### Impacto

Makarov , Valeri A. and Velarde, Manuel G. and Song, Yongli
(2009)
*Stability switches, oscillatory multistability, and spatio-temporal patterns of nonlinear oscillations in recurrently delay coupled neural networks.*
Biological Cybernetics , 101
(2).
147-167 .
ISSN 0340-1200

PDF
Restringido a Repository staff only hasta 31 December 2020. 844kB |

Official URL: http://www.springerlink.com/content/r447k4rg5283j71l/fulltext.pdf

## Abstract

A model of time-delay recurrently coupled spatially segregated neural assemblies is here proposed. We show that it operates like some of the hierarchical architectures of the brain. Each assembly is a neural network with no delay in the local couplings between the units. The delay appears in the long range feedforward and feedback inter-assemblies communications. Bifurcation analysis of a simple four-units system in the autonomous case shows the richness of the dynamical behaviors in a biophysically plausible parameter region. We find oscillatory multistability, hysteresis, and stability switches of the rest state provoked by the time delay. Then we investigate the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric local Hopf bifurcation theory of delay differential equations and derive the equation describing the flow on the center manifold that enables us determining the direction of Hopf bifurcations and stability of the bifurcating periodic orbits. We also discuss computational properties of the system due to the delay when an external drive of the network mimicks external sensory input.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Limit-cycle oscillators; Functional-differential equations; Hopf-bifurcation; Time-delay; Periodic-orbits; Identical cells; Phase-locking; Normal forms; Neurons; Ring; Neural network; Delay; Hopf bifurcation; Stability; Spatio-temporal pattern |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 16621 |

