Efficient implicit solvers for models of neuronal networks

TL;DR

This paper presents optimized implicit ODE solvers tailored for neural network simulations, significantly improving efficiency by reducing algebraic system sizes across various neuron models with complex dynamics.

Contribution

The authors develop simplified implicit solvers, specifically ESDIRK methods, that enhance simulation efficiency for neural networks with slow-fast dynamics, applicable to multiple neuron models.

Findings

Demonstrated increased efficiency in network simulations

Applicable to models like FitzHugh-Nagumo, Calcium, Hindmarsh-Rose

Effective for networks of varying sizes

Abstract

We introduce economical versions of standard implicit ODE solvers that are specifically tailored for the efficient and accurate simulation of neural networks. These reformulations allow to achieve a significant increase in the efficiency of network simulations, by reducing the size of the algebraic systems effectively solved at each time step. While we focus here specifically on Explicit first step, Diagonally Implicit Runge Kutta methods (ESDIRK), similar simplifications can also be applied to any implicit ODE solver. In order to demonstrate the capabilities of the proposed methods, we consider networks based on three different single-cell models with slow-fast dynamics, including the classical FitzHugh-Nagumo model, a Intracellular Calcium Concentration model and the Hindmarsh-Rose model. Numerical experiments on the simulation of networks of increasing size based on these models demonstrate the superior efficiency of the proposed economical methods.