Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
On contraction analysis for non-linear systems
Automatica (Journal of IFAC)
Spiking Neuron Models: An Introduction
Spiking Neuron Models: An Introduction
The quantitative single-neuron modeling competition
Biological Cybernetics - Special Issue: Quantitative Neuron Modeling
Firing patterns in the adaptive exponential integrate-and-fire model
Biological Cybernetics - Special Issue: Quantitative Neuron Modeling
Dynamics and bifurcations of the adaptive exponential integrate-and-fire model
Biological Cybernetics - Special Issue: Quantitative Neuron Modeling
Simple model of spiking neurons
IEEE Transactions on Neural Networks
Which model to use for cortical spiking neurons?
IEEE Transactions on Neural Networks
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Bidimensional spiking models are garnering a lot of attention for their simplicity and their ability to reproduce various spiking patterns of cortical neurons and are used particularly for large network simulations. These models describe the dynamics of the membrane potential by a nonlinear differential equation that blows up in finite time, coupled to a second equation for adaptation. Spikes are emitted when the membrane potential blows up or reaches a cutoff Î赂. The precise simulation of the spike times and of the adaptation variable is critical, for it governs the spike pattern produced and is hard to compute accurately because of the exploding nature of the system at the spike times. We thoroughly study the precision of fixed time-step integration schemes for this type of model and demonstrate that these methods produce systematic errors that are unbounded, as the cutoff value is increased, in the evaluation of the two crucial quantities: the spike time and the value of the adaptation variable at this time. Precise evaluation of these quantities therefore involves very small time steps and long simulation times. In order to achieve a fixed absolute precision in a reasonable computational time, we propose here a new algorithm to simulate these systems based on a variable integration step method that either integrates the original ordinary differential equation or the equation of the orbits in the phase plane, and compare this algorithm with fixed time-step Euler scheme and other more accurate simulation algorithms.