Tradeoff between accuracy and computational cost of Euler and Runge Kutta ODE solvers for the Izhikevich spiking neuron model

The Izhikevich spiking neuron model is one of the most used in neural engineering and computational neuroscience. Due to its trade-off between physiological plausibility and computational efficiency it is being used also in embedded systems with constrained computational resources. Thus, it is crucial to find a compromise between computational cost and error while numerically integrating the equations of the model. This work aims at quantifying the error produced by fixed step Ordinary Differential Equation (ODE) solvers. Our focus is to provide design hints that could be useful for embedded neural engineering applications. We evaluated three types of input and three ODE solvers: Euler, Runge Kutta 2, Runge Kutta 4. First, we generated a dataset of spike trains to draw conclusions on their general behavior while varying the discretization step. Then we showed that, within a single non-interrupting spike train, the spike delay is positive and accumulates linearly with the spike count. Finally, we introduced a robust method to assess the discretization limits. This method exploits the Victor Purpura distance and confirms that the limits depend on the spike train duration. Our results lead the way to a robust and systematic investigation of the trade-off between computational cost and discretization accuracy of fixed step ODE solvers for neuronal models.