

T. Gedeon and M. Holzer, Phase locking in integrate-and-fire models with refractory periods and modulation, J. Math. Biol., 49 (2004), 577-603. doi: 10.1007/s00285-004-0268-4.

[14]

L. GiocomoE. ZilliE. Fransén and M. Hasselmo, Temporal frequency of subthreshold oscillations scales with entorhinal grid cell field spacing, Science, 315 (2007), 1719-1722. doi: 10.1126/science.1139207.

[15]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics Amer. Math. Soc. Colloq. Publ. , Vol. 36. American Mathematical Society, Providence, R. I. , 1955.

[16]

A. GranadosL. Alsedá and M. Krupa, The period adding and incrementing bifurcations: from rotation theory to applications, SIAM Rev., 59 (2017), 225-292. doi: 10.1137/140996598.

[17]

P. Hartman, On the local linearization of differential equations, Proc. Am. Math. Soc., 14 (1963), 568-573. doi: 10.1090/S0002-9939-1963-0152718-3.

[18]

——, Ordinary Differential Equations Classics Appl. Math. , 38, SIAM, 1982. Corrected reprint of the second (1982) edition. doi: 10.1137/1.9780898719222.

[19]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.

[20]

F. Hofbauer, Periodic points for piecewise monotonic transformations, Ergodic Theory Dynam. Systems, 5 (1985), 237-256. doi: 10.1017/S014338570000287X.

[21]

E. M. Izhikevich, Simple model of spiking neurons, IEEE Trans. Neural Netw., 14 (2003), 1569-1572. doi: 10.1109/TNN.2003.820440.

[22]

——, Which model to use for cortical spiking neurons?, IEEE Trans. Neural Netw. , 15(2004), 1063-1070.

[23]

——, Dynamical Systems in Neuroscience: The Geometry of Excitability And Bursting, Comp. Neurosci. , MIT Press, 2007.

[24]

——, Resonate-and-fire neurons, Neural Netw. , 14 (2001), 883-894

[25]

E. M. Izhikevich and G. M. Edelman, Large-scale model of mammalian thalamocortical systems, Proc. Natl. Acad. Sci. USA, 105 (2008), 3593-3598.

[26]

N. D. JimenezS. MihalasR. BrownE. 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.

[27]

R. S. G. Jones, Synaptic and intrinsic properties of neurones of origin of the perforant path in layer Ⅱ of the rat entorhinal cortex in vitro, Hippocampus, 4 (1994), 335-353.

[28]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.

[29]

J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc., 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3.

[30]

J. P. KeenerF. C. Hoppensteadt and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503-517. doi: 10.1137/0141042.

[31]

I. Lampl and Y. Yarom, Subthreshold oscillations of the' membrane potential: A functional synchronizing and timing device, J. Neurophysiol., 70 (1993), 2181-2186.

[32]

L. Lapicque, Recherches quantitatifs sur l'excitation des nerfs traitee comme une polarisation, J. Physiol. Paris, 9 (1907), 620-635.

[33]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.

[34]

C. LiuM. MichaelisR. Amir and M. Devor, Spinal nerve injury enhances subthreshold membrane potential oscillations in drg neurons: Relation to neuropathic pain, J. Neurophysiol., 84 (2000), 205-215.

[35]

R. R. Llinás, The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function, Science, 242 (1988), 1654-1664.

[36]

R. R. Llinás and Y. Yarom, Electrophysiology of mammalian inferior olivary neurones in vitro. Different types of voltage-dependent ionic conductances, J. Physiol., 315 (1981), 549-567.

[37]

——, Oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: an in vitro study, J. Physiol. , 376 (1986), 163-182

[38]

A. LüthiT. Bal and D. McCormick, Periodicity of thalamic spindle waves is abolished by ZD7288, a blocker of Ih, J. Neurophysiol., 79 (1998), 3284-3289.

[39]

W. Marzantowicz and J. Signerska, On the interspike-intervals of periodically-driven integrate-and-fire models, J. Math. Anal. Appl., 423 (2015), 456-479. doi: 10.1016/j.jmaa.2014.10.013.

