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 & Continuous Dynamical Systems - A, 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.  doi: 10.1137/S0036139903430884.  Google Scholar

[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.  doi: 10.1137/050629367.  Google Scholar

[3]

P. C. Bressloff, Weakly interacting pulses in synaptically coupled neural media,, SIAM Journal Applied Mathematics, 66 (2005), 57.  doi: 10.1137/040616371.  Google Scholar

[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.  doi: 10.1162/089976602317250861.  Google Scholar

[5]

P. C. Bressloff and S. E. Folias, Solution bifurcations in an excitatory neural network,, SIAM Journal on Applied Mathematics, 65 (2004), 131.  doi: 10.1137/S0036139903434481.  Google Scholar

[6]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Advances in Differential Equations, 2 (1997), 125.   Google Scholar

[7]

S. Coombes, Waves, bumps, and patterns in neural field theories,, Biological Cybernetics, 93 (2005), 91.  doi: 10.1007/s00422-005-0574-y.  Google Scholar

[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.  doi: 10.1016/S0167-2789(03)00002-2.  Google Scholar

[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.  doi: 10.1137/040605953.  Google Scholar

[10]

G. Bard Ermentrout, Neural networks as spatio-temporal pattern-forming systems,, Report on Progress of Physics, 61 (1998), 353.  doi: 10.1088/0034-4885/61/4/002.  Google Scholar

[11]

G. Bard Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137.  doi: 10.1007/BF00336965.  Google Scholar

[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, 123 (1993), 461.  doi: 10.1017/S030821050002583X.  Google Scholar

[13]

J. W. Evans, Nerve axon equations. I. Linear approximations,, Indiana University Mathematics Journal, 21 (1972), 877.  doi: 10.1512/iumj.1973.22.22048.  Google Scholar

[14]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[15]

S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network,, SIAM Journal on Applied Dynamical Systems, 10 (2011), 744.  doi: 10.1137/100815852.  Google Scholar

[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.   Google Scholar

[17]

A. Hutt and F. M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities,, Physica D, 203 (2005), 30.  doi: 10.1016/j.physd.2005.03.002.  Google Scholar

[18]

A. Hutt and F. M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations,, Physical Review E, 73 (2006).  doi: 10.1103/PhysRevE.73.021906.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1984-0760971-6.  Google Scholar

[21]

C. R. Laing and W. C. Troy, PDE methods for nonlocal models,, SIAM Journal on Applied Dynamical Systems, 2 (2003), 487.  doi: 10.1137/030600040.  Google Scholar

[22]

C. R. Laing and W. C. Troy, Two-bump solutions of Amari-type models of neuronal pattern formation,, Physica D, 178 (2003), 190.  doi: 10.1016/S0167-2789(03)00013-7.  Google Scholar

[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.  doi: 10.3934/mbe.2010.7.421.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/040613020.  Google Scholar

[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.  doi: 10.1523/JNEUROSCI.2278-05.2005.  Google Scholar

[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.  doi: 10.1137/08074307X.  Google Scholar

[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.  doi: 10.1103/PhysRevLett.94.028103.  Google Scholar

[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.  doi: 10.1142/S0218127407018695.  Google Scholar

[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.  doi: 10.1137/S0036139999365092.  Google Scholar

[31]

H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons,, Biophysical Journal, 12 (1972), 1.  doi: 10.1016/S0006-3495(72)86068-5.  Google Scholar

[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.  doi: 10.1007/BF00288786.  Google Scholar

[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.  doi: 10.1007/s13160-010-0021-x.  Google Scholar

[34]

L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks,, Differential and Integral Equations, 16 (2003), 513.   Google Scholar

[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.  doi: 10.1016/S0022-0396(03)00170-0.  Google Scholar

[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.  doi: 10.1007/s10884-005-5404-3.  Google Scholar

[37]

L. Zhang, Dynamics of neuronal waves,, Mathematische Zeitschrift, 255 (2007), 283.  doi: 10.1007/s00209-006-0024-0.  Google Scholar

[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.  doi: 10.1137/06066789X.  Google Scholar

[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, (2010), 978.   Google Scholar

[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.  doi: 10.1016/j.physd.2009.09.022.  Google Scholar

[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, 16 (2011), 1003.  doi: 10.3934/dcdsb.2011.16.1003.  Google Scholar

[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, ().   Google Scholar

show all references

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.  doi: 10.1137/S0036139903430884.  Google Scholar

