Influence of sodium currents on speeds of traveling wave fronts in synaptically coupled neuronal networks. (English) Zbl 1197.35077

Summary: The main purpose of this paper is to apply mathematical analysis to investigate the influence of sodium currents on the speeds of traveling wave fronts. The authors use speed index functions to investigate the speeds of traveling wave fronts of some scalar integral differential equations arising from synaptically coupled neuronal networks. The mathematical model equation is
\[ u_t + f(u) = \alpha \int_{\mathbb R} K(x-y)H\left(u\left(y, t - \tfrac{1}{c}|x-y|\right)-\theta \right)\,dy, \]
where \(0<c\leq \infty\), \(\alpha> 0\) are constants, satisfying the condition \(0< 2f(\theta)< \alpha \). The function \(f(u)\) represents sodium currents, the function \(K\) denotes synaptic coupling in a neuronal network, and the Heaviside step function \(H\) is defined by \(H(u)=0\) for all \(u<0\), \(H(0)= \frac{1}{2}\) and \(H(u)= 1\) for all \(u> 0\).


35C07 Traveling wave solutions
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text: DOI


[1] Prechtl, J.C.; Cohen, L.B.; Pesaran, B.; Mitra, P.P.; Kleinfeld, D., Visual stimuli induce waves of electrical activity in turtle cortex, Proc. natl. acad. sci., USA, 94, 7621-7626, (1997)
[2] Tamas, Gabor; Buhl, Eberhard H.; Somogyi, Peter, Massive autaptic self-innervation of gabaergic neurons in cat visual cortex, J. neuroscience, 17, 6352-6364, (1997)
[3] Connors, Barry W.; Amitai, Yael, Generation of epileptiform discharge by local circuits of neocortex, (), 388-423
[4] Lance, J.W., Current concepts of migraine pathogenesis, Neurology, 43, S11-S15, (1993)
[5] Amari, Shun-ichi, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. cybern., 27, 77-87, (1977) · Zbl 0367.92005
[6] Bressloff, Paul C., Weakly interacting pulses in synaptically coupled neural media, SIAM J. appl. math., 66, 57-81, (2005) · Zbl 1091.92013
[7] Bressloff, Paul C.; Folias, Stefanos E., Front bifurcations in an excitatory neural network, SIAM J. appl. math., 65, 131-151, (2004) · Zbl 1071.92003
[8] Bressloff, Paul C.; Folias, Stefanos E.; Prat, Alain; Li, Yue-Xian, Oscillatory waves in inhomogeneous neural media, Phys. rev. lett., 91, 178101-1-178101-4, (2003)
[9] Coombes, Stephen, Waves, bumps, and patterns in neural field theories, Biol. cybern., 93, 91-108, (2005) · Zbl 1116.92012
[10] Coombes, Stephen; Lord, Gabriel J.; Owen, Markus R., Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178, 219-241, (2003) · Zbl 1013.92006
[11] Coombes, Stephen; Owen, Markus R., Evans functions for integral neural field equations with heaviside firing rate function, SIAM J. appl. dyn. syst., 3, 574-600, (2004) · Zbl 1067.92019
[12] Bard Ermentrout, G., Neural networks as spatio-temporal pattern-forming systems, Rep. progr. phys., 61, 353-430, (1998)
[13] Bard Ermentrout, G.; Bryce McLeod, J., Existence and uniqueness of travelling waves for a neural network, Proc. roy. soc. Edinburgh, A, 123, 461-478, (1993) · Zbl 0797.35072
[14] G. Bard Ermentrout, David Terman, Foundations of Mathematical Neuroscience, Cambridge University Press (in press) · Zbl 1320.92002
[15] Folias, Stefanos E.; Bressloff, Paul C., Breathing pulses in an excitatory neural network, SIAM J. appl. dyn. syst., 3, 378-407, (2004) · Zbl 1058.92010
[16] Folias, Stefanos E.; Bressloff, Paul C., Stimulus-locked traveling waves and breathers in an excitatory neural network, SIAM J. appl. math., 65, 2067-2092, (2005) · Zbl 1076.92004
[17] Pinto, David J.; Bard Ermentrout, G., Spatially structured activity in synaptically coupled neuronal networks. I. traveling fronts and pulses, II. lateral inhibition and standing pulses, SIAM J. appl. math., 62, (2001), I. 206-225, II. 226-243 · Zbl 1001.92021