[40]

S. Mihalas and E. Niebur, A generalized linear integrate-and-fire neural model produces diverse spiking behaviors, Neural Comput., 21 (2009), 704-718. doi: 10.1162/neco.2008.12-07-680.

[41]

M. Misiurewicz, Rotation intervals for a class of maps of the real line into itself, Ergodic Theory Dynam. Systems, 6 (1986), 117-132. doi: 10.1017/S0143385700003321.

[42]

——, Rotation theory, in Online Proceedings of the RIMS Workshop Dynamical Systems and Applications: Recent Progress, 2006. https://www.math.kyoto-u.ac.jp/kokubu/RIMS2006/RIMS_Online_Proceedings.html

[43]

R. NaudN. MacilleC. Clopath and W. Gerstner, Firing patterns in the adaptive exponential integrate-and-fire model, Biol. Cybernet., 99 (2008), 335-347. doi: 10.1007/s00422-008-0264-7.

[44]

F. Rhodes and Ch. L. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc., 34 (1986), 360-368. doi: 10.1112/jlms/s2-34.2.360.

[45]

——, Topologies and rotation numbers for families of monotone functions on the circle, J. London Math. Soc. , 43 (1991), 156-170. doi: 10.1112/jlms/s2-43.1.156.

[46]

H. Rotstein, Abrupt and gradual transitions between low and hyperexcited firing frequencies in neuronal models with fast synaptic excitation: A comparative study Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046104, 22pp. doi: 10.1063/1.4824320.

[47]

——, Mixed-mode oscillations in single neurons, in Encyclopedia of Computational Neuroscience, Springer, 2015,1720-1727

[48]

H. RotsteinS. Coombes and A. M. Gheorghe, Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of FitzHugh-Nagumo type, SIAM J. Appl. Dyn. Syst., 11 (2012), 135-180. doi: 10.1137/100809866.

[49]

H. RotsteinT. OppermannJ. White and N. Kopell, A reduced model for medial entorhinal cortex stellate cells: subthreshold oscillations, spiking and synchronization, J. Comput. Neurosci., 21 (2006), 271-292. doi: 10.1007/s10827-006-8096-8.

[50]

H. RotsteinM. Wechselberger and N. Kopell, Canard induced mixed-mode oscillations in a medial entorhinal cortex layer Ⅱ stellate cell model, SIAM J. Appl. Dyn. Syst., 7 (2008), 1582-1611. doi: 10.1137/070699093.

[51]

J. E. RubinJ. Signerska-RynkowskaJ. Touboul and A. Vidal, Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike adding and chaos, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3967-4002.

[52]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3d geometry of the Hodgkin-Huxley model, Biol. Cybernet., 97 (2007), 5-32. doi: 10.1007/s00422-007-0153-5.

[53]

V. S. Samovol, A necessary and sufficient condition of smooth linearization of autonomous planar systems in a neighborhood of a critical point, Math. Notes, 46 (1989), 543-550. doi: 10.1007/BF01159105.

[54]

E. Shlizerman and P. Holmes, Neural dynamics, bifurcations, and firing rates in a quadratic integrate-and-fire model with a recovery variable. Ⅰ: Deterministic behavior, Neural Comput., 24 (2012), 2078-2118. doi: 10.1162/NECO_a_00308.

[55]

J. Signerska-Rynkowska, Analysis of interspike-intervals for the general class of integrate-and-fire models with periodic drive, Math. Model. Anal., 20 (2015), 529-551. doi: 10.3846/13926292.2015.1085459.

[56]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824. doi: 10.2307/2372437.

[57]

D. Stowe, Linearization in two dimensions, J. Diff. Eq., 63 (1986), 183-226. doi: 10.1016/0022-0396(86)90047-1.

[58]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons Phys. Rev. E. , 65 (2002), 041913, 14pp. doi: 10.1103/PhysRevE.65.041913.

[59]

J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM Journal on Applied Mathematics, 68 (2008), 1045-1079. doi: 10.1137/070687268.

