September  2016, 21(7): 2211-2231. doi: 10.3934/dcdsb.2016044

Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction

1. 

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, United States

Received  July 2015 Revised  August 2015 Published  August 2016

We study the dynamics of stationary bumps in continuum neural field equations near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle-node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the lifetime of bumps increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to quantify bump extinction time both below and above the saddle-node.
Citation: Zachary P. Kilpatrick. Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2211-2231. doi: 10.3934/dcdsb.2016044
References:
[1]

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., 27 (1977), 77-87. doi: 10.1007/BF00337259.

[2]

D. Blömker, M. Hairer and G. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744. doi: 10.1088/0951-7715/20/7/009.

[3]

M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286. doi: 10.1016/S0167-2789(97)00050-X.

[4]

C. A. Brackley and M. S. Turner, Random fluctuations of the firing rate function in a continuum neural field model, Phys. Rev. E, 75 (2007), 041913. doi: 10.1103/PhysRevE.75.041913.

[5]

P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network, SIAM J Appl. Math., 65 (2004), 131-151. doi: 10.1137/S0036139903434481.

[6]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J Phys. A: Math. Theor., 45 (2012), 033001, 109pp. doi: 10.1088/1751-8113/45/3/033001.

[7]

P. C. Bressloff and Z. P. Kilpatrick, Nonlinear Langevin equations for wandering patterns in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 14 (2015), 305-334. doi: 10.1137/140990371.

[8]

P. C. Bressloff and M. A. Webber, Front propagation in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 11 (2012), 708-740. doi: 10.1137/110851031.

[9]

M. A. Buice and C. C. Chow, Dynamic finite size effects in spiking neural networks, PLoS Comput. Biol, 9 (2013), e1002872, 21pp. doi: 10.1371/journal.pcbi.1002872.

[10]

A. Compte, N. Brunel, P. S. Goldman-Rakic and X. J. Wang, Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model, Cereb. Cortex, 10 (2000), 910-923. doi: 10.1093/cercor/10.9.910.

[11]

S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybern., 93 (2005), 91-108. doi: 10.1007/s00422-005-0574-y.

[12]

S. Coombes and M. R. Owen, Bumps, breathers, and waves in a neural network with spike frequency adaptation, Phys. Rev. Lett., 94 (2005), 148102. doi: 10.1103/PhysRevLett.94.148102.

[13]

S. Coombes, H. Schmidt and I. Bojak, Interface dynamics in planar neural field models, J Math. Neurosci, 2 (2012), Art. 9, 27 pp. doi: 10.1186/2190-8567-2-9.

[14]

S. Coombes and H. Schmidt, Neural fields with sigmoidal firing rates: Approximate solutions, Discrete Contin. Dyn. Syst., 28 (2010), 1369-1379. doi: 10.3934/dcds.2010.28.1369.

[15]

S. Coombes, H. Schmidt, C. R. Laing, N. Svanstedt and J. A. Wyller, Waves in random neural media, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2951-2970. doi: 10.3934/dcds.2012.32.2951.

[16]

R. Curtu and B. Ermentrout, Oscillations in a refractory neural net, J Math. Biol., 43 (2001), 81-100. doi: 10.1007/s002850100089.

[17]

B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Rep. Prog. Phys., 61 (1998), 353-430. doi: 10.1088/0034-4885/61/4/002.

[18]

O. Faugeras, R. Veltz and F. Grimbert, Persistent neural states: Stationary localized activity patterns in the nonlinear continuous n-population, q-dimensional neural networks, Neural Comput., 21 (2009), 147-187. doi: 10.1162/neco.2009.12-07-660.

[19]

S. E. Folias and P. C. Bressloff, Breathing pulses in an excitatory neural network, SIAM J Appl. Dyn. Syst., 3 (2004), 378-407. doi: 10.1137/030602629.

[20]

S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network, SIAM J Appl. Dyn. Syst., 10 (2011), 744-787. doi: 10.1137/100815852.

