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Self-consistent formulations for stochastic nonlinear neuronal dynamics.

Jonas Stapmanns1,2, Tobias Kühn1,2, David Dahmen1

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Summary
This summary is machine-generated.

Stochastic neuron models, often overlooked by classical theories, can be analyzed using the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism. This approach systematically incorporates noise effects, revealing how nonlinearities and fluctuations create memory in neural dynamics.

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Area of Science:

  • Computational Neuroscience
  • Theoretical Physics
  • Dynamical Systems

Background:

  • Neural dynamics are crucial for brain function but are often modeled deterministically.
  • Stochasticity, arising from biological variability and network interactions, significantly impacts neural dynamics.
  • Classical bifurcation theory is insufficient for analyzing noisy, nonlinear neural systems.

Purpose of the Study:

  • To develop a systematic theoretical framework for analyzing stochastic neural dynamics.
  • To incorporate the effects of noise and nonlinearities on neuronal behavior.
  • To provide a method for calculating corrections to mean-field dynamics and time-dependent statistics.

Main Methods:

  • Formulation of stochastic neuron dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism.
  • Application of fluctuation expansion and functional renormalization group (fRG) for systematic analysis.
  • Derivation of a link between MSRDJ and Onsager-Machlup (OM) formalisms for computational advantage.
  • Development of an efficient truncation scheme for fRG flow equations.

Main Results:

  • Effective deterministic equations emerge for the first moment, explaining noise-induced memory effects.
  • An effective linear system is derived with identical power spectra and linear response.
  • The framework allows for the analysis of systems with non-Gaussian noise.
  • A novel fRG truncation scheme enhances computational efficiency.

Conclusions:

  • The MSRDJ formalism offers a powerful tool for understanding stochastic neural dynamics beyond mean-field approximations.
  • Noise and nonlinearities cooperatively shape neural memory and system responses.
  • The developed methods provide a pathway for more accurate modeling of complex brain activity.