Human brain response is the result of the overall ability of the brain in analyzing different internal and external stimuli and thus making the proper decisions. During the last decades scientists have discovered more about this phenomenon and proposed some models based on computational, biological, or neuropsychological methods. Despite some advances in studies related to this area of the brain research, there were fewer efforts which have been done on the mathematical modeling of the human brain response to external stimuli. This research is devoted to the modeling and prediction of the human EEG signal, as an alert state of overall human brain activity monitoring, upon receiving external stimuli, based on fractional diffusion equations. The results of this modeling show very good agreement with the real human EEG signal and thus this model can be used for many types of applications such as prediction of seizure onset in patient with epilepsy.
Brain as the most complex organ in the human body controls all bodies’ actions/reactions by receiving different stimuli through the nervous system. Any stimulus stronger than the threshold stimulus is translated by the number of sensory neurons generating information about the stimulus and the frequency of the action potentials. After the action potential has been generated, it travels through the neural network to the brain. In various sections of the network and the brain, integration of the signals takes place. Different areas of the brain respond depending on the kind and location of stimuli. The brain sends out signals which generate the response mechanism.
During many years, numerous studies related to the brain response to external stimuli have been reported by scientists. Some researchers studied the brain response to different kinds of stimuli without proposing any model. In case of visual stimuli, we can mention the work done by Kaneoke et al. in analyzing the effect of the visual stimulus size on the human brain response using magnetoencephalography (MEG) [
On the other hand, some scientists proposed some models of the human brain activity. On the microscopic level, the work done by Freeman in the modeling of the EEG arising from the olfactory bulb of animals during the perception of odors is noteworthy. He developed a set of nonlinear equations for this response which generates EEG like pattern [
Despite rapid advances in the studies related to the analysis of the human brain response, there has been less progress in the mathematical modeling of the human brain activity due to external stimuli. Yet, it seems that the contemporary level of developments in physics and mathematics makes establishing quantitative correlations between external physical stimuli and the brain responses to those stimuli.
This paper attempts to introduce a new mathematical model of the human brain response to external stimuli based on the fractional diffusion equation. At first we talk about the macroscopic level of brain organization and EEG signal as an alert state of human behavior monitoring. Then, by introducing fractional diffusion equations and considering the EEG signal as a fractal time series we model the EEG signal using mathematical equations. This model is then solved and after discussing different parameters in the model we provide some results and discuss about the model’s solution in details. Some concluding remarks are provided at the end.
In order to study the human neural activity, one can consider different levels of brain organization at different scales in time and space from a single neuron (microscopic level) to the whole brain organization (macroscopic level). In fact, the whole brain activity cannot be observed by measuring the activity of just a single neuron as far it is informative as it contributes to study the entire population of which it is a member.
In this research, we focus on the macroscopic level of brain organization. Macroscopic level of brain organization refers to the level of neural assemblies’ population in which each neural assembly interacts with other neural assemblies in close and distant cortical areas, exhibits spatialtemporal behaviour, and paints the human behaviour [
During many years scientists have studied the human behavior by recording and analysis of the EEG signals from different areas of the brain. The EEG signal is the composition of different frequency bands (oscillatory activitiesAlpha, Beta, …) which are structured coordinately (spatially, temporally). In fact, this signal has different characteristics that can be used in order to study the human brain response to external stimuli in our research. For instance, Figure
EEG signal of a subject in response to an external stimulus.
Here, in order to develop our model we start with a simple equation:
Equation (
On the other hand, the diffusion process can also been studied for fractals in fractal space. Fractals (such as random walk) may be defined as selfsimilar geometric objects whose scaling exponent (dimension) satisfies the Szpilrajn inequality:
Considering the diffusion process in the case of fractals, (
In (
It is now necessary to make a very important remark. It is possible to consider a Brownian motion type process as a process which takes place in an Euclidean space (see the right hand side of (
Taking all these conjectures into account, (
In this section, it is aimed to show that EEG signal as a fractal time series (but not stochastic), which represents a transient record of a random walk process, can be modeled by the solution of the fractional diffusion equation:
The direction of deflection at each moment in the signal can be studied by computing the Hurst exponent (Figure
Upward and downward deflection in the signal from one point to the next point.
As it was mentioned before the Hurst exponent can have any value between 0 and 1, where the value that it gains in each moment determines the behavior of the next deflection in the signal.
Firstly, if the Hurst exponent has a value between 0 and 0.5, it means that the process is antipersistent; that is, the trend of the value of the process at the next instant will be opposite to the trend in the previous instant. Secondly, a value of
One of the interesting points about the fractional diffusion model (
In this research we consider the brain in an informational space where
