| Citation: |
Jiazhe Lin, Rui Xu, Xiaohong Tian. PATTERN FORMATION IN REACTION-DIFFUSION NEURAL NETWORKS WITH LEAKAGE DELAY[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2224-2244. doi: 10.11948/20190001
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PATTERN FORMATION IN REACTION-DIFFUSION NEURAL NETWORKS WITH LEAKAGE DELAY
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Abstract
Due to the heterogeneity of the electromagnetic field in neural networks, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate pattern formation in a reaction-diffusion neural network with leakage delay. The existence of Hopf bifurcation, as well as the necessary and sufficient conditions for Turing instability, are studied by analyzing the corresponding characteristic equation. Based on the multiple-scale analysis, amplitude equations of the model are derived, which determine the selection and competition of Turing patterns. Numerical simulations are carried out to show the possible patterns and how these patterns evolve. In some cases, the stability performance of Turing patterns is weakened by leakage delay and synaptic transmission delay.-
Keywords:
- Pattern formation /
- reaction-diffusion /
- leakage delay /
- neural network /
- amplitude equation
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References
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Jiazhe Lin, Rui Xu, Xiaohong Tian. PATTERN FORMATION IN REACTION-DIFFUSION NEURAL NETWORKS WITH LEAKAGE DELAY[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2224-2244. doi: 10.11948/20190001
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Figure 1. Phase diagrams of system (1.2) with
$ \tau = 0.02 < \tau_0 = 0.0759 $ (see left-hand figure) and$ \tau = 0.08 > \tau_0 = 0.0681 $ (see right-hand figure), where$ c_1 = 2 $ ,$ c_2 = 4 $ ,$ a_1 = -4 $ ,$ a_2 = 2 $ ,$ b_1 = 3.6 $ ,$ b_2 = 1.2 $ and$ f_i(\cdot), g_i(\cdot) $ $ (i = 1, 2) $ are chosen as$ \tanh(\cdot) $ . -
Figure 2. The bifurcation diagram of system (1.2) with respect to the delay
$ \tau\in[0, 0.15] $ . -
Figure 3. The trajectories of
$ h_1(\tau) $ ,$ h_2(\tau) $ ,$ h_3(\tau) $ and$ h_4(\tau) $ with respect to the delay$ \tau $ , where$ d_1 = 0.1 $ ,$ d_2 = 1.6 $ ,$ c_1 = 2 $ ,$ c_2 = 4 $ ,$ \phi_1 = -4 $ ,$ \phi_2 = 2 $ ,$ \varphi_1 = 3.6 $ ,$ \varphi_2 = 1.2 $ (Fig. 3(a)) and$ d_1 = 0.1 $ ,$ d_2 = 1.6 $ ,$ c1 = 2 $ ,$ c_2 = 4 $ ,$ \phi_1 = -4 $ ,$ \phi_2 = 2 $ ,$ \varphi_1 = 3.1 $ ,$ \varphi_2 = 1.2 $ (Fig. 3(b)).$ f_i(\cdot), g_i(\cdot) $ $ (i = 1, 2) $ are chosen as$ \tanh(\cdot) $ . -
Figure 4. Snapshots of contour pictures of the time evolution of
$ u(x, y, t) $ at time$ T = 5, 50, 100, 200 $ , respectively, with parameter values$ \tau = 0.01 $ ,$ d_1 = 0.1 $ ,$ d_2 = 1.6 $ ,$ c_1 = 2 $ ,$ c_2 = 4 $ ,$ a_1 = -4 $ ,$ a_2 = 2 $ $ b_1 = 3.4 $ ,$ b_2 = 1.2 $ and initial condition$ u_0 = v_0 = \sin(x)\cos(y)+\xi(x, y) $ in which$ \xi(x, y) $ is a stochastic process with respect to the spatial variable$ (x, y) $ . -
Figure 5. The bifurcation diagram of the parameters
$ b_1 $ and$ b_2 $ with$ \tau = 0.01 $ ,$ 0.04 $ ,$ 0.08 $ , respectively, where$ d_1 = 0.1 $ ,$ d_2 = 1.6 $ ,$ c_1 = 2 $ ,$ c_2 = 4 $ ,$ a_1 = -4 $ ,$ a_2 = 2 $ . The position of the `+' is$ (3.6, 1.8) $ . -
Figure 6. Snapshots of contour pictures of the time evolution of
$ u(x, y, t) $ at time$ T = 10, 20, 40, 60 $ , respectively, with parameter values$ \tau = 0.01 $ ,$ d_1 = 0.1 $ ,$ d_2 = 1.6 $ ,$ c_1 = 2 $ ,$ c_2 = 4 $ ,$ a_1 = -4 $ ,$ a_2 = 2 $ ,$ b_1 = 3.6 $ ,$ b_2 = 1.8 $ and initial condition$ u_0 = v_0 = (x-25)^2+(y-25)^2 < 100 $ . -
Figure 7. Snapshots of contour pictures of the time evolution of
$ u(x, y, t) $ at time$ T = 10, 20, 40, 60 $ , respectively, where$ \tau = 0.08 $ and other parameter values are similar to those in Fig. 6.