# Weak multiplexing in neural networks: Switching between chimera and solitary
states

Maria Mikhaylenko  Laboratory of Solution Chemistry of Advanced Materials and
Technologies, ITMO University, 9 Lomonosova Str., Saint Petersburg 197101,
Russian Federation  Lukas Ramlow  Institut für Theoretische Physik, Technische
Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany  Sarika Jalan
Complex Systems Lab, Discipline of Physics, Indian Institute of Technology
Indore, Khandwa Road, Simrol, Indore 453552  Anna Zakharova
<EMAIL_ADDRESS>Institut für Theoretische Physik, Technische
Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

###### Abstract

We investigate spatio-temporal patterns occurring in a two-layer multiplex
network of oscillatory FitzHugh-Nagumo neurons, where each layer is
represented by a nonlocally coupled ring. We show that weak multiplexing,
i.e., when the coupling between the layers is smaller than that within the
layers, can have a significant impact on the dynamics of the neural network.
We develop control strategies based on weak multiplexing and demonstrate how
the desired state in one layer can be achieved without manipulating its
parameters, but only by adjusting the other layer. We find that for coupling
range mismatch weak multiplexing leads to the appearance of chimera states
with different shapes of the mean velocity profile for parameter ranges where
they do not exist in isolation. Moreover, we show that introducing a coupling
strength mismatch between the layers can suppress chimera states with one
incoherent domain (one-headed chimeras) and induce various other regimes such
as in-phase synchronization or two-headed chimeras. Interestingly, small
intra-layer coupling strength mismatch allows to achieve solitary states
throughout the whole network.

multiplex networks; chimera states; solitary states; FitzHugh-Nagumo system

###### pacs:

05.45.-a

> In nonlinear dynamics the paradigmatic FitzHugh-Nagumo (FHN) system is used
> to model the behavior of neurons. A single-layer network of nonlocally
> coupled oscillatory FHN neurons demonstrates various dynamic regimes
> including chimera states, that represent an intriguing mechanism of
> transition from complete coherence to complete incoherence. Here, we focus
> on a two-layer multiplex network and develop the tools for controlling the
> spatio-temporal patterns in the presence of weak coupling between the
> layers, i.e. weak multiplexing. We show that multiplexing combined with a
> mismatch between the layers, allows to induce chimera states in networks,
> where they do not appear in isolation. Our control also works in the
> opposite direction leading to the suppression of chimera states.
> Interestingly, small mismatch in the intra-layer coupling strength results
> in the formation of solitary states, that offer, compared to chimera states,
> an alternative scenario of transition from coherence to incoherence. Since
> multilayer structures naturally occur in neural networks (e.g., brain
> networks), we expect that our results can be useful for understanding and
> controlling of biological networks.

##  I Introduction

The dynamics of complex networks is one of the central issues in nonlinear
dynamics SCH16 ; PAN15 . Coupled oscillatory units can demonstrate various
types of collective behavior, including completely synchronized states,
partially synchronized patterns, oscillation suppression and desynchronized
dynamics. The transition from completely synchronized to completely irregular
behavior can occur via different mechanisms involving special types of partial
synchronization patterns. Chimera states represent an intriguing scenario,
where the system spontaneously splits into coexisting domains of coherent
(e.g., synchronized) and incoherent behavior, which are localized in space
KUR02a ; ABR04 . An alternative scenario involves solitary states where
individual “solitary” oscillators leave the synchronous cluster at random
positions in space MAI14a ; JAR15 .

Chimera states arise surprisingly in networks of completely identical units
and symmetric coupling topologies ABR04 ; PAN15 ; SCH16b . Initially detected
for nonlocally coupled rings of phase oscillators, these hybrid patterns have
been found for various other systems and network structures ranging from
globally coupled networks YEL14 ; BOE15 ; SCH14a  and networks with power-law
coupling BAN16  to irregular topologies KO08 ; SHA10 ; LAI12 ; YAO13 ; ZHU14
and hierarchical, quasi-fractal connectivities OME15 ; HIZ15 ; ULO16 ; TSI16 ;
SAW17 ; BON18 . Solitary states have been reported for local, nonlocal and
global types of coupling in networks of Kuramoto oscillators with inertia
JAR15 ; JAR18 .

