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RESEARCH ARTICLE | MA RC H 03 2 02 0
Dynamic behaviors of hyperbolic-type memristor-based
Hopfield neural network considering synaptic crosstalk
Yang Leng ; Dongsheng Yu ; Yihua Hu ; Samson Shenglong Yu ; Zongbin Ye
Chaos 30, 033108 (2020)
https://doi.org/10.1063/5.0002076
29 August 2023 15:15:38
Chaos ARTICLE scitation.org/journal/cha
Dynamic behaviors of hyperbolic-type
memristor-based Hopfield neural network
considering synaptic crosstalk
Cite as: Chaos 30, 033108 (2020); doi: 10.1063/5.0002076
Submitted: 3 February 2020 ·Accepted: 6 February 2020 ·
Published Online: 3 March 2020
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Yang Leng,1Dongsheng Yu,1Yihua Hu,1,a)Samson Shenglong Yu,2and Zongbin Ye1
AFFILIATIONS
1School of Electrical and Power Engineering, China University of Mining and Technology, No. 1 Daxue Road, Xuzhou,
Jiangsu 221116, China
2School of Engineering, Deakin University, 75 Pigdons Road, Waurn Ponds, Victoria 3216, Australia
a)Author to whom correspondence should be addressed: Y.hu35@liverpool.ac.uk
ABSTRACT
Crosstalk phenomena taking place between synapses can influence signal transmission and, in some cases, brain functions. It is thus important
to discover the dynamic behaviors of the neural network infected by synaptic crosstalk. To achieve this, in this paper, a new circuit is structured
to emulate the Coupled Hyperbolic Memristors, which is then utilized to simulate the synaptic crosstalk of a Hopfield Neural Network
(HNN). Thereafter, the HNN’s multi-stability, asymmetry attractors, and anti-monotonicity are observed with various crosstalk strengths.
The dynamic behaviors of the HNN are presented using bifurcation diagrams, dynamic maps, and Lyapunov exponent spectrums, considering
different levels of crosstalk strengths. Simulation results also reveal that different crosstalk strengths can lead to wide-ranging nonlinear
behaviors in the HNN systems.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0002076
Recently, great advancements have been made on discovering
diverse dynamic behaviors of Hopfield Neural Networks (HNNs).
Due to their unique features, such as continuous variations of
memristance, memorable information of historical charge, and
nanoscale volume, memristors (MRs) were utilized to construct
the HNN as synapse weight components, in order to obtain more
realistic emulation performances. However, the dynamic behav-
iors of HNNs due to crosstalk phenomena between two synapses
have not been fully addressed. To fill the void in this research
space, in this paper, two coupled memristors are adopted to
emulate the crosstalk actions inside the HNN, and the dynamic
behaviors are inspected with regard to the crosstalk strength. The
newly proposed HNN emulation strategy with coupled memris-
tors can be used to more accurately represent the functions of the
human brain, leading to possible further cerebral discoveries.
I. INTRODUCTION
Neural networks composed of neurons can be used to emu-
late many basic functions of the brain and have been widely used in
deep learning, information processing, and artificial intelligence.1,2
The Hopfield Neural Network was proposed by John Hopfield in
1982, after the neural network had been found in the human brain,
which is believed to be able to be created out of artificial com-
ponents. HNN possesses its unique ability of binary computation
and information storage3and hence has been widely used in many
applications such as associative memory, image identification, and
optimization.4–6
The nonlinear behaviors of hyperchaotic, chaotic, and quasi-
periodic oscillations have been observed in HNN systems, which
indicates their potential of simulating the real brain functions.7,8The
nonlinear behaviors presented by HNN systems have been reported
in many scientific articles. In Ref. 9, a novel four-dimensional
autonomous HNN system with two parameters is constructed,
which can produce extremely rich dynamics.9In order to simplify
the HNN topology and ease its implementation, numerical simu-
lations and experimental implementation on breadboard for two-
neuron-based non-autonomous HNN have been carried out, which
shows the inherent dynamics of the twin attractors.10 The connec-
tions between the frustrated chaos and the intermittency chaos in a
small HNN have been inspected in Ref. 11. Hyperchaos and hidden
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-1
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Chaos ARTICLE scitation.org/journal/cha
chaotic sets in HNNs are found in Refs. 12 and 13, respectively.
However, most of the HNN systems have been investigated by
adopting ordinary resistors for emulating the synapse weights.14–16
A memristor (MR) is a polarized two-terminal passive cir-
cuit element, whose memristance is dependent on the charge that
has passed through it.17,18 MRs have been widely applied to struc-
tural neuron networks.19 A large number of MR-based neural
networks have been implemented to replace the resistive synap-
tic weights, and their complex dynamic behaviors such as self-
excited and hidden attractors have been explicitly observed.20–23 A
new model for the two-neuron-based non-autonomous memristive
HNN is established in Ref. 24, and numerical and experimental
results from hardware experiments have validated the proposed
model.
Hyperchaos have also been discovered in MR-based HNN
systems.25 As highlighted in Ref. 25, after one weight is replaced
by an MR, complex dynamics such as quasi-periodic orbits, chaos,
and hyperchaos can be observed in an HNN system. This reveals
that the MR is crucial to the behaviors of HNNs. In Ref. 26, a novel
hyperbolic-type MR emulator with a bounded memristance is intro-
duced into an HNN system structured by three neurons, whose
multi-stability and asymmetric attractors are observed. Recently,
the coexistence of multi-stable patterns in an MR synapse-coupled
HNN with two neurons has been reported.27 Methodologically,
in this work, a static matrix was used for describing the synapse
weights, which, however, could not appropriately reflect the prop-
erties of real-world synapses.
