August  2016, 21(6): 1729-1755. doi: 10.3934/dcdsb.2016020

Existence and nonexistence of traveling pulses in a lateral inhibition neural network

1. 

Department of Mathematics, Drexel University, Philadelphia, PA 19104, United States, United States

Received  April 2015 Revised  June 2016 Published  June 2016

We study the spatial propagating dynamics in a neural network of excitatory and inhibitory populations. Our study demonstrates the existence and nonexistence of traveling pulse solutions with a nonsaturating piecewise linear gain function. We prove that traveling pulse solutions do not exist for such neural field models with even (symmetric) couplings. The neural field models only support traveling pulse solutions with asymmetric couplings. We also show that such neural field models with asymmetric couplings will lead to a system of delay differential equations. We further compute traveling 1--bump solutions using the system of delay differential equations. Finally, we develop Evans functions to assess the stability of traveling 1--bump solutions.
Citation: Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1729-1755. doi: 10.3934/dcdsb.2016020
References:
[1]

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields,, Biol. Cybernet., 27 (1977), 77. doi: 10.1007/BF00337259.

[2]

L. Bai, X. Huang, Q. Yang and J.-Y. Wu, Spatiotemporal patterns of an evoked network oscillation in neocortical slices: Coupled local oscillators,, J. Neurophysiol., 96 (2006), 2528. doi: 10.1152/jn.00645.2006.

[3]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal,, J. Math. Anal. Appl., 332 (2007), 428. doi: 10.1016/j.jmaa.2006.09.007.

[4]

G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biol. Cybern., 34 (1979), 137. doi: 10.1007/BF00336965.

[5]

P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. L. Wiener, Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex,, Phil. Trans. R. Soc. B, 356 (2001), 299. doi: 10.1098/rstb.2000.0769.

[6]

P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network,, SIAM J. Appl. Math., 65 (2004), 131. doi: 10.1137/S0036139903434481.

[7]

P. C Bressloff and J. Wilkerson, Traveling pulses in a stochastic neural field model of direction selectivity., Frontiers in Computational Neuroscience, 6 (2012). doi: 10.3389/fncom.2012.00090.

[8]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields: Invited Topical review., J. Phys. A, 45 (2012). doi: 10.1088/1751-8113/45/3/033001.

[9]

Y. Chagnac-Amitai and B. W. Connors, Synchronized excitation and inhibition driven by intrinsically bursting neurons in neocortex,, J. Neurophysiol., 62 (1989), 1149.

[10]

R. D. Chervin, P. A. Pierce and B. W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across discharges across neocortex,, J. Neurophysiol., 60 (1988), 1695.

[11]

S. Coombes, G. J. Lord and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions,, Phys. D, 178 (2003), 219. doi: 10.1016/S0167-2789(03)00002-2.

[12]

S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function,, SIAM J. Appl. Dyn. Syst., 3 (2004), 574. doi: 10.1137/040605953.

[13]

M. Enculescu, A note on traveling fronts and pulses in a firing rate model of a neuronal network,, Physica D, 196 (2004), 362. doi: 10.1016/j.physd.2004.06.005.

[14]

G. B. Ermentrout, Reduction of conductance-based models with slow synapses to neural nets,, J. Math. Biol., 6 (1994), 679. doi: 10.1162/neco.1994.6.4.679.

[15]

G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network,, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461. doi: 10.1017/S030821050002583X.

[16]

J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (1972), 877.

[17]

J. W. Evans, Nerve axon equations, II: Stability at rest,, Indiana Univ. Math. J., 22 (1972), 75. doi: 10.1512/iumj.1973.22.22009.

[18]

J. W. Evans, Nerve axon equations, III: Stability of the nerve impulse,, Indiana Univ. Math. J., 22 (1972), 577. doi: 10.1512/iumj.1973.22.22048.

[19]

J. W. Evans, Nerve axon equations, IV: The stable and unstable impulse,, Indiana Univ. Math. J., 24 (1975), 1169.

[20]

S. E. Folias and P. C. Bressloff, Stimulus-locked waves and breathers in an excitatory neural network,, SIAM J. Appl. Math., 65 (2005), 2067. doi: 10.1137/040615171.

