### 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.

Original language | English |
---|---|

Article number | 046102 |

Number of pages | 10 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 76 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2007 |

### Keywords

- partial synchronization
- oscillators

### Cite this

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,

*76*(4), [046102]. https://doi.org/10.1103/PhysRevE.76.046102

**Stability of the splay state in pulse-coupled networks.** / Zillmer, Rudiger; Livi, Roberto; Politi, Antonio; Torcini, Alessandro.

Research output: Contribution to journal › Article

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*, vol. 76, no. 4, 046102. https://doi.org/10.1103/PhysRevE.76.046102

}

TY - JOUR

T1 - Stability of the splay state in pulse-coupled networks

AU - Zillmer, Rudiger

AU - Livi, Roberto

AU - Politi, Antonio

AU - Torcini, Alessandro

PY - 2007/10

Y1 - 2007/10

N2 - 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.

AB - 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.

KW - partial synchronization

KW - oscillators

U2 - 10.1103/PhysRevE.76.046102

DO - 10.1103/PhysRevE.76.046102

M3 - Article

VL - 76

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 4

M1 - 046102

ER -