Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-09T21:47:14.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of Synchronization in a Neural Population by aPopulation Density Approach

Published online by Cambridge University Press:  10 March 2010

A. Garenne ,
J. Henry  and
C. O. Tarniceriu
A. Garenne
Affiliation:
Basal Gang, Laboratoire Mouvement, Adaptation, Cognition, CNRS-UMR 5227, Bordeaux, France Université Victor Segalen Bordeaux 2, Bordeaux, France
J. Henry*
Affiliation:
INRIA Bordeaux Sud Ouest IMB, 351, Cours de la Libération, 33405 Talence cedex, France
C. O. Tarniceriu
Affiliation:
INRIA Bordeaux Sud Ouest IMB, 351, Cours de la Libération, 33405 Talence cedex, France Department of Sciences, "Al. I. Cuza University", Iaşi, Romania
*
* Corresponding author. E-mail:[email protected]
Get access

Abstract

In this paper we deal with a model describing the evolution in time of the density of aneural population in a state space, where the state is given by Izhikevich’s two -dimensional single neuron model. The main goal is to mathematically describe theoccurrence of a significant phenomenon observed in neurons populations, thesynchronization. To this end, we are making the transition to phasedensity population, and use Malkin theorem to calculate the phase deviations of a weaklycoupled population model.

Keywords

single neuron modelpopulation density approachsynchronization
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 2: Mathematics and neuroscience , 2010 , pp. 5 - 25
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bennani, O., Chauvet, G., Chauvet, P., Dupont, J.M., Jouen, F.. A hierarchical modeling approach of hippocampus local circuit . J. Integr. Neurosci., 9 (2009), 49–76. CrossRefGoogle Scholar
Chauvet, G.A.. The use of representation and formalism in a theoretical approach to integrative neuroscience . J. Integr. Neurosci., 4 (2005), 291–312. CrossRefGoogle Scholar
Dejean, C., Gross, C.E., Bioulac, B., Boraud, T.. Dynamic changes in the cortex-basal ganglia network after dopamine depletion in the rat . J. Neurophysiol., 100 (2008), 385–396. CrossRefGoogle ScholarPubMed
Faugeras, O., Grimbert, F. Slotine, J.-J.. Absolute stability and complete synchronization in a class of neural fields models . SIAM J. Appl. Math., 61 (2008), No. 1, 205-250.CrossRefGoogle Scholar
F. C. Hoppensteadt, E. Izhikevich. Weakly connected neural networks. Springer-Verlag, New York, 1997.
E. M. Izhikevich. Dynamical Systems in Neuroscience: The geometry of excitability and bursting. The MIT Press, 2007.
Izhikevich, E. M.. Phase equations for relaxation oscillators . SIAM J. Appl. Math., 60 (2000), 1789-1804. CrossRefGoogle Scholar
Izhikevich, E.M.. Which model to use for cortical spiking neurons? . IEEE Trans Neural Netw, 15 (2004), 1063–1070. CrossRefGoogle ScholarPubMed
Koppel, N., Ermentrout, G.B.. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators. Handbook of Dynamical Systems, 2 (2002), 3–54. Google Scholar
Medvedev, G. S., Koppel, N.. Synchronization and transient dynamics in the chains of electrically coupled Fitzhugh-Nagumo oscillators . SIAM J. Appl. Math., 60 (2001), No. 5, 1762–1801. CrossRefGoogle Scholar
Meunier, C., Segev, I.. Playing the devil’s advocate: is the Hodgkin-Huxley model useful? . Trends Neurosci., 25 (2002), 558–563. CrossRefGoogle ScholarPubMed
J. Modolo. Modélisation et analyse mathématique des effets de la stimulation cérébrale profonde dans la maladie de Parkinson. Thêse 2008.
Modolo, J., Garenne, A., Henry, J., , A.. Development and validation of a neural population model based on the dynamics of discontinuous membrane potential neuron model . J. Integr. Neurosci., 6 (2007), No. 4, 625–656. CrossRefGoogle ScholarPubMed
Modolo, J., Henry, J. Beuter, A.. Dynamics of the subthalamo-pallidal complex in Parkinson’s Disease during deep brain stimulation . J. Biol. Phys., 34 (2008), 351–366.CrossRefGoogle Scholar
Modolo, J., Mosekilde, E., Beuter, A., New insights offered by a computational model of deep brain stimulation . J. Physiol. Paris, 101 (2007), 56–63. CrossRefGoogle Scholar
D. Serre. Systemes de lois the conservation I. Hyperbolicité, entropies, ondes de choc. Diederot Editeur, Paris, 1996.
Sheeba, J.H., Stefanovska, A., McClintock, P.V.. Neuronal synchrony during anesthesia: a thalamocortical model . Biophys. J., 95 (2008), 2722–2727.CrossRefGoogle ScholarPubMed