[21]

M. A. Geise, Dynamic Neural Field Theory for Motion Perception,, Dordrecht: Kluwer, (1999).

[22]

Y. Guo, Existence and stability of traveling fronts in a lateral inhibition neural network,, SIAM J. on Applied Dynamical Systems, 11 (2012), 1543. doi: 10.1137/120876903.

[23]

Y. Guo and C. C. Chow, Existence and stability of standing pulses in neural networks: I. existence,, SIAM J. on Applied Dynamical Systems, 4 (2005), 217. doi: 10.1137/040609471.

[24]

Y. Guo and C. C. Chow, Existence and stability of standing pulses in neural networks: II. stability,, SIAM J. on Applied Dynamical Systems, 4 (2005), 249. doi: 10.1137/040609483.

[25]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1.

[26]

Z. P. Kilpatrick, S. E. Folias and P. C. Bressloff, Traveling pulses and wave propagation failure in inhomogeneous neural media,, SIAM J. Appl. Dyn. Syst., 7 (2008), 161. doi: 10.1137/070699214.

[27]

K. Kishimoto and S. Amari, Existence and stability of local excitations in homogeneous neural fields,, J. Math. Biol., 7 (1979), 303. doi: 10.1007/BF00275151.

[28]

N. Laaris, G. C. Carlson and A. Keller, Thalamic-evoked synaptic interactions in barrel cortex revealed by optical imaging,, J. Neurosci., 20 (2000), 1529.

[29]

A. D. Myshkis, Differential equations, ordinary with distributed arguments,, Encyclopaedia of Mathematics, (1989), 144.

[30]

D. M. Petrich and R. E. Goldstein, Nonlocal contour dynamics model for chemical front motion,, Phys. Rev. Lett., 72 (1994), 1120. doi: 10.1103/PhysRevLett.72.1120.

[31]

D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses,, SIAM J. Appl. Math., 62 (2001), 206. doi: 10.1137/S0036139900346453.

[32]

D. J. Pinto, R. K. Jackson and C. E. Wayne, Existence and stability of traveling pulses in a continuous neuronal network,, SIAM J. Appl. Dyn. Syst., 4 (2005), 954. doi: 10.1137/040613020.

[33]

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,, J. Neurosci., 25 (2005), 8131. doi: 10.1523/JNEUROSCI.2278-05.2005.

[34]

D. J. Pinto, W. Troy and T. Kneezel, Asymmetric activity waves in synaptic cortical systems,, SIAM J. Appl. Dyn. Syst., 8 (2009), 1218. doi: 10.1137/08074307X.

[35]

P. A. Robinson, C. J. Rennie, J. J. Wright, H. Bahramali, E. Gordon and D. I. Rowe D, Prediction of electroencephalographic spectra from neurophysiology,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.021903.

[36]

D. J. T. Liley, P. J. Cadusch and M. P. Dafilis, A spatially continuous mean field theory of electrocortical activity,, Network, 13 (2002), 67.

[37]

B. Sandstede, Stability of travelling waves,, in Handbook of Dynamical Systems, 2 (2002), 983. doi: 10.1016/S1874-575X(02)80039-X.

[38]

B. Sandstede, Evans functions and nonlinear stability of travelling waves in neuronal network models,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2693. doi: 10.1142/S0218127407018695.

[39]

D. C. Somers, S. Nelson and M. Sur, An emergent model of orientation selectivity in cat visual cortical simple cells,, J. Neurosci, 15 (1995), 5448.

[40]

W. C. Troy, Traveling waves and synchrony in an excitable large-scale neuronal network with asymmetric connections,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1247. doi: 10.1137/070709888.

[41]

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

[42]

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.

[43]

J. Y. Wu, L. Guan and Y. Tsau, Propagating activation during oscillations and evoked responses in neocortical slices,, J. Neurosci., 19 (1999), 5005.

