Fractals EEG Motor Imagery | Quang N. Nguyen

Published at the 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC 2015) (Nguyen et al., 2015).

This project was carried out during a short-term cultural exchange program with the Tokyo University of Agriculture and Technology (TUAT), where I was a visiting student in the Biologically-inspired Computing Laboratory of Professor Toshiyuki Kondo.

Introduction

Conventional EEG analysis of motor imagery relies on spectral features: specifically, event-related desynchronization (ERD) and event-related synchronization (ERS), which track changes in band-limited power during imagined movement. These are well-established, linear measures that capture the oscillatory dynamics of the sensorimotor cortex. For the background on ERD/ERS and the motor imagery BCI paradigm, see the companion primer [Primer] Brain-Computer Interfaces Using EEG.

A parallel line of inquiry treats the EEG signal as a non-linear, complex system and asks what its geometric and dynamical properties reveal about brain state. Fractal dimension is one such measure: it quantifies the complexity and irregularity of a time series in a way that is orthogonal to spectral power. For the technical background on fractal dimension and Higuchi’s algorithm used here, see the companion primer [Primer] Fractal Dimension in EEG Signal Analysis.

This study asks a focused question: do the non-linear complexity characteristics of EEG signals, quantified by fractal dimension, covary with ERD during motor imagery? If so, how tightly do they track each other in time, and does this relationship generalize across subjects and datasets?

GAMMAAsys EEG active electrode system — **Figure 1:** The g.GAMMAAsys active electrode system, representative of high-density EEG acquisition hardware. The dataset used in this study was collected at the Institute for Biomedical Engineering, University of Technology Graz.

Methods

Dataset

The dataset was provided by the Department of Medical Informatics, Institute for Biomedical Engineering, University of Technology Graz, Austria, as part of the BCI Competition III (Blankertz et al., 2004). Three datasets were used:

O3VR: collected from a virtual reality experiment
S4b and X11b: collected using an adaptive basket paradigm

Virtual reality feedback paradigm — **Figure 2:** The virtual reality feedback paradigm used in the O3VR dataset. Three frames show the progression of a trial: fixation, cue onset, and feedback with cursor control.

Each dataset consisted of two bipolar EEG channels: C3 and C4, placed over the left and right sensorimotor cortices respectively.

EEG electrode placement — **Figure 3:** Bipolar electrode placement used in the Graz datasets. Two bipolar channels centered on C3 and C4 were derived from three electrode pairs on each side, 5 cm apart.

The experiment consisted of 3 sessions per subject, each with 9 runs of 40 feedback trials, for a total of 1080 trials per subject. Within each trial, a fixation cross was shown from 0 to 2 seconds, a motor imagery cue appeared at 2 seconds, feedback ran from approximately 4 to 7.5 seconds, and a pause period followed.

Experiment trial timeline — **Figure 4:** Trial structure. Motor imagery is cued at 2 seconds. The feedback period and classifier output run from approximately 4 to 7.5 seconds.

Raw EEG signal traces — **Figure 5:** Representative multi-channel EEG signal traces from a single trial, illustrating the broadband signal characteristics used as input to both the ERD/ERS and fractal dimension analyses.

ERD/ERS Quantification

ERD and ERS were quantified using the standard Pfurtscheller and Aranibar method (Pfurtscheller & Aranibar, 1979), expressed as a percentage power change relative to a pre-cue baseline interval. Positive values indicate ERS (power increase); negative values indicate ERD (power decrease):

\[\text{ERD/ERS}_j = \frac{P_j - R}{R} \times 100\%\]

where $P_j$ is the inter-trial average power at sample $j$, and $R$ is the mean baseline power over the pre-event reference interval. The method requires a reference period: the signal is first narrow-bandpass filtered (e.g., 5-10 Hz for the lower mu band), squared sample-by-sample to obtain instantaneous power, and averaged across trials before applying the formula. Two bands were analyzed: 5-10 Hz (lower mu band) and 20-25 Hz (beta band).

ERD/ERS quantification example — **Figure 6:** ERD/ERS time course (normalized power relative to baseline = 1.0). The blue shaded region shows ERD (power decrease below baseline); the red shaded region shows ERS (power rebound above baseline) after the motor event.

Fractal Dimension Estimation

Fractal dimension was estimated using Higuchi’s algorithm (Higuchi, 1988). For a time series $X(1), \ldots, X(N)$, define a family of sub-series at interval $k$ with initial offset $m$:

\[X_k^m = \left\{ X(m + ik) \right\}_{i=0}^{\lfloor(N-m)/k\rfloor}\]

The normalized length of each sub-series is:

\[L_m(k) = \frac{1}{k} \left[ \left( \sum_{i=1}^{\lfloor(N-m)/k\rfloor} \left| X(m+ik) - X(m+(i-1)k) \right| \right) \cdot \frac{N-1}{\lfloor(N-m)/k\rfloor \cdot k} \right]\]

Averaging $L_m(k)$ over all offsets $m = 1, \ldots, k$ yields $\langle L(k) \rangle$. For a fractal signal, this scales as a power law in $k$:

\[\langle L(k) \rangle \propto k^{-D}\]

The fractal dimension $D$ is the negative slope of the log-log regression of $\langle L(k) \rangle$ against $k$. Unlike ERD/ERS, the algorithm operates directly on the time-domain signal and requires no baseline period, making it an absolute measure of signal complexity in any given window. A sliding window was applied to produce a FD time course directly comparable to the ERD/ERS spectral-temporal map.