It is noteworthy that informational distance is not same as the concept of spatial distance. Two neurons, which are close together, may not exchange information which means
The exchange of information between two neurons in the brain happens when there is a potential difference along the channel. In fact, the potential difference between two neurons causes an information flux and the exchange of information from the neuron with higher informational potential value to the one with the lower value continues till the gradient becomes zero.
The fractional diffusion equation is valid for the time scale:
In order to solve the above fractional diffusion model the method proposed by Oldham and Spanier is employed here [
Upon introducing the excess value
In order to solve (
Equation (
Using the definition of fractional derivative [
Equation (
Since
An external influence can be modeled by a Gaussian pulse; that is,
In case of different types of stimuli, depending on the size and duration, the parameters in (
It is noteworthy that (
In the next section we provide a formulation for computing the effective diffusion coefficient at each moment that can be used in (
In order to analyze the behavior of fractal time series, the value and the direction of each fluctuation in the signal should be analyzed. The direction of each fluctuation can be found by computing the value of the Hurst exponent at the previous point of the signal.
In order to know about the value of the signal, we should know about different parameters which are appeared in the fractional diffusion model. By looking at (
The value of the Hurst exponent which is used in (
In order to make a relation between
In (
Considering the phaselagging model of action potential in the whole brain scale then, this model and the fractional diffusion model explain the same phenomenon. Thus, the mathematical equations that belong to these two models can be equal.
By writing this equality,
Considering
Considering a constant value for
Dependence of the dimensionless diffusivity,
Dependence of the dimensionless diffusivity,
As it can be seen in Figures
By considering (
So, in cases of Figures
As
Thus, by computing the value of
In this section using the fractional diffusion model the human brain response to a visual external stimulus is modeled and compared with the real EEG signal.
In this research the experiments were carried out on 6 voluntary healthy students (21–24 years old, 3 males and 3 females). Prior to the experiment, each subject was examined and interviewed by a physician to ensure that no neurological deficit, pain condition, or medication affects the EEG. All subjects were right handed. Informed consent was obtained from each subject after the nature of the study was fully explained.
All procedures were approved by the Internal Review Board of the University and the approval for experimentation involving human subjects was issued. It is noteworthy that the identity of all subjects remains confidential.
The EEG data used in this research were collected using Mindset 24 device, a 24channel topographic neuromapping instrument, which can measure 24 channels of data with the sampling frequency of 256 Hz.
In an electrically shielded, acoustically isolated, and dimly illuminated room a visual stimulus applied on subjects. It should be mentioned that it is endeavoured to insulate the subjects from all other external stimuli. This ensures that the response measured in the EEG signals is primarily due to the stimulus applied. In the experiment subjects were watching a checkerboard pattern (see Figure
Checkerboard pattern.
The stimulus was the checker reversal. After one second the reversed pattern (see Figure
The reversed pattern as the visual stimulus.
Mindmeld 24 software is used for the collection of data using Mindset 24 machine. The software gives data in the form of .bin files which can be processed to give text files (.txt) that are required for further processing.
Although EEG data are recorded from 24 electrodes, in this research the analysis is done on the data governed from the left occipital (O1) electrode (near to the location of the visual primary sensory area). This electrode was chosen based on the nearest place to the visual sensory area which shows the strongest response that can be seen in the signal recorded from this electrode compared to other electrodes. The electrode impedance was kept lower than 5 KΩ.
A bipolar electrooculogram (EOG, vertical and horizontal) was recorded for offline artifact rejection. After bandpass filtering in the range of 0.1–70 Hz, 2 seconds of data (256 data before stimulation and 256 data after stimulation) was saved. It means that there are 256 values of voltage collected every second. Prestimulation is defined as the status before the application of the stimulus. On the other hand, poststimulation is defined as the status after the application of the stimulus. As it was mentioned previously the stimulus was applied at
It is noteworthy that choosing higher sampling rate will result in more pre and poststimulation data. Having more data will results in more precise computation of Hurst Exponent, diffusion coefficient, and accordingly the signal and its parameters such as response initiation time.
In the first week 40 trials were collected from each subject in one day. The data collections were repeated after a week for each subject in order to examine the reproducibility of the results from experiments. By repeating the experiments in the second week totally 80 trials were collected. After visual inspection of data collected from each subject and rejection of trials with artifacts, 40 trials free of artifacts were selected for future analysis. It is noteworthy mentioning that physician monitored the subjects during all experiments.
A set of codes was written in MATLAB software in order to compute all required parameters which were discussed before.
As the recorded data were noisy, the EEG signals were filtered using the Wavelet toolbox in MATLAB and then were processed by the methodology discussed in this section.
The value of
It is required to compute the Hurst exponent value in each moment in order to analyze the generated signal. At the first step, the program computes the Hurst exponent for the recorded EEG signal and the predicted signal and generates two time series in one figure.
In each moment, the program computes
The values of some required parameters are listed in Table
Values of required parameters.
Variable  Value  Units 



