TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS

Linghai Zhang; Axel Hutt

doi:10.11948/2014001

2014 Volume 4 Issue 1

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Linghai Zhang, Axel Hutt. TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2014, 4(1): 1-68. doi: 10.11948/2014001

Citation:

Linghai Zhang, Axel Hutt. TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS[J]. Journal of Applied Analysis & Computation, 2014, 4(1): 1-68. doi: 10.11948/2014001

TRAVELING WAVE SOLUTIONS OF NONLINEAR SCALAR INTEGRAL DIFFERENTIAL EQUATIONS ARISING FROM SYNAPTICALLY COUPLED NEURONAL NETWORKS

Linghai Zhang¹,
Axel Hutt²

1 Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015, USA;
2 INRIA-Nancy Grand Est, Team NEUROSYS, 615 rue du Jardin Botanique, 54600 Villersles-Nancy, France

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Abstract

Abstract

Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks ∂u/∂t + u=α ∫₀^∞ ξ(c) [∫_R K(x -y)H (u (y, t - 1/c |x -y|) -θ) dy] dc + β ∫₀^∞ η(τ) [∫_R W(x -y)H(u(y， ,) -Θ)dy] dτ, and ∂u/∂t + u=α ∫₀^∞ ξ(c) [∫_R K(x -y)H (u (y, t - 1/c |x -y|) -θ) dy] dc + β ∫₀^∞ η(τ) [∫_R W(x -y)H(u(y， ,) -Θ)dy] dτ -w₀.These model equations generalize many important nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H=H(u -θ) represents the Heaviside step function, which is defined by H(u -θ)=0 for all u<θ, H(0)=1/2 and H(u -θ)=1 for all u>θ.
The functions ξ and η represent probability density functions defined on (0, ∞). The parameter c>0 represents the speed of an action potential and the parameter τ>0 represents a constant delay. In these equations, u=u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants α>0 and β>0 represent synaptic rates. The positive constants θ>0 and Θ>0 represent thresholds for excitation of neurons. The positive constant w₀>0 is to be given.
The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index functions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans functions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.
- synaptically coupled neuronal networks /
- nonlinear scalar integral differential equations /
- traveling wave solutions /
- existence /
- stability /
- speed index functions /
- stability index functions
MSC: 92C20

