February  2018, 38(2): 905-939. doi: 10.3934/dcds.2018039

Propagation phenomena for CNNs with asymmetric templates and distributed delays

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

* Corresponding author: Zhixian Yu

Received  January 2017 Revised  August 2017 Published  February 2018

The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward ($c_{-}^*$) and rightward ($c_{+}^*$) spreading speeds for CNNs by appealing to the theory developed in [26,27], and $c_{-}^*+c_{+}^*>0$. Especially, if cells possess the symmetric templates and the same delayed interactions, then $c_{-}^*=c_{+}^*>0$. Moreover, if the effect of the self-feedback interaction $α f'(0)$ is not less than 1, then both $c_{-}^*>0$ and $c_{+}^*>0$. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.

Citation: Zhixian Yu, Xiao-Qiang Zhao. Propagation phenomena for CNNs with asymmetric templates and distributed delays. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 905-939. doi: 10.3934/dcds.2018039
References:
[1]

D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, J. A. Goldstein, ed., Lecture Notes in Mathematics Ser. 446, Springer-Verlag, Berlin, (1975), 5-49. 

[2]

D. Aronson and H. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems-part Ⅰ, IEEE Trans. Circuits and Systems, 42 (1995), 746,752-751,756.  doi: 10.1109/81.473583.

[4]

S.-N. ChowJ. Mallet-Paret and W. Shen, Travelling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[5]

S. N. Chow and W. Shen, Stability and bifurcation of traveling wave solutions in coupled map lattices, J. Dynam. Systems Appl., 4 (1995), 1-25. 

[6]

L. Chua, CNN: A Paradigm for Complexity World Scientific Series on Nonlinear Science, Series A, Vol. 31, World Scientific, Singapore, 1998.

[7]

L. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.  doi: 10.1109/31.7600.

[8]

L. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290.  doi: 10.1109/31.7601.

[9]

P. P. CivalleriM. Gill and L. Pandolfi, On stability of cellular neural networks with delay, IEEE Trans, CAS, 40 (1993), 157-165.  doi: 10.1109/81.222796.

[10]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.

[11]

J. Fang and X. -Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.

[12]

J. FangJ. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A, 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.

[13]

D. GolombX. J. Wang and J. Rinzel, Propagation of spindle waves in a thalamic slice model, J. Neurophysiol, 75 (1996), 750-769. 

[14]

D. Golomb and Y. Amitai, Propagating neuronal discharges in neocortical slices: Computational and experimental study, J. Neurophysiol, 78 (1997), 1199-1211. 

[15]

J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985), 141--152,

[16]

J. J. Hopfield and D. W. Tank, Computing with neural circuits: A model, Science (USA), 233 (1986), 625-633.  doi: 10.1126/science.3755256.

[17]

C. HsuC. Li and S. Yang, Diversity of traveling wave solutions in delayed cellular neural networks, Internat. J. Bifur. Chaos, 18 (2008), 3515-3550.  doi: 10.1142/S0218127408022561.

[18]

C. HsuS. Lin and W. Shen, Traveling waves in cellular neural networks, Internat. J. Bifur. Chaos, 9 (1999), 1307-1319.  doi: 10.1142/S0218127499000912.

[19]

C. Hsu and S. Lin, Existence and multiplicity of traveling waves in a lattice dynamical systems, J. Differential Equations, 164 (2000), 431-450.  doi: 10.1006/jdeq.2000.3770.

[20]

C. Hsu and S. Yang, Structure of a class of traveling waves in delayed cellular neural networks, Discrete Contin. Dynam. Systems, 13 (2005), 339-359.  doi: 10.3934/dcds.2005.13.339.

[21]

C. Hsu and S. Yang, Traveling wave solutions in cellular neural networks with multiple time delays, Discrete Contin. Dynam. Systems Suppl., (2005), 410-419. 

[22]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[23]

J. Juang and S. S. Lin, Cellular neural networks: Mosaic pattern and spatial chaos, SIAM J. Appl. Math., 60 (2000), 891-915.  doi: 10.1137/S0036139997323607.

[24]

J. P. Keener, Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

[25]

B. LiH. Weinberger and M. Lewis, Spreading speeds as slowest wave speed for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[26]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[27]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[28]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, Journal of Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[29]

X. LiuP. Weng and Z. Xu, Existence of traveling wave solutions in nonlinear delayed cellular neural networks, Nonlinear Anal. RWA, 10 (2009), 277-286.  doi: 10.1016/j.nonrwa.2007.09.010.

[30]

Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection, Applicable Analysis, 89 (2010), 1067-1089.  doi: 10.1080/00036810903437804.