A multilayer approach has been recently suggested to offer a better
description of various real-world networks BOC14 ; KIV14 ; BIA14 . In
multilayer networks the nodes are distributed in different layers according to
the type of the relation they share. For example, in the case of a neuronal
network the neurons can form different layers depending on their connectivity
through a chemical link or by an ionic channel. For the investigation of the
brain, the multilayer representation allows to model its structural and
functional connectivity from a new viewpoint, i.e., combined with each other
as layers of a multilayer network DOM17 .

Chimera states and solitary states have been found in multilayer networks only
very recently GHO16a ; MAK16 ; AND17 ; BUK17 ; BUK18 . In particular, the
impact of strong multiplexing, when the strength of the coupling between the
layers is comparable with that inside each layer, has been investigated for
coupled chaotic maps GHO16 ; GHO18 . In more detail, it has been shown that
strong multiplexing can be used to control chimeras in networks of coupled
chaotic maps GHO18 . The impact of strong multiplexing with an uncoupled layer
has been investigated for a network of Hindmarsh-Rose neurons in MAJ17 , where
the strong inter-layer links represent chemical connections and weak intra-
layer couplings model electrical synapses. In the real-world networks it is
often the case that the nodes form layers (communities or populations) where
the coupling within a layer is much stronger than that across the layers. This
property of community structure is common in many social and biological
networks GIR02 , and in particular in neural networks HIZ16 .

Multilayer networks not only allow for a better representation of the topology
and dynamics for natural and man-made systems in comparison with isolated one-
layer structures, but also open up new possibilities of control. This is
especially relevant from the point of view of applications, since it is not
always possible to directly manipulate a particular layer while the network it
is multiplexed with may be accessible. For instance, in the case of the
multilayer brain network, physical connections may be adjustable while the
manipulation of the functional connectivity is much more complicated. Although
the phenomenon of synchronization LEY17a ; ZHA17 ; SIN15 ; GAM15 ; AND17  and
formation of partial synchronization patterns KOU15 ; GHO16a ; GHO16 ; JAL16 ;
MAK16 ; BER17 ; BUK17 ; AND17 ; GHO18  have been recently considered in
multilayer networks, the challenging problem of controlling chimeras by weak
multiplexing, in particular, in neuronal networks has not been yet
investigated.

There occurs a question whether weak inter-layer coupling, i.e., weak
multiplexing, can have a significant effect on the dynamics of the network. In
particular, we address the following questions: Can weak multiplexing be used
to control the spatio-temporal patterns? Does it allow to achieve desired
dynamic regimes, i.e., induce, design and suppress chimera states in neural
networks? Are other scenarios such as solitary states possible in the presence
of weak multiplexing? What are in this situation efficient control strategies?

In the present work we investigate a multi-layer network of coupled FitzHugh-
Nagumo oscillators. We demonstrate that weak multiplexing has an essential
impact on the dynamical patterns and can be used for controlling. We show that
the desired states can be achieved in a particular layer when the coupling
between the layers is rather weak. Different types of chimera states can be
induced and suppressed. Moreover, we report the occurrence of solitary states
for small intra-layer coupling strength mismatch between the layers.
Therefore, by weak multiplexing we can switch from chimera to solitary
patterns. The advantage of multiplexing control we report here is that it
allows to achieve the desired state in a certain layer without manipulating
its parameters, and it works for weak coupling between the layers.

##  II Model

We investigate a multiplex network consisting of two layers where each layer
is represented by a nonlocally coupled ring of FitzHugh-Nagumo (FHN)
oscillators. This two-dimensional system is a paradigmatic model for neural
excitability MAS17 . Previously, chimera states have been found in one-layer
networks consisting of coupled oscillatory OME13  and excitatory SEM16 ; ZAK17
; ZAK17a  FHN systems. Recently weak multiplexing has been shown to play a
significant role for coherence resonance in a network of coupled excitable FHN
units under the influence of noise SEM18 . However, the occurrence of chimera
patterns and solitary states has not yet been considered in the presence of
weak multiplexing. In the present study we focus on oscillatory FHN neurons.