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References

[1]	Shangbing Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, Journal of Differential Equations, 232(2007), 104-133. Google Scholar
[2]	Shun-ichi Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, 27(1977), 77-87. Google Scholar
[3]	Donald G. Aronson and Hans F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and related topics (Program, Tulane University, New Orleans, LA, 1974). Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, 446(1975), 5-49. Google Scholar
[4]	Donald G. Aronson and Hans F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30(1978), 33-76. Google Scholar
[5]	Fatihcan M. Atay and Axel Hutt, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM Journal on Applied Mathematics, 65(2005), 644-666. Google Scholar
[6]	Fatihcan M. Atay and Axel Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM Journal on Applied Dynamical Systems, 5(2006), 670-698. Google Scholar
[7]	Alla Borisyuk, G. Bard Ermentrout, Avner Friedman and David Terman, Tutorials in Mathematical Biosciences I Mathematical Neuroscience, Lecture Notes in Mathematics, 1860, Springer-Verlag 2005. Google Scholar
[8]	Paul C. Bressloff, Weakly interacting pulses in synaptically coupled neural media, SIAM Journal Applied Mathematics, 66(2005), 57-81. Google Scholar
[9]	P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. C. Wiener, What geometric visual hallucinations tell us about the visual cortex, Neural Computations, 14(2002), 473-491. Google Scholar
[10]	Paul C. Bressloff and Stefanos E. Folias, Solution bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, 65(2004), 131-151. Google Scholar
[11]	Paul C. Bressloff, Stefanos E. Folias, Alain Prat and Yue-Xian Li, Oscillatory waves in inhomogeneous neural media, Physical Review Letters, 91(2003), 178101-1 to 178101-4. Google Scholar
[12]	J. Bullier, J. M. Hupe, A. C. James and P. Girard, The role of feedback connections in shaping the responses of visual cortical neurons. Progress in Brain Research, 134(2001), 193-204. Google Scholar
[13]	Xinfu Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2(1997), 125-160. Google Scholar
[14]	Barry W. Connors and Yael Amitai, Generation of epileptiform discharge by local circuits of neocortex, in Epilepsy:Models, Mechanisms, and Concepts, P.A. Schwartkroin, ed., Cambridge University Press, Cambridge, UK, 1993, 388-423. Google Scholar
[15]	Stephen Coombes, Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93(2005), 91-108. Google Scholar
[16]	Stephen Coombes, Gabriel J. Lord and Markus R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178(2003), 219-241. Google Scholar
[17]	Stephen Coombes and Markus R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 3(2004), 574-600. Google Scholar
[18]	Stephen Coombes, N. A. Venkov, L. Shiau, I. Bojak, David T. J. Liley and Carlo R. Laing, Modeling electrocortical activity through improved local approximations of integral neural field equations, Physical Review E, 76(2007), 051901. Google Scholar
[19]	Peter Dayan and Laurence F Abbott, Theoretical Neuroscience:Computational and Mathematical Modeling of Neural Systems, MIT Press. Cambridge, MA. 2001. Google Scholar
[20]	John Enderle, Susan Blanchard and Joseph Bronzino, Introduction to Biomedical Engineering, Burlington MA:Elsevier Academic Press Series. Second Edition. 2005. Google Scholar
[21]	G. Bard Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Report on Progress of Physics, 61(1998), 353-430. Google Scholar
[22]	G. Bard Ermentrout and Jack D. Cowan, A mathematical theory of visual hallucination patterns, Biological Cybernetics, 34(1979), 137-150. Google Scholar
[23]	G. Bard Ermentrout and J. Bryce McLeod, Existence and uniqueness of travelling waves for a neural network, Proceedings of the Royal Society of Edinburgh, Section A, 123(1993), 461-478. Google Scholar
[24]	G. Bard Ermentrout and David Terman, Foundations of Mathematical Neuroscience, Interdisciplinary Applied Mathematics 35, Springer. New York, London, 2010, ISBN 978-0387-87707-5. Google Scholar
[25]	John W. Evans, Nerve axon equations, Indiana University Mathematics Journal, I Linear approximations, 21(1972), 877-885; 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. Google Scholar
[26]	Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, Rhode Island. 1998. Google Scholar
[27]	Stefanos E. Folias and Paul C. Bressloff, Breathing pulses in an excitatory neural network, SIAM Journal on Applied Dynamical Systems, 3(2004), 378-407. Google Scholar
[28]	Stefanos E. Folias and Paul C. Bressloff, Stimulus-locked traveling waves and breathers in an excitatory neural network, SIAM Journal on Applied Mathematics, 65(2005), 2067-2092. Google Scholar