m^{2}/s 

1 


0.002  s 

0.001  s 
As it was mentioned previously, one second of EEG data was recorded, and using these data as the reference, one second of EEG time series is predicted. After that, the modeled signal is analyzed in terms of the initiation time for the response, the response duration, and the peak to peak voltage. In this research in order to see the response fluctuations clearly, the real or the modelled signals after the poststimulation are averaged in each case. It means that for each subject 40 selected trials are used as the input to the model and, accordingly, the grand average of all modeled signals (40 signals for each subject) is presented. Then, the generated plots for the real or the predicted signals are the result of averaging over all selected trails in each case.
Here it should be mentioned that in the analyses of the EEG signals plots the initiation time for the response and the duration are chosen based on the literature notes which consider the major positive or negative pole. For instance, the fluctuations which have voltage in the range of 5 to 10
The grand average of the recorded EEG signals and the grand average of the predicted signals using the fractional diffusion model over all selected trials for 1 second poststimulation are shown in Figure
The grand average of the recorded EEG signals (black solid line) and the grand average of the predicted signals (red dashed line) for 1 second after stimulation in the case of the visual stimulus, subject 1 to subject 6.
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
Subject 6
As it can be seen for different subjects in Figure
As an example, for subject 1 in cases of the real and predicted signals the response to the stimulus starts with a positive peak (
The values of peak to peak voltage, the initiation time for the response (the initial peak of the response), and the response duration (the time difference between the first and the last peaks within the response duration) for the recorded EEG signal and the predicted signal in case of different subjects are provided in Table
Comparison between the real and the predicted signals.
Subject  The initiation time for the response (ms)  Response duration (s) 
Peak to peak voltage (  

Real  Predicted  Real  Predicted  Real  Predicted  
1  P118  P127  0.052  0.047  13.97  13.07 
2  P120  P120  0.054  0.067  12.86  13.75 
3  P110  P120  0.058  0.060  15.24  15.24 
4  P100  P120  0.078  0.057  16.28  15.50 
5  P112  P121  0.043  0.041  12.50  11.81 
6  P120  P130  0.050  0.044  13.85  15.50 
Average 






As it can be seen in Table
The grand average of the Hurst exponent variations for the recorded EEG signals (black solid line) and the grand average of the Hurst exponent variations for the predicted signals (red dashed line) for 1 second after stimulation in the case of the visual stimulus, subject 1 to subject 6.
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
Subject 6
The high correlation between the values of the real signals and also the predicted signals can be realized by looking at the values of the Hurst exponents. For instance, in the case of subject 1, the value of the Hurst exponent is distributed between 0.900 and 0.943 for the recorded EEG signal (black solid line) and between 0.900 and 0.952 for the predicted signals (red dashed line). Thus, the low uncertainty of the prediction can be confirmed, and it can be said the signal is predicted well, because the Hurst exponent values are not close to
Also, as it can be seen in the Hurst exponent plot, the value of the Hurst exponent in the case of the real EEG signal and the predicted signal experiences a sudden upward deflection. For instance, in the case of subject 1, the value of the Hurst exponent is decreasing in the time span of
By analyzing the behaviors which have been seen in Figures
All the analyses which have been done in this research show that EEG signals can be modeled by the solution of fractional partial differential equations and, thus, the behavior of system modeled by means of such equations can, in principle, not only be predicted but also quantifies.
In this paper we introduced a new mathematical model which quantifies the human brain response to external stimuli. We developed this model by applying the fractional diffusion equation to human EEG signals. The model generates a multifractal time series which shows a quantitative concurrence with the real EEG signals. Using this model we successfully predicted the EEG signals of different subjects upon receiving a visual stimulus. This model shall be further applied in case of different external stimuli where the results can be verified against the real EEG signal which means the prediction of the human behavior by forecasting the EEG signal. On the other hand this model also can be employed in order to predict different abnormal brain activities, such as epileptic seizure, by at least some seconds before the time of occurrence. If so, a seizure warning and the expected time of this epilepsy occurrence can be generated, leading to the future monitoring of this disease.
The authors declare that there is no conflict of interests regarding the publication of this paper.