[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.  doi: 10.1137/050629367.  Google Scholar

[3]

P. C. Bressloff, Weakly interacting pulses in synaptically coupled neural media,, SIAM Journal Applied Mathematics, 66 (2005), 57.  doi: 10.1137/040616371.  Google Scholar

[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.  doi: 10.1162/089976602317250861.  Google Scholar

[5]

P. C. Bressloff and S. E. Folias, Solution bifurcations in an excitatory neural network,, SIAM Journal on Applied Mathematics, 65 (2004), 131.  doi: 10.1137/S0036139903434481.  Google Scholar

[6]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations,, Advances in Differential Equations, 2 (1997), 125.   Google Scholar

[7]

S. Coombes, Waves, bumps, and patterns in neural field theories,, Biological Cybernetics, 93 (2005), 91.  doi: 10.1007/s00422-005-0574-y.  Google Scholar

[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.  doi: 10.1016/S0167-2789(03)00002-2.  Google Scholar

[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.  doi: 10.1137/040605953.  Google Scholar

[10]

G. Bard Ermentrout, Neural networks as spatio-temporal pattern-forming systems,, Report on Progress of Physics, 61 (1998), 353.  doi: 10.1088/0034-4885/61/4/002.  Google Scholar

[11]

G. Bard Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biological Cybernetics, 34 (1979), 137.  doi: 10.1007/BF00336965.  Google Scholar

[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, 123 (1993), 461.  doi: 10.1017/S030821050002583X.  Google Scholar

[13]

J. W. Evans, Nerve axon equations. I. Linear approximations,, Indiana University Mathematics Journal, 21 (1972), 877.  doi: 10.1512/iumj.1973.22.22048.  Google Scholar

[14]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[15]

S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network,, SIAM Journal on Applied Dynamical Systems, 10 (2011), 744.  doi: 10.1137/100815852.  Google Scholar

[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.   Google Scholar

[17]

A. Hutt and F. M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities,, Physica D, 203 (2005), 30.  doi: 10.1016/j.physd.2005.03.002.  Google Scholar

[18]

A. Hutt and F. M. Atay, Effects of distributed transmission speeds on propagating activity in neural populations,, Physical Review E, 73 (2006).  doi: 10.1103/PhysRevE.73.021906.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1984-0760971-6.  Google Scholar

[21]

C. R. Laing and W. C. Troy, PDE methods for nonlocal models,, SIAM Journal on Applied Dynamical Systems, 2 (2003), 487.  doi: 10.1137/030600040.  Google Scholar

[22]

C. R. Laing and W. C. Troy, Two-bump solutions of Amari-type models of neuronal pattern formation,, Physica D, 178 (2003), 190.  doi: 10.1016/S0167-2789(03)00013-7.  Google Scholar

[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.  doi: 10.3934/mbe.2010.7.421.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/040613020.  Google Scholar

[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.  doi: 10.1523/JNEUROSCI.2278-05.2005.  Google Scholar

[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.  doi: 10.1137/08074307X.  Google Scholar

[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.  doi: 10.1103/PhysRevLett.94.028103.  Google Scholar

[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.  doi: 10.1142/S0218127407018695.  Google Scholar

[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.  doi: 10.1137/S0036139999365092.  Google Scholar

[31]

H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons,, Biophysical Journal, 12 (1972), 1.  doi: 10.1016/S0006-3495(72)86068-5.  Google Scholar

[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.  doi: 10.1007/BF00288786.  Google Scholar

[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.  doi: 10.1007/s13160-010-0021-x.  Google Scholar

[34]

L. Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks,, Differential and Integral Equations, 16 (2003), 513.   Google Scholar

[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.  doi: 10.1016/S0022-0396(03)00170-0.  Google Scholar

[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.  doi: 10.1007/s10884-005-5404-3.  Google Scholar

[37]

L. Zhang, Dynamics of neuronal waves,, Mathematische Zeitschrift, 255 (2007), 283.  doi: 10.1007/s00209-006-0024-0.  Google Scholar

[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.  doi: 10.1137/06066789X.  Google Scholar

[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, (2010), 978.   Google Scholar

[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.  doi: 10.1016/j.physd.2009.09.022.  Google Scholar

[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, 16 (2011), 1003.  doi: 10.3934/dcdsb.2011.16.1003.  Google Scholar

[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, ().   Google Scholar

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