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 & 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.  doi: 10.1007/BF00337259.  Google Scholar

[2]

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

[3]

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

[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).  doi: 10.1103/PhysRevE.75.041913.  Google Scholar

[5]

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.  Google Scholar

[6]

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

[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.  doi: 10.1137/140990371.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1093/cercor/10.9.910.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.3934/dcds.2012.32.2951.  Google Scholar

[16]

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

[17]

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

[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.  doi: 10.1162/neco.2009.12-07-660.  Google Scholar

[19]

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

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[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.  doi: 10.1137/040609471.  Google Scholar

[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.   Google Scholar

[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), (1998), 499.   Google Scholar

[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.  doi: 10.1523/JNEUROSCI.2705-04.2004.  Google Scholar

[28]

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

[29]

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

[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.  doi: 10.1016/j.physd.2007.10.013.  Google Scholar

[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).  doi: 10.1103/PhysRevE.82.055701.  Google Scholar

[32]

J. P. Keener, Principles of Applied Mathematics,, Perseus Books, (2000).   Google Scholar

[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.  doi: 10.1016/j.physd.2009.06.003.  Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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.  Google Scholar

[38]

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

[39]

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

[40]

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

[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.  doi: 10.1137/S0036139901389495.  Google Scholar

[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.  doi: 10.1162/08997660360675035.  Google Scholar

[43]

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

[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.  doi: 10.1137/S0036139900346453.  Google Scholar

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

[46]

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

[47]

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

[48]

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

[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.  doi: 10.1016/0167-2789(95)00048-9.  Google Scholar

[50]

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

[51]

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

[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.  doi: 10.1137/090773611.  Google Scholar

[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.  doi: 10.1016/j.physd.2007.04.011.  Google Scholar

[54]

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.  Google Scholar

[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.  doi: 10.1038/nn.3645.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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).  doi: 10.1103/PhysRevE.75.041913.  Google Scholar

[5]

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.  Google Scholar

[6]

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

[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.  doi: 10.1137/140990371.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.1093/cercor/10.9.910.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[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.  doi: 10.3934/dcds.2012.32.2951.  Google Scholar

[16]

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

[17]

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

[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.  doi: 10.1162/neco.2009.12-07-660.  Google Scholar

[19]

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

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[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.  doi: 10.1137/040609471.  Google Scholar

[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.   Google Scholar

[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), (1998), 499.   Google Scholar

[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.  doi: 10.1523/JNEUROSCI.2705-04.2004.  Google Scholar

[28]

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

[29]

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

[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.  doi: 10.1016/j.physd.2007.10.013.  Google Scholar

[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).  doi: 10.1103/PhysRevE.82.055701.  Google Scholar

[32]

J. P. Keener, Principles of Applied Mathematics,, Perseus Books, (2000).   Google Scholar

[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.  doi: 10.1016/j.physd.2009.06.003.  Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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.  Google Scholar

[38]

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

[39]

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

[40]

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

[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.  doi: 10.1137/S0036139901389495.  Google Scholar

[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.  doi: 10.1162/08997660360675035.  Google Scholar

[43]

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

[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.  doi: 10.1137/S0036139900346453.  Google Scholar

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

[46]

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

[47]

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

[48]

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

[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.  doi: 10.1016/0167-2789(95)00048-9.  Google Scholar

[50]

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

[51]

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

[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.  doi: 10.1137/090773611.  Google Scholar

[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.  doi: 10.1016/j.physd.2007.04.011.  Google Scholar

[54]

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.  Google Scholar

[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.  doi: 10.1038/nn.3645.  Google Scholar

[1]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419

[2]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923

[3]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203

[4]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[5]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21

[6]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[7]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069

[8]

Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215

[9]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[10]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[11]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[12]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065

[13]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701

[14]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205

[15]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[16]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[17]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[18]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[19]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006

[20]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]