[44]

X. Xie and M. Giese, Nonlinear dynamics of direction-selective recurrent neural media,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.051904.

[45]

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

[46]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks,, J. Differential Equations, 197 (2004), 162. doi: 10.1016/S0022-0396(03)00170-0.

show all references

References:
[1]

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields,, Biol. Cybernet., 27 (1977), 77. doi: 10.1007/BF00337259.

[2]

L. Bai, X. Huang, Q. Yang and J.-Y. Wu, Spatiotemporal patterns of an evoked network oscillation in neocortical slices: Coupled local oscillators,, J. Neurophysiol., 96 (2006), 2528. doi: 10.1152/jn.00645.2006.

[3]

P. W. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal,, J. Math. Anal. Appl., 332 (2007), 428. doi: 10.1016/j.jmaa.2006.09.007.

[4]

G. B. Ermentrout and J. D. Cowan, A mathematical theory of visual hallucination patterns,, Biol. Cybern., 34 (1979), 137. doi: 10.1007/BF00336965.

[5]

P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas and M. L. Wiener, Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex,, Phil. Trans. R. Soc. B, 356 (2001), 299. doi: 10.1098/rstb.2000.0769.

[6]

P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network,, SIAM J. Appl. Math., 65 (2004), 131. doi: 10.1137/S0036139903434481.

[7]

P. C Bressloff and J. Wilkerson, Traveling pulses in a stochastic neural field model of direction selectivity., Frontiers in Computational Neuroscience, 6 (2012). doi: 10.3389/fncom.2012.00090.

[8]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields: Invited Topical review., J. Phys. A, 45 (2012). doi: 10.1088/1751-8113/45/3/033001.

[9]

Y. Chagnac-Amitai and B. W. Connors, Synchronized excitation and inhibition driven by intrinsically bursting neurons in neocortex,, J. Neurophysiol., 62 (1989), 1149.

[10]

R. D. Chervin, P. A. Pierce and B. W. Connors, Periodicity and directionality in the propagation of epileptiform discharges across discharges across neocortex,, J. Neurophysiol., 60 (1988), 1695.

[11]

S. Coombes, G. J. Lord and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions,, Phys. D, 178 (2003), 219. doi: 10.1016/S0167-2789(03)00002-2.

[12]

S. Coombes and M. R. Owen, Evans functions for integral neural field equations with Heaviside firing rate function,, SIAM J. Appl. Dyn. Syst., 3 (2004), 574. doi: 10.1137/040605953.

[13]

M. Enculescu, A note on traveling fronts and pulses in a firing rate model of a neuronal network,, Physica D, 196 (2004), 362. doi: 10.1016/j.physd.2004.06.005.

[14]

G. B. Ermentrout, Reduction of conductance-based models with slow synapses to neural nets,, J. Math. Biol., 6 (1994), 679. doi: 10.1162/neco.1994.6.4.679.

[15]

G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network,, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461. doi: 10.1017/S030821050002583X.

[16]

J. W. Evans, Nerve axon equations, I: Linear approximations,, Indiana Univ. Math. J., 21 (1972), 877.

[17]

J. W. Evans, Nerve axon equations, II: Stability at rest,, Indiana Univ. Math. J., 22 (1972), 75. doi: 10.1512/iumj.1973.22.22009.

[18]

J. W. Evans, Nerve axon equations, III: Stability of the nerve impulse,, Indiana Univ. Math. J., 22 (1972), 577. doi: 10.1512/iumj.1973.22.22048.

[19]

J. W. Evans, Nerve axon equations, IV: The stable and unstable impulse,, Indiana Univ. Math. J., 24 (1975), 1169.

[20]

S. E. Folias and P. C. Bressloff, Stimulus-locked waves and breathers in an excitatory neural network,, SIAM J. Appl. Math., 65 (2005), 2067. doi: 10.1137/040615171.

[21]

M. A. Geise, Dynamic Neural Field Theory for Motion Perception,, Dordrecht: Kluwer, (1999).

[22]

Y. Guo, Existence and stability of traveling fronts in a lateral inhibition neural network,, SIAM J. on Applied Dynamical Systems, 11 (2012), 1543. doi: 10.1137/120876903.

