Technical note: A fast and robust integrator of delay differential equations in DCM for electrophysiological data

Dynamic causal models (DCMs) of electrophysiological data allow, in principle, for inference on hidden, bulk synaptic function in neural circuits. The directed influences between the neuronal elements of modeled circuits are subject to delays due to the finite transmission speed of axonal connections. Ordinary differential equations are therefore not adequate to capture the ensuing circuit dynamics, and delay differential equations (DDEs) are required instead. Previous work has illustrated that the integration of DDEs in DCMs benefits from sophisticated integration schemes in order to ensure rigorous parameter estimation and correct model identification. However, integration schemes that have been proposed for DCMs either emphasize speed (at the possible expense of accuracy) or robustness (but with computational costs that are problematic in practice). In this technical note, we propose an alternative integration scheme that overcomes these shortcomings and offers high computational efficiency while correctly preserving the nature of delayed effects. This integration scheme is available as open-source code in the Translational Algorithms for Psychiatry-Advancing Science (TAPAS) toolbox and can be easily integrated into existing software (SPM) for the analysis of DCMs for electrophysiological data. While this paper focuses on its application to the convolution-based formalism of DCMs, the new integration scheme can be equally applied to more advanced formulations of DCMs (e.g. conductance based models). Our method provides a new option for electrophysiological DCMs that offers the speed required for scientific projects, but also the accuracy required for rigorous translational applications, e.g. in computational psychiatry.