Synaptic crosstalk, caused by the spillover of a neurotrans-
mitter from one synapse to another or by the lateral diffusion
of receptors from one spine to the neighboring spines, is a non-
negligible phenomenon that usually happens in neural networks.28–30
However, very few research efforts have been invested in inspect-
ing the dynamic behaviors of MR-based HNNs with synaptic
crosstalk.
In this paper, two coupled hyperbolic MRs are employed to
innovatively construct the HNN of three neurons for emulating
the synaptic crosstalk. The dynamic behaviors of this HNN with
synaptic crosstalk are inspected, considering the crosstalk strength.
The rest of the paper is organized as follows. In Sec. II, the two
coupled hyperbolic MRs are presented. In Sec. III, state equations
of the coupled MR-based HNN are detailed, and the experimen-
tal environment of the HNN system is provided. In Sec. IV, the
equilibrium point and stability behaviors are numerically calculated.
In Sec. V, the multi-stability, asymmetric attractors, and dynamic
mapping under different initial conditions are deduced in terms
of the crosstalk strength. In Sec. VI, the anti-monotonicities, espe-
cially the evolution from bubbles to trees with different connections,
and the variation of attractor structures are found by adjusting
the crosstalk strength. In Sec. VII, experimental results are given
to verify the MR-based HNN. Finally, conclusions are drawn in
Sec. VIII.
II. TWO COUPLED HYPERBOLIC MEMRISTIVE
EMULATORS
The hyperbolic tangent function is generally utilized to present
the neuron activation function. The constitutive relationship of a
FIG. 1. Circuit representing an inverting hyperbolic tangent function.
hyperbolic-type MR emulator can be described by
i=W(v0)v=[a−btanh(v0)]v, (1a)
τdv0
dt =f(v0,v)= −v0−v, (1b)
where vand irepresent the input voltage and current of the
MR emulator, respectively, v0is the intermediate state variable,
aand bare two internal positive constants of the MR emu-
lator, and τis the integral time. Memductance W(v0)can be
modeled by
W(v0)=a−btanh(v0). (2)
As shown in Fig. 1, the inverting hyperbolic tangent function circuit
is established by using two operational amplifiers, four bipolar tran-
sistors, and eleven resistors.31 The output of the hyperbolic tangent
function unit can be obtained by
v0= − tanh(vi), (3)
where viis the input voltage and v0is the output voltage.
The hyperbolic tangent function circuit is adopted to construct
the hyperbolic-type MR emulator, as shown in the rectangular frame
of Fig. 2. The emulator of the hyperbolic-type MRs is depicted
by the circuitry inside the dotted frame, which is used for emulating
the synaptic weights. The crosstalk of two synapses is emulated by
the coupling connection between the two MR emulators, as shown
in Fig. 2.
With regard to the two synaptic weights, by scaling the param-
eters to the dimensionless form, we have
W1=a1−b1tanh(u)+c1tanh(w), (4a)
W2=a2−b2tanh(w)+c2tanh(u), (4b)
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-2
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FIG. 2. Circuit of two hyperbolic-type MRs with a positive crosstalk.
where
τ=t
RC ,a1=R
Ra1
,b1=R
Rb1
,c1=gR2
Rb1Rc1
, (5a)
a2=R
Ra2
,b2=R
Rb2
,c2=gR2
Rb2Rc2
. (5b)
III. COUPLING HYPERBOLIC-TYPE MR-BASED HNN
NEURONS
An HNN can be depicted by circuit state equations. For an
HNN system with nneurons, for the ith neuron, the circuit state
equation can be expressed by
Ci
dxi
dt = − xi
Ri
+
n
X
j=1
wij tanh(xj)+Ii, (6)
where xiis the state variable standing for the voltage across the
capacitor Ci,Riis a resistor representing the membrane resistance
between the inside and outside of the neuron, Iiis the input bias
current, tanh(xj)is a neuron activation function of the voltage input
from the jth neuron, and W=(wij)is an n×nsynaptic weight
matrix indicating the strength of the connection between the ith
and jth neurons. Here, the parameters n=3, Ci=1F, Ri=1, and
Ii=0A are used in this study.
As shown in Fig. 3, an HNN system with three neurons are
adopted in this study, where two MRs coupled between Neuron 1
and Neuron 3 are established to emulate crosstalk. This memristive
FIG. 3. Hyperbolic-type MR-based HNN with three neurons.
HNN with crosstalk can be derived in a dimensionless form by
˙
x= −x+w11 tanh(x)+w12 tanh(y)−k2W2tanh(z),
˙
y= −y+w21 tanh(x)+w22 tanh(y)+w23 tanh(z),
˙
z= −z+k1W1tanh(x)+w32 tanh(y)+w33 tanh(z),
˙
u= −u+tanh(x),
˙
w= −w+tanh(z),
(7)
where
Wij =
w11 w12 w13
w21 w22 w23
w31 w32 w33
=
w11 w12 −k2W2
w21 0w23
k1W1w32 w33
. (8)
Note that W1=a1−b1tanh(u)+c1tanh(w)is the synaptic
weight impacting on Neuron 3 from Neuron 1, and W2=a2
−b2tanh(w)+c2tanh(u)is the synaptic weight impacting on Neu-
ron 1 from Neuron 3. Terms k1and k2are the strength parameters of
hyperbolic-type MRs W1and W2, respectively, and c1and c2repre-
sent the crosstalk strength of the hyperbolic-type MRs W1and W2,
respectively.
