A decision-making Fokker-Planck model in computational neuroscience

Area of Science:

• Computational Neuroscience
• Dynamical Systems Theory
• Stochastic Processes

Background:

• Decision-making models in computational neuroscience often use interacting neuron populations, leading to multi-stable dynamical systems.
• Noise is crucial for modeling finite-size effects and decision robustness in these systems.
• Analyzing stochastic dynamical systems requires understanding their associated Fokker-Planck partial differential equations.

Purpose of the Study:

• To analytically investigate the asymptotic behavior of stochastic dynamical systems towards a unique probability distribution.
• To propose and validate a numerical scheme for capturing the convergence of these systems.
• To rigorously prove the existence, positivity, and uniqueness of the probability density solution for stationary and time-evolving problems.

Main Methods:

• Analysis of Fokker-Planck partial differential equations for stochastic dynamical systems.
• Development of a numerical scheme to simulate the convergence to a stationary state.
• Analytical proofs for the existence, positivity, and uniqueness of probability density solutions.

Main Results:

• Demonstrated analytical convergence of stochastic dynamical systems to a unique probability distribution over large time scales.
• Validated deterministic moment methods using simulations of the proposed numerical scheme.
• Proved the stabilization of the system, showing convergence to a unique, positive stationary probability density.

Conclusions:

• The study confirms the stabilization of decision-making models in computational neuroscience.
• The proposed analytical and numerical approaches provide a detailed understanding of these complex systems.
• Findings support the application of these methods for studying robustness and finite-size effects in neural decision processes.