Nonlinear physics presents us with a perplexing variety of complicated fractal objects and strange sets. Naturally one wishes to characterize the objects and describe the events occurring on them. Moreover, most time series found in “real-life” applications appear quite noisy. Therefore, at almost every point in time, they cannot be approximated either by the Taylor series or by the Fourier series of just a few terms. Many experimental time series have fractal features and display singular behavior, the so-called singularities. The multifractal spectrum quantifies the degree of fractals in the processes generating the time series. A novel definition is proposed called full-width Hölder exponents that indicate maximum expansion of multifractal spectrum. The obtained results have demonstrated the multifractal structure of near-infrared spectroscopy time series and the evidence for brain imagery activities.

Neurophysiological and neuroimaging technologies have contributed much to our understanding of normative brain function. Functional magnetic resonance imaging (fMRI) is currently considered the “gold standard” for measuring functional brain activation. The limitations of fMRI include the requirement that participants must lie within the confines of the magnet bore, which limits its use for many applications. The readout gradients in the imaging pulse sequences also produce a loud noise [

In recent years, functional near-infrared spectroscopy (NIRS) has been introduced as a new neuroimaging modality with which to conduct functional brain-imaging studies. NIRS technology uses specific wavelengths of light, introduced at the scalp, to enable the noninvasive measurement of changes in the relative ratios of deoxygenated hemoglobin and oxygenated hemoglobin during brain activity. A wireless NIRS system consists of personal digital assistant software controlling the sensor circuitry, reading, saving, and sending the data via a wireless network. This technology allows the design of portable, safe, affordable, noninvasive, and minimally intrusive monitoring systems [

For such advanced features, NIRS signal processing really becomes an attractive field for computational science. Izzetoglu et al. investigated the canceling of motion artifact noise from NIRS signals by Wiener filter [

Although there are many computing analyses on NIRS biomedical signals, there is not yet any work mentioning the aspects of NIRS physics. This paper continuously explores physical aspects of NIRS following a paper mentioned about nonlinear characteristics [

Although there are a lot of papers on the multifractality of the biological signals, there are few studies that clarify the reason of multifractal and the relation between the multifractality and biological functions.

During the last decades, a number of authors have claimed not only correlations between memory span and mental speed but also with electrophysiological and hemoglobin variables of brain waves. In [

Self-affine functions are ones that are similar to themselves when transformed by anisotropic dilations. If

Fractal functions can possess multiaffine properties so that their roughness or the irregularity can fluctuate from point to point. Thus, the definition of the Hurst regularity becomes a local quantity of the velocity increment

The local the Hurst exponent

At any given point

In a signal with fractal features, an immediate question one faces is “how to quantify the fractal properties of such a signal?” The first problem is to find the set of locations of the singularities

WT is a space-scale analysis which consists in expanding signals in terms of wavelets which are constructed from a single function, the mother wavelet

The analyzing wavelet is usually well localized in both space and frequency. An interesting property of the wavelet transform is that the coefficients at these maxima are enough to encode the information contained in the signal. These maxima are defined, at each scale

The first possibility is that we find a single value

The term modulus maxima describes any point

This local maximum is a strict local maximum in either the right or the left neighborhood of

Let

The partition function

Thus, one can estimate

A linear

A novel definition is proposed in this paper called full-width the Hölder exponents that indicates maximum expansion of the Hölder exponents within spectrum

We used a multichannel NIRS instrument, OMM-3000, from Shimadzu Corporation, Japan, to acquire oxygenated hemoglobin and deoxygenated hemoglobin concentration changes. The system operated at three different wavelengths, 780 nm, 805 nm, and 830 nm, emitting an average power of 3 mW^{−2}. The illuminator and detector optodes were placed on the scalp. The detector optodes were fixed at a distance of 4 cm from the illuminator optodes. The optodes were arranged above the hemisphere on the subject’s head.