[21]

S. Funahashi, C. J. Bruce and P. S. Goldman-Rakic, Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex, J Neurophysiol., 61 (1989), 331-349.

[22]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-05389-8.

[23]

P. S. Goldman-Rakic, Cellular basis of working memory, Neuron, 14 (1995), 477-485. doi: 10.1016/0896-6273(95)90304-6.

[24]

Y. Guo and C. Chow, Existence and stability of standing pulses in neural networks: I. Existence, SIAM J Appl. Dyn. Syst., 4 (2005), 217-248. doi: 10.1137/040609471.

[25]

B. S. Gutkin, C. R. Laing, C. L. Colby, C. C. Chow and G. B. Ermentrout, Turning on and off with excitation: The role of spike-timing asynchrony and synchrony in sustained neural activity, J Comput. Neurosci., 11 (2001), 121-134.

[26]

D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits, in Methods in neuronal modeling: From ions to networks (eds. C. Koch and I. Segev), Cambridge: MIT, 1998, Chapter 13, 499-567.

[27]

X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff and J.-Y. Wu, Spiral waves in disinhibited mammalian neocortex, J Neurosci., 24 (2004), 9897-9902. doi: 10.1523/JNEUROSCI.2705-04.2004.

[28]

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.

[29]

A. Hutt, M. Bestehorn and T. Wennekers, Pattern formation in intracortical neuronal fields, Network, 14 (2003), 351-368. doi: 10.1088/0954-898X_14_2_310.

[30]

A. Hutt, A. Longtin and L. Schimansky-Geier, Additive noise-induced turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation, Physica D, 237 (2008), 755-773. doi: 10.1016/j.physd.2007.10.013.

[31]

A. Hutt and N. P. Rougier, Activity spread and breathers induced by finite transmission speeds in two-dimensional neural fields, Phys. Rev. E, 82 (2010), R055701. doi: 10.1103/PhysRevE.82.055701.

[32]

J. P. Keener, Principles of Applied Mathematics, Perseus Books, Advanced Book Program, Cambridge, MA, 2000.

[33]

Z. P. Kilpatrick and P. C. Bressloff, Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network, Physica D, 239 (2010), 547-560. doi: 10.1016/j.physd.2009.06.003.

[34]

Z. P. Kilpatrick and P. C. Bressloff, Stability of bumps in piecewise smooth neural fields with nonlinear adaptation, Physica D, 239 (2010), 1048-1060. doi: 10.1016/j.physd.2010.02.016.

[35]

Z. P. Kilpatrick and B. Ermentrout, Wandering bumps in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 12 (2013), 61-94. doi: 10.1137/120877106.

[36]

Z. P. Kilpatrick and G. Faye, Pulse bifurcations in stochastic neural fields, SIAM J Appl. Dyn. Syst., 13 (2014), 830-860. doi: 10.1137/140951369.

[37]

K. Kishimoto and S. Amari, Existence and stability of local excitations in homogeneous neural fields, J Math. Biol., 7 (1979), 303-318. doi: 10.1007/BF00275151.

[38]

C. R. Laing, Spiral waves in nonlocal equations, SIAM J Appl. Dyn. Syst., 4 (2005), 588-606. doi: 10.1137/040612890.

[39]

C. R. Laing, Derivation of a neural field model from a network of theta neurons, Phys. Rev. E, 90 (2014), 010901. doi: 10.1103/PhysRevE.90.010901.

[40]

C. R. Laing and A. Longtin, Noise-induced stabilization of bumps in systems with long-range spatial coupling, Physica D, 160 (2001), 149-172. doi: 10.1016/S0167-2789(01)00351-7.

[41]

C. R. Laing, W. C. Troy, B. Gutkin and G. B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J Appl. Math., 63 (2002), 62-97. doi: 10.1137/S0036139901389495.

[42]

B. Lindner, A. Longtin and A. Bulsara, Analytic expressions for rate and cv of a type i neuron driven by white gaussian noise, Neural Comput., 15 (2003), 1761-1788. doi: 10.1162/08997660360675035.

