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# Communications in Mathematical Sciences

## Volume 8 (2010)

### Number 2

### Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part II

### Dynamics of current-based, Poisson driven, integrate-and-fire neuronal networks

Pages: 541 – 600

DOI: http://dx.doi.org/10.4310/CMS.2010.v8.n2.a12

#### Authors

#### 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.

#### Keywords

Neuronal network; synchrony; exit-time; chaos

#### 2010 Mathematics Subject Classification

82C31, 82C32, 92B20, 94C99