Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.
Recepción: 22 Agosto 2021
Aprobación: 05 Mayo 2022
Abstract: It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns.
Keywords: fractional derivative, neural network, Turing instability, pattern formation, reaction– diffusion.
1 Introduction
In the past few years, neural networks have attracted much attention in different fields of science and engineering, owing to their valuable applications in associative memory, combinatorial optimization, image processing and so on [5, 13]. Some of the above applications are based on the dynamical behavior analysis of neural networks [2]. Actually, neural networks are realized by large-scale integrated circuits, and the density of electromagnetic fields is generally not uniform. Therefore, in factual modeling, only considering the change of time seems to be not comprehensive when electrons are moving in asymmetric and nonuniform electromagnetic fields [21,22]. Due to the reaction–diffusion effect, neural networks have rich spatial and temporal dynamical behaviors, like various Turing patterns.
Spatial dynamics in reaction–diffusion systems was originally proposed by Turing in the work of [23], which mainly focuses on chemical reaction systems. This pioneer work not only established the theoretical foundation for understanding diverse patterns occurring in the natural world, but also opened a new research field, namely, pattern dynamics, which has received extensive attention and is still a hot topic in many fields such as species dynamics [7, 27, 28], medicine [26, 31], networks [4, 14, 29]. Based on the reaction–diffusion theory of Turing, Chua and Goras¸ [4] investigated the phenomenon of pattern formation in cellular neural networks. Recently, Zhao et al. [29] studied the conditions of Hopf bifurcation and Turing instability in a reaction–diffusion neural network, where the corresponding model is described as follows:
(1)in which 
stand for state variables of neurons at time 
and spatial position
denote the rates of resetting neuronic potential to the resting state in isolation; 
represent connection weights; 
is the Laplacian operator in a two-dimensional space; 
are diffusion coefficients of electrons among neurons.
Traditional neural networks, such as system (1), are mainly established from the view of integer-order derivatives, which can be described by classical ordinary differential equations. From the perspective of the application the integer-order derivative is used to describe some properties at a certain time in physical processes or some local properties of a certain position. In recent years, experimental research indicates that fraction-order derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes [1, 15]. Generally speaking, plenty of practical objects can be described clearly by the fractional differential equations due to their more degrees of freedom and infinite memory. Thus, the research of fractional neural networks has gained much attention, and some valuable results have been referred to [9–12, 20].
Besides, due to the finite speed of signal transmission and amplifiers switching, time delay exists unavoidably in neural networks, which is known as transmission delay. Usually, the time delay is harmful to the stability performance of neural networks, causing oscillation, even chaos. Zhao et al. [30] introduced the transmission delay into a discrete- time Hopfield neural network model and showed that Hopf bifurcation occurs when the delay exceeds a critical value.
In [8], fractional calculus is used in electrical circuits that possess memory and hereditary properties, which is similar to the integrated circuits of Hopfield neural network.
Moreover, since fractional calculus is a generalization of conventional calculus, it is expected that fractional models will generally provide a more accurate description of the system dynamics than those based on classical differential equations [17]. In order to describe the memory and hereditary properties of Hopfield neural network, fractional order 
is introduced into our model. Besides, time delay 
and 
are considered, which accounts for the finite speed of signal transmission and amplifiers switching in integrated circuits. Furthermore, it is generally assumed that the reaction–diffusion terms in system (1) play a conclusive role in pattern formation. A question arises naturally whether temporal fractional derivative has some relationship with pattern formation. To be specific, can the fractional reaction–diffusion neural network produce steadily spatial patterns even if its first-derivative counterpart cannot form any steady pattern? To this end, we consider the following delayed fractional reaction–diffusion neural network:
(2)with initial condition
satisfying
and Neumann boundary condition
where 
is the fractional order; 
and 
are transmission delay that reflects the time-lag effect; 
denotes the activation function, which maps the input to the output of the neuron; a square domain 
in which 
is a positive bounded constant; 
is the outward unit normal vector of the boundary 
that is assumed to be smooth; other variables and parameters are similar to those in system (1). For simplicity, let 
The Neumann boundary condition implies that nothing enters this system and nothing exits from this system.
