Traveling waves of some integral-differential equations arising from neuronal networks with oscillatory kernels. (English) Zbl 1194.35095

Summary: We study the existence, uniqueness and profiles of traveling waves to some integral-differential equations arising from nonlinear nonlocal neuronal networks with oscillatory kernels. Our approach is fundamental ideas in differential equation and function analysis.


35C07 Traveling wave solutions
92B20 Neural networks for/in biological studies, artificial life and related topics
35B09 Positive solutions to PDEs
Full Text: DOI


[1] Amari, S., Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. cybernet., 27, 77-87, (1977) · Zbl 0367.92005
[2] Aronson, D.G.; Weiberger, H.F., Multidimensional nonlinear diffusion arising in population genetics, Adv. math., 30, 33-76, (1978) · Zbl 0407.92014
[3] Curtu, R.; Ermentrout, B., Pattern formation in a network of excitatory and inhibitory cells with adaptation, SIAM J. appl. dyn. syst., 3, 191-231, (2004) · Zbl 1090.34038
[4] Guo, Y.; Chow, C.C., Existence and stability of standing pulses in neuronal: I existence, II stability, SIAM J. appl. dyn. syst., 4, (2005), I: 217-248, II: 249-281 · Zbl 1109.34002
[5] Hutt, A.; Atay, F.M., Effects of distributed transmission speeds on propagating activity in neural populations, Phys. rev. E, 73, 021906, (2006)
[6] Hutt, A., Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay, Phys. rev. E, 70, 052902, (2004)
[7] Hidgkin, A.; Huxley, A., A quantitative description of membrane current and its application to conduction and excitation in nerve, J. phys., 117, 500-544, (1952)
[8] Rudin, W., Functional analysis, Internat. ser. pure appl. math., (1991), New York · Zbl 0867.46001
[9] Ou, C.; Wu, J.H., Persistence of wavefronts in delayed nonlocal reaction – diffusion equations, J. differential equations, 219-261, (2007) · Zbl 1117.35037
[10] Sadstede, B., Evans function and nonlinear stability of traveling waves in neuronal network models, Internat. J. bifur. chaos appl. sci. engrg., 17, 2693-2704, (2007) · Zbl 1144.35342
[11] Wilson, H.R.; Cowan, J.D., Excitatory and inhibitory interactions in localizes populations of model neurons, Biophys. J., 12, 1-24, (1972)
[12] Xiu, D.X.; Wu, Z.R.; Shu, W.C., Real analysis and function analysis, (1988), Advance Educational Publish House Beijing
[13] Zhang, L.H., How do synaptic coupling and spatial temporal delay influence traveling wave solution in nonlinear nonlocal neuronal network?, SIAM J. appl. dyn. syst., 6, 597-644, (2007) · Zbl 1210.35018
[14] Zhang, L.H., Dynamic of neuronal waves, Math. Z., 255, 283-321, (2007) · Zbl 1186.92012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.