Journal: Frontiers in Systems Neuroscience

Article Title: Distinguishing cognitive state with multifractal complexity of hippocampal interspike interval sequences

doi: 10.3389/fnsys.2015.00130

Figure Lengend Snippet: Multifractal detrended fluctuation analysis . An illustration of the MFDFA method was created using one CA1 neuron recorded on two different days: one vehicle condition (blue) and one tetrahydrocannbinol (THC) condition (green). (A) The interspike interval sequence (ISI) of each neuron is shown for the first 2000 ISIs. Five seconds were subtracted from the entire THC ISI sequence for illustration purposes only. (B) The fluctuation function F is shown at scale 16 ( s = 16) for four different q-order statistical moments. Negative q-order statistical moments amplify small fluctuations, while positive moments amplify large fluctuations. Inset: F 2 ( v, s ) is the root mean-squared residual between the fit y v (black) of one segment s from the walk-like time series Y (blue). (C) The changes in variability across scales are indicated by variable slopes at different qth powers (integer q-values from −3 to 3). The q-order Hurst exponent H(q) is the slope of each regression line. Blue lines are from the vehicle recording and green lines are from the THC recording. Dots indicate individual values from each scale (19 scales ranging from 16 to 256). (D) Multifractal complexity is visualized with the multifractal singularity spectrum. The Hurst exponent is closely related to the h -value at the apex of the singularity spectrum (black data points). The width is obtained by subtracting h -values at each end of the spectrum (independent of D(h) values) indicated by the black arrow. The singularity spectrum for the vehicle condition is wider than the THC condition, indicating THC decreases multifractality.

Article Snippet: All analyses were performed in Matlab using publicly available MFDFA code (Ihlen, ).

Techniques: Sequencing

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The time series multifractal ( upper panel ), monofractal ( middle panel ), and whitenoise ( lower panel ) with zero average ( red dashed lines ) and ±1 RMS ( red solid lines ) . All time series have equal RMS = 1, but quite different structure. RMS is only sensitive to differences in the amplitude of variation and not differences in the structure of variation. Notice the different scaling for the vertical axis of the multifractal time series.

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The computation of local fluctuations, RMS{1} , around linear (A), quadratic (B), and cubic trends (C) by Matlab code ( m = 1 , m = 2 , and m = 3 , respectively) . The red dashed line is the fitted trend, fit{v} , within eight segments of sample size 1000. The distance between the red dashed trend and the solid red lines represents ±1 RMS{1} . The local fluctuation, RMS{1} , around trends is the basic “building block” of the detrended fluctuation analysis .

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques: Blocking Assay

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The plot of overall RMS (i.e., F in Matlab code ) versus the segment sample size (i.e., scale in Matlab code ) where both F and scale are represented in log-coordinates . The scale invariant relation is indicated by the slope, H , of the regression lines, RegLine , computed by Matlab code . The slope, H , is a power law exponent called the Hurst exponent because F and scale are represented in log-coordinates. Notice that both the monofractal and multifractal time series have more apparent slow fluctuations compared to whitenoise indicated by larger amplitudes of the overall RMS on the larger scales.

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The range of Hurst exponents defines a continuum of fractal structures between white noise ( H = 0.5) and Brown noise ( H = 1.5) . The pink noise H = 1 separates between the noises H < 1 that have more apparent fast evolving fluctuations and random walks H > 1 that have more apparent slow evolving fluctuations. The examples monofractal ( red trace ) and whitenoise ( turquoise trace ) used in the present tutorial are both noise like time series. The long-range dependent structure of most biomedical signals is located within the illustrated continuum of fractal structures.

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: q -order RMS Fq(nq,:) and corresponding regression line qRegLine{nq} computed by MFDFA (i.e., Matlab code and ) for time series multifractal (A), monofractal (B), and whitenoise (C) . (A) The scaling functions Fq ( blue dots ) and corresponding regression slopes Hq ( blue dashed lines ) are q -dependent. (B,C) The scaling functions Fq ( red and turqouish dots ) and regression slope Hq ( red and turqouish dashed lines ) are q -independent. (D) The q -order Hurst exponent Hq for the time series multifractal ( blue trace ), monofractal ( red trace ) and whitenoise ( turqouish trace ) where the colored dots represents the slopes Hq for q = −3, −1, 1, and 3 illustrated in (A–C) . Notice that the intercept of Hq for multifractal and monofractal time series [intercept between blue and red trace in (D) ] are close to q = 2. This intercept reflects the similarity between their overall RMS, F , in Figure .

