We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic bursting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance or effectively control the features of the stochastic bursting. The setup can be considered a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker–Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.

1.
E. M.
Izhikevich
,
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
(
The MIT Press
,
2007
), ISBN: 9780262090438.
2.
W.
Gerstner
,
W. M.
Kistler
,
R.
Naud
, and
L.
Paninski
,
Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition
(
Cambridge University Press
,
2014
), ISBN: 9781107447615.
3.
L. F.
Abbott
and
P.
Dayan
,
Theoretical Neuroscience
(
The MIT Press
,
2005
).
4.
C.
Clopath
,
L.
Büsing
,
E.
Vasilaki
, and
W.
Gerstner
,
Nat. Neurosci.
13
,
344
(
2010
).
5.
O.
Popovych
,
S.
Yanchuk
, and
P. A. P.
Tass
,
Sci. Rep.
3
,
2926
(
2013
).
6.
R.
Lang
and
K.
Kobayashi
,
IEEE J. Quantum Electron.
16
,
347
(
1980
).
7.
K.
Lüdge
,
Nonlinear Laser Dynamics
(
Wiley-VCH Verlag GmbH & Co. KGaA
,
Weinheim
,
2011
), ISBN: 9783527639823.
8.
M. C.
Soriano
,
J.
García-Ojalvo
,
C. R.
Mirasso
, and
I.
Fischer
,
Rev. Mod. Phys.
85
,
421
(
2013
).
9.
M.
Krupa
,
B.
Sandstede
, and
P.
Szmolyan
,
J. Differ. Equ.
133
,
49
(
1997
).
10.
M.
Lichtner
,
M.
Wolfrum
, and
S.
Yanchuk
,
SIAM J. Math. Anal.
43
,
788
(
2011
).
11.
M.
Desroches
,
J.
Guckenheimer
,
B.
Krauskopf
,
C.
Kuehn
,
H. M.
Osinga
, and
M.
Wechselberger
,
SIAM Rev.
54
,
211
(
2012
).
12.
C.
Kuehn
,
Multiple Time Scale Dynamics
(
Springer-Verlag GmbH
,
2015
), Vol. 191, ISBN: 978-3-319-12315-8.
13.
H.
Jardon-Kojakhmetov
and
C.
Kuehn
, arxiv.org/abs/1901.01402 (2019).
14.
H.
Haken
,
Advanced Synergetics
(
Springer
,
Berlin
,
1985
).
15.
B.
Lindner
,
J.
García-Ojalvo
,
A.
Neiman
, and
L.
Schimansky-Geier
,
Phys. Rep.
392
,
321
(
2004
).
16.
A.
Destexhe
and
M.
Rudolph-Lilith
,
Neuronal Noise
(
Springer
,
New York
,
2012
).
17.
E.
Forgoston
and
R. O.
Moore
,
SIAM Rev.
60
,
969
(
2018
).
18.
J. D.
Murray
, Mathematical Biology, Biomathematics Vol. 19 (Springer, New York, 1989), ISBN: 0-387-19460-6 (New York), 3-540-19460-6 (Berlin).
19.
A. T.
Winfree
,
The Geometry of Biological Time
(
Springer
,
2001
), Vol. 12, ISBN: 978-1-4419-3196-2
20.
A. S.
Pikovsky
and
J.
Kurths
,
Phys. Rev. Lett.
78
,
775
(
1997
).
21.
V.-C.
Oriol
,
M.
Ronny
,
R.
Sten
, and
L.
Schimansky-Geier
,
Phys. Rev. E
83
,
036209
(
2011
).
22.
G. B.
Ermentrout
and
D.
Kleinfeld
,
Neuron
29
,
33
(
2001
).
23.
L.
Lücken
,
D. P.
Rosin
,
V. M.
Worlitzer
, and
S.
Yanchuk
,
Chaos
27
,
13114
(
2017
).
24.
I.
Franović
,
O. E.
Omel’chenko
, and
M.
Wolfrum
,
Chaos
28
,
071105
(
2018
).
25.
I.
Bačić
,
S.
Yanchuk
,
M.
Wolfrum
, and
I.
Franović
,
EPJ ST
227
,
1077
(
2018
).
26.
I.
Franović
,
K.
Todorović
,
M.
Perc
,
N.
Vasović
, and
N.
Burić
,
Phys. Rev. E
92
,
062911
(
2015
).
27.
I.
Franović
,
M.
Perc
,
K.
Todorović
,
S.
Kostić
, and
N.
Burić
,
Phys. Rev. E
92
,
062912
(
2015
).
28.
S.
Yanchuk
,
S.
Ruschel
,
J.
Sieber
, and
M.
Wolfrum
,
Phys. Rev. Lett.
123
,
053901
(
2019
).
29.
E. M.
Izhikevich
,
IEEE Trans. Neural Netw.
15
,
1063
(
2004
).
30.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
(
Addison-Wesley
,
1994
).
31.
A.
Shilnikov
and
M.
Kolomiets
,
Int. J. Bifurc. Chaos
18
,
2141
(
2008
).
32.
G.
Pavliotis
and
A.
Stuart
,
Multiscale Methods: Averaging and Homogenization
(
Springer
,
Berlin
,
2008
).
33.
M.
Galtier
and
G.
Wainrib
,
Phys. Rev. E
2
,
13
(
2012
).
34.
L.
Lücken
,
O. V.
Popovych
,
P. A.
Tass
, and
S.
Yanchuk
,
Phys. Rev. E
93
,
32210
(
2016
).
35.
A. I.
Neishtadt
,
J. Appl. Math. Mech
48
,
133
(
1984
).
36.
J.
Guckenheimer
and
P.
Holmes
,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
(
Springer-Verlag
,
New York
,
1983
).
37.
E. J.
Doedel
,
R. C.
Paffenroth
,
A. R.
Champneys
,
T. F.
Fairgrieve
,
Y. A.
Kuznetsov
,
B.
Sandstede
, and
X.
Wang
,
AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations
(
Concordia University
,
Canada
,
2007
).
38.
B.
Lindner
and
L.
Schimansky-Geier
,
Phys. Rev. E
60
,
7270
(
1999
).
39.
V. A.
Makarov
,
V. I.
Nekorkin
, and
M. G.
Velarde
,
Phys. Rev. Lett.
86
,
3431
(
2001
).
40.
R.
Aust
,
P.
Hövel
,
J.
Hizanidis
, and
E.
Schöll
,
Eur. Phys. J. Spec. Top.
187
,
77
(
2010
).
41.
N.
Kouvaris
,
L.
Schimansky-Geier
, and
E.
Schöll
,
Eur. Phys. J. Spec. Top.
191
,
29
(
2010
).
42.
N. B.
Janson
,
A. G.
Balanov
, and
E.
Schöll
,
Phys. Rev. Lett.
93
,
010601
(
2004
).
43.
P. H.
Dannenberg
,
J. C.
Neu
, and
S. W.
Teitsworth
,
Phys. Rev. Lett.
113
,
020601
(
2014
).
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