[31]

R. Lui, Biological growth and spread modeled by systems of recursions, I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.

[32]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems, Ⅱ, IEEE Trans. Circuits and Systems, 42 (1995), 752-756.  doi: 10.1109/81.473583.

[33]

J. Mallet-Paret, The global structure of traveling waves in spatial discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074.

[34]

V. Ptrez-MuiiuzuriV. Perez-Villar and L. O. Chua, Propagation failure in linear arrays of Chua's circuits, Int. J. Bifurc. and Chaos, 2 (1996), 403-406.  doi: 10.1142/S0218127492000380.

[35]

T. Roska and L. Chua, Cellular neural networks with nonlinear and delay-time template elements and nonuniform grids, Int. J. Circuit Theory Appl., 20 (1992), 469-481. 

[36]

A. Slavova, Cellular Neural Networks: Dynamics and Modelling Kluwer Academic Publishers, 2003. doi: 10.1007/978-94-017-0261-4.

[37]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Providence, RI, 1995.

[38]

H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.  doi: 10.1007/BF00279720.

[39]

H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.

[40]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[41]

P. Weng and J. Wu, Deformation of traveling waves in delayed cellular neural networks, Internat. J. Bifur. Chaos, 13 (2003), 797-813.  doi: 10.1142/S0218127403006947.

[42]

P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.

[43]

S. Wu and C. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 2785-2801.  doi: 10.1088/0951-7715/25/9/2785.

[44]

S. Wu and C. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Royal Soc. Edinb., 144 (2014), 1085-1112.  doi: 10.1017/S0308210512001412.

[45]

Z. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267.  doi: 10.1016/j.jde.2015.08.037.

[46]

Z. YuR. YuanC.-H. Hsu and Q. Jiang, Traveling waves for nonlinear cellular neural networks with distributed delays, J. Differential Equations, 251 (2011), 630-650.  doi: 10.1016/j.jde.2011.05.008.

[47]

Z. Yu, R. Yuan, C. -H. Hsu and M. Peng, Traveling waves for delayed cellular neural networks with nonmonotonic output functions Abstract and Applied Analysis 2014 (2014), ID 490161, 11pp. doi: 10.1155/2014/490161.

[48]

B. Zinner, Existence of travelling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A.

show all references

References:
[1]

D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, J. A. Goldstein, ed., Lecture Notes in Mathematics Ser. 446, Springer-Verlag, Berlin, (1975), 5-49. 

[2]

D. Aronson and H. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems-part Ⅰ, IEEE Trans. Circuits and Systems, 42 (1995), 746,752-751,756.  doi: 10.1109/81.473583.

[4]

S.-N. ChowJ. Mallet-Paret and W. Shen, Travelling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[5]

S. N. Chow and W. Shen, Stability and bifurcation of traveling wave solutions in coupled map lattices, J. Dynam. Systems Appl., 4 (1995), 1-25. 

[6]

L. Chua, CNN: A Paradigm for Complexity World Scientific Series on Nonlinear Science, Series A, Vol. 31, World Scientific, Singapore, 1998.

[7]

L. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.  doi: 10.1109/31.7600.

[8]

L. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290.  doi: 10.1109/31.7601.

[9]

P. P. CivalleriM. Gill and L. Pandolfi, On stability of cellular neural networks with delay, IEEE Trans, CAS, 40 (1993), 157-165.  doi: 10.1109/81.222796.

[10]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.

[11]

J. Fang and X. -Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.  doi: 10.1016/j.jde.2010.01.009.

[12]

J. FangJ. Wei and X.-Q. Zhao, Spreading speeds and travelling waves for non-monotone time-delayed lattice equations, Proc. R. Soc. A, 466 (2010), 1919-1934.  doi: 10.1098/rspa.2009.0577.

[13]

D. GolombX. J. Wang and J. Rinzel, Propagation of spindle waves in a thalamic slice model, J. Neurophysiol, 75 (1996), 750-769. 

[14]

D. Golomb and Y. Amitai, Propagating neuronal discharges in neocortical slices: Computational and experimental study, J. Neurophysiol, 78 (1997), 1199-1211. 

[15]

J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985), 141--152,

[16]

J. J. Hopfield and D. W. Tank, Computing with neural circuits: A model, Science (USA), 233 (1986), 625-633.  doi: 10.1126/science.3755256.

[17]

C. HsuC. Li and S. Yang, Diversity of traveling wave solutions in delayed cellular neural networks, Internat. J. Bifur. Chaos, 18 (2008), 3515-3550.  doi: 10.1142/S0218127408022561.

