Convergence of regular spiking and intrinsically bursting Izhikevich neuron models as a function of discretization time with Euler method

Abstract

This study investigates the trade-off between computational efficiency and accuracy of Izhikevich neuron models by numerically quantifying their convergence to provide design guidelines in choosing the limit time steps during a discretization procedure. This is important for bionic engineering and neuro-robotic applications where the use of embedded computational resources requires the introduction of optimality criteria. Specifically, the regular spiking (RS) and intrinsically bursting (IB) Izhikevich neuron models are evaluated with step inputs of various amplitudes. We analyze the convergence of spike sequences generated under different discretization time steps (10 µs to 10 ms), with respect to an ideal reference spike sequence approximated with a discretization time step of 1 µs. The differences between the ideal reference and the computed spike sequences were quantified by Victor–Purpura (VPd) and van Rossum (VRd) distances. For each distance, we found two limit discretization times (lower dt1 and upper dt2), as a function of the applied input and thus firing rate, beyond which the convergence is lost for each neuron model. The estimated limit time steps were found to be consistent regardless of metric used (VPd and VRd) and neuron type (RS and IB), but also to depend on the average inter-spike interval (ISI) produced by the neurons. However, in most cases, a good trade-off between the quality of the convergence of the models dynamics and the computational load required to simulate them numerically was found for values of the discretization time step between dt1 ∈ (0.1 ms, 1 ms) and dt2 ∈ (2 ms, 3 ms).