[18] Pinto, David J.; Jackson, Russell K.; Eugene Wayne, C., Existence and stability of traveling pulses in a continuous neuronal network, SIAM J. appl. dyn. syst., 4, 954-984, (2005) · Zbl 1091.45004
[19] Terman, David H.; Bard Ermentrout, G.; Yew, Alice C., Propagating activity patterns in thalamic neuronal networks, SIAM J. appl. math., 61, 1578-1604, (2001) · Zbl 0993.92006
[20] Zhang, Linghai, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential integral equations, 16, 513-536, (2003) · Zbl 1034.45012
[21] Zhang, Linghai, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. differential equations, 197, 162-196, (2004) · Zbl 1054.45005
[22] Zhang, Linghai, Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks, J. dynam. differential equations, 17, 489-522, (2005) · Zbl 1082.45009
[23] Zhang, Linghai, Dynamics of neuronal waves, Math. Z., 255, 283-321, (2007) · Zbl 1186.92012
[24] Zhang, Linghai, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM J. appl. dyn. syst., 6, 597-644, (2007) · Zbl 1210.35018
[25] Linghai Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, in: Differential Equations: Analysis and Applications. Editor-in-Chief: Frank Columbus, Nova Science Publishers, INC. New York, 2010 (in press)
[26] Atay, Fatihcan M.; Hutt, Axel, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM J. appl. math., 65, 644-666, (2004/2005) · Zbl 1068.92007
[27] Atay, Fatihcan M.; Hutt, Axel, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM J. appl. dynam. syst., 5, 670-698, (2006) · Zbl 1210.34118
[28] Coombes, Stephen; Venkov, N.A.; Shiau, L.; Bojak, I.; Liley, David T.J.; Laing, Carlo R., Modeling electrocortical activity through improved local approximations of integral neural field equations, Phys. rev. E, 76, 051901, (2007)
[29] Enculescu, Mihaela, A note on traveling fronts and pulses in a firing rate model of a neuronal network, Physica D, 196, 362-386, (2004) · Zbl 1049.92008
[30] Hutt, Axel; Atay, Fatihcan M., Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203, 30-54, (2005) · Zbl 1061.92020
[31] Hutt, Axel; Atay, Fatihcan M., Effects of distributed transmission speeds on propagating activity in neural populations, Phys. rev. E, 73, 021906, (2006) · Zbl 1210.34118
[32] Kapitula, Todd; Kutz, Nathan; Sandstede, Björn, The Evans function for nonlocal equations, Indiana univ. math. J., 53, 1095-1126, (2004) · Zbl 1059.35137
[33] Laing, Carlo R., Spiral waves in nonlocal equations, SIAM J. appl. dyn. syst., 4, 588-606, (2005) · Zbl 1090.37056
[34] Laing, Carlo R.; Troy, William C., PDE methods for nonlocal models, SIAM J. appl. dyn. syst., 2, 487-516, (2003) · Zbl 1088.34011
[35] Laing, Carlo R.; Troy, William C.; Gutkin, Boris; Bard Ermentrout, G., Multiple bumps in a neuronal model of working memory, SIAM J. appl. math., 63, 62-97, (2002) · Zbl 1017.45006
[36] Ai, Shangbing, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. differential equations, 232, 104-133, (2007) · Zbl 1113.34024
[37] Aronson, Donald G.; Weinberger, Hans F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, (), 5-49
[38] Aronson, Donald G.; Weinberger, Hans F., Multidimensional nonlinear diffusion arising in population genetics, Adv. math., 30, 33-76, (1978) · Zbl 0407.92014
[39] Chen, Xinfu, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. differential equations, 2, 125-160, (1997) · Zbl 1023.35513
[40] Evans, John W., Nerve axon equations, I linear approximations, Indiana univ. math. J., 21, 877-885, (1972), II Stability at rest, 22 (1972) 75-90, III Stability of the nerve impulse, 22 (1972) 577-593, IV The stable and the unstable impulse, 24 (1975) 1169-1190 · Zbl 0235.92002
