A variational method for analyzing limit cycle oscillations in stochastic hybrid systems.

Area of Science:

• Computational Biology
• Mathematical Biology
• Systems Biology

Background:

• Biological systems are often modeled using ordinary differential equations (ODEs) and Markov jump processes, forming stochastic hybrid systems (SHS) or piecewise deterministic Markov processes (PDMP).
• In the fast switching limit, these systems converge to deterministic ODEs.
• Understanding the dynamics of SHS is crucial for modeling complex biological phenomena, such as neuronal activity.

Purpose of the Study:

• To develop a phase reduction method for stochastic hybrid systems that exhibit a stable limit cycle in their deterministic counterpart.
• To provide an exact analytic expression for the phase dynamics of these systems.
• To demonstrate the accuracy of the developed method over extended timescales.

Main Methods:

• Developed a phase reduction method based on a variational principle for stochastic hybrid systems.
• Analyzed systems supporting a stable limit cycle in the deterministic limit, using the Morris-Lecar neuron model as a key example.
• Derived an exact analytic expression for the phase dynamics.

Main Results:

• The phase reduction method yields an exact analytic expression for the phase dynamics.
• The decomposition is accurate over timescales that are exponential in the switching rate (ϵ⁻¹).
• Demonstrated that the probability of leaving the limit cycle scales as T exp(-Ca/ϵ), confirming accuracy over long timescales.

Conclusions:

• The developed phase reduction method provides an accurate analytical tool for studying stochastic hybrid systems.
• This approach is particularly effective for systems with stable limit cycles, such as neuronal models.
• The findings offer significant insights into the long-term dynamics of complex biological systems governed by hybrid processes.