Classification of the dynamical mechanisms that support bistability between bursting oscillations and silence has not yet been clarified in detail. The purpose of this paper is to demonstrate that the coexistence of a stable equilibrium point with a state of continuous bursting can occur in a slightly modified, biophysical model that describe the dynamics of pancreatic beta-cells. To realize this form of coexistence, we have introduced an additional voltage-dependent potassium current that is activated in the region around the original, unstable equilibrium point. It is interesting to note that this modification also leads the model to display a blue-sky catastrophe in the transition region between chaotic and bursting states.
References
1.
P.
Heyward
, M.
Ennis
, A.
Keller
, and M. T.
Shipley
, “Membrane bistability in olfactory bulb mitral cells
,” J. Neurosci.
21
(14
), 5311
–5320
(2001
); available at http://www.jneurosci.org/content/21/14/5311.full.2.
Y.
Loewenstein
, S.
Mahon
, P.
Chadderton
, K.
Kitamura
, H.
Sompolinsky
, Y.
Yarom
, and M.
Häusser
, “Bistability of cerebellar Purkinje cells modulated by sensory stimulation
,” Nat. Neurosci.
8
(2
), 202
–211
(2005
).3.
E. M.
Izhikevich
, “Neural excitability, spiking and bursting
,” Int. J. Bifurcation Chaos
10
(06
), 1171
–1266
(2000
).4.
T.
Malashchenko
, A.
Shilnikov
, and G.
Cymbalyuk
, “Six types of multistability in a neuronal model based on slow calcium current
,” PloS One
6
(7
), e21782
(2011
).5.
A. N.
Pisarchik
and U.
Feudel
, “Control of multistability
,” Phys. Rep.
540
, 167
–218
(2014
).6.
J. R.
Pomerening
, E. D.
Sontag
, and J. E.
Ferrell
, “Building a cell cycle oscillator: Hysteresis and bistability in the activation of Cdc2
,” Nat. Cell Biol.
5
(4
), 346
–351
(2003
).7.
K.
Tsaneva-Atanasova
, C. L.
Zimliki
, R.
Bertram
, and A.
Sherman
, “Diffusion of calcium and metabolites in pancreatic islets: Killing oscillations with a pitchfork
,” Biophys. J.
90
(10
), 3434
–3446
(2006
).8.
D. K.
Wells
, W. L.
Kath
, and A. E.
Motter
, “Control of stochastic and induced switching in biophysical networks
,” Phys. Rev. X
5
, 031036
(2015
).9.
B. N.
Kholodenko
, “Cell-signalling dynamics in time and space
,” Nat. Rev. Mol. Cell Biol.
7
(3
), 165
–176
(2006
).10.
P. A.
Tass
, Phase Resetting in Medicine and Biology: Stochastic Modelling and Data Analysis
(Springer Science & Business Media
, 2007
).11.
J. G.
Milton
, “Epilepsy as a dynamic disease: A tutorial of the past with an eye to the future
,” Epilepsy Behav.
18
(1
), 33
–44
(2010
).12.
G.
Buzsáki
and A.
Draguhn
, “Neuronal oscillations in cortical networks
,” Science
304
(5679
), 1926
–1929
(2004
).13.
H.
Bergman
, A.
Feingold
, A.
Nini
, A.
Raz
, H.
Slovin
, M.
Abeles
, and E.
Vaadia
, “Physiological aspects of information processing in the basal ganglia of normal and parkinsonian primates
,” Trends Neurosci.
21
(1
), 32
–38
(1998
).14.
J.
Sarnthein
, A.
Morel
, A.
Von Stein
, and D.
Jeanmonod
, “Thalamic theta field potentials and EEG: High thalamocortical coherence in patients with neurogenic pain, epilepsy and movement disorders
,” Thalamus Relat. Syst.
2
(03
), 231
–238
(2003
).15.
A.
Koseska
, E.
Volkov
, and J.
Kurths
, “Oscillation quenching mechanisms: Amplitude vs. oscillation death
,” Phys. Rep.
531
(4
), 173
–199
(2013
).16.
G. A.
Leonov
and N. V.
Kuznetsov
, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits
,” Int. J. Bifurcation Chaos
23
(1
), 1330002
(2013
).17.
G. A.
Leonov
, N. V.
Kuznetsov
, and T. N.
Mokaev
, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
,” Eur. Phys. J. Spec. Top.
224
, 1421
(2015
).18.
