Automatically Selecting a Suitable Integration Scheme for Systems of Differential Equations in Neuron Models - Blundell, Inga; Plotnikov, Dimitri; Eppler, Jochen M.; Morrison, Abigail (Frontiers in Neuroinformatics, 2018)
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.