AUTHOR=Blundell Inga , Plotnikov Dimitri , Eppler Jochen M. , Morrison Abigail 

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

JOURNAL=Frontiers in Neuroinformatics

VOLUME=Volume 12 - 2018

YEAR=2018

URL=https://www.frontiersin.org/journals/neuroinformatics/articles/10.3389/fninf.2018.00050

DOI=10.3389/fninf.2018.00050

ISSN=1662-5196

ABSTRACT=On the level of the spiking activity, the
integrate-and-fire neuron is one of the most
commonly used descriptions of neural activ-
ity. A multitude of variants has been proposed
to cope with the huge diversity of behaviors
observed in biological nerve cells. The main
appeal of this class of model is that it can be
defined in terms of a hybrid model, where a
set of mathematical equations describes the
sub-threshold dynamics of the membrane po-
tential and the generation of action potentials
is often only added algorithmically without the
shape of spikes being part of the equations. In
contrast to more detailed biophysical models,
this simple description of neuron models allows
the routine simulation of large biological neu-
ronal networks on standard hardware widely
available in most laboratories these days.

The time evolution of the relevant state vari-
ables is usually defined by a small set of
ordinary differential equations (ODEs). Evolu-
tion schemes for the corresponding systems of
ODEs are well known for many neuron mod-
els and most of them are both efficient and
accurate. Such schemes are the basis of the
neuron model implementations built into com-
monly used simulators like Brian, NEST and
NEURON. 
However, an often neglected problem is that
the implemented evolution schemes are only
rarely selected through a structured process
based on numerical criteria. This practice can-
not guarantee accurate and stable solutions for
the equations and the actual quality of the so-
lution depends largely on the parametrization
of the model. 
In this article, we give an overview of typical
equations and state descriptions for the dynam-
ics of the relevant variables in integrate-and-fire
models. We then describe a formal mathemati-
cal process to automate the design or selection
of a suitable evolution scheme for this large
class of models. Finally, we present the refer-
ence implementation of our symbolic analysis
toolbox for ODEs that can guide modelers 
during the implementation of custom neuron
models.