[18]

C. HsuS. Lin and W. Shen, Traveling waves in cellular neural networks, Internat. J. Bifur. Chaos, 9 (1999), 1307-1319.  doi: 10.1142/S0218127499000912.

[19]

C. Hsu and S. Lin, Existence and multiplicity of traveling waves in a lattice dynamical systems, J. Differential Equations, 164 (2000), 431-450.  doi: 10.1006/jdeq.2000.3770.

[20]

C. Hsu and S. Yang, Structure of a class of traveling waves in delayed cellular neural networks, Discrete Contin. Dynam. Systems, 13 (2005), 339-359.  doi: 10.3934/dcds.2005.13.339.

[21]

C. Hsu and S. Yang, Traveling wave solutions in cellular neural networks with multiple time delays, Discrete Contin. Dynam. Systems Suppl., (2005), 410-419. 

[22]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[23]

J. Juang and S. S. Lin, Cellular neural networks: Mosaic pattern and spatial chaos, SIAM J. Appl. Math., 60 (2000), 891-915.  doi: 10.1137/S0036139997323607.

[24]

J. P. Keener, Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

[25]

B. LiH. Weinberger and M. Lewis, Spreading speeds as slowest wave speed for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[26]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[27]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[28]

X. LiangY. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, Journal of Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.

[29]

X. LiuP. Weng and Z. Xu, Existence of traveling wave solutions in nonlinear delayed cellular neural networks, Nonlinear Anal. RWA, 10 (2009), 277-286.  doi: 10.1016/j.nonrwa.2007.09.010.

[30]

Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection, Applicable Analysis, 89 (2010), 1067-1089.  doi: 10.1080/00036810903437804.

[31]

R. Lui, Biological growth and spread modeled by systems of recursions, I. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.

[32]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems, Ⅱ, IEEE Trans. Circuits and Systems, 42 (1995), 752-756.  doi: 10.1109/81.473583.

[33]

J. Mallet-Paret, The global structure of traveling waves in spatial discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074.

[34]

V. Ptrez-MuiiuzuriV. Perez-Villar and L. O. Chua, Propagation failure in linear arrays of Chua's circuits, Int. J. Bifurc. and Chaos, 2 (1996), 403-406.  doi: 10.1142/S0218127492000380.

[35]

T. Roska and L. Chua, Cellular neural networks with nonlinear and delay-time template elements and nonuniform grids, Int. J. Circuit Theory Appl., 20 (1992), 469-481. 

[36]

A. Slavova, Cellular Neural Networks: Dynamics and Modelling Kluwer Academic Publishers, 2003. doi: 10.1007/978-94-017-0261-4.

[37]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Providence, RI, 1995.

[38]

H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.  doi: 10.1007/BF00279720.

[39]

H. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.

[40]

H. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[41]

P. Weng and J. Wu, Deformation of traveling waves in delayed cellular neural networks, Internat. J. Bifur. Chaos, 13 (2003), 797-813.  doi: 10.1142/S0218127403006947.

[42]

P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.

[43]

S. Wu and C. Hsu, Entire solutions of nonlinear cellular neural networks with distributed time delays, Nonlinearity, 25 (2012), 2785-2801.  doi: 10.1088/0951-7715/25/9/2785.

[44]

S. Wu and C. Hsu, Entire solutions of non-quasi-monotone delayed reaction-diffusion equations with applications, Proc. Royal Soc. Edinb., 144 (2014), 1085-1112.  doi: 10.1017/S0308210512001412.

[45]

Z. Yu and M. Mei, Uniqueness and stability of traveling waves for cellular neural networks with multiple delays, J. Differential Equations, 260 (2016), 241-267.  doi: 10.1016/j.jde.2015.08.037.

[46]

Z. YuR. YuanC.-H. Hsu and Q. Jiang, Traveling waves for nonlinear cellular neural networks with distributed delays, J. Differential Equations, 251 (2011), 630-650.  doi: 10.1016/j.jde.2011.05.008.

[47]

Z. Yu, R. Yuan, C. -H. Hsu and M. Peng, Traveling waves for delayed cellular neural networks with nonmonotonic output functions Abstract and Applied Analysis 2014 (2014), ID 490161, 11pp. doi: 10.1155/2014/490161.

[48]

B. Zinner, Existence of travelling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A.

Figure 1.  The spread of wn, and the left plot shows 3-D graph of wn, and the right one indicates that projection of wn is on the plane (n; t)-plane.
Figure 2.  The rightward traveling waves observed for wn(t) in different views.
Figure 3.  The leftward traveling waves observed for wn(t) in different views.
Figure 4.  The leftward traveling waves observed for $w_n(t)$ in different views.
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