Linear noise approximation for oscillations in a stochastic inhibitory network with delay

Grégory Dumont, Georg Franz Josef Northoff, André Longtin

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Understanding neural variability is currently one of the biggest challenges in neuroscience. Using theory and computational modeling, we study the behavior of a globally coupled inhibitory neural network, in which each neuron follows a purely stochastic two-state spiking process. We investigate the role of both this intrinsic randomness and the conduction delay on the emergence of fast (e.g., gamma) oscillations. Toward that end, we expand the recently proposed linear noise approximation (LNA) technique to this non-Markovian "delay" case. The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity. The LNA then yields two coupled DDEs, one of which is driven by additive Gaussian white noise. These equations on their own provide an excellent approximation to the full network dynamics, which are much longer to integrate. They further allow us to compute a theoretical expression for the power spectrum of the population activity. Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity and the conduction delay are necessary for the occurrence of oscillations. The intrinsic noise arises from the probabilistic description of each neuron, yet it is expressed at the system activity level, and it can only be controlled by the system size. In fact, its effect on the fluctuations in system activity disappears in the infinite network size limit, but the characteristics of the oscillatory activity depend on all model parameters including the system size. Using the Hilbert transform, we further show that the intrinsic noise causes sporadic strong fluctuations in the phase of the gamma rhythm. © Published by the American Physical Society.
Original languageEnglish
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number1
Publication statusPublished - 2014
Externally publishedYes


  • Computation theory
  • Differential equations
  • Neural networks
  • Power spectrum
  • Stochastic systems
  • Additive Gaussian white noise
  • Delay differential equations
  • Infinite network size limit
  • Intrinsic randomness
  • Linear noise approximation
  • Population activities
  • Probabilistic descriptions
  • Theoretical expression
  • Low noise amplifiers
  • biological model
  • cytology
  • nerve cell
  • nerve cell inhibition
  • nerve cell network
  • physiology
  • statistical model
  • statistics
  • Linear Models
  • Models, Neurological
  • Nerve Net
  • Neural Inhibition
  • Neurons
  • Stochastic Processes


Dive into the research topics of 'Linear noise approximation for oscillations in a stochastic inhibitory network with delay'. Together they form a unique fingerprint.

Cite this