Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

Accurate and Efficient Simulation of Very High-Dimensional Neural Mass Models with Distributed-Delay Connectome Tensors

This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass model, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen-Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.

On predictive inference for intractable models via approximate Bayesian computation

Approximate Bayesian computation (ABC) is commonly used for parameter estimation and model comparison for intractable simulator-based statistical models whose likelihood function cannot be evaluated. In this paper we instead investigate the feasibility of ABC as a generic approximate method for predictive inference, in particular, for computing the posterior predictive distribution of future observations or missing data of interest. We consider three complementary ABC approaches for this goal, each based on different assumptions regarding which predictive density of the intractable model can be sampled from. The case where only simulation from the joint density of the observed and future data given the model parameters can be used for inference is given particular attention and it is shown that the ideal summary statistic in this setting is minimal predictive sufficient instead of merely minimal sufficient (in the ordinary sense). An ABC prediction approach that takes advantage of a certain latent variable representation is also investigated. We additionally show how common ABC sampling algorithms can be used in the predictive settings considered. Our main results are first illustrated by using simple time-series models that facilitate analytical treatment, and later by using two common intractable dynamic models.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11222-022-10163-6.

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.

Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

Brain-Computer Interfaces Systems for Upper and Lower Limb Rehabilitation: A Systematic Review

In recent years, various studies have demonstrated the potential of electroencephalographic (EEG) signals for the development of brain-computer interfaces (BCIs) in the rehabilitation of human limbs. This article is a systematic review of the state of the art and opportunities in the development of BCIs for the rehabilitation of upper and lower limbs of the human body. The systematic review was conducted in databases considering using EEG signals, interface proposals to rehabilitate upper/lower limbs using motor intention or movement assistance and utilizing virtual environments in feedback. Studies that did not specify which processing system was used were excluded. Analyses of the design processing or reviews were excluded as well. It was identified that 11 corresponded to applications to rehabilitate upper limbs, six to lower limbs, and one to both. Likewise, six combined visual/auditory feedback, two haptic/visual, and two visual/auditory/haptic. In addition, four had fully immersive virtual reality (VR), three semi-immersive VR, and 11 non-immersive VR. In summary, the studies have demonstrated that using EEG signals, and user feedback offer benefits including cost, effectiveness, better training, user motivation and there is a need to continue developing interfaces that are accessible to users, and that integrate feedback techniques.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.