Nonlinear Dyn
https://doi.org/10.1007/s11071-018-4385-9
ORIGINAL PAPER
Elimination of spiral waves in excitable media by magnetic
induction
Zahra Rostami · Sajad Jafari · Matjaž Perc ·
Mitja Slavinec
Received: 2 March 2018 / Accepted: 19 May 2018
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract The formation of spiral waves in excitable
media is a fascinating example of the beauty of nonlin-
ear dynamics in spatiotemporal systems. Apart from the
beauty of the patterns, the subject also has many prac-
tical application. For example, the emergence of spiral
waves in cardiac tissue can lead to arrhythmias. Corti-
cal spiral waves are also involved in epileptic seizures.
Motivated by this, we here study the effects of magnetic
induction on the formation of spiral waves in excitable
media. An external sinusoidal magnetic induction with
different amplitudes and angular frequencies is applied
in order to study whether spiral waves could be elimi-
nated. We use a network of coupled neurons as a model
Z. Rostami · S. Jafari
Biomedical Engineering Department, Amirkabir University
of Technology, Tehran 15875-4413, Iran
e-mail: z.rostami.zrmi@gmail.com
S. Jafari
e-mail: sajadjafari83@gmail.com
M. Perc (
B
) · M. Slavinec
Faculty of Natural Sciences and Mathematics, University
of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
e-mail: matjaz.perc@uni-mb.si
M. Slavinec
e-mail: mitja.slavinec@um.si
M. Perc
CAMTP – Center for Applied Mathematics and
Theoretical Physics, University of Maribor, Mladinska 3,
2000 Maribor, Slovenia
M. Perc
Complexity Science Hub, Josefstädterstraße 39, 1080 Vienna,
Austria
for the excitable medium. The four-variable magnetic
Hindmarsh–Rose model is used for the local dynam-
ics of each isolated neuron. The distribution of the cell
membrane potential over time, affected by magnetic
induction, is determined and the results are depicted
as snapshots of the 2D network. Our research reveals
that the continuance of rotating spiral seeds is impaired
by high-amplitude magnetic induction. Moreover, we
show that low-frequency induction is not capable of
breaking the reorganizing rhythm of the spiral seeds,
while much higher frequencies can be too fast to over-
come this special rhythm.
Keywords Spiral wave · Spatiotemporal pattern ·
Magnetic flux · Neuronal network · Magnetic
Hindmarsh–Rose model
1 Introduction
Neuronal system is a complex system that exhibits col-
lective behaviors. Collective behavior is one of the most
dominant characteristics of complex systems and can-
not simply be inferred from the properties of the indi-
vidual components [1–3]. More detailed definitions of
the complexity and the complex systems can be found
in Ref. [4]. Neuronal network, an assembly of a large
number of neurons interacting with each other [5,6],
is capable of representing complex demonstrations [7].
These complexities can be in the form of synchronized
or asynchronized behaviors, which are the important
properties of the dynamical networks [8]. They are also
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