Near-infrared rays leave each illuminator, pass through the skull and the brain tissue of the cortex, and are received by the detector optodes. The photomultiplier cycles through all the illuminator-detector pairings to acquire data at every sampling period. The data were digitized by the 16-bit analog-to-digital converter. Because oxygenated and deoxygenated hemoglobin types have characteristic optical properties in the visible and near-infrared light range, the change in concentration of these molecules during neurovascular coupling can be measured using optical methods. By measuring absorption changes at two (or more) wavelengths, one of which is more sensitive to Oxy-Hb and the other to Deox-Hb, changes in the relative concentrations of these chromophores can be calculated. Using these principles, researchers have demonstrated that it is possible to assess brain activity through the intact skull in adult humans.

The NIRS instrument was capable of storing the raw signals for each of the channels, one of which consists of the intensity values of 3 wavelengths, and also the derived values of oxygenated hemoglobin [Ox-Hb], deoxygenated hemoglobin [Deox-Hb], and total hemoglobin [Total-Hb] = [Ox-Hb] + [Deox-Hb] concentration changes for all time points in an output file in a prespecified format.

In this work, we investigate an experiment brain response on imagery moving tasks. The stimulus is a computer screen with arrows indicating left turn or right turn. The subject is a normal 30-year-old man measured during 2 mins, with the sampling time of 25 ms. In terms of optode placement, there is currently no standardized placement scheme for NIRS measurements. With such a standardized placement of electroencephalography (EEG), we have proposed 2 positions number 8 and number 9 in primary motor cortex of Brodmann’s areas as shown in Figure

Measured positions based on international 10–20 system.

Experiment imagery moving tasks.

This section included illustrated results in three tests, testing monofractal of fractional Brownian motion (fBM) signals, detecting singularities throughout artificial signals, and detecting singularities of real-life NIRS signals.

Figure

(a) A fractional Brownian with the Hurst exponent

Data

Hölder exponent

Multifractal Spectrum

Figure

(a) Data testing singularities. (b) Local maxima line. (c) Partition functions. (d) Wavelet coefficients at the maximum scale. (e) Scaling exponents. (f) Multifractal spectrum.

Data

Local Maxima Lines-Singularity

Sum over local maxima

Coefficiens at maximal scale

Hölder exponent

Multifractal spectrum

The objective of this paper is detection of the singularities on NIRS time series and then finding the active periods of human brain. Figures

(a) Data of changes in concentrations of Oxy-Hb (b) Local maxima line (c) Partition functions (d) Wavelet coefficiens at the maximum scale (e) Scaling exponents (f) Multifractal spectrum.

Data

Local maxima lines-singularity

Sum over local maxima

Coefficients at maximal scale

Hölder exponent

Multifractal spectrum

(a) Data of changes in concentrations of DeOxy-Hb. (b) Local maxima line. (c) Partition functions. (d) Wavelet coefficients at the maximum scale. (e) Scaling exponents. (f) Multifractal spectrum.

Data

Local maxima lines-singularity

Sum over local maxima

Coefficients at maximal scale

Hölder exponent

Multifractal spectrum

Figure

Average full-width Hölder exponents during right-hand motor imagery.

Average full-width-Hölder exponents during left hand motor imagery.

The advantages of NIRS are well demonstrated in many recent reports, although quantification of the changes of NIRS responses is still being developed. In the present paper, we have focused mainly on detection of multifractal characteristics of NIRS time series to identify the active-state period of human brain. Multifractal parameters are regarded as a flexibility of human brain activities to understand brain activities. Different functional states of brain are probably governed by different degrees of multifractality. Further investigations into applications of NIRS signals could carry out meaningful contributions in medical and biological engineering.

The authors would like to acknowledge the Research Grant from the Vietnam National University in Ho Chi Minh City and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Research Grant no. 106.99-2010.11. Furthermore, They are deeply grateful for the support from our volunteers and friend.