Adder circuits, integral circuits, and two nonlinearly coupled
MRs and three nonlinear inverting hyperbolic activation function
circuits are used to physically realize the HNN with the hyperbolic
coupled MRs. The schematic diagram is shown in Fig. 4. Note that
for the purpose of maintaining the internal constitutive relationship
of hyperbolic-type MRs, the input of W2must be vcrather than −vc,
and in order to make the front-end symbol of W2negative in system
state equations. A current mirror inverter is used to achieve inver-
sion, which is marked by the frame with label “−1” in Fig. 4. We can
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-3
Published under license by AIP Publishing.
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FIG. 4. Circuit realization of HNN with two hyperbolic-type MRs.
have
τdv1
dt = −v1−R
R1
tanh(v1)+R
R2
tanh(v2)−tanh(v3)
×R
Ra2
−gR
Rb2
tanh(v5)+gR2
Rb2Rc2
tanh(v4),
τdv2
dt = −v2−R
R3
tanh(v1)+R
R4
tanh(v3),
τdv3
dt = −v3−R
R5
tanh(v2)+R
R6
tanh(v3)+tanh(v1)
×R
Ra1
−gR
Rb1
tanh(v4)+gR2
Rb1Rc1
tanh(v5),
τdv4
dt = −v4+tanh(v1),
τdv5
dt = −v5+tanh(v3). (9)
By assuming the time constant τ=RC =1 ms and configuring
R=10 k,C=100 nF, g=0.1, based on Eq. (8), the values of the
resistors can be obtained by
R1=R/|w11|,R2=R/|w12 |,R3=R/|w21|,
R4=R/|w23|,R5=R/|w32 |,R6=R/|w33|,
Ra1=R/(k1a1),Rb1=gR/(k1b1),
Rc1=gR2/(Rb1k1|c1|)=b1R/|c1|,
Ra2=R/(k2a2),Rb2=gR/(k2b2),
Rc2=gR2/(Rb2k2|c2|)=b2R/|c2|.
It can be proved that the orbits of this system including periodic and
chaotic traces are confined into a bounded region. By introducing a
Lyapunov function, we have
V(x,y,z,u,w)=1
2x2+1
2y2+1
2z2+1
2u2+1
2w2. (10)
Taking the time derivative of Eq. (10) yields
˙
V(x,y,z,u,w)=x˙
x+y˙
y+z˙
z+u˙
u+w˙
w
= −x2−y2−z2−u2−w2
+(w11x+w21 y+k1W1z+u)tanh(x)
+(w12x+w22 y+w32z)tanh(y)
+(−k2W2x+w23y+w33 z+w)tanh(z). (11)
By denoting
V(x,y,z,u,w)=(w11x+w21 y+k1W1z+u)tanh(x)
+(w12x+w22 y+w32z)tanh(y)
+(−k2W2x+w23y+w33 z+w)tanh(z), (12)
Eq. (11) can be rewritten as
˙
V(x,y,z,u,w)= −2V(x,y,z,u,w)+v(x,y,z,u,w). (13)
For u,w∈R,−1<tanh(u) < 1, −1<tanh(w) < 1, we have
a1−b1− |c1|<W1<a1+b1+ |c1|, (14a)
a2−b2− |c2|<W2<a2+b2+ |c2|, (14b)
and
|W1| ≤ M1=max{|a1−b1− |c1||,|a1+b1+ |c1||}, (15a)
|W2| ≤ M2=max{|a2−b2− |c2||,|a2+b2+ |c2||}. (15b)
For any variable x, condition |tanh(x)|<1 can be satisfied. Thus,
Eq. (12) can be simplified as
V(x,y,z,u,w)≤ |(w11x+w21 y+k1W1z+u)tanh(x)|
+ |(w12x+w22 y+w32z)tanh(y)|
+ |(−k2W2x+w23y+w33 z+w)tanh(z)|
≤ |w11x+w21 y+k1W1z+u|
+ |w12x+w22 y+w32z|
+ | − k2W2x+w23y+w33 z+w|
≤(|w11| + |w21 | + k2M2)|x|
+(|w12| + |w22 | + |w32|)|y|
+(−k1M1+ |w23| + |w33 |)|z|
+ |u| + |w|, (16)
where M1and M2are constants. Assuming D0>0 is a sufficiently
large region, for all (x,y,z,u,w)satisfying V(x,y,z,u,w)=Dwith
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-4
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D>D0, the following condition holds:
v(x,y,z,u,w)≤(|w11| + |w21 | + k2M2)|x|
+(|w12| + |w22 | + |w32|)|y|
+(k1M1x+ |w23| + |w33 |)|z| + |u|
+ |w| ≤ x2+y2+z2+u2+w2, (17)
where (|w11| + |w21 | + k2M2)and (|w23| + |w33 | + k1M1)are posi-
tive constants. Consequently, on the surface, we have
{(x,y,z,u,w)|V(x,y,z,u,w)=D}. (18)
For D>D0, we have
˙
V(x,y,z,u,w)= −2V(x,y,z,u,w)+v(x,y,z,u,w) < 0, (19)
which means that set
(x,y,z,u,w)|V(x,y,z,u,w)≤D(20)
is a confined region of all solutions of Eq. (10) and thus the
memristive crosstalk system is also bounded.
