1. Fractal Theory And Complexity

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Fractal is a wide growing concept that is used to quantify the complexity of a process. The process that is quantified by fractal theory can be time series or pattern. A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale (self-similar). The scaling rules are characterized by "scaling exponents" (dimension). "Simple" regular fractals have integer-scaling dimensions [1]. Complex self-similar objects have non-integer dimension. In fact, fractal is the measure of complexity of the process where its greater value stands for a more-complex process. Fractal dimension is the scaling exponent to quantify the complexity of process. It should be noted here that fractal dimension is a form of power law exponent.

During years, many works reported in literatures that employed fractal theory for their analysis. More works be satisfied with the analysis, and less works proceeded with the fractal modelling. The reported works on analysis of DNA [2-4], EEG signal [5-8], s-ABR signal [9], heart rate signal [10], human respiration time series [11,12] eye movement time series [13], human stride time series [14], human face pattern [15], spider brain signal [16,17] are noteworthy to be mentioned. In all reported studies, the variation of complexity of time series or pattern analyzed in different conditions. The fractal dimension can be computed using different methods that all of them are based on entropy concept [18].

2. Hurst Exponent, Approximate Entropy And Randomness

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Hurst exponent is another exponent that is defined under fractal theory. Hurst exponent that is the measure of randomness of signal, is used to quantify the memory concept of process. Hurst exponent can vary in the range of 0 to 1. The value of 0.5 indicates the Brownian motion that is complete chaotic condition. The system that has Hurst exponent closer to 0.5, contains greater level of randomness. For instance, a signal with the Hurst exponent of 0.6 is more random than a signal with the Hurst exponent of 0.8. There are very limited works in literature that employed Hurst exponent to discuss about the randomness and its memory feature. In [19], Namazi et al. showed that memory of EEG signal, which is quantified with the Hurst exponent, is correlated with the memory of applied music to subjects.

Approximate entropy is another exponent that can quantify the randomness of system. In general, the system that is less random, has smaller value of approximate entropy. Approximate entropy as the measure of system randomness also received less attention by researchers. There are limited works that employed approximate entropy to talk about randomness of bio-signals .

Since both Hurst exponent and approximate entropy defines the randomness of time series, therefore we can link them together. In other words, the time series that has greater randomness should have greater approximate entropy and also the Hurst exponent closer to 0.5.

3. Brain Memory, Randomness And Complexity

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Now, the question that rises here is that can we make a link between complexity, randomness and memory? The link between complexity and randomness can be made by relating the fractal dimension and the Hurst exponent. For a time series, the Hurst exponent has a direct relation with fractal dimension by H=2-F, where H and F stand for the Hurst exponent and fractal dimension respectively. On the other hand, in the last section we mentioned that the Hurst exponent and approximate entropy are linked together using randomness concept. Therefore, it can be said that we can correlate complexity, randomness and memory to each other. In order to this jobwe can use mathematics.

Since brain is a complex system that contains memory[20-22], we can investigate about the relation between memory, complexity and randomness in the brain. For this purpose, we can apply the mathematics to EEG signal. EEG signal as a random time series has complex structure. On the other hand, we can define the memory for EEG signal using the Hurst exponent. Therefore, we can say that by making the relation between the Hurst exponent, fractal dimension and approximate entropy, we make a link between EEG signal's memory, complexity and randomness. In further steps, we can link between EEG signal and human memories to see is there any direct relation between them. If so, the complexity, randomness and memory of the brain are linked to each other that is a great achievement in human neuroscience. In further steps, we can employ the mathematical equations [23] to develop a model for this relation. The developed model can be linked to other mathematical models in human body [24] to solve the big questions about the interacting between human brain and other organisms. On the hand, the developed model can be applied to other branch of engineering and science [25], where relating the memory and complexity of system is the important concept.