Aperiodic neural activity has been the subject of intense research interest lately as it could reflect on the cortical excitation/inhibition ratio, which is suspected to be affected in numerous clinical conditions. This phenomenon is characterized via the aperiodic scaling exponent β, equal to the spectral slope following log-log transformation of power spectra. Despite recent progress, however, most current methods do not take into consideration the plausible multimodal nature in the power spectra of neurophysiological recordings - i.e., β might be different in low- (β_lo) and high-frequency (β_hi) regimes -, especially in case of |β_lo|>|β_hi|. Here we propose an algorithm, the multi-modal spectral parametrization method (MMSPM) that aims to account for this issue. MMSPM estimates β_lo and β_hi separately using a constrained, piece-wise regression technique, and also assesses if they are significantly different or instead the spectrum is indeed unimodal and can be characterized simply with broadband β. Here we present the MMSPM algorithm and evaluate its performance in silico on simulated power spectra. Then, we use MMSPM on resting-state electroencephalography (EEG) data collected from 19 young, healthy volunteers, as well as on a separate dataset of EEG recordings from 30 schizophrenia patients and 31 healthy controls, and demonstrate that broadband (0.1-100 Hz and 0.5-45 Hz) EEG spectra can indeed present a bimodality pattern with significantly steeper low-range (<∼2 Hz) and flatter high-range scaling regimes (i.e., |β_lo|>|β_hi|). Clinical relevance: The MMSPM method characterizes aperiodic neural activity in distinct scaling regimes, which can be relevant in numerous pathological conditions such as dementia or schizophrenia.

A planar stick balancing task was investigated using stabilometry parameters (SP); a concept initially developed to assess the stability of human postural sway. Two subject groups were investigated: 6 subjects (MD) with many days of balancing a 90 cm stick on a linear track and 25 subjects (OD) with only one day of balancing experience. The underlying mechanical model is a pendulum-cart system. Two control force models were investigated by means of numerical simulations: (1) delayed state feedback (DSF); and (2) delay-compensating predictor feedback (PF). Both models require an internal model and are subject to certainty thresholds with delayed switching. Measured and simulated time histories were compared quantitatively using a cost function in terms of some essential SPs for all subjects. Minimization of the cost function showed that the control strategy of both OD and MD subjects can better be described by DSF. The control mechanism for the MD subjects was superior in two aspects: (1) they devoted less energy to controlling the cart’s position; and (2) their perception threshold for the stick’s angular velocity was found to be smaller. Findings support the concept that when sufficient sensory information is readily available, a delay-compensating PF strategy is not necessary.

Background Ordinal scales based on qualitative observation are the mainstay in the clinical assessment of tremor, but are limited by inter-rater reliability, measurement precision, range, and ceiling effects. Quantitative tremor evaluation is well-developed in research, but clinical application has lagged, in part due to cumbersome mathematical application and lack of established standards. Objectives To develop a novel method for evaluating tremor that integrates a standardized clinical exam, wrist-watch accelerometers, and a software framework for data analysis that does not require advanced mathematical or computing skills. The utility of the method was tested in a sequential cohort of patients with predominant postural and action tremor presenting to a specialized surgical clinic with the presumptive diagnosis of Essential Tremor (ET). Methods Wristwatch accelerometry was integrated with a standardized clinical exam. A MATLAB application was developed for automated data analysis and graphical representation of tremor. Measures from the power spectrum of acceleration of tremor in different upper limb postures were derived in 25 consecutive patients. The linear results from accelerometry were correlated with the commonly used non-linear Clinical Rating Scale for Tremor (CRST). Results The acceleration power spectrum was reliably produced in all consecutive patients. Tremor frequency was stable in different postures and across patients. Both total and peak power of acceleration during postural conditions correlated well with the CRST. The standardized clinical examination with integrated accelerometry measures was therefore effective at characterizing tremor in a population with predominant postural and action tremor. The protocol is also illustrated on repeated measures in an ET patient who underwent Magnetic Resonance-Guided Focused Ultrasound thalamotomy. Conclusion Quantitative assessment of tremor as a continuous variable using wristwatch accelerometry is readily applicable as a clinical tool when integrated with a standardized clinical exam and a user-friendly software framework for analysis. The method is validated for patients with predominant postural and action tremor, and can be adopted for characterizing tremor of different etiologies with dissemination in a wide variety of clinical and research contexts in ageing populations.