We consider a two-layer multiplex network, where each layer is given by a ring
of $N$ nonlocally coupled FHN oscillators:

$\
\begin{array}[]{c}\varepsilon\frac{du_{1i}}{dt}=u_{1i}-\frac{u^{3}_{1i}}{3}-v_{1i}+\frac{\sigma_{1}}{2R_{1}}\sum\limits_{j=i-R_{1}}^{i+R_{1}}[b_{uu}(u_{1j}-u_{1i})+\\\
+b_{uv}(v_{1j}-v_{1i})]+\sigma_{12}(u_{2i}-u_{1i}),\\\
\frac{dv_{1i}}{dt}=u_{1i}+a_{i}+\frac{\sigma_{1}}{2R_{1}}\sum\limits_{j=i-R_{1}}^{i+R_{1}}[b_{vu}(u_{1j}-u_{1i})+\\\
+b_{vv}(v_{1j}-v_{1i})],\\\
\varepsilon\frac{du_{2i}}{dt}=u_{2i}-\frac{u^{3}_{2i}}{3}-v_{2i}+\frac{\sigma_{2}}{2R_{2}}\sum\limits_{j=i-R_{2}}^{i+R_{2}}[b_{uu}(u_{2j}-u_{2i})+\\\
+b_{uv}(v_{2j}-v_{2i})]+\sigma_{12}(u_{1i}-u_{2i}),\\\
\frac{dv_{2i}}{dt}=u_{2i}+a_{i}+\frac{\sigma_{2}}{2R_{2}}\sum\limits_{j=i-R_{2}}^{i+R_{2}}[b_{vu}(u_{2j}-u_{2i})+\\\
+b_{vv}(v_{2j}-v_{2i})],\end{array}$ (1)

where $u_{1i}$ and $v_{1i}$ are the activator and inhibitor variables in the
first(upper) layer, respectively, $i=1,...,N$ with $N$ being the total number
of elements in the network. All indices are modulo $N$. In a similar way
$u_{2i}$ and $v_{2i}$ represent the activator and inhibitor variables in the
second (lower) layer, respectively. The strength of the coupling within the
layer (intra-layer coupling) is given by $\sigma_{1}$ for the first layer and
$\sigma_{2}$ for the second layer. The parameters $R_{1}$ and $R_{2}$ indicate
the number of nearest neighbors in each direction on a ring for the first and
second layer, respectively. The coupling between the layers (inter-layer
coupling) is bidirectional, diffusive and its strength is characterized by
$\sigma_{12}$. Here we are mainly interested in the impact of weak
multiplexing, i.e., when the inter-layer coupling $\sigma_{12}$ is smaller
than the strength of the intra-layer connections $\sigma_{1}$ and
$\sigma_{2}$. We also introduce a coupling range for both layers. It is
represented by the normalized number of nearest neighbours for the first
(upper) layer $r_{1}=R_{1}/N$ and for the second (lower) layer
$r_{2}=R_{2}/N$. A small parameter responsible for the time scale separation
of fast activator and slow inhibitor is given by $\varepsilon>0$ and $a_{i}$
defines the excitability threshold. For an individual FHN element it
determines whether the system is in the excitable ($|a_{i}|>1$), or
oscillatory ($|a_{i}|<1$) regime. In the present study we assume that all
elements are in the oscillatory regime ($a_{i}\equiv a=0.5$).

Figure 1: Schematic diagram showing a multiplex network consisting of two
layers with different coupling range.

Eq. (1) contains not only direct, but also cross couplings between activator
($u$) and inhibitor ($v$) variables, which is modeled by a rotational coupling
matrix OME13 :

$B=\left(\begin{array}[]{ccc}b_{\mathrm{uu}}&&b_{\mathrm{uv}}\\\
b_{\mathrm{vu}}&&b_{\mathrm{vv}}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\phi&&\sin\phi\\\
-\sin\phi&&\cos\phi\end{array}\right),$ (2)

where $\phi\in[-\pi;\pi)$. Here we fix the parameter $\phi=\pi/2-0.1$. Chimera
states have been found for this value of $\phi$ in both the deterministic
oscillatory OME13  and the noisy excitable regime SEM16 ; ZAK17 . Moreover, it
has been shown that chimera states occurring in the excitable regime SEM16 ;
ZAK17  are different from those detected in the oscillatory regime OME13 .

