research article fractional diffusion based modelling and...

Research ArticleFractional Diffusion Based Modelling and Prediction ofHuman Brain Response to External Stimuli

Hamidreza Namazi1 and Vladimir V Kulish2

1Department of Mechanical Engineering Faculty of Engineering Universiti Malaysia Sarawak 94300 Kuching Sarawak Malaysia2School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore 639798

Correspondence should be addressed to Hamidreza Namazi nhamidrezafengunimasmy

Received 2 November 2014 Accepted 17 January 2015

Academic Editor Shahram Shirani

Copyright copy 2015 H Namazi and V V Kulish This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Human brain response is the result of the overall ability of the brain in analyzing different internal and external stimuli and thusmaking the proper decisions During the last decades scientists have discovered more about this phenomenon and proposed somemodels based on computational biological or neuropsychological methods Despite some advances in studies related to this areaof the brain research there were fewer efforts which have been done on the mathematical modeling of the human brain response toexternal stimuliThis research is devoted to themodeling and prediction of the human EEG signal as an alert state of overall humanbrain activity monitoring upon receiving external stimuli based on fractional diffusion equations The results of this modelingshow very good agreement with the real human EEG signal and thus this model can be used for many types of applications such asprediction of seizure onset in patient with epilepsy

1 Introduction

Brain as the most complex organ in the human body controlsall bodiesrsquo actionsreactions by receiving different stimulithrough the nervous system Any stimulus stronger than thethreshold stimulus is translated by the number of sensoryneurons generating information about the stimulus and thefrequency of the action potentials After the action potentialhas been generated it travels through the neural network tothe brain In various sections of the network and the brainintegration of the signals takes place Different areas of thebrain respond depending on the kind and location of stimuliThe brain sends out signals which generate the responsemechanism

Duringmany years numerous studies related to the brainresponse to external stimuli have been reported by scientistsSome researchers studied the brain response to differentkinds of stimuli without proposing any model In case ofvisual stimuli we can mention the work done by Kaneokeet al in analyzing the effect of the visual stimulus size onthe human brain response using magnetoencephalography(MEG) [1] see also [2 3] Other groups of researchers

investigated the effect of auditory stimuli on the brainresponse For instance Will and Berg studied and comparedthe brain responses to periodic stimulations silence andrandom noise using electroencephalography (EEG) [4] seealso [5 6] Olfactory stimuli also were the main focus ofsome researchers Sutani et al investigated the brain responseto pleasant and unpleasant olfactory stimuli using MEGsignals They found out that the MEG signals have recordedfrom frontalprefrontal cortical areas of the brain has somedifferences in case of pleasant versus unpleasant stimuli [7]see also [8 9] Different works have been reported on theinvestigation of the brain response to other kinds of stimulussuch as emotional stimuli [10 11] and pain stimuli [12 13]

On the other hand some scientists proposed somemodels of the human brain activity On the microscopiclevel the work done by Freeman in the modeling of theEEG arising from the olfactory bulb of animals during theperception of odors is noteworthy He developed a set ofnonlinear equations for this response which generates EEGlike pattern [14 15] In another work Seetharaman et al pro-posed a mathematical model for generation and propagationof action potential in a node of Ranvier and they called it as

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 148534 11 pageshttpdxdoiorg1011552015148534

2 Computational and Mathematical Methods in Medicine

the phase-lagging model of single action potential [16]When the microscopic models are extended to a macrolevelthen different methods are employed Many of these modelsassume the cortical region to be a continuum Liley et aldeveloped a set of nonlinear continuumfield equationswhichdescribed the macroscopic dynamics of neural activity in thecortical region [17] These equations were used by Steyn-Ross et al who introduced noise terms into them to give aset of stochastic partial differential equations (SPDEs) Theyalso converted the equations governed by Liley et al intolinearized ODEs This model could predict the substantialincrease in low frequency power at the critical points ofinduction and emergence They later used this model tostudy the electrical activity of an anaesthetized cortex [18ndash21] Kramer et al started with the equations given by Steyn-Ross and coworkers and neglected the spatial variation andthe stochastic input They believed that this gave rise to a setof ordinary differential equations (ODEs) for the modelingof the cortical activity They showed that the results obtainedfrom the SPDE model agree with clinical data in an approx-imate way [22] but they also stated that the spatial samplingof the cortex was poor because of inherent shortcomingsin the equipment used Kulish and Chan have suggested anovel method for the modeling of the brain response usingfundamental laws of nature like energy conservation and theleast action principle The model equation obtained has beensolved and the results show a good agreement with real EEGs[23]