IV. EQUILIBRIUM POINT AND STABILITY ANALYSIS
Taking the synaptic weight matrix W0for an example, the char-
acteristics of the equilibrium point and stability can be quantitatively
known, and
W0=
−1.4 1.16 −k2W2
1.1 0 2.82
k1W1−2 4
. (21)
A. Symmetry
The system is invariant under the transformation (x,y,z,u,w,
b1,b2,c1,c2) to (−x,−y,−z,−u,−w,−b1,−b2,−c1,−c2), and
therefore, the HNN with crosstalk is symmetric about parameters
b1,b2,c1, and c2.
The system parameters used for obtaining the attractors shown
in Figs. 5(a) and 5(b) are b1=0.02, b2=0.03, c1=0.2, c2=0.2
and b1= −0.02, b2= −0.03, c1= −0.2, c2= −0.2, respectively.
The initial conditions of Figs. 5(a) and 5(b) are (0, 0.1, 0, 0, 0)and
(0, −0.1, 0, 0, 0), respectively. When both parameters and initial con-
ditions are reversed, it can be found that the chaotic phase-orbit
diagram has an inversed central symmetry.
B. Equilibrium point analysis
Setting the left-hand side of state equations to zero, the
equilibrium points of the HNN with crosstalk can be numeri-
cally determined, namely one zero equilibrium point P0(0, 0, 0, 0, 0)
and two non-zero equilibrium points P1(η1,η2,η3,η4,η5)and
P2(µ1,µ2,µ3,µ4,µ5). Let η1=x1,η3=z1,µ1=x2,µ3=z2for the
undefined constant variables, the nonzero equilibrium points can be
FIG. 5. Dynamic behaviors under symmetric parameters.
calculated by solving the following two equation sets:
η1=x1,
η2=1.1 tanh(x1)+2.82 tanh(z1),
η3=z1,
η4=tanh(x1),
η5=tanh(z1),
(22a)
and
µ1=x2,
µ2=1.1 tanh(x2)+2.82 tanh(z2),
µ3=z2,
µ4=tanh(x2),
µ5=tanh(z2).
(22b)
Two curves can be derived from Eq. (23) by substituting equation
sets Eq. (22) into Eq. (7),
h1(x,z)= −x−1.4 tanh(x)
+1.16 tanh[1.1 tanh(x)+2.82 tanh(z)]−k2tanh(z)
× {a2−b2tanh[tanh(z)]+c2tanh[tanh(x)]}, (23a)
h2(x,z)= −z+4 tanh(z)
−2 tanh[1.1 tanh(x)+2.82 tanh(z)]+k1tanh(x)
× {a1−b1tanh[tanh(x)]+c1tanh[tanh(z)]}, (23b)
where (x1,z1),(x2,z2)are the two intersection points of the above
two function curves, and (x1,z1),(x2,z2)can be determined by
graphical analytic methods. Choosing k1=k2=1, a1=1, a2=7,
b1=0.02, b2=0.03, and c1=c2=0.2 as an example, the two
function curves are plotted in Fig. 6, where the two points are
P1(1.6612, −0.4919)and P2(−1.1452, 0.3628). Apparently, the two
non-zero equilibrium points are asymmetric. Maintaining other
parameters unchanged and letting c1= −2, it can be found that
equilibrium point P2evolves farther from the origin and the equi-
librium point P1moves toward and coincide with the origin. Then,
let c1=2, it can be found that equilibrium point P1moves farther
from the origin and equilibrium point P2almost travels to and coin-
cide with the origin. Although the first crosstalk strength parameter
c1has a great influence on the shape of function h2(x,z), the sec-
ond crosstalk strength parameter c2can also influence the shape
of the function h1(x,z)but only within a narrow range. Therefore,
crosstalk strength coefficients can be used for altering the position
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-5
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FIG. 6. Two function curves and intersections under different crosstalk strengths:
(a) c1=0.2, c2=0.2, (b) c1= −2, c2=0.2, and (c) c1=2, c2=0.2.
of the asymmetric equilibrium point. The Jacobian matrix at the
equilibrium point can be derived by
J=
−1−1.4h11.16h2−k2W2h3α β
1.1h1−1 2.82h30 0
k1W1h1−2h2−1+4h3γ δ
h10 0 −1 0
0 0 h30−1
, (24)
where
h1=sech2(¯
x),h2=sech2(¯
y),
h3=sech2(¯
z),h4=sech2(¯
u),h5=sech2(¯
w),
˙
W1=a1−b1tanh(¯
u)+c1tanh(¯
w),
˙
W2=a2−b2tanh(¯
w)+c2tanh(¯
u),
α= −k2c2tanh(¯
z)h4,β=k2b2tanh(¯
z)h5,
γ= −k1b1tanh(¯
x)h4,δ=k1c1tanh(¯
x)h5.
At zero equilibrium point P0(0, 0, 0, 0, 0), the characteristic
polynomial equation can be derived as
P(λ) =det(λI−J)
=(λ +1)2[λ3+0.4λ2+(a1a2k1k2−3.436)λ
+a1a2k1k2−3.2712a1k1−2.2a2k2+10.164]. (25)
The coefficients in the characteristic polynomial equation are
only related to a1k1and a2k2, instead of the internal MR parame-
ters b1and b2, as well as the crosstalk strength coefficients c1and c2.