##  III Results

We study a multiplex network composed of two non-identical layers. In
particular, we investigate the following two cases: (i) the layers are
characterized by different coupling range $r_{1}\neq r_{2}$ while the coupling
strength within the layers is the same $\sigma_{1}=\sigma_{2}$; (ii) the
layers have an intra-layer coupling strength mismatch, i.e.,
$\sigma_{1}\neq\sigma_{2}$ while the coupling range is fixed and the same for
both layers $r_{1}=r_{2}$.

###  III.1 Dynamics of isolated layers

First, we consider the dynamics of the disconnected non-identical layers
$\sigma_{12}=0$. Both of them are represented by a nonlocally coupled ring of
$N=300$ identical elements and the intra-layer coupling strength is fixed
$\sigma_{1}=\sigma_{2}=0.1$. We introduce the coupling range mismatch by
choosing $r_{1}=0.2$ for the first (upper) layer and $r_{2}=0.35$ for the
second (lower) layer. Therefore, the second layer is characterized by the
higher link density compared with the first layer. The isolated layer with the
smaller coupling range exhibits desynchronized dynamics (Fig. 2(a),(b),(c)).
This becomes evident from the space-time plot that shows incoherent pattern
(Fig. 2(a)). In the other layer chimera states are observed (Fig.
2(d),(e),(f)). Coexistence in space of well-separated synchronized and
desynchronized groups of oscillating FHN neurons is clearly seen from the
space-time plot (Fig. 2(d)) and the snapshot (Fig. 2(e)). The mean phase
velocity profile has a typical arc-shaped profile, a characteristic signature
of chimera states (Fig. 2(f)). The map of regimes in the ($r$, $\sigma$)
parameter plane for an isolated nonlocally coupled ring of oscillatory FHN
elements has been analyzed in detail in OME13 .

Figure 2: Two disconnected layers ($\sigma_{12}=0$) with different coupling
range. (a), (b), (c): first (upper) layer, $r_{1}=0.2$; (d), (e), (f): second
(lower) layer $r_{2}=0.35$; (a), (d): space-time plots for the variable
$u_{k}$; (b), (e): snapshots for the variable $u_{k}$; (c), (f): mean phase
velocity profiles. Initial conditions: randomly distributed on the circle
$u^{2}+v^{2}=4$. Other parameters: $N=300$, $\varepsilon=0.05$,
$\phi=\pi/2-0.1$, $a=0.5$, $\sigma_{1}=\sigma_{2}=0.1$.

###  III.2 Multiplex network: coupling range mismatch.

Next, we introduce the coupling between the non-identical layers, and
investigate the impact of the inter-layer coupling strength $\sigma_{12}$ on
the dynamics of the network. Interestingly, even if the inter-layer coupling
$\sigma_{12}=0.01$ is much smaller than that within the layers
$\sigma_{1}=\sigma_{2}=0.1$, chimera states are observed for both rings (Fig.
3). One can clearly distinguish coherent and incoherent groups in the
snapshots (Fig. 3(a),(b)) and identify the typical arc-shape of the mean phase
velocity profiles (Fig. 3(c)(d)). Interestingly, the location in space of
coherent and incoherent domains of the chimera pattern coincides in the two
layers of the multiplex network (Fig. 3).

Therefore, weak multiplexing with a denser layer allows to induce chimera
states in the layer with the lower link density that does not demonstrate
chimera states in isolation. Consequently, in a weakly multiplexed neural
network the control of the dynamics in one of the layers can be realized
without manipulating the internal parameters of its elements or the couplings
between them. The control is achieved by adjusting the topology (coupling
range in this particular case) of the other layer. This is relevant for the
applications, since it is not always possible to directly access the desired
layer while the network it is multiplexed with may be adaptable. It is
important to note that the same effect can be achieved in the presence of
strong multiplexing. In particular, when the strength of the coupling between
the layers is equal to the inter-layer coupling
$\sigma_{12}=\sigma_{1}=\sigma_{2}=0.1$, multiplexing with a denser layer
induces chimera patterns in the sparser layer (Fig. 4).