Despite rapid advances in the studies related to theanalysis of the human brain response there has been lessprogress in the mathematical modeling of the human brainactivity due to external stimuli Yet it seems that the con-temporary level of developments in physics and mathematicsmakes establishing quantitative correlations between externalphysical stimuli and the brain responses to those stimuli

This paper attempts to introduce a new mathematicalmodel of the human brain response to external stimuli basedon the fractional diffusion equation At first we talk aboutthe macroscopic level of brain organization and EEG signalas an alert state of human behavior monitoring Then byintroducing fractional diffusion equations and consideringthe EEG signal as a fractal time series we model the EEGsignal using mathematical equations This model is thensolved and after discussing different parameters in the modelwe provide some results and discuss about the modelrsquossolution in details Some concluding remarks are provided atthe end

2 Macroscopic Level of Brain Organizationand the EEG Signal

In order to study the human neural activity one can considerdifferent levels of brain organization at different scales intime and space from a single neuron (microscopic level) tothe whole brain organization (macroscopic level) In fact thewhole brain activity cannot be observed by measuring theactivity of just a single neuron as far it is informative as itcontributes to study the entire population of which it is amember

40

30

20

10

0

minus10

minus20

minus301 11 12 13 14 15 16 17 18 19 2

Time (s)

Volta

ge (120583

V)

Figure 1 EEG signal of a subject in response to an external stimulus

In this research we focus on the macroscopic level ofbrain organization Macroscopic level of brain organizationrefers to the level of neural assembliesrsquo population in whicheach neural assembly interacts with other neural assem-blies in close and distant cortical areas exhibits spatial-temporal behaviour and paints the human behaviour [24]The functional behaviour of the brain is encoded in thesespatial-temporal structures and can be extracted from themacroscopic quantities dynamics observed by EEG signalmostly [25]

During many years scientists have studied the humanbehavior by recording and analysis of the EEG signals fromdifferent areas of the brain The EEG signal is the com-position of different frequency bands (oscillatory activities-Alpha Beta ) which are structured coordinately (spatiallytemporally) In fact this signal has different characteristicsthat can be used in order to study the human brain responseto external stimuli in our research For instance Figure 1shows the EEG signal of a subject who received an externalstimulus at 119905 = 1 s In this figure the brain response to externalstimulus can be seen as a sudden upward deflection at about119905 = 125 s after the application of stimulus at 119905 = 1 s

3 Fractional Diffusion Equation

Here in order to develop our model we start with a simpleequation

120597119865

120597119905

= 119863nabla

2119865

(1)

where

nabla

2119865 =

120597

2119865

120597119909

2

(2)

Equation (1) is the well-known diffusion equation wherethe coefficient 119863 is the diffusivity of the medium for theproperty 119865 In fact this equation arises in many descriptionsof biological and physical phenomena including Brownianmotion [26] gradient driven chemical diffusion (with Fickrsquoslaw) and heat transfer (heat diffusion with Fourierrsquos Law) Itis noteworthy that (1) is written in the Euclidean space

On the other hand the diffusion process can also beenstudied for fractals in fractal space Fractals (such as randomwalk) may be defined as self-similar geometric objects whosescaling exponent (dimension) satisfies the Szpilrajn inequal-ity

alefsym ge 119863

119879 (3)

Computational and Mathematical Methods in Medicine 3

wherealefsym is the scaling exponent (dimension) of the object and119863

119879is its topological dimension that is Euclidean dimension

of units from which the fractal object is built In fact Fractaland Euclidean geometries are conjugate approaches to thegeometry of natural forms Fractal geometry builds complexobjects by applying simple processes to complex buildingblocks Euclidean geometry uses simpler building blocks butfrequently requires complex building processes

Considering the diffusion process in the case of fractals(1) is changed to (4) which is called the fractional diffusionequation [27]

120597

2119867119865

120597119905

2119867= 119862

2(2119867minus1)119863

2(1minus119867)nabla

2119865

(4)

where nabla2119865 = 120597

2119865120597119909

2In (4) 119862 is the speed of propagation and 119867 is the Hurst

exponent with the value within the range 0 le 119867 le 1 thatbrings predictability of signal into account In fact the Hurstexponent can be viewed as the probability of the diffusionprocess being persistent in a certain given directionNote thatthe case119867 = 12 which corresponds to a nonfractal diffusionprocess leads to the well-known classical equation Observealso that if119867 = 0 (4) degenerates into the Poisson equation120597

