This paper introduces the Cross-frequency Amplitude Transfer Function (CATF), a model-free method for quantifying nonlinear stimulus-response interaction based on phase-locked amplitude relationship. The CATF estimates the amplitude transfer from input frequencies at stimulation signal to their harmonics/intermodulation at the response signal. We first verified the performance of CATF in simulation tests with systems containing a static nonlinear function and a linear dynamic, i.e., Hammerstein and Wiener systems. We then applied the CATF to investigate the second-order nonlinear amplitude transfer in the human proprioceptive system from the periphery to the cortex. The simulation demonstrated that the CATF is a general method which can well quantify nonlinear stimulus-response amplitude transfer for different orders of nonlinearity in Wiener or Hammerstein system configurations. Applied to the human proprioceptive system, we found a complicated nonlinear system behavior with substantial amplitude transfer from the periphery stimulation to cortical response signals in the alpha band. This complicated system behavior may be associated with the nonlinear behavior of the muscle spindle and the dynamic interaction in the thalamocortical radiation. This paper provides a new tool to identify nonlinear interaction in the nervous system. The results provide novel insight of nonlinear dynamics in the human proprioceptive system@en

The human nervous system is an ensemble of connected neuronal networks. Modeling and system identification of the human nervous system helps us understand how the brain processes sensory input and controls responses at the systems level. This study aims to propose an advanced approach based on a hierarchical neural network and non-linear system identification method to model neural activity in the nervous system in response to an external somatosensory input. The proposed approach incorporates basic concepts of Non-linear AutoRegressive Moving Average Model with eXogenous input (NARMAX) and neural network to acknowledge non-linear closed-loop neural interactions. Different from the commonly used polynomial NARMAX method, the proposed approach replaced the polynomial non-linear terms with a hierarchical neural network. The hierarchical neural network is built based on known neuroanatomical connections and corresponding transmission delays in neural pathways. The proposed method is applied to an experimental dataset, where cortical activities from ten young able-bodied individuals are extracted from electroencephalographic signals while applying mechanical perturbations to their wrist joint. The results yielded by the proposed method were compared with those obtained by the polynomial NARMAX and Volterra methods, evaluated by the variance accounted for (VAF). Both the proposed and polynomial NARMAX methods yielded much better modeling results than the Volterra model. Furthermore, the proposed method modeled cortical responded with a mean VAF of 69.35% for a three-step ahead prediction, which is significantly better than the VAF from a polynomial NARMAX model (mean VAF 47.09%). This study provides a novel approach for precise modeling of cortical responses to sensory input. The results indicate that the incorporation of knowledge of neuroanatomical connections in building a realistic model greatly improves the performance of system identification of the human nervous system.

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In hemiparetic stroke, functional recovery of paretic limb may occur with the reorganization of neural networks in the brain. Neuroimaging techniques, such as magnetic resonance imaging (MRI), have a high spatial resolution which can be used to reveal anatomical changes in the brain following a stroke. However, low temporal resolution of MRI provides less insight of dynamic changes of brain activity. In contrast, electro-neurophysiological techniques, such as electroencephalography (EEG), have an excellent temporal resolution to measure such transient events, however are hindered by its low spatial resolution. This proof-of-principle study assessed a novel multimodal brain imaging technique namely Variational Bayesian Multimodal Encephalography (VBMEG), which aims to improve the spatial resolution of EEG for tracking the information flow inside the brain and its changes following a stroke. The limitations of EEG are complemented by constraints derived from anatomical MRI and diffusion weighted imaging (DWI). EEG data were acquired from individuals suffering from a stroke as well as able-bodied participants while electrical stimuli were delivered sequentially at their index finger in the left and right hand, respectively. The locations of active sources related to this stimulus were precisely identified, resulting in high Variance Accounted For (VAF above 80%). An accurate estimation of dynamic information flow between sources was achieved in this study, showing a high VAF (above 90%) in the cross-validation test. The estimated dynamic information flow was compared between chronic hemiparetic stroke and able-bodied individuals. The results demonstrate the feasibility of VBMEG method in revealing the changes of information flow in the brain after stroke. This study verified the VBMEG method as an advanced computational approach to track the dynamic information flow in the brain following a stroke. This may lead to the development of a quantitative tool for monitoring functional changes of the cortical neural networks after a unilateral brain injury and therefore facilitate the research into, and the practice of stroke rehabilitation.

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Objectives: Visual system abnormalities in migraine are linked to symptoms like photophobia and the visual aura. Little is known about the mechanisms contributing to these visual system alterations. Processing of visual input by the brain is a highly nonlinear operation, involving complex interactions among cortical and subcortical neuronal networks. Timing of this process can be estimated by analysing the cortical response to external light input at different frequencies. Using a sum-of-sinusoid light signal, instead of the classic pulse train, as input and novel EEG analyses it is possible to assess the time delay and frequency domain response. Here we investigate nonlinear visual processing in subgroups of migraine patients and headache-free participants.