When k1=1, a1=1, and a2=7, according to the Routh–Hurwitz
criterion, it can be confirmed that P0is stable if 0.7381 <k2
<0.8206. Taking a1=1, a2=7, k1=1, and k2=0.8 as an exam-
ple, it can be calculated that the eigenvalues of P0are λ1=λ2
= −1, λ3,4 = −0.1596 ±j1.4535, and λ5= −0.0808, and thus P0is
a stable point. Taking k1=1, k2=1, a1=1, a2=7 and k1=0.8,
k2=1, a1=1, a2=7 as an example, it can be derived that the
eigenvalues of P0are λ1=λ2= −1, λ3,4 = −0.3947 ±j1.9276, λ5
=0.3893 and λ1=λ2= −1, λ3,4 = −0.5731±j1.6404, λ5=0.7462,
respectively. Therefore, P0is an unstable saddle point for these
scenarios.
For the two non-zero equilibrium points P1and P2, the cor-
responding eigenvalues can be solved by MATLAB embedded
solvers. When k1=k2=1, a1=1, a2=7, b1=0.02, b2=0.03 and
k1=0.8, k2=1, a1=1, a2=7, b1=0.02, b2=0.03 are chosen as
different crosstalk strength coefficients, two non-zero equilibrium
points, corresponding eigenvalues, and types of chaotic attractors
are shown in Tables I and II, respectively.
Positive eigenvalues lead to unstable manifolds for the equi-
librium points, and a bigger positive eigenvalue of nonzero equi-
libriums results in more severe instability. In order to illustrate
the relationship between the crosstalk parameters and system sta-
bility, the positive eigenvalues in terms of non-zero equilibrium
TABLE I. Nonzero equilibrium points, the corresponding eigenvalues, and the generating attractor types (k1=1).
(c1,c2) Nonzero equilibrium points P1,P2Eigenvalues Attractor type
(−0.28,0,2) (1.3157,−0.0821,−0.3847,0.8657,−0.3668) 0.4164 ±j1.5419,−1.0684,−0.6532,−1 Left:chaos, right:period-1.
(−1.4741,0.2561,0.4748,−0.9004,0.4420) 0.3690 ±j1.2105,−0.8925 ±j0.0979,−1
(0.2,0.2) (1.6612,−0.2604,−0.4913,0.9304,−0.4553) 0.4090 ±j1.2391,−1.0057,−0.8295,−1 Left:Period-1, right:chaos.
(−1.1452,0.0827,0.3628,−0.8162,0.3477) 0.3310 ±j1.5388,−0.9952,−0.6177,−1
(0.335,0.2) (1.7910,−0.3330,−0.5322,0.9459,−0.4871) 0.3884 ±j1.1352,−0.9822,−0.8909,−1 Left:period-1, right:period-8.
(−1.0722,0.0505,0.3385,−0.7903,0.3262) 0.3107 ±j1.6250,−1.0168,−0.5556,−1
(0.2,−0.95) (1.1336,−0.2244,−0.4194,0.8122,−0.3964) 0.3107 ±j1.6250,−1.0168,−0.5556,−1 Left:period-1, right:chaos.
(−1.5960,0.0809,0.4094,−0.9211,0.3880) 0.4878 ±j1.4927,−1.1098,−0.6802,−1
(0.2,1) (2.0069,−0.2705,−0.5128,0.9645,−0.4721) 0.4561 ±j1.2403,−1.0749,−0.8266,−1 Left:period-1, right:chaos.
(−0.8419,0.0811,0.3059,−0.6868,0.2967) 0.1682 ±j1.5606,−0.7141 ±j0.1886,−1
(0.2,2.2) (2.4961,−0.2768,−0.5268,0.9865,−0.4830) 0.4916 ±j1.2400,−1.0869,−0.8667,−1 Left:stable point, right:period-3
(−0.5197,0.0693,0.2140,−0.4775,0.2108) −0.0705 ±j1.5240,−0.5587 ±j0.5170,−1
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TABLE II. Nonzero equilibrium points, the corresponding eigenvalues, and the generating attractor types (k1=0.8).
(c1,c2) Nonzero equilibrium points P1,P2Eigenvalues Attractor type
(−0.25,0,2) (1.7087,−0.2868,−0.5062,0.9365,−0.4670) 0.4116 ±j1.3204,−1.0562, −0.8114,−1 Double-scroll chaotic attractor.
(−1.8484,0.4778,0.6050,−0.9516,0.5406) 0.3062 ±j1.0100,−0.9568 ±j0.0842,−1
(0.2,0.2) (2.0737,−0.4927,−0.6222,0.9689,−0.5526) 0.3146 ±j1.0200,−1.0185,−0.9181,−1 Double-scroll chaotic attractor.
(−1.5086,0.2761,0.4867,−0.9067,0.4516) 0.3803 ±j1.3132,−0.9757,−0.8498,−1
(0.2,−0.7) (1.6271,−0.4690,−0.5865,0.9256,−0.5274) 0.2857 ±j1.0009,−0.9421 ±j0.1763,−1 Double-scroll chaotic attractor.
(−1.8955,0.2839,0.5146,−0.9559,0.4735) 0.4401 ±j1.3187,−1.0870,−0.8110,−1
(0.2,0.6) (2.2644,−4.979,−0.6304,0.9786,−0.5583) 0.3229 ±j1.0240,−1.0530,−0.8988,−1 Single-scroll chaotic attractor.