2119865120597119905

2= 119862

minus2119863

2nabla

2119865 that is there is no preferred direction

of randomwalk in this case while the case119867 = 1 leads to thewave equation 12059721198651205971199052 = 119862

2nabla

2119865

It is now necessary to make a very important remark Itis possible to consider a Brownian motion type process asa process which takes place in an Euclidean space (see theright hand side of (4)) considering the temporal dimension119905 of the process as fractal time which for the same diffusioncoefficient 119863 either slows down (in case of 119867 lt 12) orspeeds up (in case of 119867 gt 12) the process in question Thiscan be described with the generalized diffusion equation (4)inwhich the time coordinate appears as a fractal quantityThegeneralized diffusion equation is a fractional PDEof order 2119867with respect to time

Taking all these conjectures into account (4) is a general-ized form for describing the diffusion process when the timebecomes fractal

4 Fractional Diffusion Model of EEG Signal

In this section it is aimed to show that EEG signal as a fractaltime series (but not stochastic) which represents a transientrecord of a random walk process can be modeled by thesolution of the fractional diffusion equation

120597

2119867119881

120597119905

2119867= 119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119881

120597120578

2 (5)

where the term 119881 which stands for the brain response isthe voltage fluctuations resulting from ionic current flowswithin the neurons of the brain The impulse propagationspeed within the neural network which is a finite quantity isrepresented by 119862 The term 119863eff the effective diffusion coef-ficient as the property of neural tissue is related to neuronrsquosresistance to the electrical impulse as it travels over the nerve

Downward deflectionUpward deflection

Figure 2 Upward and downward deflection in the signal from onepoint to the next point

The term119867 as a time variable parameter corresponds to theHurst exponent that brings the predictability of signal intoaccount

The direction of deflection at each moment in the signalcan be studied by computing the Hurst exponent (Figure 2)The Hurst exponent is an indicator of the long term memoryof the process generating the signal and thus it is the measureof the predictability of the signal

As it was mentioned before the Hurst exponent can haveany value between 0 and 1 where the value that it gains ineach moment determines the behavior of the next deflectionin the signal

Firstly if the Hurst exponent has a value between 0 and05 it means that the process is antipersistent that is thetrend of the value of the process at the next instant will beopposite to the trend in the previous instant Secondly a valueof 119867 between 05 and 1 means that the process is persistentthat is the trend of the value of the process at the next instantwill be the same as the trend in the previous instant Finallyif 119867 = 05 the process is considered to be truly random(eg Brownian motion) It means that there is absolutely nocorrelation between any values of the process

One of the interesting points about the fractional dif-fusion model (5) is that it accounts for a finite time lag(reaction time) between any given disturbance (stimulus) andthe brain response to it (human actionreaction) based onthe assumption that no instantaneous propagation of infor-mation is possible within the brain This effect is consideredduring the derivation of this model but it is substituted by 119862119863 according to the equation

120591 =

119863eff119862

2 (6)

Equation (6) is the formula for relaxation time in hyperbolicreaction diffusion equations [28]

In this research we consider the brain in an informa-tional space where 120578 is the spatial dimension of this spaceIn fact each two neurons in informational space whichhave informational interlink make an informational channelbetween themselves and the length of this channel is calledthe informational distance 120578 where

0 le 120578 lt infin (7)

It is noteworthy that informational distance is not sameas the concept of spatial distance Two neurons which areclose together may not exchange information which means120578 sim infin On the other hand two neurons which are far apartmay be closely interlinked to each other and exchange a lot ofinformation which makes 120578 very small Thus 120578 has value inthe range 0 le 120578 le infin [29]


The exchange of information between two neurons in thebrain happens when there is a potential difference along thechannel In fact the potential difference between two neuronscauses an information flux and the exchange of informationfrom the neuron with higher informational potential valueto the one with the lower value continues till the gradientbecomes zero

The fractional diffusion equation is valid for the timescale

119905 ge 0 (8)

Also regarding (5)