D.
Dudkowski
, S.
Jafari
, T.
Kapitaniak
, N. V.
Kuznetsov
, G. A.
Leonov
, and A.
Prasad
, “Hidden attractors in dynamical systems
,” Phys. Rep.
637
, 1
–50
(2016
).19.
T. R.
Chay
and J.
Keizer
, “Minimal model for membrane oscillations in the pancreatic beta-cell
,” Biophys. J.
42
(2
), 181
–189
(1983
).20.
A.
Sherman
, J.
Rinzel
, and J.
Keizer
, “Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing
,” Biophys. J.
54
(3
), 411
–425
(1988
).21.
P.
Smolen
, J.
Rinzel
, and A.
Sherman
, “Why pancreatic islets burst but single beta cells do not. The heterogeneity hypothesis
,” Biophys. J.
64
(6
), 1668
–1680
(1993
).22.
E.
Gylfe
, E.
Grapengiesser
, and B.
Hellman
, “Propagation of cytoplasmic Ca2+ oscillations in clusters of pancreatic β-cells exposed to glucose
,” Cell Calcium
12
(2-3
), 229
–240
(1991
).23.
E.
Mosekilde
, B.
Lading
, S.
Yanchuk
, and Y.
Maistrenko
, “Bifurcation structure of a model of bursting pancreatic cells
,” BioSystems
63
(1
), 3
–13
(2001
).24.
B.
Lading
, E.
Mosekilde
, S.
Yanchuk
, and Y.
Maistrenko
, “Chaotic synchronization between coupled pancreatic β-cells
,” Prog. Theor. Phys. Suppl.
139
, 164
–177
(2000
).25.
D. E.
Postnov
, O. V.
Sosnovtseva
, S. Y.
Malova
, and E.
Mosekilde
, “Complex phase dynamics in coupled bursters
,” Phys. Rev. E
67
(1
), 016215
(2003
).26.
A. P.
Kuznetsov
, S. P.
Kuznetsov
, E.
Mosekilde
, and N. V.
Stankevich
, “Co-existing hidden attractors in a radio-physical oscillator system
,” J. Phys. A: Math. Theor.
48
(12
), 125101
(2015
).27.
M. G.
Pedersen
, E.
Mosekilde
, K. S.
Polonsky
, and D. S.
Luciani
, “Complex patterns of metabolic and Ca2+ entrainment in pancreatic islets by oscillatory glucose
,” Biophys. J.
105
(1
), 29
–39
(2013
).28.
T. R.
Chay
, “On the effect of the intracellular calcium-sensitive K+ channel in the bursting pancreatic beta-cell
,” Biophys. J.
50
(5
), 765
–777
(1986
).29.
E.
Heart
and P. J.
Smith
, “Rhythm of the β-cell oscillator is not governed by a single regulator: Multiple systems contribute to oscillatory behavior
,” Am. J. Physiol.-Endocrinol. Metab.
292
(5
), E1295
–E1300
(2007
).30.
G. R.
Monteith
, D.
McAndrew
, H. M.
Faddy
, and S. J.
Roberts-Thomson
, “Calcium and cancer: Targeting Ca2+ transport
,” Nat. Rev. Cancer
7
(7
), 519
–530
(2007
).31.
X.
Huang
and L. Y.
Jan
, “Targeting potassium channels in cancer
,” J. Cell Biol.
206
(2
), 151
–162
(2014
).32.
A.
Litan
and S. A.
Langhans
, “Cancer as a channelopathy: Ion channels and pumps in tumor development and progression
,” Front. Cell. Neurosci.
9
, 86
(2015
).33.
B. A.
Simms
and G. W.
Zamponi
, “Neuronal voltage-gated calcium channels: Structure, function, and dysfunction
,” Neuron
82
(1
), 24
–45
(2014
).34.
A.
Shilnikov
and G.
Cymbalyuk
, “Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe
,” Phys. Rev. Lett.
94
(4
), 048101
(2005
).35.
J.
Rinzel
, “On repetitive activity in nerve
,” Fed. Proc.
37
(14
), 2793
–2802
(1978
); available at https://nyuscholars.nyu.edu/en/publications/on-repetitive-activity-in-nerve.36.
R.
Guttman
, S.
Lewis
, and J.
Rinzel
, “Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator
,” J. Physiol.
305
, 377
–395
(1980
).© 2017 Author(s).
2017
Author(s)
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