Population density equations for stochastic processes with memory kernels.

Area of Science:

• Computational Neuroscience
• Mathematical Neuroscience
• Statistical Physics

Background:

• Population density equations (PDEs) are crucial for modeling large populations of neurons.
• Existing methods often struggle with non-Markovian noise and arbitrary jump size distributions.
• Integrating computational neuroscience and random network theory offers novel solutions.

Purpose of the Study:

• To present a novel numerical method for solving population density equations (PDEs).
• To handle non-Markovian noise with arbitrary jump size distributions in neuronal populations.
• To combine techniques from computational neuroscience and random network theory.

Main Methods:

• A geometric binning scheme based on the method of characteristics captures deterministic neurodynamics.
• Separates deterministic and stochastic processes for a modular numerical solution.
• Replaces the traditional master equation with the generalized Montroll-Weiss equation for non-Markovian processes.

Main Results:

• The method accurately models neuronal populations subject to non-Markov noise.
• Demonstrated accuracy for leaky- and quadratic-integrate and fire neuron models.
• Successfully modeled responses to both excitatory and inhibitory inputs from renewal processes.

Conclusions:

• The proposed method offers a flexible and accurate approach to solving population density equations.
• It effectively handles complex noise patterns, advancing the simulation of neuronal networks.
• This interdisciplinary approach provides a powerful tool for computational neuroscience research.