119881 (120578 0) = 119881

0 (9)

is the initial condition at 119905 = 0 Also the solution of themodel is required to be bounded as 120578 rarr infin Otherwise theconservation of energy principle would be violated In otherwords

lim120578rarr+infin

119881 (120578 119905) = const lt infin (10)

is the boundary condition in the case of infinite domainIn order to solve the above fractional diffusion model the

method proposed by Oldham and Spanier is employed here[30]

Upon introducing the excess value 119881(120578 119905) = 119881(120578 119905) minus 119881

0

so the initial condition imposed on

119881 is 119881(120578 0) = 0In order to solve (5) we apply the Laplace transform with

respect to time 119905 then we have

119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119884

120597120578

2minus 119904

2119867119884 = 0 (11)

where 119904 is the Laplace transform variable and 119884 denotes theLaplace transform of the excess value 119881

Equation (11) is a second-order ordinary differentialequation (ODE) where the general solution is calculated as

119884 (120578 119904) = 119860

1(119904) 119890

minus120578119904119867[119862(2119867minus1)119863eff(1minus119867)]

+ 119860

2(119904) 119890

120578119904119867[119862(2119867minus1)119863eff(1minus119867)]

(12)

where1198601(119904) and119860

2(119904) are two arbitrary functions However

since the solution is to be bounded for all 120578 the secondarbitrary function 119860

2(119904) must be identically zero so the

solution (12) is changed to

119884 (120578 119904) = 119860

1(119904) 119890


(13)

Upon differentiating (13) with respect to 120578

119889119884 (120578 119904)

119889120578

= minus

119860

1(119904) 119904

119867119890


[119862

(2119867minus1)119863eff(1minus119867)

]

(14)

After comparing (13) and (14) 1198601(119904) can be eliminated then

it can be written as

119884 (120578 119904) = minus119904

minus119867119862

(2119867minus1)119863eff(1minus119867)

119889119884 (120578 119904)

119889120578

(15)

By taking the inverse Laplace transform of (15) and restoringthe original variables then we have

119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 120597

minus119867

120597119905

minus119867(

120597119881

120597120578

) (16)

which is written in terms of fractional derivative of order minus119867with respect to 119905

Using the definition of fractional derivative [30] namely

120597

120572119891

120597119905

120572=

1

Γ (minus120572)

int

119905

0

119891 (120585) 119889120585

(119905 minus 120585)

120572+1 Re (120572) lt 0 (17)

where Γ(120572) is the Gamma function and noticing the Fickrsquoslaw

minus

120597119881

120597120578

=

120593

119863eff (18)

where 120593 represents the flux and (16) can be written as

119881 (120578 119905) = 119881

0+ 119862

(2119867minus1)119863eff(minus119867) 1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (19)

Since the function 120593(120578 119905) represents the flux it can be equatedwith the external influence acting on the system (externalstimulus)

Equation (19) provides the relationship between thenonequilibrium value 119881(120578 119905) and the external influenceacting on the system 120593(120578 120585) This equation is valid for everylocationwithin the domain (including the boundary) at everymoment

Since 119867 is a nonnegative parameter it follows from (19)that the value of 119881 on the average increases with the timeaccording to the power law as 119905119867 provided of course thatthe fluctuations are small in comparison with the averagedinfluence Note that for 119867 = 12 Γ(12) = 120587

12 and (19)yields the well-known diffusion (random walk) growth givenby 11990512

An external influence can be modeled by a Gaussianpulse that is

120593 (119909 119905) = 120593

0(119909 119905) exp[minus

(119905 minus 119905

lowast)

2

120590

2] (20)

where 119905lowast denotes the moment at which the Gaussian pulsereaches its maximal value 120593

0(119909 119905) whereas 120590 is the standard

deviation of the Gaussian pulseIn case of different types of stimuli depending on the

size and duration the parameters in (20) will have differentvalues In the case of many concurrently external stimuli aseries of Gaussian pulses is considered

It is noteworthy that (20) is not a compulsory formof external stimulus where other formulas might be usedprovided that they meet the principle demands discussed inthis research This mathematical form is chosen because itscapability was examined in the modeling of generation ofaction potential in a neuron [16]

In the next section we provide a formulation for com-puting the effective diffusion coefficient at each moment thatcan be used in (19) in order to compute the brain response toexternal stimuli


5 A Phase-Lagging Diffusion Based Modelof the Diffusion Coefficient

In order to analyze the behavior of fractal time series thevalue and the direction of each fluctuation in the signalshould be analyzed The direction of each fluctuation can befound by computing the value of the Hurst exponent at theprevious point of the signal

