The Inverse First Passage time method for a two dimensional Ornstein Uhlenbeck process with neuronal application

Area of Science:

• Stochastic processes
• Computational mathematics
• Mathematical physics

Background:

• The Inverse First Passage time problem is crucial for understanding stochastic processes.
• It involves determining a boundary from a process and a first passage time distribution.
• Gauss-Markov diffusion processes are widely used in modeling complex systems.

Purpose of the Study:

• To numerically solve the Inverse First Passage time problem for a two-dimensional Gauss-Markov diffusion process.
• To investigate how boundary shapes vary with different first passage time distributions (Inverse Gaussian, Gamma).
• To explore the impact of parameter choices, including heavy and light tails, on boundary characteristics.

Main Methods:

• Numerical solution techniques for inverse problems.
• Analysis of two-dimensional Gauss-Markov diffusion processes.
• Characterization of boundary shapes for specified probability distributions.

Main Results:

• The study provides numerical solutions for the Inverse First Passage time problem in the specified context.
• Identified distinct boundary shapes corresponding to Inverse Gaussian and Gamma distributions.
• Demonstrated sensitivity of boundary shapes to distribution parameters and tail behaviors.

Conclusions:

• The numerical approach effectively determines boundaries for the Inverse First Passage time problem.
• Boundary shapes are significantly influenced by the choice of first passage time distribution and its parameters.
• Findings have potential applications in neuroscience for modeling neural dynamics.