# Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 5, Issue: 2, page 67-99
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topMa, J., and Wu, J.. "Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops." Mathematical Modelling of Natural Phenomena 5.2 (2010): 67-99. <http://eudml.org/doc/197647>.

@article{Ma2010,

abstract = {We study the coexistence of multiple periodic solutions for an analogue of the
integrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayed
feedback, which incorporates the firing process and absolute refractory period. Upon
receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a
spike with a pattern-related delay, in addition to the synaptic delay. We present a
theoretical framework to view the inhibitory signal from the inhibitory neuron as a
self-feedback of the excitatory neuron with this additional delay. Our analysis shows that
the inhibitory feedbacks with firing and the absolute refractory period can generate four
basic types of oscillations, and the complicated interaction among these basic
oscillations leads to a large class of periodic patterns and the occurrence of
multistability in the recurrent inhibitory loop. We also introduce the average time of
convergence to a periodic pattern to determine which periodic patterns have the potential
to be used for neural information transmission and cognition processing in the nervous
system. },

author = {Ma, J., Wu, J.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {multistability; periodic pattern; neural network; time delay; pattern formation; recurrent inhibitory loops; integrate-and-fire neuron model},

language = {eng},

month = {3},

number = {2},

pages = {67-99},

publisher = {EDP Sciences},

title = {Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops},

url = {http://eudml.org/doc/197647},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Ma, J.

AU - Wu, J.

TI - Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/3//

PB - EDP Sciences

VL - 5

IS - 2

SP - 67

EP - 99

AB - We study the coexistence of multiple periodic solutions for an analogue of the
integrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayed
feedback, which incorporates the firing process and absolute refractory period. Upon
receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a
spike with a pattern-related delay, in addition to the synaptic delay. We present a
theoretical framework to view the inhibitory signal from the inhibitory neuron as a
self-feedback of the excitatory neuron with this additional delay. Our analysis shows that
the inhibitory feedbacks with firing and the absolute refractory period can generate four
basic types of oscillations, and the complicated interaction among these basic
oscillations leads to a large class of periodic patterns and the occurrence of
multistability in the recurrent inhibitory loop. We also introduce the average time of
convergence to a periodic pattern to determine which periodic patterns have the potential
to be used for neural information transmission and cognition processing in the nervous
system.

LA - eng

KW - multistability; periodic pattern; neural network; time delay; pattern formation; recurrent inhibitory loops; integrate-and-fire neuron model

UR - http://eudml.org/doc/197647

ER -

## References

top- C. A. Bares, M. S. Suster, J. Shen B. L. McNaughton. Multistability of cognitive maps in the hippocampus of old rats. Nature, 388, 272-275 (1997).
- A. Beuter, J. G. Milton, C. Labrie, L. Glass. Complex motor dynamics and control in multi-loop negative feedback systems. Proc IEEE Systems Man Cybern. 899-902. (1989).
- R. M. Borisyuk A. Kirillov. Bifurcation analysis of a neural network model. Biological Cybernetics, 66319-325 (1992).
- C. Canavier, D. Baxter, J. Clark J. Byrne. Multiple modes of activity in a neuron model suggest a novel mechanism for the effects of neuromodulators. J. Neurophysiol., 72, 872-882 (1994).
- C. C. Chow, J. A. White, J. Ritt N. Kopell. Frequency control in synchronized networks of inhibitory neurons. Neural Comput., 5, 407-420 (1998).
- D. Cotreras, A. Destexhe, T. J. Sejnowski M. Steraide. Control of spatiotemporal coherence of a thalamic oscillation by corticothalamic feedback. Science, 274, 771-774 (1996).
- G. B. Ermentrout N. Kopell. Fine structure of neural spiking and synchronization in the presence of conduction delays. Proc. Nat. Acad. Sci., 95, 1259-1264 (1998).
- J. Foss, A. Longtin, B. Mensour J. Milton. Multistability and delayed recurrent loops. Phys. Rev. Lett., 76, 708-711 (1996).
- J. Foss, F. Moss J. Milton. Noise, multistability, and delayed recurrent loops. Phys. Rev. E55, 4536-4543 (1997).
- J. Foss, J. Milton. Multistability in recurrent neural loops arising from delay. J. Neurophysiol., 84(2) 975-985 (2000).
- M. J. Gutnick D. A. Prince. Thalamocortical relay neurons: antidromic invasion of spikes from a cortical epileptogenic focus. Science, 176, 424-426 (1972).
- A. C. Guyton. Textbook of medical physiology. Saunders, Toronto, 1976.
- J. J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci., 79, 2554-2558 (1982).
- J. J. Hopfield. Neurons with grades response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci., 81, 3088-3092 (1984).
- N. Kopell, G. B. Ermentrout, M. A. Whittington R. D. Traub. Gamma rhythms and beta rhythms have different synchronization properties. PNAS, 97, 1867-1872 (2000).
- N. Kopell, D. Pervouchine, H. G. Rotstein, T. Netoff, M. Whittington, T. Gloveli. Multiple rhythms and switches in the nervous system. In press.
- J. Ma J. Wu. Multistability in spiking neuron models of delayed recurrent neural loops. Neural Comput., 19, 2124-2148 (2007).
- J. Ma, J. Wu. Transition and coexistence of periodic patterns in spiking neuron models of delayed recurrent inhibitory loops. Submitted to SIAM J. Appl. Math..
- J. Milller. What is the contribution of axonal conduction delay to temporal structure in brain dynamics? 53–57. In: Oscillatory event-related brain dynamics. C. Pantev, Ed. Plenum, New York, 1994.
- J. Milton. Epilepsy: Multistability in a dynamic disease. In: Self-organized biological dynamics and nonlinear control. J. Walleczek, Ed. Cambridge University Press, Cambridge, 374-386, 2000.
- J. Milton. Insights into seizure propagation from axonal conduction times. In: Epilepsy as a dynamic disease. J. Milton, P. Jung, Eds. New York. Springer-Verlag 15-23 (2002).
- M. Morita. Associative memory with non-monotone dynamics. Neural Networks, 6, 115-123 (1993).
- M. Proctor, K. Gale. Basal Ganglia and Brainstem Anatomy and Physiology, In: Epilepsy: A comprehensive textbook. J. Engel, T. A. Pedley, Eds. Philadelphia, PA: Lippincott-Raven 353-368 (1997).
- P. A. Schwartzkroin, D. C. McIntyre. Limbic anatomy and physiology. In: Epilepsy: a comprehensive textbook. J. Engel, T. A. Pedley, Eds. Philadelphia, PA: Lippincott-Raven 323-340 (1997).
- P. Tiňo, B. G. Horne C. L. Giles. Attractive periodic sets in discrete-time recurrent networks with emphasis on fixed-point stability and bifurcations in two-neuron networks. Neural Comput., 13, 1379-1414 (2001).
- R.D. Traub, R. Miles. Neuronal networks of the hippocampus. Cambridge University Press, New York, 1991.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.