# American Institute of Mathematical Sciences

May  2014, 34(5): 2405-2450. doi: 10.3934/dcds.2014.34.2405

## Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks

 1 Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015

Received  January 2013 Revised  August 2013 Published  October 2013

Consider the following nonlinear scalar integral differential equation arising from delayed synaptically coupled neuronal networks \begin{eqnarray*} \frac{\partial u}{\partial t}+f(u) &=&\alpha\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y)H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+&\beta\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H(u(y,t-\tau)-\Theta){\rm d}y\right]{\rm d}\tau. \end{eqnarray*} This model equation generalizes 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), but also lateral inhibitions (modeled with Mexican hat kernel functions) and lateral excitations (modeled with upside down Mexican hat kernel functions). In this nonlinear scalar integral differential equation, $u=u(x,t)$ stands for the membrane potential of a neuron at position $x$ and time $t$. The integrals represent nonlocal spatio-temporal interactions between neurons.
We have accomplished the existence and stability of three traveling wave fronts $u(x,t)=U_k(x+\mu_kt)$ of the nonlinear scalar integral differential equation in an earlier work [42], where $\mu_k$ denotes the wave speed and $z=x+\mu_kt$ denotes the moving coordinate, $k=1,2,3$. In this paper, we will investigate how the neurobiological mechanisms represented by the synaptic couplings $(K,W)$, by the probability density functions $(\xi,\eta)$, by the synaptic rate constants $(\alpha,\beta)$ and by the firing thresholds $(\theta,\Theta)$ influence the wave speeds $\mu_k$ of the traveling wave fronts. We will define several speed index functions and use rigorous mathematical analysis to investigate the influence of the neurobiological mechanisms on the wave speeds. In particular, we will compare wave speeds of the traveling wave fronts of the nonlinear scalar integral differential equation with different synaptic couplings and with different probability density functions; we will accomplish new asymptotic behaviors of the wave speeds; we will compare wave speeds of traveling wave fronts of many reduced forms of nonlinear scalar integral differential equations of the above model equation; we will establish new estimates of the wave speeds. All these will greatly improve results obtained in previous work [38], [40] and [41].
Citation: Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
##### References:
 [1] F. M. Atay and A. Hutt, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM Journal on Applied Mathematics, 65 (2005), 644-666. doi: 10.1137/S0036139903430884. [2] F. M. Atay and A. Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM Journal on Applied Dynamical Systems, 5 (2006), 670-698. doi: 10.1137/050629367. [3] P. C. Bressloff, Weakly interacting pulses in synaptically coupled neural media, SIAM Journal Applied Mathematics, 66 (2005), 57-81. doi: 10.1137/040616371. [4] 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. doi: 10.1162/089976602317250861. [5] P. C. Bressloff and S. E. Folias, Solution bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, 65 (2004), 131-151. doi: 10.1137/S0036139903434481. [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160. [7] S. Coombes, Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93 (2005), 91-108. doi: 10.1007/s00422-005-0574-y. [8] S. Coombes, G. J. Lord and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178 (2003), 219-241. doi: 10.1016/S0167-2789(03)00002-2. [9] S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 3 (2004), 574-600. doi: 10.1137/040605953. [10] G. Bard Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Report on Progress of Physics, 61 (1998), 353-430. doi: 10.1088/0034-4885/61/4/002. [11] G. Bard Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns, Biological Cybernetics, 34 (1979), 137-150. doi: 10.1007/BF00336965. [12] 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. doi: 10.1017/S030821050002583X. [13] J. W. Evans, Nerve axon equations. I. Linear approximations, Indiana University Mathematics Journal, 21 (1972), 877-885; II Stability at rest, 22 (1972), 75-90 (MR0323372); III Stability of the nerve impulse, 22(1972), 577-593 (MR0393890); IV The stable and the unstable impulse, 24 (1975), 1169-1190 (MR0393891). doi: 10.1512/iumj.1973.22.22048. [14] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, Rhode Island. 