Comparing the Two Methods

Both ERD/ERS and Higuchi’s FD are feature extraction methods targeting the same neural event, but they approach it from fundamentally different angles:

Property	ERD/ERS	Higuchi's FD
What it measures	Relative change in band-limited power	Complexity / irregularity of the time-domain signal
Signal model	Linear (spectral power)	Non-linear (scaling law)
Frequency specificity	Band-specific: requires bandpass filtering first	Broadband aggregate; no frequency transform needed
Baseline dependence	Mandatory: relative to a reference period $$R$$	None: $$D$$ is absolute within any window
Sensitivity	Oscillatory synchronization in a defined band	Signal structure across all timescales
Real-time suitability	Moderate: FFT or filtering pipeline required	High: operates on raw samples, no spectral transform

Table 1: Comparison of ERD/ERS and Higuchi's fractal dimension as EEG feature extraction methods.

A practical consequence of these differences is that ERD/ERS is well-suited to verifying that a subject is correctly performing the motor imagery task, since power suppression in the mu and beta bands is the canonical marker of sensorimotor cortical engagement. Higuchi’s FD, on the other hand, does not require a known baseline, generalizes across frequency bands, and tends to capture hidden dynamical structure that linear power metrics can miss. This makes it attractive as an input feature for BCI classifiers, where non-linear complexity measures can complement spectral features and improve robustness.

One further advantage of HFD that is particularly relevant for real-time BCI applications is computational cost. HFD operates entirely in the time domain: the algorithm computes absolute differences between samples and sums them, followed by a single linear regression on a log-log plot. These are elementary arithmetic operations, and the overall complexity scales as $O(N)$ in the number of samples $N$ per window.

ERD/ERS quantification requires first isolating a frequency band, either by computing a Fast Fourier Transform ($O(N \log N)$) or by applying a digital bandpass filter (FIR or IIR). Filtering involves repeated convolutions, which are substantially heavier than the additions and subtractions in HFD. This difference compounds in a real-time BCI loop where features must be recomputed every few hundred milliseconds across multiple channels simultaneously.

In short, HFD extracts a meaningful neural feature with less compute and no spectral transform, which is why it is a natural candidate for low-latency, embedded, or resource-constrained BCI systems.

Results

The main result is shown below: for all six channels across the three datasets, the global minimum in FD over time corresponds to the peak ERD event in the spectral-temporal map.

Time-frequency maps and fractal dimension — **Figure 7:** Time-frequency maps (top row) and fractal dimension time courses for the 5-10 Hz band (middle row) and 20-25 Hz band (bottom row), for all six channel-dataset combinations (O3VR-C3, O3VR-C4, S4b-C3, S4b-C4, X11b-C3, X11b-C4). Blue areas in the spectral map indicate ERD; red areas indicate ERS. In every case, the global minimum of FD aligns with the ERD peak.

The correspondence between the ERD peak (maximum power suppression) and the FD minimum (lowest complexity) held consistently across datasets:

Subject	Channel	Hand Imagery	ERD latency (s)	FD latency (s)
O3VR	C3	Left	4.92	4.21
	C3	Right	4.63	3.85
	C4	Left	4.59	3.97
	C4	Right	5.50	4.33
S4b	C3	Left	6.26	5.65
	C3	Right	6.87	5.65
	C4	Left	6.21	5.65
	C4	Right	6.28	5.89
X11b	C3	Left	6.64	6.85
	C3	Right	7.43	7.81
	C4	Left	7.63	6.73
	C4	Right	6.78	5.65
Mean			6.145	5.52
SD			1.03	1.24

Table 2: Latency of minimum ERD and FD in 5-10 Hz band across all channels and tasks.

The alignment is consistent with the interpretation that both measures track the same underlying neural event: the onset and peak of motor cortical engagement during motor imagery. The FD minimum corresponds to a moment of reduced signal complexity, coinciding with the maximal suppression of the mu rhythm.

Conclusion

This study establishes a consistent relationship between ERD patterns and the fractal complexity of EEG during motor imagery:

The global minimum in fractal dimension aligns with the ERD peak across all datasets and channels examined. This suggests that the two measures are tracking the same underlying neural transition.
When the brain enters an active motor imagery state, EEG signals become less random and more structured in a particular way, reflected as decreased fractal dimension. This is consistent with the interpretation that motor cortical activity during imagery is not random but organized around specific neural dynamics.
Low FD values during motor imagery may serve as a strong indicator for the occurrence of motor imagery, offering a non-linear complement to spectral ERD features for BCI classification.

A limitation of this work is that the relationship between fractal dimension and ERS, the post-movement power rebound, was not fully characterized. The FD time course during the ERS period showed less consistent behavior, and the mechanisms governing FD during cortical synchronization remain an open question.

These findings motivate further use of non-linear complexity measures alongside conventional spectral features in motor imagery BCI systems, where they may provide complementary information that improves classification robustness.