The main contributions of this paper can be summarized as follows: (i) By theoretical analysis and numerical simulations, the paper illustrates that temporal fractional derivative contributes to pattern formation in system (2); (ii) Fractional derivative can soften the stability conditions of the homogeneous steady state, which implies that when fractional-order 
Hopf bifurcation occurs in system (2), but once 
the homogeneous steady state may become stable; (iii) To avoid complicated and tedious MATLAB programming, numerical simulations are mainly completed by Simulink, which is more efficient and can visually represent the interrelationships of neuron states; (IV) Numerical simulations indicate that the fractional derivative is related to the shape of the Turing pattern. Besides, as the fractional derivative decreases, it softens the Turing instability conditions of system (2).
This paper is organized as follows. In Section 2, we first analyze the existence of Hopf bifurcation and obtain the parameter sets, where the homogeneous steady state of system (2) is stable. Furthermore, we present and discuss the results of Turing instability and pattern formation, which will be illustrated by numerical simulations in Section 3. The paper ends with a conclusion in Section 4.
2 Hopf bifurcation and Turing instability
Throughout this paper, we address the following assumption holds.
(H1) 
2.1 Hopf bifurcation
The homogeneous steady state of system (2) is 
In this subsection, we focus on the local asymptotic stability of 
, which is useful for Turing instability analysis. The linear perturbation equations with respect to 
are
(3)where 
Expand the perturbation variables in the Fourier space
(4)where 
is the growth rate of perturbations in time 
, i is the imaginary unit and 
In order to facilitate the analysis, let 
Substituting (4) into (3), we obtain the characteristic equation as follows:
(5)where
To find possible periodic solutions, which may bifurcate from a Hopf bifurcation point, let 
is a real number) be a root of (5). Separating the real and imaginary parts yields
(6)It follows that
(6)From the first equation of (6) we have for 
Define the bifurcation point of system (2) as 
To establish the main results of this section, we list the following assumptions.
(H2) Equation (7) has no positive real root.
(H3) Equation (7) has at least one positive real root.
(H4) 
where 
will be defined in Lemma 2.
Define a parameter set
where 
is locally asymptotically stable when 
and 
For 
and 
, we first introduce the following lemma.
Lemma 1. (See [16].) The following autonomous system
where 
is asymptotically stable if and only if 
In this case, each component of the states decays towards 0 like 
Also, this system is asymptotically stable if and only if 
and those critical eigenvalues that satisfy 
have geometric multiplicity one.
From Lemma 1 we obtain that the homogeneous steady state 
is locally asymptotically stable if the following inequalities hold: 
Similarly, we define the following parameter set:
where 
is locally asymptotically stable for 
and 
.
In conclusion, when 
, the homogeneous steady state 
is stable if the parameters 
and 
are in the set 
Lemma 2. Let 
be the root of Eq. (5)near 
satisfying 
then the following transversality condition holds:
Proof. Based on implicit function theorem, we calculate the derivative of Eq. (5) with respect to 
as follows: 
, where 
The real part of 
is 
, where 
and 
are the real and imaginary parts of 
crespectively; 
and 
are the real and imaginary parts of 
respectively. Based on assumption (H4), the transversality condition is met.
Theorem 1. For system (2), the following results hold:
-
If
stays in the set 
and (H2) holds, then the homogeneous steady state is locally asymptotically stable for 
-
If
stays in the set 
and (H3), (H4) hold, then-
The homogeneous steady state is locally asymptotically stable for
-
System (2) undergoes a Hopf bifurcation at the homogeneous steady state when
, i.e., it has a branch of periodic solutions bifurcating from the homogeneous steady state near 
.
-
Choose 
as free parameters and project the parameter sets 
and 
into a two-dimensional space with respect to 
. From the conditions of parameter sets 
and 
we draw the stability regions of ... In Fig. 1, the parameter set 
corresponds to region A (green), while 
corresponds to the region B (yellow). Region B implies that temporal fractional derivative can enlarge the stability region of 
.