[43]

E. Montbrió, D. Pazó and A. Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X, 5 (2015), 021028.

[44]

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), 206-225. doi: 10.1137/S0036139900346453.

[45]

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), 8131-8140. doi: 10.1523/JNEUROSCI.2278-05.2005.

[46]

S. Qiu and C. Chow, Field theory for biophysical neural networks,, arXiv preprint, (). 

[47]

K. A. Richardson, S. J. Schiff and B. J. Gluckman, Control of traveling waves in the mammalian cortex, Phys. Rev. Lett., 94 (2005), 028103. doi: 10.1103/PhysRevLett.94.028103.

[48]

F. Sagues, J. M. Sancho and J. Garcia-Ojalvo, Spatiotemporal order out of noise, Rev. Mod. Phys., 79 (2007), 829-882. doi: 10.1103/RevModPhys.79.829.

[49]

P. Schütz, M. Bode and H.-G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 382-397. doi: 10.1016/0167-2789(95)00048-9.

[50]

D. Sigeti and W. Horsthemke, Pseudo-regular oscillations induced by external noise, J Stat. Phys., 54 (1989), 1217-1222. doi: 10.1007/BF01044713.

[51]

S. H. Strogatz, Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, Westview press, 2014.

[52]

R. Veltz and O. Faugeras, Local/global analysis of the stationary solutions of some neural field equations, SIAM J Appl. Dyn. Syst., 9 (2010), 954-998. doi: 10.1137/090773611.

[53]

N. A. Venkov, S. Coombes and P. C. Matthews, Dynamic instabilities in scalar neural field equations with space-dependent delays, Physica D, 232 (2007), 1-15. doi: 10.1016/j.physd.2007.04.011.

[54]

H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5.

[55]

K. Wimmer, D. Q. Nykamp, C. Constantinidis and A. Compte, Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory, Nat. Neurosci., 17 (2014), 431-439. doi: 10.1038/nn.3645.

show all references

References:
[1]

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., 27 (1977), 77-87. doi: 10.1007/BF00337259.

[2]

D. Blömker, M. Hairer and G. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744. doi: 10.1088/0951-7715/20/7/009.

[3]

M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286. doi: 10.1016/S0167-2789(97)00050-X.

[4]

C. A. Brackley and M. S. Turner, Random fluctuations of the firing rate function in a continuum neural field model, Phys. Rev. E, 75 (2007), 041913. doi: 10.1103/PhysRevE.75.041913.

[5]

P. C. Bressloff and S. E. Folias, Front bifurcations in an excitatory neural network, SIAM J Appl. Math., 65 (2004), 131-151. doi: 10.1137/S0036139903434481.

[6]

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, J Phys. A: Math. Theor., 45 (2012), 033001, 109pp. doi: 10.1088/1751-8113/45/3/033001.

[7]

P. C. Bressloff and Z. P. Kilpatrick, Nonlinear Langevin equations for wandering patterns in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 14 (2015), 305-334. doi: 10.1137/140990371.

[8]

P. C. Bressloff and M. A. Webber, Front propagation in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 11 (2012), 708-740. doi: 10.1137/110851031.

[9]

M. A. Buice and C. C. Chow, Dynamic finite size effects in spiking neural networks, PLoS Comput. Biol, 9 (2013), e1002872, 21pp. doi: 10.1371/journal.pcbi.1002872.

[10]

A. Compte, N. Brunel, P. S. Goldman-Rakic and X. J. Wang, Synaptic mechanisms and network dynamics underlying spatial working memory in a cortical network model, Cereb. Cortex, 10 (2000), 910-923. doi: 10.1093/cercor/10.9.910.

[11]

S. Coombes, Waves, bumps, and patterns in neural field theories, Biol. Cybern., 93 (2005), 91-108. doi: 10.1007/s00422-005-0574-y.

[12]

S. Coombes and M. R. Owen, Bumps, breathers, and waves in a neural network with spike frequency adaptation, Phys. Rev. Lett., 94 (2005), 148102. doi: 10.1103/PhysRevLett.94.148102.