(−1.3330,0.2696,0.4660,−0.8700,0.4350) 0.3415 ±j1.3048,-0.8900 ±j0.1423, −1
points P1and P2on the right and left plane have been drawn in
a map, as shown in Figs. 7(a) and 7(b), respectively. According to
the number of positive roots, the maps are colored by red, green,
and yellow, which represent zero, one, and two positive eigenvalues,
respectively.
V. COEXISTENCE OF ASYMMETRIC ATTRACTORS AND
MULTI-STABILITY BEHAVIORS WITH VARYING
CROSSTALK STRENGTH PARAMETERS
A. Dynamic behaviors when
c
1changes
When k1=k2=1, a1=1, a2=7, b1=0.02, b2=0.03,
c2=0.2, and the initial simulation condition is set to (0, 0.1, 0, 0, 0),
the bifurcation diagram of state variable zwith the change of first
crosstalk strength parameter c1is shown in Fig. 8(a). It can be
observed that the system generates a period-doubling bifurcation
and then enters a chaotic state, which is followed by a narrow peri-
odic window and chaotic state. After the narrow chaotic window
disappears, four bubbles emerge which are transferred to the period-
four and period-two limit cycles with a reverse period-doubling
route. Then, the chaotic state appears again with tangent bifurca-
tion routes, and it finally degrades to the period-one state by the
reverse period-doubling bifurcation. With the initial condition of
(0, −0.1, 0, 0, 0), the bifurcation diagram of the state variable zis
shown in Fig. 8(b). It can be found that the two bifurcation dia-
grams are not symmetrical, and the most obvious characteristic is
the disappearance of the period-three window and bubbles when
comparing the blue bifurcation diagram with the red one. Several
FIG. 7. Maps of the number of positive eigenvalues for (a) the right non-zero point
P1and (b) the left non-zero point P2.
phase portraits of coexisting asymmetric attractors in the x−zplane
are plotted in Fig. 9, where the phase portraits with initial condi-
tions (0, 0.1, 0, 0, 0)and (0, −0.1, 0, 0, 0)are drawn in colors red and
blue, respectively. These results show the existence of asymmetric
attractors and multi-stability behaviors.
It can be found that the phase portraits basically match
the bifurcation diagram. The initial condition is selected as
(0, 0.1, 0, 0, 0), and Wolf’s method is used to calculate the finite time
Lyapunov exponents, where the time step is set to 0.5s, and the
finish time 10 000s. Figure 10 shows the spectra of Lyapunov expo-
nents by changing the crosstalk strength coefficient c1, which agrees
with the bifurcation diagrams. Taking c1= −0.2 as an example,
the corresponding Lyapunov exponents are calculated as LE1=0,
LE2= −0.0947, LE3= −0.0969, LE4= −1.0003, LE5= −1.0504,
which means the system is periodic. Taking c1=0.2 as an
example, the Lyapunov exponents are LE1=0.0901, LE2=0,
LE3= −0.3828, LE4= −1.0001, LE5= −1.0108 and the corre-
sponding Lyapunov dimension DL=2.2354. In this case, we can see
that the current system is chaotic.
B. Dynamic behaviors when
c
2changes
When k1=k2=1, a1=1, a2=7, b1=0.02, b2=0.03,
c1=0.2, the bifurcation diagram of state variable zwith the change
of c2under the initial condition (0, 0.1, 0, 0, 0)is shown in Fig. 11(a).
As c2increases, the system first undergoes two Feigenbaum-tree
routes and then enters the wide chaotic state by tangent bifurcation.
When c2is positive, in the wide chaotic state, a narrow window of
FIG. 8. Bifurcation diagram of state variable zwith coefficient c1change: (a) initial
condition (0,0.1,0,0,0) and (b) initial condition (0,−0.1,0,0,0).
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FIG. 9. Phase por traits in the x−zplane with different c1: (a) Left-chaotic
spiral attractor and Right-period-1 limit cycle (c1= −0.28), (b) Left-period-4
and Right-period-1 (c1= −0.2), (c) Period-2 limit cycle (c1= −0.05), (d)
Left-period-1 limit cycle and Right-chaotic spiral attractor (c1=0.2), (e) Left-
-period-1 and Right-period-3 limit cycle (c1=0.292), and (f) Left-period-1 and
Right-period-8 limit cycle (c1=0.335).
period-3 can be observed and then the system enters the next chaotic
state with period-doubling routes.
Under initial condition (0, −0.1, 0, 0, 0), the bifurcation dia-
gram of state variable zwith changing crosstalk strength c2is shown
in Fig. 11(b), which shows that the system only changes between
FIG. 10. Lyapunov exponent spectrums: (a) initial condition (0, 0.1, 0, 0, 0)and
(b) initial condition (0, −0.1, 0, 0, 0).
FIG. 11. Bifurcation diagram of state variable zwith changing c2: (a) initial
condition (0, 0.1, 0, 0, 0)and (b) initial condition (0, −0.1, 0, 0, 0).
different periodic states without displaying chaotic behaviors.
Several phase portraits of coexisting asymmetric attractors in the
x−zplane are plotted in Fig. 12, where the phase portraits with
initial conditions (0, 0.1, 0, 0, 0)and (0, −0.1, 0, 0, 0)are drawn in red
and blue, respectively.