1998. [15] S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network, SIAM Journal on Applied Dynamical Systems, 10 (2011), 744-787. doi: 10.1137/100815852. [16] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117 (1952), 500-544. [17] A. Hutt and F. M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203 (2005), 30-54. doi: 10.1016/j.physd.2005.03.002. [18] A. Hutt and F. 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, 5 pp. doi: 10.1103/PhysRevE.73.021906. [19] A. Hutt and L. Zhang, Distributed nonlocal feedback delays may destabilize fronts in neural fields, distributed transmission delays do not,, The Journal of Mathematical Neuroscience, ().  doi: 10.1186/2190-8567-3-9. [20] C. K. R. T. Jones, Stability of the traveling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. [21] C. R. Laing and W. C. Troy, PDE methods for nonlocal models, SIAM Journal on Applied Dynamical Systems, 2 (2003), 487-516. doi: 10.1137/030600040. [22] C. R. Laing and W. C. Troy, Two-bump solutions of Amari-type models of neuronal pattern formation, Physica D, 178 (2003), 190-218. doi: 10.1016/S0167-2789(03)00013-7. [23] F. Maria, G. Magpantay and X. 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. doi: 10.3934/mbe.2010.7.421. [24] D. 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), 206-225; II. 226-243 (MR1857543). [25] D. J. Pinto, R. 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. doi: 10.1137/040613020. [26] D. J. Pinto, S. L. Patrick, W. C. Huang and B. W. Connors, Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms, Journal of Neuroscience, 25 (2005), 8131-8140. doi: 10.1523/JNEUROSCI.2278-05.2005. [27] D. J. Pinto, W. C. Troy and T. Kneezel, Asymmetric activity waves in synaptic cortical systems, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1218-1233. doi: 10.1137/08074307X. [28] K. A. Richardson, S. J. Schiff and B. J. Gluckman, Control of traveling waves in the mammalian cortex, Physical Review Letters, 94 (2005), 028103-1 to 028103-4. doi: 10.1103/PhysRevLett.94.028103. [29] B. 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. doi: 10.1142/S0218127407018695. [30] D. Terman, G. Bard Ermentrout and A. C. Yew, Propagating activity patterns in thalamic neuronal networks, SIAM Journal on Applied Mathematics, 61 (2001), 1578-1604. doi: 10.1137/S0036139999365092. [31] H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5. [32] H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13 (1973), 55-80. doi: 10.1007/BF00288786. [33] E. Yanagida and L. Zhang, Speeds of traveling waves of some integral differential equations, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 347-373. doi: 10.1007/s13160-010-0021-x. [34] L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16 (2003), 513-536. [35] L. 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. doi: 10.1016/S0022-0396(03)00170-0. [36] L. Zhang, Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks, Journal of Dynamics and Differential Equations, 17 (2005), 489-522. doi: 10.1007/s10884-005-5404-3. [37] L. Zhang, Dynamics of neuronal waves, Mathematische Zeitschrift, 255 (2007), 283-321. doi: 10.1007/s00209-006-0024-0. [38] L. 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. doi: 10.1137/06066789X. [39] L. Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, in Advances in Mathematics Research. Volume 10. Editor-in-Chief: Albert R. Baswell. Nova Science Publishers, INC. New York. 2010. ISBN 978-1-60876-265-1. pages 53-204. [40] 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. doi: 10.1016/j.physd.2009.09.022. [41] L. Zhang, P.-S. Wu and M. A. Stoner, Influence of neurobiological mechanisms on speeds of traveling waves in mathematical neuroscience, Discrete and Continuous Dynamical Systems, Series B, 16 (2011), 1003-1037. doi: 10.3934/dcdsb.2011.16.1003. [42] L. Zhang and A. Hutt, Traveling wave solutions of nonlinear scalar integral differential equations arising from delayed synaptically coupled neuronal networks,, Journal of Differential Equations, ().