[60]

——, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Comput. , 21 (2009), 2114-2122. doi: 10.1162/neco.2009.09-08-853.

[61]

J. Touboul and R. Brette, Dynamics and bifurcations of the adaptive exponential integrate-and-fire model, Biol. Cybernet., 99 (2008), 319-334. doi: 10.1007/s00422-008-0267-4.

[62]

——, Spiking dynamics of bidimensional integrate-and-fire neurons, SIAM J. Appl. Dyn. Syst. , 8 (2009), 1462-1506. doi: 10.1137/080742762.

[63]

M. Yoshida and A. Alonso, Cell-type-specific modulation of intrinsic firing properties and subthreshold membrane oscillations by the M (Kv7)-current in neurons of the entorhinal cortex, Journal of Neurophysiol, 98 (2007), 2779-2794. doi: 10.1152/jn.00033.2007.

[64]

L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

show all references

References:
[1]

A. Alonso and R. Klink, Differential electroresponsiveness of stellate and pyramidal-like cells of medial entorhinal cortex layer Ⅱ, J. Neurophysiol., 70 (1993), 128-143.

[2]

A. Alonso and R. Llinás, Subthreshold Na+-dependent theta-like rhythmicity in stellate cells of entorhinal cortex layer Ⅱ, Nature, 342 (1989), 175-177. doi: 10.1038/342175a0.

[3]

L. AlsedáJ. LlibreM. Misiurewicz and Ch. Tresser, Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier (Grenoble), 39 (1989), 929-952. doi: 10.5802/aif.1195.

[4]

L. Alsedá, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One (second ed. ), Advanced Series in Nonlinear Dynamics, 5. World Scientific Publishing Co. , Inc. , River Edge, NJ, 1993. doi: 10.1142/4205.

[5]

R. AmirM. Michaelis and M. Devor, Membrane potential oscillations in dorsal root ganglion neurons: Role in normal electrogenesis and neuropathic pain, J. Neurosci., 19 (1999), 8589-8596.

[6]

L. S. Bernardo and R. E. Foster, Oscillatory behavior in inferior olive neurons: Mechanism, modulation, cell agregates, Brain Research Bulletin, 17 (1986), 773-784.

[7]

R. Brette, Rotation numbers of discontinuous orientation-preserving circle maps, Set-Valued Anal., 11 (2003), 359-371. doi: 10.1023/A:1025644532200.

[8]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642.

[9]

N. Brunel and P. Latham, Firing rate of noisy quadratic integrate-and-fire neurons, Neural Comput., 15 (2006), 2281-2306. doi: 10.1162/089976603322362365.

[10]

S. Coombes and P. Bressloff, Mode locking and Arnold tongues in integrate-and-fire oscillators, Phys. Rev. E., 60 (1999), 2086-2096. doi: 10.1103/PhysRevE.60.2086.

[11]

N. Fourcaud-Trocme, D. Hansel, C. van Vreeswijk and N. Brunel, How spike generation mechanisms determine the neuronal response to fluctuating inputs, J. Neurosci. , 23 (2003), 11628.

[12]

E. FoxallR. EdwardsS. 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.

[13]

T. Gedeon and M. Holzer, Phase locking in integrate-and-fire models with refractory periods and modulation, J. Math. Biol., 49 (2004), 577-603. doi: 10.1007/s00285-004-0268-4.

[14]

L. GiocomoE. ZilliE. Fransén and M. Hasselmo, Temporal frequency of subthreshold oscillations scales with entorhinal grid cell field spacing, Science, 315 (2007), 1719-1722. doi: 10.1126/science.1139207.

[15]

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics Amer. Math. Soc. Colloq. Publ. , Vol. 36. American Mathematical Society, Providence, R. I. , 1955.

[16]

A. GranadosL. Alsedá and M. Krupa, The period adding and incrementing bifurcations: from rotation theory to applications, SIAM Rev., 59 (2017), 225-292. doi: 10.1137/140996598.

