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Title: Stability of the splay state in pulse-coupled networks

Abstract

The stability of the dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse width and the interspike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott and van Vreesvijk, Phys. Rev. E 48, 1483 (1993)], (ii) a component measuring the instability of 'finite-frequency' modes, (iii) a number of 'isolated' eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.

Authors:
 [1];  [2];  [1];  [1]
  1. Istituto dei Sistemi Complessi, CNR, via Madonna del Piano 10, I-50019 Sesto Fiorentino (Italy)
  2. Dipartimento di Fisica, Universita di Firenze, and Unita INFM via Sansone, 1-I-50019 Sesto Fiorentino (Italy)
Publication Date:
OSTI Identifier:
21076248
Resource Type:
Journal Article
Journal Name:
Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Additional Journal Information:
Journal Volume: 76; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevE.76.046102; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1063-651X
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; DIFFERENTIAL EQUATIONS; EIGENVALUES; INSTABILITY; MEAN-FIELD THEORY; NERVE CELLS; NEURAL NETWORKS; PULSES; STABILITY

Citation Formats

Zillmer, Ruediger, Laboratoire de neurophysique et de physiologie du systeme moteur, CNRS-EP 1848, 45 Rue des Saints-Peres, 75270 Paris, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Livi, Roberto, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Politi, Antonio, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Torcini, Alessandro, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino. Stability of the splay state in pulse-coupled networks. United States: N. p., 2007. Web. doi:10.1103/PHYSREVE.76.046102.
Zillmer, Ruediger, Laboratoire de neurophysique et de physiologie du systeme moteur, CNRS-EP 1848, 45 Rue des Saints-Peres, 75270 Paris, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Livi, Roberto, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Politi, Antonio, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Torcini, Alessandro, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, & Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino. Stability of the splay state in pulse-coupled networks. United States. https://doi.org/10.1103/PHYSREVE.76.046102
Zillmer, Ruediger, Laboratoire de neurophysique et de physiologie du systeme moteur, CNRS-EP 1848, 45 Rue des Saints-Peres, 75270 Paris, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Livi, Roberto, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Politi, Antonio, Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino, Torcini, Alessandro, INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino, and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino. 2007. "Stability of the splay state in pulse-coupled networks". United States. https://doi.org/10.1103/PHYSREVE.76.046102.
@article{osti_21076248,
title = {Stability of the splay state in pulse-coupled networks},
author = {Zillmer, Ruediger and Laboratoire de neurophysique et de physiologie du systeme moteur, CNRS-EP 1848, 45 Rue des Saints-Peres, 75270 Paris and INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino and Livi, Roberto and INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino and Politi, Antonio and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino and Torcini, Alessandro and INFN Sez. Firenze, via Sansone, 1-I-50019 Sesto Fiorentino and Centro Interdipartimentale per lo Studio delle Dinamiche Complesse, via Sansone, 1-I-50019 Sesto Fiorentino},
abstractNote = {The stability of the dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of globally coupled leaky integrate-and-fire neurons. This is done by reducing the set of differential equations to a map that is investigated in the limit of large network size. We show that the stability of the splay state depends crucially on the ratio between the pulse width and the interspike interval. More precisely, the spectrum of Floquet exponents turns out to consist of three components: (i) one that coincides with the predictions of the mean-field analysis [Abbott and van Vreesvijk, Phys. Rev. E 48, 1483 (1993)], (ii) a component measuring the instability of 'finite-frequency' modes, (iii) a number of 'isolated' eigenvalues that are connected to the characteristics of the single pulse and may give rise to strong instabilities (the Floquet exponent being proportional to the network size). Finally, as a side result, we find that the splay state can be stable even for inhibitory coupling.},
doi = {10.1103/PHYSREVE.76.046102},
url = {https://www.osti.gov/biblio/21076248}, journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
issn = {1063-651X},
number = 4,
volume = 76,
place = {United States},
year = {Mon Oct 15 00:00:00 EDT 2007},
month = {Mon Oct 15 00:00:00 EDT 2007}
}