Figure 1
The bifurcation diagram of the parameters ϕ1 and ϕ2 with c1 = 2, c2 = 4, φ1 = 4, φ2 = 2, τ = 0, where the position of the mark “ ” is (2, 3), and the one of “ ” is (3, 4). The two solid (blue) lines come from the conditions of Π1, while the dashed (orange) line comes from the condition of Π2.
2.2Turing instability
Next, we will verify that system (2) can form steadily spatial patterns even if its first- derivative counterpart cannot develop any steady pattern. Firstly, we discuss the Turing instability of system (2). The unbalanced changes of phases, corresponding to Turing branches, are the transitions of system (2) from the uniform state to the oscillatory state [29, 32]. After the process, the formed patterns are called Turing patterns. From (5) we obtain the necessary conditions for causing Turing instability when 
:
(8)
(9)
(10)Conditions (8)–(10) indicate that system (2) is unstable for some perturbations to the wave number 
. Thus, we obtain that 
at the critical value. That is to say, Turing bifurcation occurs when Im 
Re 
When Turing patterns come into being, the wave number 
satisfies
Based on above discussions, we get the following result.
Lemma 3. If the parameters 
and 
are in the set 
and the following conditions hold
(11)
then Turing instability in system (2)occurs when 
.
When 
Turing instability occurs when 
namely, 
and then we have the following Turing instability curve:
Letting 
yields
Denote
We set 
as free parameters, then project the parameter set 
into a two- dimensional space with respect to 
. From Fig. 2 we observe that 
corresponds with regions C (green) and D (yellow), where the Turing instability occurs.
Figure 2
The bifurcation diagram of the parameters ϕ1 and ϕ2 with d1 = 0.1, d2 = 1.6, c1 = 2, c2 = 4, φ1 = −4, φ2 = 2, where the position of the mark “∗” is (4, 1) and the one of “+” is (3.5, 3). The dashed (red) lines come from the conditions of Π3, while the dot-and-dash (green) lines corresponds to ϕ2 (ϕ1) when k = 2.
When the fraction order 
the Turing instability region only consists of region C. That is to say, region D is the enlarged region, where Turing instability is caused by fractional derivatives and reaction–diffusion terms together.
Theorem 2. If 
stays in 
, we obtain the following results:
-
For
system (2) will undergoes k-mode Turing bifurcation at 
-
When
mode Turing–Turing bifurcation occurs at 
Remark 1. Usually, the curves of 
with 
may reduce the area of Turing instability region. In Fig. 2, regions C and D are exactly below the curve of 
with 
and the Turing–Turing bifurcation point with 
is on the left side of regions C and D. Thus, in this paper, the curves of 
with 
make no difference with the Turing instability region.
3 Numerical simulations
Numerical algorithms for reaction–diffusion systems are often complicated, which need tedious MATLAB programming. Garvie [6] proposed a semiimplicit (in time) finite- difference scheme to approximate the solutions of reaction–diffusion systems. The semi- implicit method means this algorithm involves approximations at the current time level 
and at the previous time level 
The corresponding algorithm is reduced to a sparse, banded and linear system of algebraic equations. In this section, we use Simulink to simplify the programming, which is more efficient and visually represents the interrelationships of neuron states. Simulink is graphical programming that is easy for nonprofessional scholars to program and realize the equivalent effect gained by MATLAB code. Besides, we can utilize user-defined MATLAB function blocks in Simulink, which takes full advantage of MATLAB code and Simulink.
