Detrended fluctuation analysis: a scale-free view on neuronal oscillations
Recent years of research have shown that the complex temporal structure of ongoing oscillations is scale-free and characterized by long-range temporal correlations. Detrended fluctuation analysis (DFA) has proven particularly useful, revealing that genetic variation, normal development, or disease can lead to differences in the scale-free amplitude modulation of oscillations. Furthermore, amplitude dynamics is remarkably independent of the time-averaged oscillation power, indicating that the DFA provides unique insights into the functional organization of neuronal systems. To facilitate understanding and encourage wider use of scaling analysis of neuronal oscillations, we provide a pedagogical explanation of the DFA algorithm and its underlying theory. Practical advice on applying DFA to oscillations is supported by MATLAB scripts from the Neurophysiological Biomarker Toolbox (NBT) and links to the NBT tutorial website http://www.nbtwiki.net/. Finally, we provide a brief overview of insights derived from the application of DFA to ongoing oscillations in health and disease, and discuss the putative relevance of criticality for understanding the mechanism underlying scale-free modulation of oscillations.
Neuronal activity in the subthalamic nucleus (STN) of patients with Parkinson's disease (PD) is characterised by excessive neuronal synchronization, particularly in the beta frequency range. However, less is known about the temporal dynamics of neuronal oscillations in PD. In this respect long-range temporal correlations (LRTC) are of special interest as they quantify the neuronal dynamics on different timescales and have been shown to be relevant for optimal information processing in the brain. While the presence of LRTC has been demonstrated in cortical data, their existence in deep brain structures remains an open question. We investigated (i) whether LRTC are present in local field potentials (LFP) recorded bilaterally from the STN at wakeful rest in ten patients with PD after overnight withdrawal of levodopa (OFF) and (ii) whether LRTC can be modulated by levodopa treatment (ON). Detrended fluctuation analysis was utilised in order to quantify the temporal dynamics in the amplitude fluctuations of LFP oscillations. We demonstrated for the first time the presence of LRTC (extending up to 50 s) in the STN. Importantly, the ON state was characterised by significantly stronger LRTC than the OFF state, both in beta (13-35 Hz) and high-frequency (> 200 Hz) oscillations. The existence of LRTC in subcortical structures such as STN provides further evidence for their ubiquitous nature in the brain. The weaker LRTC in the OFF state might indicate limited information processing in the dopamine-depleted basal ganglia. The present results implicate LRTC as a potential biomarker of pathological neuronal processes in PD.
Although thalamic deep brain stimulation is an effective treatment for patients with essential tremor, little is known about its effect on cortical neural dynamics. Therefore, we investigated long-range temporal correlations and spectral power in electroencephalographic recordings of patients during OFF versus ON bilateral thalamic deep brain stimulation in comparison with healthy controls. Cortical dynamics were analyzed in the range of 6-30 Hz. We found the presence of long-range temporal correlations up to 20 s in patients and controls. Thalamic deep brain stimulation was associated with increased long-range temporal correlations in the high beta band (21-30 Hz) and decreased power in the low beta band (13-20 Hz) compared with OFF stimulation and healthy controls. Long-range temporal correlations in the 6-10 Hz range were increased with OFF stimulation compared with the controls. Importantly, deep brain stimulation-induced changes in long-range temporal correlations within 6-10 Hz and in the beta ranges (13-20, 21-30 Hz) were correlated with OFF-ON changes in the tremor severity and with the disease duration, respectively. The differential reactivity of long-range temporal correlations and spectral power to deep brain stimulation might suggest that both measures reflect distinct aspects of cortical dynamics and might represent biomarkers for stimulation-induced modulations of neural dynamics in electroencephalography. The fact that long-range temporal correlations, but not spectral power, were correlated with clinical information might suggest long-range temporal correlations as a potential marker for disease severity in essential tremor.
Although the long-range temporal correlation (LRTC) of the amplitude fluctuations of neuronal EEG/MEG oscillations is widely acknowledged, the majority of studies to date have been performed in sensor space, disregarding the mixing effects implied by volume conduction and confounding noise. While the effect of mixing on the evaluation of evoked responses and connectivity measures has been extensively studied, there are, to date, no studies reporting on the differences in the values of the estimated Hurst exponents when moving between sensor and source space representations of the multivariate data or on the effect of noise. Such differences, if not duly acknowledged, may lead to erroneous data interpretations. We show in simulations and in theory that measuring Hurst exponents in sensor space may lead to an incomplete picture of the LRTC properties of the underlying data and that noise may significantly bias the estimate of the Hurst exponent of the underlying signal. Moreover, these predictions are confirmed in real data, where we analyze the amplitude dynamics of neuronal oscillations in the resting state from EEG data. By moving either to an independent components representation or to a source representation which maximizes the signal to noise ratio in the alpha frequency range, we observe greater variance, skewness and kurtosis over measured Hurst exponents than in sensor space. We confirm the suitability of conventional source separation methodology by introducing a novel algorithm HeMax which obtains a source maximizing the Hurst exponent in the amplitude dynamics of narrow band oscillations. Our findings imply that the long-range correlative properties of the EEG should be studied in source space, in such a way that the SNR is maximized, or at least with spatial decomposition techniques approximating source activities, rather than in sensor space.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.