[23]

Y. Guo and C. C. Chow, Existence and stability of standing pulses in neural networks: I. existence,, SIAM J. on Applied Dynamical Systems, 4 (2005), 217. doi: 10.1137/040609471.

[24]

Y. Guo and C. C. Chow, Existence and stability of standing pulses in neural networks: II. stability,, SIAM J. on Applied Dynamical Systems, 4 (2005), 249. doi: 10.1137/040609483.

[25]

V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1.

[26]

Z. P. Kilpatrick, S. E. Folias and P. C. Bressloff, Traveling pulses and wave propagation failure in inhomogeneous neural media,, SIAM J. Appl. Dyn. Syst., 7 (2008), 161. doi: 10.1137/070699214.

[27]

K. Kishimoto and S. Amari, Existence and stability of local excitations in homogeneous neural fields,, J. Math. Biol., 7 (1979), 303. doi: 10.1007/BF00275151.

[28]

N. Laaris, G. C. Carlson and A. Keller, Thalamic-evoked synaptic interactions in barrel cortex revealed by optical imaging,, J. Neurosci., 20 (2000), 1529.

[29]

A. D. Myshkis, Differential equations, ordinary with distributed arguments,, Encyclopaedia of Mathematics, (1989), 144.

[30]

D. M. Petrich and R. E. Goldstein, Nonlocal contour dynamics model for chemical front motion,, Phys. Rev. Lett., 72 (1994), 1120. doi: 10.1103/PhysRevLett.72.1120.

[31]

D. J. Pinto and G. B. Ermentrout, Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses,, SIAM J. Appl. Math., 62 (2001), 206. doi: 10.1137/S0036139900346453.

[32]

D. J. Pinto, R. K. Jackson and C. E. Wayne, Existence and stability of traveling pulses in a continuous neuronal network,, SIAM J. Appl. Dyn. Syst., 4 (2005), 954. doi: 10.1137/040613020.

[33]

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,, J. Neurosci., 25 (2005), 8131. doi: 10.1523/JNEUROSCI.2278-05.2005.

[34]

D. J. Pinto, W. Troy and T. Kneezel, Asymmetric activity waves in synaptic cortical systems,, SIAM J. Appl. Dyn. Syst., 8 (2009), 1218. doi: 10.1137/08074307X.

[35]

P. A. Robinson, C. J. Rennie, J. J. Wright, H. Bahramali, E. Gordon and D. I. Rowe D, Prediction of electroencephalographic spectra from neurophysiology,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.021903.

[36]

D. J. T. Liley, P. J. Cadusch and M. P. Dafilis, A spatially continuous mean field theory of electrocortical activity,, Network, 13 (2002), 67.

[37]

B. Sandstede, Stability of travelling waves,, in Handbook of Dynamical Systems, 2 (2002), 983. doi: 10.1016/S1874-575X(02)80039-X.

[38]

B. Sandstede, Evans functions and nonlinear stability of travelling waves in neuronal network models,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2693. doi: 10.1142/S0218127407018695.

[39]

D. C. Somers, S. Nelson and M. Sur, An emergent model of orientation selectivity in cat visual cortical simple cells,, J. Neurosci, 15 (1995), 5448.

[40]

W. C. Troy, Traveling waves and synchrony in an excitable large-scale neuronal network with asymmetric connections,, SIAM J. Appl. Dyn. Syst., 7 (2008), 1247. doi: 10.1137/070709888.

[41]

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

[42]

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.

[43]

J. Y. Wu, L. Guan and Y. Tsau, Propagating activation during oscillations and evoked responses in neocortical slices,, J. Neurosci., 19 (1999), 5005.

[44]

X. Xie and M. Giese, Nonlinear dynamics of direction-selective recurrent neural media,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.051904.

[45]

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

[46]

L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks,, J. Differential Equations, 197 (2004), 162. doi: 10.1016/S0022-0396(03)00170-0.

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