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

Katherine A. Newhall, Gregor Kovačič, Peter R. Kramer, Doug Zhou, Aaditya Rangan, David Cai

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)541-600
Number of pages60
JournalCommunications in Mathematical Sciences
Volume8
Issue number2
StatePublished - Jun 2010

Fingerprint

Neuronal Network
Siméon Denis Poisson
Fires
Integrate
Probability density function
Neurons
Electric potential
Neuron
Voltage
Self-consistency
Exit Time
Trajectories
Curve
Diffusion Approximation
Asymptotic Solution
Chaotic Dynamics
Spike
Central limit theorem
Impulse
Computer simulation

Keywords

  • Chaos
  • Exit-time
  • Neuronal network
  • Synchrony

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Newhall, K. A., Kovačič, G., Kramer, P. R., Zhou, D., Rangan, A., & Cai, D. (2010). Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks. 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.

In: Communications in Mathematical Sciences, Vol. 8, No. 2, 06.2010, p. 541-600.

Research output: Contribution to journalArticle

Newhall, KA, Kovačič, G, Kramer, PR, Zhou, D, Rangan, A & Cai, D 2010, 'Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks', Communications in Mathematical Sciences, vol. 8, no. 2, pp. 541-600.
Newhall, Katherine A. ; Kovačič, Gregor ; Kramer, Peter R. ; Zhou, Doug ; Rangan, Aaditya ; Cai, David. / Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks. In: Communications in Mathematical Sciences. 2010 ; Vol. 8, No. 2. pp. 541-600.
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