Andreas Daffertshofer

Phase lag index: Assessment of functional connectivity from multi channel EEG and MEG with diminished bias from common sources

To address the problem of volume conduction and active reference electrodes in the assessment of functional connectivity, we propose a novel measure to quantify phase synchronization, the phase lag index (PLI), and compare its performance to the well‐known phase coherence (PC), and to the imaginary component of coherency (IC).

Comparing brain networks of different size and connectivity density using graph theory

Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.

Dynamical structure of center-of-pressure trajectories in patients recovering from stroke

In a recent study, De Haart et al. (Arch Phys Med Rehabil 85:886–895, 2004) investigated the recovery of balance in stroke patients using traditional analyses of center-of-pressure (COP) trajectories to assess the effects of health status, rehabilitation, and task conditions like standing with eyes open or closed and standing while performing a cognitive dual task. To unravel the underlying control processes, we reanalyzed these data in terms of stochastic dynamics using more advanced analyses. Dimensionality, local stability, regularity, and scaling behavior of COP trajectories were determined and compared with shuffled and phase-randomized surrogate data. The presence of long-range correlations discarded the possibility that the COP trajectories were purely random. Compared to the healthy controls, the COP trajectories of the stroke patients were characterized by increased dimensionality and instability, but greater regularity in the frontal plane. These findings were taken to imply that the stroke patients actively (i.e., cognitively) coped with the stroke-induced impairment of posture, as reflected in the increased regularity and decreased local stability, by recruiting additional control processes (i.e., more degrees of freedom) and/or by tightening the present control structure while releasing non-essential degrees of freedom from postural control. In the course of rehabilitation, dimensionality stayed fairly constant, whereas local stability increased and regularity decreased. The progressively less regular COP trajectories were interpreted to indicate a reduction of cognitive involvement in postural control as recovery from stroke progressed. Consistent with this interpretation, the dual task condition resulted in less regular COP trajectories of greater dimensionality, reflecting a task-related decrease of active, cognitive contributions to postural control. In comparison with conventional posturography, our results show a clear surplus value of dynamical measures in studying postural control.

Generative Models of Cortical Oscillations: Neurobiological Implications of the Kuramoto Model

Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker–Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity.

Characteristics of instructed and uninstructed interpersonal coordination while walking side-by-side.

We examined how people synchronize their leg movements while walking side-by-side on a treadmill. Walker pairs were either instructed to synchronize their steps in in-phase or in antiphase or received no coordination instructions. Frequency and phase analysis revealed that instructed in-phase and antiphase coordination were equally stable and independent of walking speed and the difference in individually preferred stride frequencies. Without instruction we found episodes of frequency locking in three pairs and episodes of phase locking in four pairs, albeit not always at (or near) 0 degrees or 180 degrees. Again, we found no difference in the stability of in-phase and antiphase coordination and no systematic effects of walking speed and the difference in individually preferred stride frequencies. These results suggest that the Haken-Kelso-Bunz model for rhythmic interlimb coordination does not apply to interpersonal coordination during gait in a straightforward manner. When the typically involved parameter constraints are relaxed, however, this model may largely account for the observed dynamical characteristics.

Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces

Abstract Recently, Jirsa et al. and Haken discussed a theory comprising the brain wave equation proposed by Nunez, the Wilson–Cowan/Ermentrout–Cowan model, and Hopfield networks. This theory was applied to model findings obtained in an experiment that relates brain activity and behavior, the so-called Julliard experiment. In previous works of Jirsa et al. and Frank et al. the focus was on the brain wave aspect. Recently, we conducted similar experiments. The results obtained are modeled in this paper in terms of coupled oscillator systems. Coupled oscillator systems are considered to represent the Wilson–Cowan/Ermentrout–Cowan-model aspect of the unifying theory. Building on a model proposed by Haken, Kelso, and Bunz and the theory of weakly coupled oscillators established by Winfree and by Kuramoto, we derive a nonlinear Fokker–Planck equation whose stationary solutions mimic the neocortical brain activity observed in our experiments.