Figure 3: Two weakly multiplexed layers ($\sigma_{12}=0.01$) with different
coupling range. (a), (c) first (upper) layer, $r_{1}=0.2$; (b), (d): second
(lower) layer, $r_{2}=0.35$; (a),(b): snapshots of variable $u_{k}$; (c),(d):
mean phase velocity profiles. Other parameters: $N=300$, $\varepsilon=0.05$,
$\phi=\pi/2-0.1$, $a=0.5$, $\sigma_{1}=\sigma_{2}=0.1$. Figure 4: Two strongly
multiplexed layers ($\sigma_{12}=\sigma_{1}=\sigma_{2}=0.1$) with different
coupling range. (a), (c): first (upper) layer, $r_{1}=0.2$; (b), (d): second
(lower) layer, $r_{2}=0.35$; (a),(b): snapshots of variable $u_{k}$; (c),(d):
mean phase velocity profiles. Other parameters: $N=300$, $\varepsilon=0.05$,
$\phi=\pi/2-0.1$, $a=0.5$.

To get an overall view on the dynamics of the network we calculate the map of
regimes in each layer for varying coupling range $r_{2}$ and inter-layer
coupling strength $\sigma_{12}$ and keeping all other parameters fixed (Fig.
5). In other words, we only manipulate the second (lower) layer while keeping
the parameters of the first (upper) layer unchanged.

Figure 5: Map of regimes in the case of coupling range mismatch for (a) the
first (upper) layer and (b) the second (lower) layer in the
($r_{2}$,$\sigma_{12}$) parameter plane. The parameters $[r_{2},\sigma_{12}]$
for the selected points: A=[0.12,0.06], B=[0.323,0.06], C=[0.35,0.06],
D=[0.44,0.06]. The colors stand for different chimera types; white region
indicates incoherent patterns. (c): Mean phase velocity profiles for the upper
layer (lower layer behaves similarly) for points A,B,C,D. Other parameters:
$N=300$, $\varepsilon=0.05$, $\phi=\pi/2-0.1$, $a=0.5$,
$\sigma_{1}=\sigma_{2}=0.1$,$r_{1}=0.2$.

It turns out that even very weak coupling between the layers forces them to
behave the same way ($\sigma_{12}>0.007$). We detect different types of
chimera states (points A,B,C,D in Fig. 5(a),(b)) and incoherent patterns
(white region in Fig. 5(a),(b)) depending on coupling parameters.
Interestingly, chimeras can be induced in the first (upper) layer by
multiplexing it with not exclusively denser layer as shown in Figs. 3,4 and
Fig. 5 (see point D in Fig. 5(a),(b) and panel D in (Fig. 5(c)), but also with
the layer characterized by lower density (point A in Fig. 5(a),(b) and panel A
in (Fig. 5(c)). Depending on the coupling range in the second (lower) layer
$r_{2}$, we can observe different chimera patterns. For small coupling range
$r_{2}$ (point A in Fig. 5(a),(b) and panel A in (Fig. 5(c)) the chimera
pattern is characterized by a smaller incoherent domain compared with those
observed for larger values of $r_{2}$ (points B,C,D in Fig. 5(a),(b) and
panels B,C,D in (Fig. 5(c)). Moreover, for the patterns in region A (Fig.
5(a),(b)) the difference between the maximum frequency from the incoherent
domain and that of coherent domain is smaller. The latter property also
applies to the patterns in region B (Fig. 5(a),(b)). Furthermore, the mean
phase velocity of the chimeras in region B (panel B in Fig. 5(c)) has a
plateau-shaped profile while chimeras in regions A and D (Fig. 5(a),(b))
demonstrate a classical arc-shaped profile (panels A and D in Fig. 5(c)). A
distinguishing feature of chimeras in region C (panel C in (Fig. 5 (c)) is a
step-like structure of the mean phase velocity profile. The formation of such
a structure can be explained by the complex synchronization cascade mechanisms
recently analyzed in OME18b . It is important to note that the coupling range
where chimeras are observed in the second (lower) layer in the multiplex
network is shifted towards higher values if compared with the isolated second
(lower) layer (Fig. 5(b)). Therefore, weak multiplexing induces chimeras not
only in the first (upper) layer, but also in the parameter range of the second
(lower) layer where no chimeras are observed in isolation.