FIG. 12. Phase portraits in the x−zplane under different coefficient c2:
(a) Left-period-3 and right period-1 limit cycle (c2= −1.5), (b) Left-period-1 limit
cycle and Right-Feigenbaum chaotic attractor (c2= −0.95), (c) Left-period-1 and
Right-period-4 limit cycle (c2= −0.8), (d) Left-period-1 and Right-period-2 limit
cycle (c2= −0.35), (e) Left-period-1 limit cycle and Right-chaotic spiral attractor
(c2=1), and (f) Left-stable point and Right-period-3 limit cycle (c2=2.2).
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FIG. 13. Bifurcation diagram of state variable zwith changing the initial condition
factor y(0).
According to the phase portraits, it matches the bifurcation
diagram as c2changes. Selecting c1=c2=0.2 as the typical chaotic
crosstalk strength parameter, the bifurcation diagram drawn with
the initial condition y(0)is shown in Fig. 13. It can be found that
there is a period-1 window between the chaotic states, and the
dynamic mapping with initial conditions x(0)and y(0)under typ-
ical chaotic parameters c1=c2=0.2 is plotted in Fig. 14. The
chaotic state is marked with green, while the area of periodic state
with red, it is obvious that the period region is encircled by a
chaotic area, and numerous periodic points exist the chaotic region,
which clearly illustrates the coexistence of asymmetric attractors
and multi-stability behaviors. Overall, this system has rich dynamic
behaviors.
Figure 15 is the Lyapunov exponents spectrum with chang-
ing crosstalk strength c2, which matches the bifurcation diagram.
Taking c2= −2 as an example, the corresponding Lyapunov expo-
nents are calculated as LE1=0, LE2= −0.0484, LE3= −0.5340,
FIG. 14. Dynamic mapping of initial state variables.
FIG. 15. Lyapunov exponent spectrum with changing crosstalk strength
parameter c2.
LE4= −0.6534, LE5= −1.0001, which means that the system
is periodic. Taking c2=1 as an example, the Lyapunov expo-
nents are LE1=0.0756, LE2=0, LE3= −0.2903, LE4= −1.0003,
LE5= −1.1023, and the corresponding Lyapunov dimension DL
is equal to 2.2604, which infers that the current system is
chaotic.
Under initial condition (0, 0.1, 0, 0, 0), by taking the crosstalk
strength parameters c1and c2as horizontal and vertical axis vari-
ables, respectively, the dynamic mapping of crosstalk strength
parameters is drawn in Fig. 16. The chaotic state is marked with
yellow, while the area of periodic state is colored purple.
The two-dimensional bifurcation diagram of state variable zby
changing the two crosstalk strength parameters is shown in Fig. 17.
It can be noticed that there is extensive space of chaotic state, and
the boundary between the states of period and chaos is clear. These
two features indicate that this HNN with crosstalk is suitable for
encrypted communication.
FIG. 16. Dynamic mapping of coupling parameters.
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FIG. 17. Two-dimensional bifurcation diagram.
VI. ANTI-MONOTONICITY AND VARYING ATTRACTOR
BEHAVIORS WITH CROSSTALK STRENGTH
PARAMETERS CHANGE
A. Remerging Feigenbaum trees with changing
crosstalk strength parameters
Taking the synapse weight matrix W0as an example, when
k1=k2=1, a1=1, a2=7, b2=0.03, c2=0.2, the bifurcation
diagrams by changing b1under different values of c1can be
obtained. Figure 18 shows the evolution from bubbles to trees by
decreasing c1.
FIG. 18. Evolution from bubbles to trees: (a) c1=0.3, (b) c1=0.27, (c)
c1=0.25, (d) c1=0.23, (e) c1=0.15, and (f) c1=0.12.
FIG. 19. Evolution from bubbles to trees: (a) c2=0.13, (b) c2=0.15,
(c) c2=0.16, (d) c2=0.18, (e) c2=0.23, and (f) c2=0.27.
Choosing the synaptic weight matrix W0as an example, when
k1=k2=1, a1=1, a2=7, b2=0.03, c1=0.2, by increasing c2
monotonically, the bifurcation diagram demonstrates the evolution
from bubbles to trees, and the serial connections between bubbles
and trees and two different trees are shown in Fig. 19.
It can be found that the dynamic behaviors of the Feigen-
baum trees are very rich when varying the two crosstalk strength
parameters. Figures 18 and 19, reflect similar evolutionary processes,
which start from four branches of two-cycle bubbles in parallel, and
subsequently bifurcate to four branches of four-cycle bubbles and
small trees. As the crosstalk strength parameters further change, the
trees in different branches amalgamate, and the serial connections
between bubbles and trees and two different trees then appear. Due
to the convergence effect of the trees, the accompanying narrow
periodic windows appear.
A group of bifurcation diagrams when changing one of the
crosstalk strengths drawn here is highly representative for describ-
ing the relationship between the remerging Feigenbaum trees and
the crosstalk parameters. Therefore, as shown in Fig. 20, with grow-
ing c1, two bubbles which only connect with each left side, are shown
in Fig. 20(a). Due to Feigenbaum bifurcation, the bubbles evolve into
trees and continue to fuse. In Fig. 20(d), a period-5 window is gener-
ated due to the fusion of trees. Finally, after the two trees completely
merge, the tangent bifurcation route appears, which follows multiple
sets of bubbles and the new trees.