Methods: Migraine patients with aura, without aura and healthy participants (N = 10/group) were subjected to bi-sinusoidal light stimulation for 320 1 sec-epochs, while scalp EEG was recorded at the occipital, parietal and frontal lobes. Light stimulus frequencies were chosen to guarantee no overlap of their harmonic and intermodulation frequencies for different orders of nonlinearity. Nonlinear interactions and time delay from stimulus to cortical EEG response were analysed in the frequency domain using novel phase clustering measures and amplitude spectral measures.

Results: Higher harmonic and intermodulation interactions were detected between visual input and cortical responses. Amplitude spectrum and phase clustering responses differed per order and group. Migraine patients with aura showed a decreased time delay only at the occipital lobe compared to healthy controls and migraine patients without aura.

Conclusion: Visual processing is altered in migraine patients with aura compared to healthy controls and patients without aura. Furthermore, we demonstrated the potential of quantifying nonlinear interactions and temporal dynamics in the visual system using sum-of-sinusoid light stimulation. We are able to uncover alterations in visual processing in the context of neurological disease.@en

The essential task of a motor imagery brain–computer interface (BCI) is to extract the motor imagery-related features from electroencephalogram (EEG) signals for classifying motor intentions. However, the optimal frequency band and time segment for extracting such features differ from subject to subject. In this work, we aim to improve the multi-class classification and to reduce the required EEG channel in motor imagery-based BCI by subject-specific time-frequency selection. Our method is based on a criterion namely Fisher discriminant analysis-type F-score to simultaneously select the optimal frequency band and time segment for multi-class classification. The proposed method uses only few Laplacian EEG channels (C3, Cz and C4) located around the sensorimotor area for classification. Applied to a standard multi-class BCI dataset (BCI competition III dataset IIIa), our method leads to better classification performance and smaller standard deviation across subjects compared to the state-of-art methods. Moreover, adding artifacts contaminated trials to the training dataset does not necessarily deteriorate our classification results, indicating that our method is tolerant to artifacts.

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Mechanical perturbations applied to the wrist joint typically evoke a stereotypical sequence of cortical and muscle responses. The early cortical responses (<100 ms) are thought be involved in the “rapid” transcortical reaction to the perturbation while the late cortical responses (>100 ms) are related to the “slow” transcortical reaction. Although previous studies indicated that both responses involve the primary motor cortex, it remains unclear if both responses are engaged by the same effective connectivity in the cortical network. To answer this question, we investigated the effective connectivity cortical network after a “ramp-and-hold” mechanical perturbation, in both the early (<100 ms) and late (>100 ms) periods, using dynamic causal modeling. Ramp-and-hold perturbations were applied to the wrist joint while the subject maintained an isometric wrist flexion. Cortical activity was recorded using a 128-channel electroencephalogram (EEG). We investigated how the perturbation modulated the effective connectivity for the early and late periods. Bayesian model comparisons suggested that different effective connectivity networks are engaged in these two periods. For the early period, we found that only a few cortico-cortical connections were modulated, while more complicated connectivity was identified in the cortical network during the late period with multiple modulated cortico-cortical connections. The limited early cortical network likely allows for a rapid muscle response without involving high-level cognitive processes, while the complexity of the late network may facilitate coordinated responses.@en

Neural coupling between the central nervous system and the periphery is essential for the neural control of movement. Corticomuscular coherence is a popular linear technique to assess synchronised oscillatory activity in the sensorimotor system. This oscillatory coupling originates from ascending somatosensory feedback and descending motor commands. However, corticomuscular coherence cannot separate this bidirectionality. Furthermore, the sensorimotor system is nonlinear, resulting in cross-frequency
coupling. Cross-frequency oscillations cannot be assessed nor exploited by linear measures. Here, we emphasise the need of novel coupling measures, which provide directionality and acknowledge nonlinearity, to unveil neural coupling in the sensorimotor system. We highlight recent advances in the field and argue that assessing directionality and nonlinearity of neural coupling
will break new ground in the study of the control of movement in healthy and neurologically impaired individuals.@en

Coupling between cortical oscillations and muscle activity facilitates neuronal communication during motor control. The linear part of this coupling, known as corticomuscular coherence, has received substantial attention, even though neuronal communication underlying motor control has been demonstrated to be highly nonlinear. A full assessment of corticomuscular coupling, including the nonlinear part, is essential to understand the neuronal communication within the sensorimotor system. In this study, we applied the recently developed n:m coherence method to assess nonlinear corticomuscular coupling during isotonic wrist flexion. The n:m coherence is a generalized metric for quantifying nonlinear cross-frequency coupling as well as linear iso-frequency coupling. By using independent component analysis (ICA) and equivalent current dipole source localization, we identify four sensorimotor related brain areas based on the locations of the dipoles, i.e., the contralateral primary sensorimotor areas, supplementary motor area (SMA), prefrontal area (PFA) and posterior parietal cortex (PPC). For all these areas, linear coupling between electroencephalogram (EEG) and electromyogram (EMG) is present with peaks in the beta band (15–35 Hz), while nonlinear coupling is detected with both integer (1:2, 1:3, 1:4) and non-integer (2:3) harmonics. Significant differences between brain areas is shown in linear coupling with stronger coherence for the primary sensorimotor areas and motor association cortices (SMA, PFA) compared to the sensory association area (PPC); but not for the nonlinear coupling. Moreover, the detected nonlinear coupling is similar to previously reported nonlinear coupling of cortical activity to somatosensory stimuli. We suggest that the descending motor pathways mainly contribute to linear corticomuscular coupling, while nonlinear coupling likely originates from sensory feedback.@en