###  III.3 Multiplex network: coupling strength mismatch.

Figure 6: Two-layer multiplex network with intra-layer coupling strength
mismatch for different values of inter-layer coupling strength $\sigma_{12}$.
Snapshots of variable $u_{k}$ (top row) and mean phase velocity profiles
(bottom row). (a),(b),(g),(h): two disconnected layers $\sigma_{12}=0$
((a),(g) - first (upper) layer and (b),(h) - second (lower) layer);
(c),(d),(i),(j): weakly coupled layers $\sigma_{12}=0.01$ ((c),(i) - first
(upper) layer and (d),(j) - second (lower) layer); (e),(f),(k),(l): weakly
coupled layers $\sigma_{12}=0.05$ ((e),(k) - first (upper) layer and (f),(l) -
second (lower) layer). Other parameters: $N=300$, $\varepsilon=0.05$, $a=0.5$,
$\phi=\pi/2-0.1$, $r_{1}=r_{2}=0.35$, $\sigma_{1}=0.1$, $\sigma_{2}=0.4$.

Further, we consider a different type of a multiplex network with non-
identical layers. We fix the link density, i.e. the coupling range $r$, in
both rings $r_{1}=r_{2}=0.35$ and introduce a mismatch in the intra-layer
coupling strength $\sigma_{1}\neq\sigma_{2}$. In more detail, the elements in
the first (upper) layer are coupled more weakly than the nodes in the second
(lower) layer $\sigma_{1}<\sigma_{2}$. Without multiplexing $\sigma_{12}=0$,
the first (upper) layer exhibits a chimera state (Fig. 6(a),(g)). In contrast,
the strongly coupled layer ($\sigma_{2}=0.4$) demonstrates coherent behavior:
both the snapshot of the variable $u_{k}$ (Fig. 6(b)) and the flat mean phase
velocity profile (Fig. 6(h)) indicate synchronization.

Once the layers are connected, chimera states in the first (upper) layer
become less pronounced if compared with the isolated case. In more detail,
from the snapshots of variable $u_{k}$ it can be seen that the size of the
incoherent domain decreases (see Fig. 6(a)) for the isolated case and (Fig.
6(c)) for the multiplexed case). In the mean phase velocity profile, the
difference between the maximum frequency of incoherent domain and that of
coherent domain decreases and at the same time a dip is formed in the middle
of the incoherent domain (Fig. 6(i)). The occurrence of the dip can be
explained by the fact that in the isolated nonlocally coupled ring the
increase of the coupling strength for the fixed coupling range leads to the
formation of multichimera states. In other words, the transition from a
classical chimera state with one incoherent domain to a chimera pattern with
two incoherent domains occurs through the formation of the dip in the
incoherent domain of the mean phase velocity profile OME13 . Here such a
modification of the mean phase velocity profile is caused by weak multiplexing
with the strongly coupled layer. The increase of the inter-layer coupling
(within the weak multiplexing range) destroys chimera states in the first
(upper) layer and induces synchronization throughout the whole network: both
within and across the layers (Fig. 6(e),(f),(k),(l)). Therefore, weak
multiplexing allows not only to induce chimeras (as shown in Sec. III B), but
also to suppress them.

Figure 7: Map of regimes in the case of coupling strength mismatch for (a) the
first (upper) layer and (b) the second (lower) layer in the
($\sigma_{2}$,$\sigma_{12}$) parameter plane. The parameters
$[\sigma_{2},\sigma_{12}]$ for the selected points:
A=[0.125,0.03],B=[0.225,0.03],C=[0.330,0.03],D=[0.450,0.03], 1=[0.3,0.04],
2=[0.3,0.06],3=[0.3,0.08]. Hatching marks the regions where the layers behave
differently; colors stand for different chimera types; white region indicates
incoherent patterns. (c): Mean phase velocity profiles for the first (upper)
layer (lower layer behaves similarly) for points A,B,C,D. (d): Mean phase
velocity profiles for the first (upper) layer (top row) and the second (lower)
layer (bottom row) for points 1,2,3. Other parameters: $N=300$,
$\varepsilon=0.05$, $\phi=\pi/2-0.1$, $a=0.5$,
$\sigma_{1}=0.1$,$r_{1}=r_{2}=0.35$.