B. Effect of changing crosstalk strength on attractor
types
When k1=0.8, k2=1, a1=1, a2=7, b1=0.02, b2=0.03,
c2=0.2, the bifurcation diagram of state variable zby increasing c1
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FIG. 20. Evolution from bubbles to trees: (a) c1=0.18, (b) c1=0.2,
(c) c1=0.22, (d) c1=0.24, and (e) c1=0.26.
is shown in Fig. 21(a). The corresponding Lyapunov exponent spec-
trum is depicted in Fig. 21(b). The phase portraits are presented in
Fig. 22. It can be found that the periodic states, double scroll chaos,
and single scroll chaos are generated in this system. The attractor
types can vary as the crosstalk strength parameters alter.
When k1=0.8, k2=1, a1=1, a2=7, b1=0.02, b2=0.03,
and c1=0.2, the bifurcation diagram of the state variable zby
changing c2can be obtained. The intermittent chaos, as well as
the dynamic behaviors changing between chaos and periodic oscil-
lations, can be observed in Fig. 23, whose orbital graphs are also
presented in Fig. 24. It can be seen that the system generates double
scroll chaos, periodic oscillations, and single scroll chaos, with bub-
bles appearing in the periodic state. As the second crosstalk strength
parameter c2increases, the attractor type of this system changes
from single to double and back to single scroll.
FIG. 21. Varied attractor type with increasing c1: (a) bifurcation diagram of state
variable zand (b) Lyapunov exponents spectrum.
FIG. 22. Several phase portraits with different crosstalk strength parameter
c1: (a) c1= −0.25, (b) c1=0.15, (c) c1=0.2, and (d) c1=0.3.
VII. SIMULATION
The coupling strength parameters k1and k2, MR internal
parameters, and crosstalk strength coefficients can be adjusted
by the hyperbolic precision potentiometer. Taking a1=1, a2=7,
k1=1, k2=1, b1=0.02, b2=0.03, and c1=c2=0.2 as an exam-
ple, it can be derived that Ra1=10 k,Rb1=50 k,
Rc1=1 k,Ra2=1.429 k,Rb2=33.33 k, and Rc2=1.5 k. The
circuit parameters of the crosstalk MR-based emulator are listed in
Table III, and the resulting resistances are shown in Table IV.
The resistances of ordinary resistors in the synaptic weight
matrix can be calculated as
R1=R/1.4 =7143,R2=R/1.16 =8621,
R3=R/1.1 =9091,R4=R/2.82 =3546,
R5=R/2=5000,R6=R/4=2500.
FIG. 23. Varied attractor type with increasing c2(a) The bifurcation diagram of
the state variable z, and (b) Lyapunov exponent spectrums.
Chaos 30, 033108 (2020); doi: 10.1063/5.0002076 30, 033108-11
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FIG. 24. Several phase portraits with different crosstalk strength parameter
c2: (a) c2= −0.8, (b) c2= −0.7, (c) c2= −0.15, and (d) c2=0.6.
TABLE III. Circuit parameters of the coupled MR emulator.
Parameter Signification Value
RResistance 10 k
CCapacitance 100 nF
Ra1Resistance 10 k
Ra2Resistance 1.429 k
Rb1Resistance 50 k
Rb2Resistance 33.33 k
gMultiplier gain 0.1
RFResistance 520
RcResistance 1 k
RTResistance 2 k
RWResistance 9.8 k
TABLE IV. Rc1and Rc2under different (c1,c2).
Order (c1,c2)Rc1() Rc2()
a (−0.2,0.2) 1000 1500
b (−0.05,0.2) 4000 1500
c (0.2,0.2) 1000 1500
d (0.292,0.2) 684.9 1500
e (0.335,0.2) 597 1500
f (0.2,−1.5) 1000 200
g (0.2,−0.95) 1000 315.8
h (0.2,−0.8) 1000 375
i (0.2,1) 1000 300
j (0.2,2.2) 1000 136.4
FIG. 25. Several phase portraits observed via PSIM simulation.
In Fig. 25, the system orbits are presented using PSIM under ini-
tial condition (0, 0.1, 0, 0, 0). It can be found that the experimental
results are consistent with the theoretical analysis in this study.
The dynamic behaviors of the hyperbolic-type MR-based emula-
tor against the change of crosstalk strengths have thus far been
effectively demonstrated.
VIII. CONCLUSION
In this paper, we have proposed a novel HNN by coupling
two hyperbolic-type MRs for emulating the synaptic crosstalk. We
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have found that the positions of non-zero equilibrium points as
well as the type of attractors can be intentionally altered by chang-
ing crosstalk strength parameters. For different crosstalk strengths,
when different initial conditions are applied, different coexisting
behaviors of asymmetric attractors will emerge, hence revealing the
multi-stability. The direction of evolution from bubbles to remerg-
ing Feigenbaum trees can be also changed by adjusting the two
crosstalk strength parameters. Through theoretical analyses and
mathematical derivations, the HNN’s phase portraits, bifurcation
diagrams, Lyapunov exponent spectrums, and dynamical mapping
have been presented in this study. PSIM simulations have also been
conducted, which verified the validity of the HNN-based emulation
model. Considering the synaptic crosstalk strengths, the dynamic
behaviors including the coexisting asymmetric attractors and anti-
monotonicity can appear by changing the crosstalk strength param-
eters. The above results constitute the main contribution of this
study, which should benefit future investigations in developing
MR-based HNN emulation platforms for accurate representation of
the human brain functions.
ACKNOWLEDGMENTS
This work was supported in part by the Fundamental Research
Funds for the Central Universities of China under Grant No.
2019XKQYMS36.
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