Feedback and delays in neurological diseases: a modeling study using dynamical systems

Cortical beta oscillations reflect the contextual gating of visual action feedback

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We decouple seen and felt hand postures during action via virtual reality.

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Vision of the hand is either task-relevant or a distractor.

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Task-relevance of vision is reflected by in- or decreases of occipital beta power.

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DCM suggests underlying changes in cortical (visual) excitability.

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Occipital beta may indicate the contextual gating of visual action feedback.

In sensorimotor integration, the brain needs to decide how its predictions should accommodate novel evidence by ‘gating’ sensory data depending on the current context. Here, we examined the oscillatory correlates of this process by recording magnetoencephalography (MEG) data during a new task requiring action under intersensory conflict. We used virtual reality to decouple visual (virtual) and proprioceptive (real) hand postures during a task in which the phase of grasping movements tracked a target (in either modality). Thus, we rendered visual information either task-relevant or a (to-be-ignored) distractor. Under visuo-proprioceptive incongruence, occipital beta power decreased (relative to congruence) when vision was task-relevant but increased when it had to be ignored. Dynamic causal modeling (DCM) revealed that this interaction was best explained by diametrical, task-dependent changes in visual gain. These results suggest a crucial role for beta oscillations in the contextual gating (i.e., gain or precision control) of visual vs proprioceptive action feedback, depending on current behavioral demands.

Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

The Mechanical Representation of Temporal Delays

When we knock on a door, we perceive the impact as a collection of simultaneous events, combining sound, sight, and tactile sensation. In reality, information from different modalities but from a single source is flowing inside the brain along different pathways, reaching processing centers at different times. Therefore, interpreting different sensory modalities which seem to occur simultaneously requires information processing that accounts for these different delays. As in a computer-based robotic system, does the brain use some explicit estimation of the time delay, to realign the sensory flows? Or does it compensate for temporal delays by representing them as changes in the body/environment mechanics? Using delayed-state or an approximation for delayed-state manipulations between visual and proprioceptive feedback during a tracking task, we show that tracking errors, grip forces, and learning curves are consistent with predictions of a representation that is based on approximation for delay, refuting an explicit delayed-state representation. Delayed-state representations are based on estimating the time elapsed between the movement commands and their observed consequences. In contrast, an approximation for delay representations result from estimating the instantaneous relation between the expected and observed motion variables, without explicit reference to time.

Phase-locking swallows in coupled oscillators with delayed feedback

We show that a nonlinear coupling with delayed feedback between two limit-cycle oscillators can lead to phase-locked, periodically modulated, and chaotic phase synchronization as well as to desynchronization. Parameter regions with stable phase-locked states attain the well-known form of the swallows or shrimps found and studied for nonlinear maps. We demonstrate that the swallow regions can be accompanied by a different bifurcation scenario where the periodic orbits of the phase-locked states undergo a torus bifurcation instead of a previously reported period-doubling bifurcation. This property has an impact on the spatial organization of the swallows in the parameter space. The swallow regions contribute to the synchronization domain of the considered system, and we analytically approximate the parameter synchronization threshold.

Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity

We investigate the dynamics of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity. We perform a linearized stability analysis and multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the time delay as the bifurcation parameter, the presence of steady-state bifurcation, Bogdanov-Takens bifurcation, triple zero, and Hopf-zero singularities is demonstrated. In the case when the system has a simple zero eigenvalue, center manifold reduction and normal form theory are used to investigate the stability and the types of steady-state bifurcation. The stability of the zero solution of the system near the simple zero eigenvalue singularity is completely solved.