[13]

S. Coombes, H. Schmidt and I. Bojak, Interface dynamics in planar neural field models, J Math. Neurosci, 2 (2012), Art. 9, 27 pp. doi: 10.1186/2190-8567-2-9.

[14]

S. Coombes and H. Schmidt, Neural fields with sigmoidal firing rates: Approximate solutions, Discrete Contin. Dyn. Syst., 28 (2010), 1369-1379. doi: 10.3934/dcds.2010.28.1369.

[15]

S. Coombes, H. Schmidt, C. R. Laing, N. Svanstedt and J. A. Wyller, Waves in random neural media, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 2951-2970. doi: 10.3934/dcds.2012.32.2951.

[16]

R. Curtu and B. Ermentrout, Oscillations in a refractory neural net, J Math. Biol., 43 (2001), 81-100. doi: 10.1007/s002850100089.

[17]

B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Rep. Prog. Phys., 61 (1998), 353-430. doi: 10.1088/0034-4885/61/4/002.

[18]

O. Faugeras, R. Veltz and F. Grimbert, Persistent neural states: Stationary localized activity patterns in the nonlinear continuous n-population, q-dimensional neural networks, Neural Comput., 21 (2009), 147-187. doi: 10.1162/neco.2009.12-07-660.

[19]

S. E. Folias and P. C. Bressloff, Breathing pulses in an excitatory neural network, SIAM J Appl. Dyn. Syst., 3 (2004), 378-407. doi: 10.1137/030602629.

[20]

S. E. Folias, Nonlinear analysis of breathing pulses in a synaptically coupled neural network, SIAM J Appl. Dyn. Syst., 10 (2011), 744-787. doi: 10.1137/100815852.

[21]

S. Funahashi, C. J. Bruce and P. S. Goldman-Rakic, Mnemonic coding of visual space in the monkey's dorsolateral prefrontal cortex, J Neurophysiol., 61 (1989), 331-349.

[22]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd edition, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-05389-8.

[23]

P. S. Goldman-Rakic, Cellular basis of working memory, Neuron, 14 (1995), 477-485. doi: 10.1016/0896-6273(95)90304-6.

[24]

Y. Guo and C. Chow, Existence and stability of standing pulses in neural networks: I. Existence, SIAM J Appl. Dyn. Syst., 4 (2005), 217-248. doi: 10.1137/040609471.

[25]

B. S. Gutkin, C. R. Laing, C. L. Colby, C. C. Chow and G. B. Ermentrout, Turning on and off with excitation: The role of spike-timing asynchrony and synchrony in sustained neural activity, J Comput. Neurosci., 11 (2001), 121-134.

[26]

D. Hansel and H. Sompolinsky, Modeling feature selectivity in local cortical circuits, in Methods in neuronal modeling: From ions to networks (eds. C. Koch and I. Segev), Cambridge: MIT, 1998, Chapter 13, 499-567.

[27]

X. Huang, W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff and J.-Y. Wu, Spiral waves in disinhibited mammalian neocortex, J Neurosci., 24 (2004), 9897-9902. doi: 10.1523/JNEUROSCI.2705-04.2004.

[28]

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.

[29]

A. Hutt, M. Bestehorn and T. Wennekers, Pattern formation in intracortical neuronal fields, Network, 14 (2003), 351-368. doi: 10.1088/0954-898X_14_2_310.

[30]

A. Hutt, A. Longtin and L. Schimansky-Geier, Additive noise-induced turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation, Physica D, 237 (2008), 755-773. doi: 10.1016/j.physd.2007.10.013.

[31]

A. Hutt and N. P. Rougier, Activity spread and breathers induced by finite transmission speeds in two-dimensional neural fields, Phys. Rev. E, 82 (2010), R055701. doi: 10.1103/PhysRevE.82.055701.