[17]

P. Hartman, On the local linearization of differential equations, Proc. Am. Math. Soc., 14 (1963), 568-573. doi: 10.1090/S0002-9939-1963-0152718-3.

[18]

——, Ordinary Differential Equations Classics Appl. Math. , 38, SIAM, 1982. Corrected reprint of the second (1982) edition. doi: 10.1137/1.9780898719222.

[19]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.

[20]

F. Hofbauer, Periodic points for piecewise monotonic transformations, Ergodic Theory Dynam. Systems, 5 (1985), 237-256. doi: 10.1017/S014338570000287X.

[21]

E. M. Izhikevich, Simple model of spiking neurons, IEEE Trans. Neural Netw., 14 (2003), 1569-1572. doi: 10.1109/TNN.2003.820440.

[22]

——, Which model to use for cortical spiking neurons?, IEEE Trans. Neural Netw. , 15(2004), 1063-1070.

[23]

——, Dynamical Systems in Neuroscience: The Geometry of Excitability And Bursting, Comp. Neurosci. , MIT Press, 2007.

[24]

——, Resonate-and-fire neurons, Neural Netw. , 14 (2001), 883-894

[25]

E. M. Izhikevich and G. M. Edelman, Large-scale model of mammalian thalamocortical systems, Proc. Natl. Acad. Sci. USA, 105 (2008), 3593-3598.

[26]

N. D. JimenezS. MihalasR. BrownE. 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.

[27]

R. S. G. Jones, Synaptic and intrinsic properties of neurones of origin of the perforant path in layer Ⅱ of the rat entorhinal cortex in vitro, Hippocampus, 4 (1994), 335-353.

[28]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.

[29]

J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc., 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3.

[30]

J. P. KeenerF. C. Hoppensteadt and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math., 41 (1981), 503-517. doi: 10.1137/0141042.

[31]

I. Lampl and Y. Yarom, Subthreshold oscillations of the' membrane potential: A functional synchronizing and timing device, J. Neurophysiol., 70 (1993), 2181-2186.

[32]

L. Lapicque, Recherches quantitatifs sur l'excitation des nerfs traitee comme une polarisation, J. Physiol. Paris, 9 (1907), 620-635.

[33]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.

[34]

C. LiuM. MichaelisR. Amir and M. Devor, Spinal nerve injury enhances subthreshold membrane potential oscillations in drg neurons: Relation to neuropathic pain, J. Neurophysiol., 84 (2000), 205-215.

[35]

R. R. Llinás, The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function, Science, 242 (1988), 1654-1664.

[36]

R. R. Llinás and Y. Yarom, Electrophysiology of mammalian inferior olivary neurones in vitro. Different types of voltage-dependent ionic conductances, J. Physiol., 315 (1981), 549-567.

[37]

——, Oscillatory properties of guinea-pig inferior olivary neurones and their pharmacological modulation: an in vitro study, J. Physiol. , 376 (1986), 163-182

[38]

A. LüthiT. Bal and D. McCormick, Periodicity of thalamic spindle waves is abolished by ZD7288, a blocker of Ih, J. Neurophysiol., 79 (1998), 3284-3289.

[39]

W. Marzantowicz and J. Signerska, On the interspike-intervals of periodically-driven integrate-and-fire models, J. Math. Anal. Appl., 423 (2015), 456-479. doi: 10.1016/j.jmaa.2014.10.013.

[40]

S. Mihalas and E. Niebur, A generalized linear integrate-and-fire neural model produces diverse spiking behaviors, Neural Comput., 21 (2009), 704-718. doi: 10.1162/neco.2008.12-07-680.

[41]

M. Misiurewicz, Rotation intervals for a class of maps of the real line into itself, Ergodic Theory Dynam. Systems, 6 (1986), 117-132. doi: 10.1017/S0143385700003321.

[42]

——, Rotation theory, in Online Proceedings of the RIMS Workshop Dynamical Systems and Applications: Recent Progress, 2006. https://www.math.kyoto-u.ac.jp/kokubu/RIMS2006/RIMS_Online_Proceedings.html

[43]

R. NaudN. MacilleC. Clopath and W. Gerstner, Firing patterns in the adaptive exponential integrate-and-fire model, Biol. Cybernet., 99 (2008), 335-347. doi: 10.1007/s00422-008-0264-7.

