A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics

Area of Science:

• Computational neuroscience
• Mathematical modeling of neural systems
• Stochastic dynamical systems

Background:

• Neural mass models are crucial for simulating large-scale brain activity.
• The Jansen and Rit model (JR-NMM) is a foundational tool for mesoscopic neural dynamics.
• Existing models often lack the incorporation of stochasticity inherent in biological systems.

Purpose of the Study:

• To develop and analyze a stochastic version of the Jansen and Rit neural mass model (JR-NMM).
• To investigate the mathematical properties, including path characteristics and moment bounds, of the stochastic JR-NMM.
• To establish the long-time stability and convergence properties of the proposed stochastic model.

Main Methods:

• Formulation of a stochastic JR-NMM incorporating random input, resulting in a damped stochastic Hamiltonian system.
• Analysis of path properties and moment bounds for the stochastic model.
• Establishment of geometric ergodicity to prove long-time stability and convergence to an invariant measure.
• Development of an efficient numerical splitting scheme for simulating the stochastic JR-NMM.

Main Results:

• The stochastic JR-NMM exhibits properties of a damped stochastic Hamiltonian system with nonlinear displacement.
• Path properties and moment bounds for the stochastic model were rigorously investigated.
• The system was proven to be geometrically ergodic, ensuring convergence to a unique invariant measure regardless of initial conditions.
• An efficient numerical scheme was developed, preserving the qualitative behavior of the model's solutions.

Conclusions:

• The stochastic JR-NMM provides a robust framework for modeling mesoscopic neural dynamics with inherent randomness.
• The established long-time stability guarantees reliable predictions of neural system behavior.
• The efficient numerical simulation method facilitates further empirical and theoretical investigations of stochastic neural mass models.