[29]	W. Gerstner and W. Kistler, Spiking Neuron Models, Cambridge University Press, 2002. Google Scholar
[30]	Alan Lloyd Hodgkin and Andrew Fielding Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117(1952), 500-544. Google Scholar
[31]	Jonathan C. Horton and David H. Hubel, Regular patchy distribution of cytochrome oxidase staining in primary visual cortex of macaque monkey, Nature, 292(1981)(5825), 762-764. Google Scholar
[32]	Xiaoying Huang, William C. Troy, Qian Yang, Hongtao Ma, Carlo R. Laing, Steven J. Schiff and Jian-Young Wu, Spiral waves in disinhibited mammalian neocortex, The Journal of Neuroscience, 24(2004), 9897-9902. Google Scholar
[33]	David H. Hubel and Torsten N. Wiesel, Receptive fields of cells in striate cortex of very young, visually unexperienced kittens, Journal of Physiology, 26(1963), 994-1002. Google Scholar
[34]	Axel Hutt, Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds, Physical Review E, 75(2007), 026214. Google Scholar
[35]	Axel Hutt, The anaesthetic propofol shifts the frequency of maximum spectral power in EEG during general anaesthesia:analytical insights from a linear model, Frontiers in Computational Neuroscience, 7(2013), 2. Google Scholar
[36]	Axel Hutt and Fatihcan M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203(2005), 30-54. Google Scholar
[37]	Axel Hutt and Fatihcan M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations, Physical Review E, Statistical, nonlinear, and soft matter physics, 73(2006), 021906. Google Scholar
[38]	Axel Hutt and Fatihcan M. Atay, Spontaneous and evoked activity in extended neural populations with gamma-distributed spatial interactions and transmission delay, Chaos, Solitons and Fractals, 32(2007), 547-560. Google Scholar
[39]	Axel Hutt and Linghai Zhang, Distributed nonlocal feedback delays may destabilize fronts in neural fields, distributed transmission delays do not, The Journal of Mathematical Neuroscience, 9(2013), 1-21. Google Scholar
[40]	Viktor K. Jirsa and Hermann Haken, Field theory of electromagnetic brain activity, Physical Review Letters, 77(1996), 960-963. Google Scholar
[41]	Christopher K. R. T. Jones, Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286(1984), 431-469. Google Scholar
[42]	Todd Kapitula, Nathan Kutz and Björn Sandstede, The Evans function for nonlocal equations, Indiana University Mathematics Journal, 53(2004), 1095-1126. Google Scholar
[43]	J. W. Lance, Current concepts of migraine pathogenesis, Neurology, 43(1993), S11-S15. Google Scholar
[44]	Carlo R. Laing, Spiral waves in nonlocal equations, SIAM Journal on Applied Dynamical Systems, 4(2005), 588-606. Google Scholar
[45]	Carlo R. Laing and William C. Troy, PDE methods for nonlocal models, SIAM Journal on Applied Dynamical Systems, 2(2003), 487-516. Google Scholar
[46]	Carlo R. Laing, William C. Troy, Boris Gutkin, and G. Bard Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM Journal on Applied Mathematics, 63(2002), 62-97. Google Scholar
[47]	Felicia Maria G. Magpantay and Xingfu Zou, Wave solutions in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections, Mathematical Biosciences and Engineering, 7(2010), 421-442. Google Scholar
[48]	P. L. Nunez, Neocortical Dynamics and Human EEG Rhythms, Oxford University Press, New York. 1995. Google Scholar
[49]	David J. Pinto and G. Bard Ermentrout, Spatially structured activity in synaptically coupled neuronal networks. I. traveling solutions and pulses, Ⅱ. Lateral inhibition and standing pulses, SIAM Journal on Applied Mathematics, 62(2001), I. 206-225, Ⅱ. 226-243. Google Scholar
[50]	David J. Pinto, Russell K. Jackson, C. Eugene Wayne, Existence and stability of traveling pulses in a continuous neuronal network, SIAM Journal on Applied Dynamical Systems, 4(2005), 954-984. Google Scholar
[51]	David J. Pinto, Saundra L. Patrick, W. C. Huang and Barry W. Connors, Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms, Journal of Neuroscience, 25(2005), 8131-8140. Google Scholar
[52]	J. C. Prechtl, L. B. Cohen, B. Pesaran, P. P. Mitra, D. Kleinfeld, Visual stimuli induce waves of electrical activity in turtle cortex, Proceedings of the National Academy of Sciences, USA, 94(1997), 7621-7626. Google Scholar
[53]	J A Reggia and D Montgomery, A computational model of visual hallucinations in migraine, Comput. Biol. Med, 26(1996), 133-139. Google Scholar
[54]	C. J. Rennie, P. A. Robinson and J. J. Wright, Unified neurophysical model of EEG spectra and evoked potentials, Biological Cybernetics, 86(2002), 457-471. Google Scholar