To get an overall view on the dynamics of the network we calculate the map of
regimes in each layer for varying coupling strength $\sigma_{2}$ and inter-
layer coupling strength $\sigma_{12}$ and keeping all other parameters fixed
(Fig. 7). Therefore, we again manipulate only the second (lower) layer while
keeping the parameters of the first (upper) layer unchanged. The difference
from the Sect. III B is that here we adjust the strength of the intra-layer
coupling and the range. In comparison with the previous case, where the
coupling range was varied (see Fig. 5), here we observe a more complex
structure of regimes (Fig. 7). Depending on coupling parameters, the layers
may behave the same way or differently (hatched region in Fig. 7(a),(b)). For
a fixed value of the inter-layer coupling strength ($\sigma_{12}=0.03$), by
changing the intra-layer coupling strength ($\sigma_{2}$) we observe a change
in the mean phase velocity profile of chimeras (points A,B,C,D in Fig.
7(a),(b)). For relatively small values of $\sigma_{2}$ we observe classical
chimeras with one incoherent domain (for example, point A in Fig. 7(a),(b) and
panel A in Fig. 7(c)). The increase of $\sigma_{2}$ leads to the formation of
a dip in the mean phase velocity profile (panel B in (Fig. 7(c)). When we
increase $\sigma_{2}$ even further, the dip reaches the frequency level of the
coherent domain resulting in the formation of two-headed chimera (panel C in
Fig. 7(c)). Further increase of $\sigma_{2}$ leads to an in-phase synchronized
regime (panel D in Fig. 7(c)). Therefore, we can suppress chimeras in the
first (lower) layer not only by changing the inter-layer coupling strength
($\sigma_{12}$) (as shown in Fig. 6, but also by increasing the intra-layer
coupling strength in the second (lower) layer ($\sigma_{2}$) for a fixed value
of $\sigma_{12}$ (Fig. 7(a)).

Compared with Sect. III B, where both layers show similar behavior, here the
layers may behave differently depending on the coupling parameters. For the
fixed value of intra-layer coupling ($\sigma_{2}=0.3$), the increase in inter-
layer coupling strength $\sigma_{12}$ makes the layers demonstrate different
mean phase velocity profiles (point 1,2,3 in Fig. 7(a),(b) and panels 1,2,3 in
Fig. 7(d)). Although, the position in space for coherent and incoherent
domains in both layers stays the same and the coherent domains are
synchronized, the incoherent domains start demonstrating different behavior.
For example, the maximum value of mean phase velocity becomes higher in the
second (upper) layer (panel 2 in Fig. 7(d)). Further increase of the inter-
layer coupling strength $\sigma_{12}$, when the two-headed chimera is formed,
results in a better pronounced incoherent domain in the second (lower) layer,
while in the first (upper) layer, the chimera is almost suppressed (panel 3 in
Fig. 7(d)). This result is intriguing since the two-headed chimeras are better
pronounced in the second (lower) layer for the range of $\sigma_{2}$ values
that correspond to no chimera (in-phase synchronization) for this layer in
isolation. Therefore, weak multiplexing can have a dramatic effect on the
dynamic regime of the network.

###  III.4 Multiplex network: switching to solitary states.

Figure 8: Two weakly coupled layers ($\sigma_{12}=0.05$) with small intra-
layer coupling strength mismatch. (a), (b), (c): first (upper) layer,
$\sigma_{1}=0.4$; (d), (e), (f): second (lower) layer $\sigma_{2}=0.3$; (a),
(d): space-time plots for the variable $u_{k}$; (b), (e): snapshots for the
variable $u_{k}$; (c), (f): mean phase velocity profiles. Other parameters:
$N=300$, $\varepsilon=0.05$, $\phi=\pi/2-0.1$, $a=0.5$, $r=0.35$.

A solitary state is a partial synchronization pattern whose formation
mechanism is different from that of a chimera state. In what follows we show
that weak multiplexing allows not only for control of chimeras, but can also
lead to the occurrence of solitary states. In particular, we demonstrate that
for small intra-layer coupling strength mismatch, multiplexing induces
solitary states in nonlocally coupled rings, which do not show these patterns
in isolation.