In order to know about the value of the signal we shouldknow about different parameters which are appeared in thefractional diffusion model By looking at (19) and replacingthe value of 120593 = minus119863eff(120597119881120597120578) we have

119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(21)

The value of the Hurst exponent which is used in (21)can be computed using MATLAB based on Rescaled RangeAnalysis method [31 32]Then in order to compute the valueof the signal in (21) twoparameters119862 and119863 should be knownIn order to do this first a relationship between119863 and119867 canbe made and then by using the formula for relaxation time(6) 119862 can be replaced by119863 and 120591

In order tomake a relation between119863eff and119867 the phase-lagging model of action potential can be used [16]

119881 (119909 119905) = 119881

0minus (

119863

120591

)

12

int

119905

0

120597119881

120597119909

T0(

119905 minus 120585

2120591

) exp(minus119905 minus 120585

2120591

) 119889120585

(22)

In (22) the diffusivity term119863 is related to the resistance ofthe neuron to the electrical impulse 119863 is the property of theneural tissue and will dampen the impulse as it travels overthe nerve The term 120591 is the reaction time and T

0(119911) is the

zero-order modified Bessel functionConsidering the phase-lagging model of action potential

in the whole brain scale then this model and the fractionaldiffusion model explain the same phenomenon Thus themathematical equations that belong to these two models canbe equal

By writing this equality

119881

0minus

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

= 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(23)

where 119863 is the diffusion coefficient in the phase laggingmodel of action potential and 119863eff is the diffusion coefficientin the fractional diffusion model

Considering 119862 = (119863eff120591)12 and removing 119881

0from both

sides of the equation

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

minus 120591

12minus119867radic119863eff

1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867= 0

(24)

Then we can write

int

119905

0

radic

119863

120591

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905 minus 120585)

1minus119867119889120585 = 0

(25)

Because (25) is valid for all values of 119905 gt 0 thenwe can removethe integral

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867= 0

(26)

Thus

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

= 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867

(27)

Diving both sides of (27) by 120597119881120597120578 = 0 and introducing anew variable 119911 = 1199052120591 we have

119863eff119863

= [2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(28)

Considering a constant value for 119863 in order to study thevariation of 119863eff119863 versus 119911 Figures 3 and 4 are provided Itis noteworthy in order to analyze the value of the signal that itis only needed to concentrate on the calculations of119863eff andaccordingly 119881 in 0 le 119867 le 05 or 05 le 119867 le 1 because as itis known having the probability of 119867 on one side yields theprobability of1198671015840 = 1minus119867 in the reverse side butwith the samevalue of fluctuation In this research the span 05 le 119867 le 1 istaken as the reference So for instance in order to computethe value of the signal in a point with119867 = 02 it is needed justto compute the value of the signal with119867 = 08

As it can be seen in Figures 3 and 4 first the value ofdimensionless diffusivity increases but after that as timeincreases (119911 rarr infin) dimensionless diffusivity tends aconstant value In order to describe this behavior (28) canbe analyzed when 119911 rarr infin

lim119911rarrinfin

119863eff119863

= lim119911rarrinfin

[2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(29)

lim119911rarrinfin

T0(119911) is computed as

lim119911rarrinfin

T0(119911) =

exp (119911)radic2120587119911

= exp (119911) 2minus12120587minus12119911minus12 (30)

By substituting (30) into (29)

lim119911rarrinfin

119863eff119863


[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] (31)


14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD

Figure 3 Dependence of the dimensionless diffusivity 119863eff119863 onthe dimensionless temporal variable 119911 for119867 = 05

1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD

Figure 4 Dependence of the dimensionless diffusivity 119863eff119863 onthe dimensionless temporal variable 119911 in case of 05 lt 119867 le 1

Considering (31) in case of119867 = 05

lim119911rarrinfin

119863eff119863


[1] = 1(32)

which is the same as the trends observed in Figure 3By considering (31) in case of 05 lt 119867 le 1 we have

lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] = lim119911rarrinfin

[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)

which is the same as the trend observed in Figure 4 for valuesof 119867 where 05 lt 119867 le 1 So in both cases (119867 = 05 and05 lt 119867 le 1) the dimensionless diffusivity tends to a constantvalue as time increase

