Automatically Selecting a Suitable Integration Scheme for Systems of Differential Equations in Neuron Models

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Neural Modeling

Background:

• Integrate-and-fire neuron models are widely used for simulating neural activity due to their computational efficiency.
• Existing implementations often lack a structured approach for selecting numerical evolution schemes, potentially compromising solution accuracy and stability.
• The diversity of biological neuron behaviors necessitates flexible and reliable modeling approaches.

Purpose of the Study:

• To provide an overview of common equations and state descriptions for integrate-and-fire neuron dynamics.
• To present a formal mathematical process for automating the design or selection of appropriate numerical evolution schemes.
• To introduce a symbolic analysis toolbox for ordinary differential equations (ODEs) to aid modelers in implementing custom neuron models.

Main Methods:

• Review of typical ODEs and state descriptions for integrate-and-fire neuron models.
• Development of a formal mathematical framework for selecting numerical integration methods based on numerical criteria.
• Implementation of a symbolic analysis toolbox for ODEs to guide the selection process.

Main Results:

• A systematic approach to choosing numerical evolution schemes for integrate-and-fire models is formally described.
• The proposed methodology aims to improve the accuracy and stability of neural network simulations.
• A reference implementation of a symbolic analysis toolbox is presented to assist researchers.

Conclusions:

• Automating the selection of numerical schemes enhances the reliability of integrate-and-fire neuron simulations.
• The developed toolbox can guide modelers in implementing custom neuron models with greater confidence.
• This work addresses a critical, often overlooked, aspect of computational neuroscience model development.