The provided code implements algorithms for estimating the Hurst exponent and performing Detrended Fluctuation Analysis (DFA) on time series data. These methods are used to analyze the fractal properties and long-range correlations of data, particularly in the context of biological signals. ### Biological Basis 1. **Fractality and Long-Range Correlations in Neural Activity:** - Neuronal systems exhibit complex dynamics that are often non-linear and display long-range correlations over time. These properties manifest in signals as fractal-like patterns. In neuroscience, the Hurst exponent and DFA are commonly used to quantify these characteristics, providing insights into the self-similarity and persistence of fluctuations within neural signals, such as brain activity recorded via EEG or fMRI. 2. **Hurst Exponent:** - The Hurst exponent (`H`) is a measure of the long-term memory of time series data. In the context of neural signals, a Hurst exponent greater than 0.5 indicates persistent behavior (trends to continue in the same direction), while an exponent less than 0.5 suggests anti-persistence (tends to revert to the mean). A value of 0.5 indicates a Brownian motion or random walk. 3. **Detrended Fluctuation Analysis (DFA):** - DFA is a method to assess the fractal scaling properties of a time series. It detrends non-stationary data by segmenting the data into equal bins, subtracting the local trend in each bin, and analyzing how the remaining fluctuations scale with increasing bin size. This approach helps characterize the statistical self-affinity of neural signals, distinguishing between intrinsic noise and meaningful patterns. 4. **Applications in Neuroscience:** - By applying these analytical techniques, researchers can investigate dynamic features across different states of brain function, such as sleep, anesthesia, attention, and neurological disorders. Observing alterations in fractal properties might help in identifying pathological conditions like epilepsy, Alzheimer's disease, or other neurodegenerative diseases by differentiating between normal and abnormal brain states. 5. **Surrogate Data Analysis:** - The code includes surrogate data analysis using randomly shuffled time series. This method creates confidence intervals for the computed Hurst and DFA values by comparing them against randomly generated noise, allowing differentiation between structured biological phenomena and random signals. The code focuses on assessing how neuronal time series exhibit fractal characteristics, a feature significantly relevant to understanding the complex, scale-free nature of biological systems. These analyses provide crucial insights into how the brain processes information over various temporal and spatial scales, reflecting its adaptability and robustness.