[41] Evans, Lawrence C., ()
[42] Fife, Paul C., Mathematical aspects of reacting and diffusing systems, () · Zbl 0403.92004
[43] Fife, Paul C.; Bryce McLeod, J., The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. ration. mech. anal., 65, 335-361, (1977) · Zbl 0361.35035
[44] Guo, Yixin; Chow, Carson C., Existence and stability of standing pulses in neural networks: I. existence. II stability, SIAM J. appl. dyn. syst., 4, (2005), I: 217-248, II 249-281 · Zbl 1109.34002
[45] Huang, Xiaoying; Troy, William C.; Yang, Qian; Ma, Hongtao; Laing, Carlo R.; Schiff, Steven J.; Wu, Jian-Young, Spiral waves in disinhibited Mammalian neocortex, J. neurosci., 24, 9897-9902, (2004)
[46] Richardson, Kristen A.; Schiff, Steven J.; Gluckman, Bruce J., Control of traveling waves in the Mammalian cortex, Phys. rev. lett., 94, 028103-1-028103-4, (2005)
[47] Rinzel, John; Keller, Joseph B., Traveling wave solutions of a nerve conduction equation, Biophys. J., 13, 1313-1337, (1973)
[48] Rinzel, John; Terman, David, Propagation phenomena in a bistable reaction-diffusion system, SIAM J. appl. math., 42, 1111-1137, (1982) · Zbl 0522.92004
[49] Rubin, Jonathan E.; Troy, William C., Sustained spatial patterns of activity in neuronal populations without recurrent excitation, SIAM J. appl. math., 64, 1609-1635, (2004) · Zbl 1075.45011
[50] Sandstede, Björn, Evans functions and nonlinear stability of travelling waves in neuronal network models, Internat J. bifurc. chaos appl. sci. eng., 17, 2693-2704, (2007) · Zbl 1144.35342
[51] Wilson, Hugh R.; Cowan, Jack D., Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., 12, 1-24, (1972)
[52] Wilson, Hugh R.; Cowan, Jack D., A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13, 55-80, (1973) · Zbl 0281.92003
[53] Borisyuk, Alla; Ermentrout, G. Bard; Friedman, Avner; Terman, David, ()
[54] Hodgkin, Alan Lloyd; Huxley, Andrew Fielding, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol., 117, 500-544, (1952)
[55] Hutt, Axel, Generalization of the reaction-diffusion, swift – hohenberg, and kuramoto – sivashinsky equations and effects of finite propagation speeds, Phys. rev. E, 75, 026214, (2007)
[56] Jirsa, Viktor K.; Haken, Hermann, Field theory of electromagnetic brain activity, Phys. rev. lett., 77, 960-963, (1996)
[57] John Enderle, Susan Blanchard, Joseph Bronzino, Introduction to Biomedical Engineering, second ed. Elsevier Academic Press Series, Burlington, MA, 2005
[58] Pinto, David J.; Patrick, Saundra L.; Huang, W.C.; Connors, Barry W., Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms, J. neuroscience, 25, 8131-8140, (2005)
[59] Wu, Jian Young; Guan, Li; Tsau, Yang, Propagating activation during oscillations and evoked responses in neocortical slices, J. neuroscience, 19, 5005-5015, (1999)
[60] Xu, Weifeng; Huang, Xiaoying; Takagaki, Kentaroh; Wu, Jian-young, Compression and reflection of visually evoked cortical waves, Neuron, 55, 119-129, (2007)
[61] Jones, Christopher K.R.T., Stability of the travelling wave solution of the fitzhugh – nagumo system, Trans. amer. math. soc., 286, 431-469, (1984) · Zbl 0567.35044
[62] Chen, Xinfu; Qi, Yuanwei, Sharp estimates on minimum travelling wave speed of reaction diffusion systems modeling autocatalysis, SIAM J. math. anal., 39, 437-448, (2007) · Zbl 1360.34065
[63] Coddington, Earl A.; Levinson, Norman, Theory of ordinary differential equations, (1955), McGraw-Hill Book Company, Inc. New York, Toronto, London · Zbl 0064.33002
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