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##### References:
 [1] F. M. Atay and A. Hutt, Stability and bifurcations in neural fields with finite propagation speed and general connectivity, SIAM Journal on Applied Mathematics, 65 (2005), 644-666. doi: 10.1137/S0036139903430884. [2] F. M. Atay and A. Hutt, Neural fields with distributed transmission speeds and long-range feedback delays, SIAM Journal on Applied Dynamical Systems, 5 (2006), 670-698. doi: 10.1137/050629367. [3] P. C. Bressloff, Weakly interacting pulses in synaptically coupled neural media, SIAM Journal Applied Mathematics, 66 (2005), 57-81. doi: 10.1137/040616371. [4] 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. doi: 10.1162/089976602317250861. [5] P. C. Bressloff and S. E. Folias, Solution bifurcations in an excitatory neural network, SIAM Journal on Applied Mathematics, 65 (2004), 131-151. doi: 10.1137/S0036139903434481. [6] X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160. [7] S. Coombes, Waves, bumps, and patterns in neural field theories, Biological Cybernetics, 93 (2005), 91-108. doi: 10.1007/s00422-005-0574-y. [8] S. Coombes, G. J. Lord and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Physica D, 178 (2003), 219-241. doi: 10.1016/S0167-2789(03)00002-2. [9] S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function, SIAM Journal on Applied Dynamical Systems, 3 (2004), 574-600. doi: 10.1137/040605953. [10] G. Bard Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Report on Progress of Physics, 61 (1998), 353-430. doi: 10.1088/0034-4885/61/4/002. [11] G. Bard Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns, Biological Cybernetics, 34 (1979), 137-150. doi: 10.1007/BF00336965. [12] 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. doi: 10.1017/S030821050002583X. [13] J. W. Evans, Nerve axon equations. I. Linear approximations, Indiana University Mathematics Journal, 21 (1972), 877-885; II Stability at rest, 22 (1972), 75-90 (MR0323372); III Stability of the nerve impulse, 22(1972), 577-593 (MR0393890); IV The stable and the unstable impulse, 24 (1975), 1169-1190 (MR0393891). doi: 10.1512/iumj.1973.22.22048. [14] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, Rhode Island. 1998. [15] S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network, SIAM Journal on Applied Dynamical Systems, 10 (2011), 744-787. doi: 10.1137/100815852. [16] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117 (1952), 500-544. [17] A. Hutt and F. M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Physica D, 203 (2005), 30-54. doi: 10.1016/j.physd.2005.03.002. [18] A. Hutt and F. 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, 5 pp. doi: 10.1103/PhysRevE.73.021906. [19] A. Hutt and L. Zhang, Distributed nonlocal feedback delays may destabilize fronts in neural fields, distributed transmission delays do not,, The Journal of Mathematical Neuroscience, ().  doi: 10.1186/2190-8567-3-9. [20] C. K. R. T. Jones, Stability of the traveling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469. doi: 10.1090/S0002-9947-1984-0760971-6. [21] C. R. Laing and W. C. Troy, PDE methods for nonlocal models, SIAM Journal on Applied Dynamical Systems, 2 (2003), 487-516. doi: 10.1137/030600040. [22] C. R. Laing and W. C. Troy, Two-bump solutions of Amari-type models of neuronal pattern formation, Physica D, 178 (2003), 190-218. doi: 10.1016/S0167-2789(03)00013-7. [23] F. Maria, G. Magpantay and X. 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. doi: 10.3934/mbe.2010.7.421. [24] D. 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), 206-225; II. 226-243 (MR1857543). [25] D. J. Pinto, R. 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. doi: 10.1137/040613020. [26] D. J. Pinto, S. L. Patrick, W. C. Huang and B. W. Connors, Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms, Journal of Neuroscience, 25 (2005), 8131-8140. doi: 10.1523/JNEUROSCI.2278-05.2005. [27] D. J. Pinto, W. C. Troy and T. Kneezel, Asymmetric activity waves in synaptic cortical systems, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1218-1233. doi: 10.1137/08074307X. [28] K. A. Richardson, S. J. Schiff and B. J. Gluckman, Control of traveling waves in the mammalian cortex, Physical Review Letters, 94 (2005), 028103-1 to 028103-4. doi: 10.1103/PhysRevLett.94.028103. [29] B. 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. doi: 10.1142/S0218127407018695. [30] D. Terman, G. Bard Ermentrout and A. C. Yew, Propagating activity patterns in thalamic neuronal networks, SIAM Journal on Applied Mathematics, 61 (2001), 1578-1604. doi: 10.1137/S0036139999365092. [31] H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5. [32] H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetic, 13 (1973), 55-80. doi: 10.1007/BF00288786. [33] E. Yanagida and L. Zhang, Speeds of traveling waves of some integral differential equations, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 347-373. doi: 10.1007/s13160-010-0021-x. [34] L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16 (2003), 513-536. [35] L. 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. doi: 10.1016/S0022-0396(03)00170-0. [36] L. Zhang, Traveling waves of a singularly perturbed system of integral-differential equations arising from neuronal networks, Journal of Dynamics and Differential Equations, 17 (2005), 489-522. doi: 10.1007/s10884-005-5404-3. [37] L. Zhang, Dynamics of neuronal waves, Mathematische Zeitschrift, 255 (2007), 283-321. doi: 10.1007/s00209-006-0024-0. [38] L. 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. doi: 10.1137/06066789X. [39] L. Zhang, Traveling Waves Arising from Synaptically Coupled Neuronal Networks, in Advances in Mathematics Research. Volume 10. Editor-in-Chief: Albert R. Baswell. Nova Science Publishers, INC. New York. 2010. ISBN 978-1-60876-265-1. pages 53-204. [40] 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. doi: 10.1016/j.physd.2009.09.022. [41] L. Zhang, P.-S. Wu and M. A. Stoner, Influence of neurobiological mechanisms on speeds of traveling waves in mathematical neuroscience, Discrete and Continuous Dynamical Systems, Series B, 16 (2011), 1003-1037. doi: 10.3934/dcdsb.2011.16.1003. [42] L. Zhang and A. Hutt, Traveling wave solutions of nonlinear scalar integral differential equations arising from delayed synaptically coupled neuronal networks,, Journal of Differential Equations, ().
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