Interaction between distant neuronal populations is essential for communication within the nervous system and can occur as a highly nonlinear process. To better understand the functional role of neural interactions, it is important to quantify the nonlinear connectivity in the nervous system. We introduce a general approach to measure nonlinear connectivity through phase coupling: the multi-spectral phase coherence (MSPC). Using simulated data, we compare MSPC with existing phase coupling measures, namely n : m synchronization index and bi-phase locking value. MSPC provides a system description, including (i) the order of the nonlinearity, (ii) the direction of interaction, (iii) the time delay in the system, and both (iv) harmonic and (v) intermodulation coupling beyond the second order; which are only partly revealed by other methods. We apply MSPC to analyze data from a motor control experiment, where subjects performed isotonic wrist flexions while receiving movement perturbations. MSPC between the perturbation, EEG and EMG was calculated. Our results reveal directional nonlinear connectivity in the afferent and efferent pathways, as well as the time delay (43±8ms) between the perturbation and the brain response. In conclusion, MSPC is a novel approach capable to assess high-order nonlinear interaction and timing in the nervous system.@en

Communication between neuronal populations is facilitated by synchronization of their oscillatory activity. Although nonlinearity has been observed in the sensorimotor system, its nonlinear connectivity has not been widely investigated yet. This study investigates nonlinear connectivity during the human stretch reflex based on neuronal synchronization. Healthy participants generated isotonic wrist flexion while receiving a periodic mechanical perturbation to the wrist. Using a novel cross-frequency phase coupling metric, we estimate directional nonlinear connectivity, including time delay, from the perturbation to brain and to muscle, as well as from brain to muscle. Nonlinear phase coupling is significantly stronger from the perturbation to the muscle than to the brain, with a shorter time delay. The time delay from the perturbation to the muscle is 33 ms, similar to the reported latency of the spinal stretch reflex at the wrist. Source localization of nonlinear phase coupling from the brain to the muscle suggests activity originating from the motor cortex, although its effect on the stretch reflex is weak. As such nonlinear phase coupling between the perturbation and muscle activity is dominated by the spinal reflex loop. This study provides new evidence of nonlinear neuronal synchronization in the stretch reflex at the wrist joint with respect to spinal and transcortical loops.

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Objective: This paper introduces a generalized coherence framework for detecting and characterizing nonlinear interactions in the nervous system, namely cross-spectral coherence (CSC). CSC can detect different types of nonlinear interactions including harmonic and intermodulation coupling as present in static nonlinearities and also subharmonic coupling, which only occurs with dynamic nonlinearities. Methods: We verified the performance of CSC in model simulations with both static and dynamic nonlinear systems. We applied CSC to investigate nonlinear stimulus–response interactions in the human proprioceptive system. A periodic movement perturbation was imposed to the wrist when the subjects performed an isotonic wrist flexion. CSC analysis was performed between the perturbation and brain responses (electroencephalogram, EEG). Results: Both the simulation and the application demonstrated that CSC successfully detected different types of nonlinear interactions. High-order nonlinearities were revealed in the proprioceptive system, shown in harmonic and intermodulation coupling between the perturbation and EEG for all subjects. Subharmonic coupling was found in some subjects but not all. Conclusion: This paper provides a general tool to detect and characterize nonlinear nature and dynamics of the nervous system. The application of CSC on the experimental dataset indicates a complex nonlinear dynamics in the proprioceptive system. Significance: This novel framework 1) unveils the nonlinear neural dynamics in a more complete way than the existing coherence measures, and 2) is more suitable for estimating the input–output relation regarding both phase and amplitude compared to phase synchrony measures (which only consider phase coupling). Subharmonic coupling is reported in human proprioceptive system for the first time.@en

Neural systems can present various types of nonlinear input-output relationships, such as harmonic, subharmonic, and/or intermodulation coupling. This paper aims to introduce a general framework in frequency domain for detecting and characterizing nonlinear coupling in neural systems, called the cross-frequency coherence framework (CFCF). CFCF is an extension of classic coherence based on higher-order statistics. We demonstrate an application of CFCF for identifying nonlinear interactions in human motion control. Our results indicate that CFCF can effectively characterize nonlinear properties of the afferent sensory pathway. We conclude that CFCF contributes to identifying nonlinear transfer in neural systems.

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