Mechanical models of two human balancing tasks, quiet standing and stick balancing on the fingertip, are discussed with special attention to the reaction time delay. As in many control systems, time delay sets limitation to performance during balancing tasks. Human subjects cannot balance an arbitrarily short stick on the fingertip, because a short stick falls faster than the time needed to make a corrective motion. Also, increased reaction time delay is one of the main reasons of instability during quiet standing, which can lead to falls especially among the elderly. The governing equation of the two models are second-order delay differential equations. In this chapter, stabilizability issues are discussed in terms of the critical delay for different feedback concepts, such as proportional-derivative, proportional-derivative-acceleration and predictor feedback.

A dynamic disease of the nervous system is a disease that arises from abnormalities in neural control mechanisms. Whereas traditional approaches for classifying neurological diseases are based on (static) anatomical, cellular, and molecular abnormalities, the focus here is on dynamics, namely, the variation of signs and symptoms of disease as a function of time. The hallmarks of dynamic diseases are sudden, qualitative changes in the temporal pattern of clinical signs. Identifying a neurological disorder as a dynamic disease has two major implications: (1) the observed dynamics and their responses to various manipulations provide important insights into the nature and abnormality of neural control, and (2) based on computational models of the abnormalities, it may be possible to devise novel treatment strategies for dynamic diseases of the brain.

A new mathematical model for biological rhythms in the hypothalamic-pituitary-adrenal (HPA) axis is proposed. This model takes the form of a system of impulsive time-delay differential equations which include pulsatile release of adrenocorticotropin (ACTH) by the pituitary gland and a time delay for the release of glucocorticoid hormones by the adrenal gland. Numerical simulations demonstrate that the model's response to periodic and circadian inputs from the hypothalamus are consistent with those generated by recent models which do not include a pulsatile pituitary. In contrast the oscillatory phenomena generated by the impulsive delay equation mode occur even if the time delay is zero. The observation that the time delay merely introduces a small phase shift suggesting that the effects of the adrenal gland are "downstream" to the origin of pulsativity. In addition, the model accounts for the occurrence of ultradian oscillations in an isolated pituitary gland. These observations suggest that principles of pulse modulated control, familiar to control engineers, may have an increasing role to play in understanding the HPA axis.

Postural sway is a result of a complex action–reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the response to large enough perturbations during quiet standing such that the resulting responses can clearly be distinguished from the local postural sway. Measurements show that typical responses very closely resemble those of a critically damped oscillator. The recovery dynamics are modelled by an inverted pendulum subject to delayed state feedback and is described in the space of the control parameters. We hypothesize that the control gains are tuned such that (H1) the response is at the border of oscillatory and nonoscillatory motion similarly to the critically damped oscillator; (H2) the response is the fastest possible; (H3) the response is a result of a combined optimization of fast response and robustness to sensory perturbations. Parameter fitting shows that H1 and H3 are accepted while H2 is rejected. Thus, the responses of human postural balance to “large” perturbations matches a delayed feedback mechanism that is optimized for a combination of performance and robustness.

We develop the formal mathematics of the lognormal random walk model. We start by discussing continuous compounding for risk-free investments. We introduce a random variable to model the uncertainty of a risky investment. We apply the Central Limit Theorem to argue that under three strong assumptions, the values of risky investments at any time horizon are lognormally distributed. We model the random walk using a stochastic differential equation. We define the notion of an " Ito process " and prove that it is equivalent to our formulation. We apply the model to the S&P 500 stock market index as an example. We learn how to do parameter estimation for the model using historical time series data and how to do calculations in the model in computer programs. We discuss how uncertainty and risk increase with time horizon when investing in volatile assets like stocks, contrary to popular opinion. We conclude by asking the all-important question of how well the simple random walk model describes how financial markets actually work. We mention known failings of the model and conclude that at best it is a rough approximation to reality and should be used for real-life financial planning with caution. We assume that the reader is familiar with the normal and lognormal probability distributions as presented in reference [8].