So in cases of Figures 3 and 4 before the maximum pointof graph as time goes on the value of the dimensionlessdiffusivity increases but after passing the maximum pointthe dimensionless diffusivity shows the opposite behaviorand after some time the dimensionless diffusivity tends toa constant value which is 0 and 1 in the cases of 05 lt 119867 le 1

and119867 = 05 respectivelyAs 119863eff has been considered as a time dependent param-

eter and 119863 as a constant thus from previous discussion itcan be concluded that when human senses a stimulus thediffusion of this stimulus to human brain increases but aftersome time the diffusion of the stimulus decreases In factthese results validate and show the importance of (28) forcomputing the value of119863eff

Thus by computing the value of119863eff at each timemomentand substituting its value in the fractional diffusionmodel thevalue of the signal at each time moment can be computed

6 Result and Discussion

In this section using the fractional diffusionmodel the humanbrain response to a visual external stimulus is modeled andcompared with the real EEG signal

61 Subjects In this research the experiments were carriedout on 6 voluntary healthy students (21ndash24 years old 3 malesand 3 females) Prior to the experiment each subject wasexamined and interviewed by a physician to ensure that noneurological deficit pain condition or medication affects theEEG All subjects were right handed Informed consent wasobtained from each subject after the nature of the study wasfully explained

All procedures were approved by the Internal ReviewBoard of the University and the approval for experimentationinvolving human subjects was issued It is noteworthy that theidentity of all subjects remains confidential

62 EEG Recording The EEG data used in this research werecollected using Mindset 24 device a 24-channel topographicneuromapping instrument which can measure 24 channelsof data with the sampling frequency of 256Hz

In an electrically shielded acoustically isolated and dimlyilluminated room a visual stimulus applied on subjects Itshould be mentioned that it is endeavoured to insulate thesubjects from all other external stimuli This ensures that theresponse measured in the EEG signals is primarily due to thestimulus applied In the experiment subjects were watchinga checkerboard pattern (see Figure 5) on the monitor of acomputer from the distance of 130 cm

The stimulus was the checker reversal After one secondthe reversed pattern (see Figure 6) was displayed and then theoriginal pattern was displayed again Thus the stimulus wasapplied at 119905 = 1 sThe interstimulus interval of 5 s was chosenin these experiments

Mindmeld 24 software is used for the collection of datausing Mindset 24 machine The software gives data in theform of bin files which can be processed to give text files(txt) that are required for further processing


Figure 5 Checkerboard pattern

Figure 6 The reversed pattern as the visual stimulus

Although EEG data are recorded from 24 electrodesin this research the analysis is done on the data governedfrom the left occipital (O1) electrode (near to the location ofthe visual primary sensory area) This electrode was chosenbased on the nearest place to the visual sensory area whichshows the strongest response that can be seen in the signalrecorded from this electrode compared to other electrodesThe electrode impedance was kept lower than 5KΩ

A bipolar electrooculogram (EOG vertical and hori-zontal) was recorded for off-line artifact rejection Afterbandpass filtering in the range of 01ndash70Hz 2 seconds of data(256 data before stimulation and 256 data after stimulation)was saved It means that there are 256 values of voltagecollected every second Prestimulation is defined as the statusbefore the application of the stimulus On the other handpoststimulation is defined as the status after the application

of the stimulus As it was mentioned previously the stimuluswas applied at 119905 = 1 s

It is noteworthy that choosing higher sampling ratewill result in more pre- and poststimulation data Havingmore data will results in more precise computation of HurstExponent diffusion coefficient and accordingly the signaland its parameters such as response initiation time

In the first week 40 trials were collected from each subjectin one day The data collections were repeated after a weekfor each subject in order to examine the reproducibility ofthe results from experiments By repeating the experimentsin the secondweek totally 80 trials were collected After visualinspection of data collected from each subject and rejectionof trials with artifacts 40 trials free of artifacts were selectedfor future analysis It is noteworthymentioning that physicianmonitored the subjects during all experiments

63 Data Analysis A set of codes was written in MATLABsoftware in order to compute all required parameters whichwere discussed before

As the recorded data were noisy the EEG signals werefiltered using theWavelet toolbox inMATLAB and then wereprocessed by the methodology discussed in this section

The value of 1198810can be read from the record of the signal

at the moment the stimulus is applied to the subject 119905 = 1 sThe initial value of119867 is computed for 1 second of the recordeddata before the application of the stimulus to the subject Inorder to compute the Hurst exponent as it was mentionedpreviously the Rescaled Range Analysis method is employedwhich is widely used by statisticians

