First passage time densities in resonate-and-fire models

Area of Science:

• Computational Neuroscience
• Stochastic Processes
• Statistical Physics

Background:

• First passage time (FPT) densities are crucial for understanding the dynamics of stochastic processes, particularly in neural modeling.
• Non-Markovian processes, characterized by memory effects, present significant analytical challenges for FPT calculations.
• Existing Markovian approximations often fail to capture the complex behaviors observed in realistic neural systems.

Purpose of the Study:

• To derive and validate approximations for first passage time (FPT) densities of non-Markovian differentiable random processes.
• To investigate the properties of FPT densities in systems exhibiting resonant dynamics, such as resonate-and-fire neurons.
• To provide accurate analytical tools for non-Markovian systems where Markovian approximations are insufficient.

Main Methods:

• Derivation of an exact FPT density expression as an infinite series of integrals over joint level-crossing densities.
• Application of truncation and approximate summation techniques to the derived series for practical approximations.
• Modeling resonate-and-fire neurons with different damping (underdamped, moderately damped) and noise types (white Gaussian, Ornstein-Uhlenbeck) as a case study.

Main Results:

• Short-time approximations of FPT densities are accurately obtained using the first few terms of the series.
• Decoupling approximations provide good performance across the entire time domain for processes with rapidly decaying correlations.
• The approximations successfully reproduce diverse FPT density structures, including monomodal and multimodal patterns with decaying peaks.
• The methods are applicable to systems of any dimension and are particularly effective for processes with narrow spectral densities.

Conclusions:

• The developed approximation methods offer accurate insights into FPT densities for non-Markovian processes, especially in resonant neuronal dynamics.
• These techniques overcome limitations of Markovian approximations, providing a more robust framework for analyzing complex stochastic systems.
• The study offers valuable tools for computational neuroscience and statistical physics, enhancing the understanding of neural firing mechanisms and other non-Markovian phenomena.