Firstly, we give the following result about the Laplacian operator, which is used for the approximate calculation of reaction–diffusion terms. By means of the finite-difference scheme mentioned in [6, 25], the Laplacian with respect to the concentration field 
in the node 
is calculated along the 
and 
directions simultaneously:
where
and 
in Eq. (13) are 
and 
grid spacings, respectively. Assume
and discretize Laplacian operation in a two-dimensional domain 
:
(14)From (14) the two-dimensional Laplacian operator is
Following the approach of the modified Adams–Bashforth–Moulton predictor–corrector scheme mentioned in [3], we derive the approximate numerical values of the fractional derivatives with delay. Denote 
as the time-step. Consider a uniform grid
where 
and 
are integers such that 
and 
Discretize the initial condition
Besides, the delayed terms can be rewritten as follows:
where 
The iteration formulas from 
are
where the discrete right-hand terms in system (2) are
and the predictor terms are
In addition,
and 
are defined as
System (2) is simulated numerically in a 
two-dimensional square region. Time step and space step are set as 0.005 and 0.5, respectively, which needs large amount of computation but ensures the accuracy of numerical simulations. Diffusion coefficients
are chosen as (0.1, 1.6). Besides, 
and 
are chosen as free parameters. Simulink construction of system (2), as we can see in Fig. 3, consists of some blocks and lines. Each block can realize different functions and is connected with others by data flow lines. The main blocks are 2D Convolution and user- defined MATLAB function. The 2D convolution completes the approximate calculation of the reaction–diffusion terms by two-dimensional Laplacian operator 
. The user-defined MATLAB function realizes the integration of the right-hand terms in system (2) by the above modified Adams–Bashforth–Moulton predictor–corrector scheme.
When fractional order 
and 
, system (2) becomes a first-derivative system. Activation function 
is selected as tanh 
, where 
and 
Set as (4,1) (marked by 
) and (3.5, 3) (marked by 
), respectively (as we can see in Fig. 2). Figures 4 and 5 are the two-dimension spatial states of neurons 
and 
at some moments, respectively. On account of diffusion, neuronic states not only vary in time, but also change in space. What is interesting is that spatial states take
Figure 3
Simulink construction of system 2 with q ∈ 0 1
on particular shapes. Comparing the figures of 
, 
with the figures of 
, 
in Fig. 4, we observe that system (2) forms steady Turing patterns. Furthermore, when 
is (3.5, 3), we observe in Fig. 5 that as the time 
increases from 50 to 100, the figures of 
, 
are different from the figures of 
, 
, which means that under this condition, system (2) cannot induce any steady Turing pattern.
As we can observe in Fig. 6, system (2) with 
can form steady spatial patterns. Comparing it with Fig. 5, we know that system (2) form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern. Hence, these patterns are induced by temporal fractional derivative and reaction–diffusion terms together, which corresponds with region D in Fig. 2. Furthermore, to investigate the relationship between fractional derivative and Turing pattern, we change the fractional order 
as 0.72 or 0.87. From Figs. 7 and 8 we observe that the shape of Turing patterns varies with the fractional order 
. Obviously, when 
, the patterns of 
and 
become various steady types. When
the stability performance of the patterns of 
and 
is a little weak. In Fig. 8, when the time . increases from 50 to 100, the center patterns change to some extent.
It rises a question that whether temporal fractional derivative has some relationship with the locally asymptotical stability region or Turing instability region with respect to 
. From the left figure in Fig. 9 we obtain that the stability region of 
varies with fractional order 
. As 
increases from 0.72 to 0.79, the stability region of 
loses region D instead. Increasing 
from 0.79 to 0.87 continuously, the stability region does not contain region C, namely, when 
is stable when
only stays in regions A and B. Besides, we wonder how the Turing instability region induced by fractional derivatives and reaction–diffusion terms together varies with fractional order 
. From the right figure in Fig. 9, when 
, the Turing instability region induced by fractional derivatives and reaction–diffusion terms together consists of regions E, F and G.
Figure 4
The steadily spatial patterns of u(x, y, t) and v(x, y, t) of system (2) in two-dimension square domain with τ = 0, q = 1, d1 = 0.1, d2 = 1.6, c1 = 2, c2 = 4, a1 = −4, a2 = 2, b1 = 4 and b2 = 1. The initial condition is chosen as u(0) = ((x−50)2 +0.2(y −50)2) < 250, v(0) = (0.2(x−50)2 +(y −50)2) < 250.
Figure 5
The unsteadily spatial patterns of u(x, y, t) and v(x, y, t) of system (2) in two-dimension square domain with q = 1, b1 = 3.5 and b2 = 3 and other parameters are similar to those in Fig. 4.