References: | Braitenberg V, Schuz A (1998) Cortex: statistics and geometry of neuronal connectivity. Springer, Berlin Bruce IC, Tatton WG (1980) Sequential output-input maturation of kitten motor cortex. Exp Brain Res 39: 411–419 Bungay S, Campbell SA (2007) Patterns of oscillation in a ring of identical cells with delayed coupling. Int J Bifurcat Chaos 17: 3109–3125 Campbell SA, Edwards R, Van den Driessche P (2004) Delayed coupling between two neural network loops. SIAM J Appl Math 65: 316–335 Campbell SA, Yuan Y, Bungay SD (2005) Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18: 2827–2846 Castellanos NP, Malmierca E, Nuñez A, Makarov VA (2007) Corticofugal modulation of the tactile response coherence of projecting neurons in the gracilis nucleus. J Neurophysiol 98: 2537–2549 Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, New York Engelborghs K, Luzyanina T, Samaey G (2001) DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium Ernst U, Pawelzik K, Geisel T (1995) Synchronization induced by temporal delays in pulse-coupled oscillators. Phys Rev Lett 74: 1570–1573 Faria T, Magalháes LT (1995a) Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation. J Differ Equ 122: 181–200 Faria T, Magalháes LT (1995b) Normal form for retarded functional differential equations and applications to Bogdanov–Takens singularity. J Differ Equ 122: 201–224 Fuchs A, Haken H (1988) Pattern recognition and associative memory as dynamical processes in nonlinear systems. Neural Netw 1: 217–224 Gerstner W (1996) Rapid phase locking in systems of pulse-coupled oscillators with delay. Phys Rev Lett 76: 1755–1758 Golubitsky M, Stewart I, Schaeffer D (2003) Singularities and groups in bifurcation theory, vol II. Springer, New York Guo S (2005) Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18: 2391–2407 Guo S (2007) Stability of nonlinear waves in a ring of neurons with delays. J Differ Equ 236: 343–374 Guo S, Huang L (2003) Hopf bifurcating periodic orbits in a ring of neurons with delays. Physica D 183: 19–44 Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York Hildebrand C, Skoglund S (1971) Calibre spectra of some fibre tract in the feline central nervous system during postnatal development. Acta Physiol Scand (Suppl) 364: 5–41 Huang L, Wu J (2003) Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation. SIAM J Math Anal 34: 836–860 Ikeda K, Matsumoto K (1987) High-dimensional chaotic behavior in systems with time-delayed feedback. Physica D 29: 223–235 Jabbur SJ, Towe AL (1961) Analysis of the antidromic cortical response following stimulation at the medullary pyramids. J Physiol 155: 148–160 Kandel ER, Schwartz JH, Jessell TM (2000) Principles of neural science, 4th edn. McGraw-Hill, New York Kuang Y (1994) Delay differential equations with applications in population dynamics. Academic Press, Boston Makarov VA, Makarova J, Herreras O (2009) Deconvolution of local field potential sources by independent component analysis. J Comput Neurosci (submitted) Mallet-Paret J, Yorke JA (1982) Oriented families of periodic orbits, their sources, sinks and continuation. J Differ Equ 43: 419–450 Malmierca E, Castellanos NP, Nunez A, Makarov VA, Nuñez A (2009) Neuron synchronization in the rat gracilis nucleus facilitates sensory transmission in the somatosensory pathway. Eur J Neurosci (accepted) Marcus CM, Westervelt RM (1989) Stability of analog neural networks with delay. Phys Rev A 39: 347–359 Niebur E, Schuster HG, Kammen D (1991) Collective frequencies and metastability in networks of limit-cycle oscillators with time delay. Phys Rev Lett 67: 2753–2756 Nuñez A, Malmierca E (2007) Corticofugal modulation of sensory information. Advances in anatomy embryology and cell biology, vol 187. Springer, Berlin Purpura DP (1973) Analysis of morphophysiological developmental processes in mammalian brain. Res Publ Assoc Nerv Ment Dis 51: 79–110 Ramana Reddy DV, Sen A, Johnston GL (1998) Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 80: 5109–5112 Ramana Reddy DV, Sen A, Johnston GL (1999) Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 129: 15–34 Ruan S (2001) Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Q Appl Math 59: 159–173 Schuster HG, Wagner P (1989) Mutual entrainment of two limit cycle oscillators with time delayed coupling. Prog Theor Phys 81: 939–945 Seunghwan K, Seon H, Ryu CS (1997) Multistability in coupled oscillator systems with time delay. Phys Rev Lett 79: 2911–2914 Song Y, Han M, Peng Y (2004) Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays. Chaos Solitons Fractals 22: 1139–1148 Song Y, Han M, Wei J (2005) Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays. Physica D 200: 185–204 Song Y, Wei J, Yuan Y (2007) Stability switches and Hopf bifurcations in a pair of delay-coupled oscillators. J Nonlinear Sci 17: 145–166 Timme M, Wolf F, Geisel T (2002a) Prevalence of unstable attractors in networks of pulse-coupled oscillators. Phys Rev Lett 89: 154105-1-4 Timme M, Wolf F, Geisel T (2002b) Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators. Phys Rev Lett 89: 258701-1-4 Valverde F (1966) The pyramidal tract in rodents. A study of its relations with the posterior column nuclei, dorsolateral reticular formation of the medulla, and cervical spinal cord (Golgi and E.M. observations). Zeit f Zellforsch 71: 297–363 Wei J, Velarde MG (2004) Bifurcation analysis and existence of periodic solutions in a simple neural network with delays. Chaos 14: 940–953 Wei J, Velarde MG, Makarov VA (2002) Oscillatory phenomena and stability of periodic solutions in a simple neural network with delay. Nonlinear Phenom Complex Syst 5: 407–417 Wen Q, Chklovskii DB (2005) Segregation of the brain into gray and white matter: A design minimizing conduction delays. PLoS Comput Biol 1: 617–630 Wu J (1998) Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 350: 4799–4838 Wu J, Faria T, Huang YS (1999) Synchronization and stable phase-locking in a network of neurons with memory. Math Comput Model 30: 117–138 Yordanova J, Kolev V (1997) Developmental changes in the event-related EEG theta response and P300. Electroencephalogr Clin Neurophysiol 104: 418–430 Yuan Y (2007) Dynamics in a delayed-neural network. Chaos Solitons Fractals 33: 443–454 Yuan Y, Campbell SA (2004) Stability and synchronization of a ring of identical cells with delayed coupling. J Dyn Differ Equ 16: 709–744 |

Deposited On: | 04 Oct 2012 08:46 |

Last Modified: | 07 Feb 2014 09:32 |

Repository Staff Only: item control page