[44]

F. Rhodes and Ch. L. Thompson, Rotation numbers for monotone functions on the circle, J. London Math. Soc., 34 (1986), 360-368. doi: 10.1112/jlms/s2-34.2.360.

[45]

——, Topologies and rotation numbers for families of monotone functions on the circle, J. London Math. Soc. , 43 (1991), 156-170. doi: 10.1112/jlms/s2-43.1.156.

[46]

H. Rotstein, Abrupt and gradual transitions between low and hyperexcited firing frequencies in neuronal models with fast synaptic excitation: A comparative study Chaos: An Interdisciplinary Journal of Nonlinear Science, 23 (2013), 046104, 22pp. doi: 10.1063/1.4824320.

[47]

——, Mixed-mode oscillations in single neurons, in Encyclopedia of Computational Neuroscience, Springer, 2015,1720-1727

[48]

H. RotsteinS. Coombes and A. M. Gheorghe, Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of FitzHugh-Nagumo type, SIAM J. Appl. Dyn. Syst., 11 (2012), 135-180. doi: 10.1137/100809866.

[49]

H. RotsteinT. OppermannJ. White and N. Kopell, A reduced model for medial entorhinal cortex stellate cells: subthreshold oscillations, spiking and synchronization, J. Comput. Neurosci., 21 (2006), 271-292. doi: 10.1007/s10827-006-8096-8.

[50]

H. RotsteinM. Wechselberger and N. Kopell, Canard induced mixed-mode oscillations in a medial entorhinal cortex layer Ⅱ stellate cell model, SIAM J. Appl. Dyn. Syst., 7 (2008), 1582-1611. doi: 10.1137/070699093.

[51]

J. E. RubinJ. Signerska-RynkowskaJ. Touboul and A. Vidal, Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike adding and chaos, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3967-4002.

[52]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3d geometry of the Hodgkin-Huxley model, Biol. Cybernet., 97 (2007), 5-32. doi: 10.1007/s00422-007-0153-5.

[53]

V. S. Samovol, A necessary and sufficient condition of smooth linearization of autonomous planar systems in a neighborhood of a critical point, Math. Notes, 46 (1989), 543-550. doi: 10.1007/BF01159105.

[54]

E. Shlizerman and P. Holmes, Neural dynamics, bifurcations, and firing rates in a quadratic integrate-and-fire model with a recovery variable. Ⅰ: Deterministic behavior, Neural Comput., 24 (2012), 2078-2118. doi: 10.1162/NECO_a_00308.

[55]

J. Signerska-Rynkowska, Analysis of interspike-intervals for the general class of integrate-and-fire models with periodic drive, Math. Model. Anal., 20 (2015), 529-551. doi: 10.3846/13926292.2015.1085459.

[56]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824. doi: 10.2307/2372437.

[57]

D. Stowe, Linearization in two dimensions, J. Diff. Eq., 63 (1986), 183-226. doi: 10.1016/0022-0396(86)90047-1.

[58]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons Phys. Rev. E. , 65 (2002), 041913, 14pp. doi: 10.1103/PhysRevE.65.041913.

[59]

J. Touboul, Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons, SIAM Journal on Applied Mathematics, 68 (2008), 1045-1079. doi: 10.1137/070687268.

[60]

——, Importance of the cutoff value in the quadratic adaptive integrate-and-fire model, Neural Comput. , 21 (2009), 2114-2122. doi: 10.1162/neco.2009.09-08-853.

[61]

J. Touboul and R. Brette, Dynamics and bifurcations of the adaptive exponential integrate-and-fire model, Biol. Cybernet., 99 (2008), 319-334. doi: 10.1007/s00422-008-0267-4.

[62]

——, Spiking dynamics of bidimensional integrate-and-fire neurons, SIAM J. Appl. Dyn. Syst. , 8 (2009), 1462-1506. doi: 10.1137/080742762.

