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Abstract

Synchronous and asynchronous dynamics in all-to-all coupled networks
of identical excitatory, current-based, integrate-and-fire
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.

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This work was partly supported by the National Science
Foundation through grants DMS-0506287 and DMS-0636358.

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