[32]

J. P. Keener, Principles of Applied Mathematics, Perseus Books, Advanced Book Program, Cambridge, MA, 2000.

[33]

Z. P. Kilpatrick and P. C. Bressloff, Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network, Physica D, 239 (2010), 547-560. doi: 10.1016/j.physd.2009.06.003.

[34]

Z. P. Kilpatrick and P. C. Bressloff, Stability of bumps in piecewise smooth neural fields with nonlinear adaptation, Physica D, 239 (2010), 1048-1060. doi: 10.1016/j.physd.2010.02.016.

[35]

Z. P. Kilpatrick and B. Ermentrout, Wandering bumps in stochastic neural fields, SIAM J. Appl. Dyn. Syst., 12 (2013), 61-94. doi: 10.1137/120877106.

[36]

Z. P. Kilpatrick and G. Faye, Pulse bifurcations in stochastic neural fields, SIAM J Appl. Dyn. Syst., 13 (2014), 830-860. doi: 10.1137/140951369.

[37]

K. Kishimoto and S. Amari, Existence and stability of local excitations in homogeneous neural fields, J Math. Biol., 7 (1979), 303-318. doi: 10.1007/BF00275151.

[38]

C. R. Laing, Spiral waves in nonlocal equations, SIAM J Appl. Dyn. Syst., 4 (2005), 588-606. doi: 10.1137/040612890.

[39]

C. R. Laing, Derivation of a neural field model from a network of theta neurons, Phys. Rev. E, 90 (2014), 010901. doi: 10.1103/PhysRevE.90.010901.

[40]

C. R. Laing and A. Longtin, Noise-induced stabilization of bumps in systems with long-range spatial coupling, Physica D, 160 (2001), 149-172. doi: 10.1016/S0167-2789(01)00351-7.

[41]

C. R. Laing, W. C. Troy, B. Gutkin and G. B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J Appl. Math., 63 (2002), 62-97. doi: 10.1137/S0036139901389495.

[42]

B. Lindner, A. Longtin and A. Bulsara, Analytic expressions for rate and cv of a type i neuron driven by white gaussian noise, Neural Comput., 15 (2003), 1761-1788. doi: 10.1162/08997660360675035.

[43]

E. Montbrió, D. Pazó and A. Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X, 5 (2015), 021028.

[44]

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), 206-225. doi: 10.1137/S0036139900346453.

[45]

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), 8131-8140. doi: 10.1523/JNEUROSCI.2278-05.2005.

[46]

S. Qiu and C. Chow, Field theory for biophysical neural networks,, arXiv preprint, (). 

[47]

K. A. Richardson, S. J. Schiff and B. J. Gluckman, Control of traveling waves in the mammalian cortex, Phys. Rev. Lett., 94 (2005), 028103. doi: 10.1103/PhysRevLett.94.028103.

[48]

F. Sagues, J. M. Sancho and J. Garcia-Ojalvo, Spatiotemporal order out of noise, Rev. Mod. Phys., 79 (2007), 829-882. doi: 10.1103/RevModPhys.79.829.

[49]

P. Schütz, M. Bode and H.-G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 382-397. doi: 10.1016/0167-2789(95)00048-9.

[50]

D. Sigeti and W. Horsthemke, Pseudo-regular oscillations induced by external noise, J Stat. Phys., 54 (1989), 1217-1222. doi: 10.1007/BF01044713.

[51]

S. H. Strogatz, Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, Westview press, 2014.

[52]

R. Veltz and O. Faugeras, Local/global analysis of the stationary solutions of some neural field equations, SIAM J Appl. Dyn. Syst., 9 (2010), 954-998. doi: 10.1137/090773611.

[53]

N. A. Venkov, S. Coombes and P. C. Matthews, Dynamic instabilities in scalar neural field equations with space-dependent delays, Physica D, 232 (2007), 1-15. doi: 10.1016/j.physd.2007.04.011.

[54]

H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5.

[55]

K. Wimmer, D. Q. Nykamp, C. Constantinidis and A. Compte, Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory, Nat. Neurosci., 17 (2014), 431-439. doi: 10.1038/nn.3645.

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