The stability radius provides a scalar measure of the distance from instability in the space of the system and control parameters. It is not difficult to imagine that the risk of falling might be inversely proportional to the stability radius, i.e the smaller the stability radius the higher the risk of falling due to sensory uncertainty.

The effect of reaction delay, temporal sampling, sensory quantization, and control torque saturation is investigated numerically for a single-degree-of-freedom model of postural sway with respect to stability, stabilizability, and control effort. It is known that reaction delay has a destabilizing effect on the balancing process: the later one reacts to a perturbation, the larger the possibility of falling. If the delay is larger than a critical value, then stabilization is not even possible. In contrast, numerical analysis showed that quantization and control torque saturation have a stabilizing effect: the region of stabilizing control gains is greater than that of the linear model. Control torque saturation allows the application of larger control gains without overcontrol while sensory quantization plays a role of a kind of filter when sensory noise is present. These beneficial effects are reflected in the energy demand of the control process. On the other hand, neither control torque saturation nor sensory quantization improves stabilizability properties. In particular, the critical delay cannot be increased by adding saturation and/or sensory quantization.

Postural sway is a result of a complex action-reaction feedback mechanism generated by the interplay between the environment, the sensory perception, the neural system and the musculation. Postural oscillations are complex, possibly even chaotic. Therefore fitting deterministic models on measured time signals is ambiguous. Here we analyse the response to large enough perturbations during quiet standing such that the resulting responses can clearly be distinguished from the local postural sway. Measurements show that typical responses very closely resemble those of a critically damped oscillator. The recovery dynamics is modelled by an inverted pendulum subject to delayed state feedback and is described in the space of the control parameters. We hypothesize that the control gains are tuned such that (H1) the response is at the border of oscillatory and nonoscillatory motion; (H2) the response is the fastest possible; (H3) the response is a result of a combined optimization of fast response and robustness to sensory perturbations. Parameter fitting shows that H1 and H3 are accepted while H2 is rejected. Thus, the responses of human postural balance to “large” perturbations matches a delayed feedback mechanism that is optimized for a combination of performance and robustness.

An expert pole balancer cannot balance a 0.3 m stick at their fingertip for longer than 5 s if one eye is patched! This is a consequence of the limited ability of human depth perception in the sagittal (anterior-posterior) plane.

The concept of Dynamical Diseases provides a framework to understand physiological control systems in pathological states due to their operating in an abnormal range of control parameters: this allows for the possibility of a return to normal condition by a redress of the values of the governing parameters. The analogy with bifurcations in dynamical systems opens the possibility of mathematically modeling clinical conditions and investigating possible parameter changes that lead to avoidance of their pathological states. Since its introduction, this concept has been applied to a number of physiological systems, most notably cardiac, hematological, and neurological. A quarter century after the inaugural meeting on dynamical diseases held in Mont Tremblant, Québec [Bélair et al., Dynamical Diseases: Mathematical Analysis of Human Illness (American Institute of Physics, Woodbury, NY, 1995)], this Focus Issue offers an opportunity to reflect on the evolution of the field in traditional areas as well as contemporary data-based methods.

Microchaos is a low amplitude type of chaos that arises in time-delayed dynamical systems from the interplay between a sensory dead zone and time-sampled feedback. In the previous chapter we suggested that the Weibull-type survival curves observed for pole balancing at the fingertip were a manifestation of transient microchaos.

Postural sway refers to the fluctuations in the horizontal movements of the center of mass (COM) while a human stands quietly. The fluctuations in the center of pressure (COP) measured with a force platform provide a measure of the neuromuscular response to the motions of the COM.