### Abstract

Synchronous and asynchronous dynamics in all-to-all coupled networks of identical, excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves-graphs depicting the dependence of the network firing rate on the external drive strength-are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

Original language | English (US) |
---|---|

Pages (from-to) | 541-600 |

Number of pages | 60 |

Journal | Communications in Mathematical Sciences |

Volume | 8 |

Issue number | 2 |

State | Published - Jun 2010 |

### Fingerprint

### Keywords

- Chaos
- Exit-time
- Neuronal network
- Synchrony

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*8*(2), 541-600.

**Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks.** / Newhall, Katherine A.; Kovačič, Gregor; Kramer, Peter R.; Zhou, Doug; Rangan, Aaditya; Cai, David.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 8, no. 2, pp. 541-600.

}

TY - JOUR

T1 - Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks

AU - Newhall, Katherine A.

AU - Kovačič, Gregor

AU - Kramer, Peter R.

AU - Zhou, Doug

AU - Rangan, Aaditya

AU - Cai, David

PY - 2010/6

Y1 - 2010/6

N2 - Synchronous and asynchronous dynamics in all-to-all coupled networks of identical, excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves-graphs depicting the dependence of the network firing rate on the external drive strength-are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

AB - Synchronous and asynchronous dynamics in all-to-all coupled networks of identical, excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves-graphs depicting the dependence of the network firing rate on the external drive strength-are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

KW - Chaos

KW - Exit-time

KW - Neuronal network

KW - Synchrony

UR - http://www.scopus.com/inward/record.url?scp=77954652851&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954652851&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77954652851

VL - 8

SP - 541

EP - 600

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 2

ER -