Abstract
On the level of the spiking activity, the integrate-and-fire neuron is one of the most commonly used descriptions of neural activity. 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 potential 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 neuronal networks on standard hardware widely available in most laboratories these days. The time evolution of the relevant state variables is usually defined by a small set of ordinary differential equations (ODEs). A small number of evolution schemes for the corresponding systems of ODEs are commonly used for many neuron models, and form the basis of the neuron model implementations built into commonly 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 cannot guarantee accurate and stable solutions for the equations and the actual quality of the solution depends largely on the parametrization of the model. In this article, we give an overview of typical equations and state descriptions for the dynamics of the relevant variables in integrate-and-fire models. We then describe a formal mathematical process to automate the design or selection of a suitable evolution scheme for this large class of models. Finally, we present the reference implementation of our symbolic analysis toolbox for ODEs that can guide modelers during the implementation of custom neuron models.
Highlights
In common with all body cells, nerve cells are delimited by a bi-lipid layer which is largely impermeable for ions and bigger molecules
One was to apply the stiffness tester to the neuron models currently implemented in the NEST Modeling Language (NESTML; Plotnikov et al, 2016), the other was to compare runtimes of explicit and implicit evolution schemes applied to two commonly used simplified versions of the Hodgkin-Huxley model
Neuron models in NEST are usually rather simple point neurons or models with a few electrical compartments instead of rich compartmental neurons built from morphologically detailed reconstructions
Summary
In common with all body cells, nerve cells (neurons) are delimited by a bi-lipid layer (the cell membrane) which is largely impermeable for ions and bigger molecules. Active ion pumps and passive channels embedded into the membrane allow the selective passage of certain ions. Through these transporter molecules, neurons maintain a gradient of different ion types across the membrane, which leads to the membrane potential (Kandel et al, 2013). In the absence of input, the membrane potential fluctuates around the resting potential EL (typically at around −70 mV). Excitatory input depolarizes the membrane, driving the membrane potential closer to zero, while inhibitory input hyperpolarizes the neuron, driving the membrane potential away from zero. If the membrane potential crosses the spiking threshold θ (typically at around −55 mV), the neuron fires an action potential (spike), which is transmitted to all downstream (postsynaptic) neurons, where it in turn elicits excursions of their membrane potentials
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