Modeling rhythmic interlimb coordination. Beyond the Haken-Kelso-Bunz model

Although the Haken-Kelso-Bunz (HKB) model was originally formulated to account for phase transitions in bimanual movements, it evolved, through experimentation and conceptual elaboration, into a fundamental formal construct for the experimental study of rhythmically coordinated movements in general. The model consists of two levels of formalization: a potential defining the stability properties of relative phase and a system of coupled limit cycle oscillators defining the individual limb movements and their interactions. Whereas the empirical validity of the potential is well established, the validity of the formalization in terms of coupled oscillators is questionable, both with regard to the assumption that individual limb movements are limit cycle oscillators with (only) two active degrees of freedom and with regard to the postulated coupling. To remedy these limitations a more elaborate system of coupled oscillators is outlined, comprising two coupled limit cycle oscillators at the neural level, each of which is coupled to a linearly damped oscillator, representing the end-effectors.

Generalizability of the Proportional Recovery Model for the Upper Extremity After an Ischemic Stroke

Background and objective. Spontaneous neurological recovery after stroke is a poorly understood process. The aim of the present article was to test the proportional recovery model for the upper extremity poststroke and to identify clinical characteristics of patients who do not fit this model. Methods. A change in the Fugl-Meyer Assessment Upper Extremity score (FMA-UE) measured within 72 hours and at 6 months poststroke served to define motor recovery. Recovery on FMA-UE was predicted using the proportional recovery model: ΔFMA-UEpredicted = 0.7·(66 − FMA-UEinitial) + 0.4. Hierarchical cluster analysis on 211 patients was used to separate nonfitters (outliers) from fitters, and differences between these groups were studied using clinical determinants measured within 72 hours poststroke. Subsequent logistic regression analysis served to predict patients who may not fit the model. Results. The majority of patients (~70%; n = 146) showed a fixed proportional upper extremity motor recovery of about 78%; 65 patients had substantially less improvement than predicted. These nonfitters had more severe neurological impairments within 72 hours poststroke (P values <.01). Logistic regression analysis revealed that absence of finger extension, presence of facial palsy, more severe lower extremity paresis, and more severe type of stroke as defined by the Bamford classification were significant predictors of not fitting the proportional recovery model. Conclusions. These results confirm in an independent sample that stroke patients with mild to moderate initial impairments show an almost fixed proportional upper extremity motor recovery. Patients who will most likely not achieve the predicted amount of recovery were identified using clinical determinants measured within 72 hours poststroke.

A model for phase transitions in human hand movements during multifrequency tapping

In bimanual tapping, abrupt transitions between frequency ratios were observed when movement frequency was gradually increased. The transition routes showed individual tendencies, not necessarily in agreement with predictions derived from the sine circle map. Therefore, a more detailed theoretical model of coupled oscillators was developed. In the model the interaction function is a polynomial of coupling terms which allow for specific frequency locks. The magnitudes of these coupling terms are related to the amplitude of oscillation and the order of the frequency lock. Because increase in movement frequency is associated with a drop in amplitude, it results in differential loss of stability of the allowed frequency ratios. New frequency-locked states may be attained by detuning the stiffness parameters of the component oscillators. The model accounts for both free-running solutions and individual tendencies in transition routes. The relative weights of the coupling terms are influenced by practice and intention.

Keeping with the beat: movement trajectories contribute to movement timing

Previous studies of paced repetitive movements with respect to an external beat have either emphasised (a) the form of movement trajectories or (b) timing errors made with respect to the external beat. The question of what kinds of movement trajectories assist timing accuracy has not previously been addressed. In an experiment involving synchronisation or syncopation with an external auditory metronome we show that the nervous system produces trajectories that are asymmetric with respect to time and velocity in the out and return phases of the repeating movement cycle. This asymmetry is task specific and is independent of motor implementation details (finger flexion vs. extension). Additionally, we found that timed trajectories are less smooth (higher mean squared jerk) than unpaced ones. The degree of asymmetry in the flexion and extension movement times is positively correlated with timing accuracy. Negative correlations were observed between synchronisation timing error and the movement time of the ensuing return phase, suggesting that late arrival of the finger is compensated by a shorter return phase and conversely for early arrival. We suggest that movement asymmetry in repetitive timing tasks helps satisfy requirements of precision and accuracy relative to a target event.