[63]

M. Yoshida and A. Alonso, Cell-type-specific modulation of intrinsic firing properties and subthreshold membrane oscillations by the M (Kv7)-current in neurons of the entorhinal cortex, Journal of Neurophysiol, 98 (2007), 2779-2794. doi: 10.1152/jn.00033.2007.

[64]

L.-S. Young, What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

Figure 1.  The geometry of MMOs: (Upper row) Phase plane with $v$ and $w$ nullclines (dashed black) and stable (red) and unstable (blue) manifolds of the saddle; the stable manifold winds around the repulsive singular point. The reset line $\{v=v_R\}$ (solid vertical line) intersects the stable manifold, separating out regions such that trajectories emanating from each undergo a specific number of small oscillations (colored segments, here from 0 to 3 below the $w$-nullcline and from $3.5$ to $0.5$ above). (Lower rows) The solution for one given initial condition in each segment. Note that the time interval varies in the different plots (indicated on the $x$-axis). Simulations had initial conditions $v=v_R=0.012$ and $w$ chosen within the different intervals on the reset line.
Figure 2.  Geometry of the phase plane with indication of the points relevant in the characterization of the adaptation map $\Phi$. In this example, there are only $p=2$ intersections of $\{ v=v_R\}$ with $\mathcal{W}^s$ (thus $p_1=1$).
Figure 3.  Typical topology of manifolds and sections in Lemma 3.2: we consider the correspondence map between sections $S_s$ (red) and $S_u$ (orange) transverse, respectively, to the stable and unstable manifolds (black lines) of the saddle (orange circle). Typical trajectories are plotted in blue. The key arguments are the characterization of correspondence maps associated with the linearized system (upper left inset) between two transverse sections $S_s'$ and $S_u'$, and the smooth conjugacy between the nonlinear flow and its linearization.
Figure 4.  Partitions of $(d, \gamma)$ parameter space (for fixed values of the other parameters) according to geometric properties of the map $\Phi$ for the quartic model ($F=v^4+2av$, $a=\varepsilon =0.1$, $b=1$, $I=0.1175$ and $v_R=0.1158$) assuming only two intersections of the reset line with the stable manifold (see text for further information).
Figure 8.  The orientation-preserving maps $\Psi_l$ (green) and $\Psi_r$ (blue) enveloping the lift $\Psi$ (red line), which is non-monotonic and admits negative jumps, for the adaptation map $\Phi$ (blue dashed curve) in the overlapping case.
Figure 5.  Phase plane structure, $v$ signal generated along attractive periodic orbits and sequence of $w$ reset values for two sets of parameter values for which the map $\Phi$ is in the non-overlapping case (C4). In both cases, $v_R=0.1$ and $\gamma=0.05$. The top case ($d=0.08$) illustrates the regular spiking behavior corresponding to the rotation number $\varrho = 0$. The bottom case ($d=0.08657$) displays a complex MMBO periodic orbit with associated rational rotation number.
Figure 6.  Phase plane (inset) and adaptation map (top) fulfilling condition (C4) and the additional condition $\Phi(\alpha)<w_1<\Phi(\beta)$, along with the associated MMBO orbit of system (1) (bottom). The rotation number is equal to $0.5$, hence the $v$ signal along the orbit is a periodic alternation of a pair of spikes and one small oscillation. The parameter values of the system corresponding to this simulation are $v_R=0.1$, $\gamma=0.05$ and $d=0.087$.
Figure 7.  Rotation number as a function of $d$. The parameter values $v_R=0.1$ and $\gamma=0.05$ have been chosen such that the adaptive map $\Phi$ fulfills condition (C4) for any value of $d \in [0.08, 0.092]$. Theorem 4.3 applies here, and the rotation number varies as a devil's staircase, as shown in the bottom plot. The top panels show the adaptation map and corresponding attractive periodic orbit at the $d$ values labelled correspondingly in the rotation number plot; note that the rotation number for case (b) is a rational number between 1/3 and 1/2
Figure 9.  Rotation intervals for the lifts $\Psi_{d, l}, \Psi_{d, r}$ of the adaptation maps $\Phi_d$ for a range of $d$. The parameter value $\gamma=0.05$ has been chosen so that $\Phi_d$ remains in the overlapping case for all $d \in [0.0745, 0.0825]$.
Figure 10.  Rotation numbers according to $(d, \gamma)$. Left panel: rotation number of the point $w=0$ together with the boundaries of the regions A to E corresponding to the different subcases when $w_1$ is the unique discontinuity of the adaptation map lying in the interval $[\beta, \alpha]$ (see text for more details). Right panel: rotation numbers of the left and right lifts $\Psi_l$ and $\Psi_r$ associated with $\Phi$ for $(d, \gamma)$ varying along the blue segment drawn in the inset
[1]