Figure 6
The steadily spatial patterns of u(x, y, t) and v(x, y, t) for system (2) in two-dimension square domain with q = 0.79, b1 = 3.5 and b2 = 3 and other parameters are similar to those in Fig. 5.
Figure 7
The steadily spatial patterns of u(x, y, t) and v(x, y, t) for system (2) in two-dimension square domain with q = 0.72 and other parameters are similar to those in Fig. 6.
Figure 8
The unsteadily spatial patterns of u(x, y, t) and v(x, y, t) for system (2) in two-dimension square domain with q = 0.87 and other parameters are similar to those in Fig. 6.
Figure 9
The bifurcation diagram of the parameters ϕ1 and ϕ2 with different fractional orders.
As 
increases, the corresponding Turing instability region shrinks to region E, that is to say, when 
stays in regions F and G, system (2) with 
cannot form steady Turing patterns.
Remark 2. Due to that time delay widely exists in the neural network, we investigate the impact of transmission delay on Turing patterns. As we observe in Fig. 10, the patterns of 
and 
are more complicated than the patterns of those in Figs. 4–5 (a nondelay integer-order subsystem of system (2)) and Figs. 6–8 (a nondelay fractional- order subsystem of system (2)). Compared with Fig. 6, as 
increase from 0 to 0.6, Turing patterns change unpredictably and become unstable.
Figure 10
The unsteadily spatial patterns of u(x, y, t) and v(x, y, t) for system (2) in two-dimension square domain with q = 0.79, τ = 0.6 and other parameters are similar to those in Fig. 6.
Figure 11
The bifurcation diagrams of u(0, 0, ·) and v(0, 0, ·) with respect to the parameter τ ∈ [0, 1].
Remark 3. As shown in Fig. 11, the central points of 
and 
are selected to study the bifurcation diagrams of system (2) with respect to transmission delay 
and 
stay in stable state when
and oscillate greatly as 
exceeds 
, where 
can be seen as the critical value of Hopf bifurcation. Hence, as the transmission delay increases, Turing patterns lose its stability, and state variables of neurons vary in both time and space. Furthermore, when
and fractional order is in a certain section, a stable Turing pattern may occur, which is induced by temporal
fractional derivative and reaction–diffusion terms together; when
, no matter how fractional order changes, the stable Turing pattern is hard to form.
4 Conclusion
In this paper, we proposed a delayed reaction–diffusion neural network with the Caputo- type fractional derivative. The condition of Hopf bifurcation was obtained firstly through analyzing relevant characteristic equation, while the condition of Turing instability was derived following the pattern dynamics theory proposed by [23]. Especially, we found that fractional derivative has a certain relationship with Hopf bifurcation and Turing pattern. To be specific, the fractional derivative could enlarge the stability region of homogeneous steady state 
concerning 
, which means that when the first-derivative counterpart of system (2) is unstable at 
, but once introducing fractional derivative, 
may become stable again. Besides, as the fraction derivative decreases, it softens the Turing instability conditions of system (2). With the help of Simulink, we obtained the approximation solutions of system (2). Numerical simulations show that temporal fractional derivative contributes to Turing patterns in system (2), which accords with the previous theoretical analysis. Meanwhile, we found that the shape of the Turing pattern is related to the fraction order and the time delay.
Pattern dynamics in neural networks with diffusion has been investigated in [4,14,29], but previous research does not consider fractional derivative. In our study, we illustrate the impact of fractional derivative on spatial and temporal dynamics for system (2) and obtain plenty of meaningful findings. [29] obtained the amplitude equations for system (1) and studied the selection of Turing patterns. Due to the existence of fractional derivative, it is difficult for us to obtain the amplitude equations of system (2). In our further research, we will try to improve some classic methods, such as multiple-scale analysis, to obtain the amplitude equations for fractional reaction–diffusion systems.
The spatial dynamics analysis of reaction–diffusion neural networks not only reveals some properties of integrated circuits that are used to build neural networks, where a Turing pattern corresponds to a steady state that leads to nonuniformly spatial oscillation, but also illustrates many biological phenomena such as normal neuron firing in the brain [18] or biological disorders such as fibrillation [24].
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