In more detail, the isolated layers of nonlocally coupled FHN oscillators for
the chosen set of coupling parameters exhibit complete synchronization. Once
the two rings are coupled (and rather weak inter-layer coupling is enough)
solitary states are observed in both layers (Fig. 8). We can see from the
space-time plots that solitary nodes are distributed randomly along the
network, a characteristic signature of solitary states (8 (a)(d)). The
snapshots indicate the presence of two groups of FHN elements: the coherent
cluster and solitary nodes split from the synchronized group (8(b)(e)).
Therefore, multiplexing allows to achieve solitary states throughout the whole
network without manipulating the intra-layer coupling parameters.

Figure 9: Solitary state in first (upper) layer of the multiplex network for
$\sigma_{12}=0.05$. (a) phase portrait and (b) space-time plot for two
selected nodes: solitary node (blu) and a node from the synchronized cluster
(red). Other parameters: $N=300$, $\varepsilon=0.05$, $\phi=\pi/2-0.1$,
$a=0.5$, $r=0.35$ $\sigma_{1}=0.4$, $\sigma_{2}=0.3$.

It is important to point out that the solitary states we observe here for FHN
neurons are different from those observed in networks of Kuramoto phase
oscillators. They have a rather flat mean phase velocity profile indicating
that the frequency of the solitary nodes is close to that of the synchronized
cluster (Fig. 8(c),(f)). Further, the solitary nodes and the synchronized
elements have different amplitudes (Fig. 9(a)). Another distinctive feature of
the solitary patterns in FHN model is that the solitary nodes and the
synchronized cluster are characterized by a small phase shift (Fig. 9(b)). The
occurrence of the phase shift has been previously reported for globally
coupled networks combining repulsive and attractive interactions MAI14a . In
contrast to that, in our case the solitary states are observed in the network
where the interactions are all of the same type.

##  IV Conclusions

In conclusion, weak multiplexing can have a dramatic effect on the behavior of
the network. One possible explanation for this fact can be provided by drawing
an analogy with classical synchronization theory of periodic oscillations. For
example, mutual synchronization of two bidirectionally coupled periodic
oscillators can be achieved for a weaker connection between them if compared
with external synchronization of unidirectionally coupled oscillating systems.
Here we deal with two layers that are coupled mutually, therefore, weak
multiplexing is enough to make them essentially influence the dynamics of each
other.

Weak multiplexing represents a powerful tool for controlling dynamic patterns
in neural networks. It allows to adjust the dynamics of one layer without
manipulating its parameters. In more detail, we demonstrate two control
strategies: (i) by tuning the coupling range in one layer, chimeras with
desired mean phase velocity profile can be induced in the other layer.
Moreover, the same dynamical patterns across the layers are achieved for even
very weak coupling between them. (ii) By tuning the intra-layer coupling
strength we can suppress chimera states with one incoherent domain and induce
a variety of other regimes, including in-phase synchronization and two-headed
chimeras. Furthermore, we can make the layers behave differently.
Interestingly, for small intra-layer coupling strength mismatch between the
layers we can switch from a chimera to a solitary state.

In both cases the control of the dynamics for one of the layers can be
realized without manipulating the internal parameters of its elements or the
connections between them. The control is achieved by adjusting the coupling
parameters (coupling range or coupling strength) of the other layer and
varying the coupling strength between the layers within the weak multiplexing
range. This is important from the point of view of applications, since it is
not always possible to directly access the desired layer while the network it
is multiplexed with may be adaptable. We believe that our results can be
especially useful for the modeling of brain multiplex networks, where the
adjustment of physical connection is feasible due to advances of modern brain
surgery, while the manipulation of the functional connectivity appears to be
much more complicated.

##  V Acknowledgments

We thank Eckehard Schöll, Vadim Anishchenko, Yuri Maistrenko and Iryna
Omelchenko for fruitful discussions. This work was supported by the German
Academic Exchange Service (DAAD) and the Department of Science and Technology
of India (DST) within the PPP project (INT/FRG/DAAD/P-06/2018). AZ
acknowledges support by DFG in the framework of SFB 910.

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