[55]	Kristen A. Richardson, Steven J. Schiff and Bruce J. Gluckman, Control of traveling waves in the mammalian cortex, Physical Review Letters, 94(2005), 028103-1 to 028103-4. Google Scholar
[56]	J L Ringo, R W Doty, S Dementer and P Y Simard, Time is of the essence:a conjecture that hemispheric specialization arises from interhemispheric conduction delay, Cereb. Cortex, 4(1994), 331-343. Google Scholar
[57]	John Rinzel and Joseph B. Keller, Traveling wave solutions of a nerve conduction equation, Biophysics Journal, 13(1973), 1313-1337. Google Scholar
[58]	John Rinzel and David Terman, Propagation phenomena in a bistable reactiondiffusion system, SIAM Journal on Applied Mathematics, 42(1982), 1111-1137. Google Scholar
[59]	Jonathan E. Rubin and William C. Troy, Sustained spatial patterns of activity in neuronal populations without recurrent excitation. SIAM Journal on Applied Mathematics, 64(2004), 1609-1635. Google Scholar
[60]	Björn Sandstede, Evans functions and nonlinear stability of travelling waves in neuronal network models, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 17(2007), 2693-2704. Google Scholar
[61]	L. Schwabe, K. Obermayer, A. Angelucci and P.C. Bressloff, The role of feedback in shaping the extra-classical receptive field of cortical neurons:a recurrent network model, Journal of Neuroscience, 26(2006), 9117-9126. Google Scholar
[62]	Hagai Agmon-Snir and Idan Segev, Signal delay and input synchronisation in passive dendritic structures, Journal of Neurophysiology, 70(1993), 2066-2085. Google Scholar
[63]	Elias M. Stein, Harmonic Analysis:Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press. 1993. ISBN 0691-03216-5. Google Scholar
[64]	M. L. Steyn-Ross, D. A. Steyn-Ross, J. W. Sleigh and L. C. Wilcocks, Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex:I. A thermodynamic analogy, Physics Review E, 64(2001), 011917. Google Scholar
[65]	H A. Swadlow, Impulse conduction in the mammalian brain:Physiological properties of individual axons monitored for several months, Science, 218(1982), 911-913. Google Scholar
[66]	Gabor Tamas, Eberhard H. Buhl and Peter Somogyi, Massive autaptic selfinnervation of GABAergic neurons in cat visual cortex, Journal of Neuroscience, 17(1997), 6352-6364. Google Scholar
[67]	J. G. Taylor, Neural "bubble" dynamics in two dimensions:Foundations, Biological Cybernetics, 80(1999), 393-409. Google Scholar
[68]	David Terman, G. Bard Ermentrout and Alice C. Yew, Propagating activity patterns in thalamic neuronal networks, SIAM Journal on Applied Mathematics, 61(2001), 1578-1604. Google Scholar
[69]	Hugh R. Wilson and Jack D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12(1972), 1-24. Google Scholar
[70]	Hugh R. Wilson and Jack D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13(1973), 55-80. Google Scholar
[71]	Jian Young Wu, Li Guan and Yang Tsau, Propagating activation during oscillations and evoked responses in neocortical slices, Journal of Neuroscience, 19(1999), 5005-5015. Google Scholar
[72]	Weifeng Xu, Xiaoying Huang, Kentaroh Takagaki and Jian-young Wu, Compression and reflection of visually evoked cortical waves, Neuron, 55(2007), 119-129. Google Scholar
[73]	Linghai Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16(2003), 513-536. Google Scholar
[74]	Linghai Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, Journal of Differential Equations, 197(2004), 162-196. Google Scholar
[75]	Linghai Zhang, Traveling waves of a singularly perturbed system of integraldifferential equations arising from neuronal networks, Journal of Dynamics and Differential Equations, 17(2005), 489-522. Google Scholar
[76]	Linghai Zhang, Dynamics of neuronal waves, Mathematische Zeitschrift, 255(2007), 283-321. Google Scholar
[77]	Linghai Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves in nonlinear nonlocal neuronal networks?, SIAM Journal on Applied Dynamical Systems, 6(2007), 597-644. Google Scholar
[78]	Linghai Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, Advances in Mathematics Research, Nova Science Publishers, INC. New York, ISBN 978-1-60876-265-1, 10(2010), 53-204. Google Scholar
[79]	Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of sodium currents on speeds of traveling wave solutions in synaptically coupled neuronal networks, Physica D, 239(2010), 9-32. Google Scholar
[80]	Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner, Influence of neurobiological mechanisms on speeds of traveling waves in mathematical neuroscience, Discrete and Continuous Dynamical Systems, Series B, 16(2011), 1003-1037. Google Scholar
[81]	Linghai Zhang, Wave speed analysis of traveling wave fronts in synaptically coupled neuronal networks, Discrete and Continuous Dynamical Systems, Series A, 34(2014). Google Scholar