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: Multiple representations of multifractal spectrum for multifractal ( blue traces ), monofractal ( red traces ), and whitenoise ( turquoise trace ) time series . (A) q -order Hurst exponent Hq computed in Matlab code . (B) Mass exponent tq computed in Matlab code . (C) Multifractal spectrum of Dq and hq ( upper right panels ) computed in Matlab code and plotted against each other. The arrow indicates the difference between the maximum and minimum hq that are called the multifractal spectrum width. Notice that the constant Hq for monofractal and whitenoise times series leads to a linear tq that further leads to a constant hq and Dq that, finally, are joined to become only tiny arcs in (C) .

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: (A) The multifractal , monofractal, and whitenoise time series (upper panel) and their local Hurst exponents Ht(:,5) computed by Matlab code (lower panel). The multifractal time series have a larger variation in the local Hurst exponents Ht(5,:) compared with the monofractal and whitenoise time series. The period with the local fluctuation of the smallest magnitude in multifractal time series contains the maximum Ht(5,:) (see Ht max in period between the black dashed lines ) whereas the period with the local fluctuation of the largest magnitudes contains the smallest Ht(5,:) (see Ht min in the period between red dashed lines ). (B) The probability distribution Ph of the local Hurst exponents Ht estimated as histograms by Matlab code for the multifractal , monofractal, and whitenoise time series. (C) The multifractal spectrum Dh estimated from distribution Ph by Matlab code for the same time series. The distribution Ph and spectrum Dh have a larger width for the multifractal time series compared to the monofractal and whitenoise time series.

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The time series multifractal (upper panel), monofractal (middle panel), and whitenoise (lower panel) are shown as blue traces . They are examples of noise like time series used in the present tutorial. All time series contain 8000 data samples each where the sample numbers are indicated by the horizontal axis. Matlab code converts the noises ( blue traces ) to random walks ( red traces ) that have a picture-in-picture similarity (subplot in the upper panel). Notice that the time series multifractal has distinct periods with small and large fluctuations in contrast to time series monofractal and whitenoise . The aim of this section is to introduce MFDFA that quantify the structure of fluctuations within the periods with small and large fluctuations.

Article Snippet: A possible solution suggested by Eke et al. ( ) is to run a monofractal DFA (i.e., Matlab code and ) before running MFDFA1 and MFDFA2 .

Techniques: Introduce

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: q -order RMS Fq(nq,:) and corresponding regression line qRegLine{nq} computed by MFDFA (i.e., Matlab code and ) for time series multifractal (A), monofractal (B), and whitenoise (C) . (A) The scaling functions Fq ( blue dots ) and corresponding regression slopes Hq ( blue dashed lines ) are q -dependent. (B,C) The scaling functions Fq ( red and turqouish dots ) and regression slope Hq ( red and turqouish dashed lines ) are q -independent. (D) The q -order Hurst exponent Hq for the time series multifractal ( blue trace ), monofractal ( red trace ) and whitenoise ( turqouish trace ) where the colored dots represents the slopes Hq for q = −3, −1, 1, and 3 illustrated in (A–C) . Notice that the intercept of Hq for multifractal and monofractal time series [intercept between blue and red trace in (D) ] are close to q = 2. This intercept reflects the similarity between their overall RMS, F , in Figure .

Article Snippet: The construction of the Matlab code for MFDFA is represented as Matlab code boxes within the text.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The time series multifractal (upper panel), monofractal (middle panel), and whitenoise (lower panel) are shown as blue traces . They are examples of noise like time series used in the present tutorial. All time series contain 8000 data samples each where the sample numbers are indicated by the horizontal axis. Matlab code converts the noises ( blue traces ) to random walks ( red traces ) that have a picture-in-picture similarity (subplot in the upper panel). Notice that the time series multifractal has distinct periods with small and large fluctuations in contrast to time series monofractal and whitenoise . The aim of this section is to introduce MFDFA that quantify the structure of fluctuations within the periods with small and large fluctuations.

Article Snippet: The construction of the Matlab code for MFDFA is represented as Matlab code boxes within the text.

Techniques: Introduce