Chaotic theories, tools, and the concept of information can be used to understand the dynamics of the brain and develop new methods for the diagnosis of brain pathologies. Different processes in the universe are carried out due to coexistence co-ordination between complementary and yet opposite actions. In this regard, one can refer to a series of biological activities and processes within the body for the regulation of heart rate that accelerates the sympathetic nerves and decelerate parasympathetic nerves [1,2]. The Mandala pattern used in this study is an archetype used in many religious traditions (Figure 1) [3].
According to the global definition, ASD is characterized by problems and challenges in social interaction, communication, and the tendency to perform repetitive behaviors [4].
Complementary charts depict a pattern in all NC subjects during the heartbeat intervals (Figure 2) and large numbers of cardiac and psychiatric patients. No Mandala pattern was observed in the heart rate record of patients with severe cardiac dysfunction or some psychiatric patients regardless of treatment with antipsychotic medicine [5,6].
 

Figure 1. Mandala archetypes: Aztec sundial (top). The wheel in the Indian flag (left) is originated from Buddhism and Gandhi’s Spinning wheel. Chartres Cathedral Labyrinth (right)

Figure 2. There are six rings in the RRI series of a healthy person (top left) and the circle inside is empty. Mandala pattern is often shown for patients with coronary artery disease (top right); however, it is absent in patients with severe psychosis (down in the middle).

 
In the past few years, some studies have been carried out using the Mandala pattern on brain data including that conducted by Sadeghi Bajestani et al. (2017) in which two groups of control and ASD were used and the results were consistent with those of the present study. In this study, the Mandala pattern was used as the main strategy for distinguishing ASD from NC individuals, since this pattern was not formed in the NC group [7].
 
Materials and Methods
No filters have been used in the present study since based on the authors’ viewpoint, there is no standard method to eliminate undesirable items from the vital signal without jeopardizing the information and creativity. It is worth mentioning that part of the important information was lost due to the application of conventional filters [8,9,10]. The block diagram of this study is presented in Figure 3.
Recording Protocol in Research
The recording was carried out using a 10-20 system, a well-known international method. This system consists of several electrodes, the positions of which are shown in Figure 4.
In total, 10 individuals out of 20 participants in this study had ASD and the other 10 were included in the NC group. The mean ±SD age of participants was estimated at 8±5 years old. The recording was performed in an acoustic room equipped with a chair and a monitor to display cartoons for both groups.
Recording steps; content and time specifications; method of implementation of triple recording steps
In the first stage (or baseline), the child sat motionless on a chair for about one minute in a completely normal form. In the next stage, a cartoon was played for five minutes with sound. Afterward, the same cartoon was displayed again soundless. Recording for all NC children was performed similarly. In the stage when cartoons were displayed with sound, such questions as word completion and word guesses were raised for autistic children. In the third stage, the same images (including objects, animals, and fruits) were displayed without background sound.
 
Recording data tool
The recording device and software used in this study included FlexComp Infiniti Encoder and BioGraph Infiniti, respectively. The device (256 Hz, 12-bit ADC) had the capability to record and collect biological data from brain activity online and synchronously.
 
Opposite concept
Although the universality of duality has been widely recognized, there are divergent views as to the nature of the opposition. Oppositions change performance and hence produce information. Information is encoded in opposition, examples of which include positive and negative states of computer components, true and false in logic, action, and resting potential in neurons [11]. The computation of sine and cosine of each term in a time series allows one to explore the opposite components of a process when only a one-time series is available. The interaction of opposites plays a fundamental role in natural and human processes. The sine and cosine of each term in the series were calculated to investigate the opposition. As a result, a single time series was decomposed into two complementary surrogate series (sine and cosine transform). Sine and cosine are an example of complementary opposites, they wax and wane out of phase (displayed by pi/2). It should be noted that such rising and falling do not occur independently. Sine and cosine transform lead to the provision of a two-dimensional framework to project a time series and examine the relationship between the opposite components. Trigonometric transformations were introduced as a strategy to develop numerical analyses and plots [12].
 
Trigonometric model
Trigonometric analysis decomposes a complex series into two orthogonal and opposite components. Trigonometric transformations are widely used in time series analysis. Fourier analysis decomposes a complex time series into a (potentially infinite) series of sine and cosine components. The trigonometric decomposition into sine and cosine components portrays informational oppositions. The relation between sine and cosine functions is considered as a mathematical model for complementary opposition since they include reciprocity and orthogonal relations [12]. Linear (diagonal), orthogonal (two linear orthogonal opposite), and partial (dotted lines in each semicircle) opposites are presented in Figure 5.
Complementary plots
Complementary plots are generated by plotting the cosine and sine transforms in X and Y axes as well as drawing a straight line between successive points to represent transitions. In the complementary plot, the value of each term in the time series corresponds to a point in a circumference (Figure 6). The vertical axis represents the sine scale while the horizontal axis denotes the cosine scale, with zero in both scales corresponding to the center of circumference [12]. The Mandala pattern was achieved from the successive transitions between the sinus and cosine. The evaluation of brain data between the NC and ASD groups was carried out based on this pattern.

 
 Table 1. Values of four features for NC and ASD groups in the first stage of recording on both channels

 
 Table 2. Values of the four features for the NC and the ASD groups in the second recording stage on both channels

 Table 3. Values of the four features for the NC and the ASD groups in the third recording stage on both channels

 

  Table 4. Maximum values obtained in Tables 1-3

  ¹ Number (feature)
 
  Table 5. A summary of t-test results in three stages of recording

 

Conclusions                                        
The EEG signal appears to be composed of delayed potentials, which are classified into excitatory and inhibitory categories [13]. ASD is a software disorder (in the interaction of various brain regions) that results in secondary hardware problems [14]. The results indicate that all features in the ASD group were maximal compared to the NC group. According to the statistical analysis of these features, the PDC was introduced as an optimal feature to differentiate NC from ASD groups. In general, these results indicate that there are more periodic episodes in the brain signal of autistic children and that the brain does not have the normal symmetry. In other words, the excitatory and inhibitory imbalance is the origin of many abnormalities in people with autism [14].
 
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no conflict of interest regarding the publication of the present study.
 
Ethical approval
All procedures performed in studies involving human participants were conducted base on the ethical standards of the respective institutions and/or national research committee and in accordance with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.
 
Informed consent
Informed consent was obtained from all participants in the present study.
 
References