Bo Lu, Shenquan Liu, Xiaofang Jiang, Jing Wang, Xiaohui Wang. The mixed-mode oscillations in Av-Ron-Parnas-Segel model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 487-504. doi: 10.3934/dcdss.2017024

[2]

Michele Barbi, Angelo Di Garbo, Rita Balocchi. Improved integrate-and-fire model for RSA. Mathematical Biosciences & Engineering, 2007, 4 (4) : 609-615. doi: 10.3934/mbe.2007.4.609

[3]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A leaky integrate-and-fire model with adaptation for the generation of a spike train. Mathematical Biosciences & Engineering, 2016, 13 (3) : 483-493. doi: 10.3934/mbe.2016002

[4]

Mathieu Desroches, Bernd Krauskopf, Hinke M. Osinga. The geometry of mixed-mode oscillations in the Olsen model for the Peroxidase-Oxidase reaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 807-827. doi: 10.3934/dcdss.2009.2.807

[5]

Shyan-Shiou Chen, Chang-Yuan Cheng. Delay-induced mixed-mode oscillations in a 2D Hindmarsh-Rose-type model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 37-53. doi: 10.3934/dcdsb.2016.21.37

[6]

Tomáš Roubíček, V. Mantič, C. G. Panagiotopoulos. A quasistatic mixed-mode delamination model. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 591-610. doi: 10.3934/dcdss.2013.6.591

[7]

Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3967-4002. doi: 10.3934/dcdsb.2017204

[8]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (1) : 1-10. doi: 10.3934/mbe.2014.11.1

[9]

Theodore Vo, Richard Bertram, Martin Wechselberger. Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2879-2912. doi: 10.3934/dcds.2012.32.2879

[10]

Timothy J. Lewis. Phase-locking in electrically coupled non-leaky integrate-and-fire neurons. Conference Publications, 2003, 2003 (Special) : 554-562. doi: 10.3934/proc.2003.2003.554

[11]

Feng Zhang, Alice Lubbe, Qishao Lu, Jianzhong Su. On bursting solutions near chaotic regimes in a neuron model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1363-1383. doi: 10.3934/dcdss.2014.7.1363

[12]

Benoît Perthame, Delphine Salort. On a voltage-conductance kinetic system for integrate & fire neural networks. Kinetic & Related Models, 2013, 6 (4) : 841-864. doi: 10.3934/krm.2013.6.841

[13]

Roberta Sirovich, Luisa Testa. A new firing paradigm for integrate and fire stochastic neuronal models. Mathematical Biosciences & Engineering, 2016, 13 (3) : 597-611. doi: 10.3934/mbe.2016010

[14]

Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239

[15]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187

[16]

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391

[17]

Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019056

[18]

Fanchao Kong, Juan J. Nieto. Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019107

[19]

Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633

[20]

Achilleas Koutsou, Jacob Kanev, Maria Economidou, Chris Christodoulou. Integrator or coincidence detector --- what shapes the relation of stimulus synchrony and the operational mode of a neuron?. Mathematical Biosciences & Engineering, 2016, 13 (3) : 521-535. doi: 10.3934/mbe.2016005

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (24)
  • HTML views (8)
  • Cited by (1)

[Back to Top]