It is required to compute the Hurst exponent value ineach moment in order to analyze the generated signal At thefirst step the program computes the Hurst exponent for therecorded EEG signal and the predicted signal and generatestwo time series in one figure

In each moment the program computes 119863eff using (28)Also as it was mentioned previously that for all analysisperformed here a single stimulus is considered and thus asingle Gaussian pulse is modeled using (20) By substitutingthe required parameters in (35) the program computes thevalue of the signal in each moment and then plots themodeled signal in a figure together with the recorded EEGsignal (after stimulation)

119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)

It is noteworthy that (35) is governed by substituting119862 = (119863eff120591)

12 into (19)The values of some required parameters are listed in

Table 1As it was mentioned previously one second of EEG

data was recorded and using these data as the referenceone second of EEG time series is predicted After that themodeled signal is analyzed in terms of the initiation timefor the response the response duration and the peak topeak voltage In this research in order to see the responsefluctuations clearly the real or the modelled signals after thepoststimulation are averaged in each case It means that for


Table 1 Values of required parameters

Variable Value Units119863 65 times 10

minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s

each subject 40 selected trials are used as the input to themodel and accordingly the grand average of all modeledsignals (40 signals for each subject) is presented Then thegenerated plots for the real or the predicted signals are theresult of averaging over all selected trails in each case

Here it should be mentioned that in the analyses of theEEG signals plots the initiation time for the response andthe duration are chosen based on the literature notes whichconsider themajor positive or negative pole For instance thefluctuations which have voltage in the range of 5 to 10120583V orminus5 to minus10 120583V are related with the response to the stimulusThus for instance in order to have a feature of the responseduration the time span between the first peak and the lastpeak which have the voltage values within one of these tworanges is considered

The grand average of the recorded EEG signals and thegrand average of the predicted signals using the fractionaldiffusion model over all selected trials for 1 second post-stimulation are shown in Figure 7

As it can be seen for different subjects in Figure 7 therecorded signal (black solid line) and the predicted signal(red dashed line) show the similar behavior In both casesthe brain response to the stimulus starts with a positive peak(119875) after the application of the stimulus to the subject whereits amplitude goes further than 5 120583V This response causesthe signalrsquos voltage to fluctuate in a bigger span Followingthe positive peak a negative rebound (119873) can be seen in theplot In fact the response to the stimulus terminates at thisnegative peak after which the brain goes back to its normalstatus during rest without any big deflection in the signal

As an example for subject 1 in cases of the real andpredicted signals the response to the stimulus starts with apositive peak (119875) at about 118ms and about 127ms respec-tively after the application of the stimulus to the subjectThisresponse causes the signalrsquos voltage to fluctuate in a biggerspan Following the positive peak a negative rebound (119873) atabout 119905 = 1170 s and 119905 = 1174 s can be seen in the plot incases of the real and predicted signals respectively In fact theresponse to the stimulus terminates at this point after whichthe brain goes back to its normal status during rest withoutany big deflection in the signal

The values of peak to peak voltage the initiation timefor the response (the initial peak of the response) and theresponse duration (the time difference between the first andthe last peaks within the response duration) for the recordedEEG signal and the predicted signal in case of differentsubjects are provided in Table 2

As it can be seen in Table 2 for all subjects the predictedinitiation time for the response the response duration andthe peak to peak voltage have very close values with their

related values in the recorded EEG signalsThus it can be saidthat the predicted signal resembles the real EEG signal withinthe response duration in the cases of the initiation time for theresponse the peak to peak voltage and the response durationMoreover in order to study the uncertainty and predictabilityof the modelrsquos solution the Hurst exponent variations forthe recorded EEG signals and the predicted signals over allselected trials for 1 second after stimulation are shown inFigure 8

The high correlation between the values of the real signalsand also the predicted signals can be realized by looking atthe values of the Hurst exponents For instance in the caseof subject 1 the value of the Hurst exponent is distributedbetween 0900 and 0943 for the recorded EEG signal (blacksolid line) and between 0900 and 0952 for the predictedsignals (red dashed line) Thus the low uncertainty of theprediction can be confirmed and it can be said the signalis predicted well because the Hurst exponent values are notclose to 119867 = 05 which stand for a truly random processThis behavior can be seen in the Hurst exponent plots for allsubjects

Also as it can be seen in the Hurst exponent plot thevalue of the Hurst exponent in the case of the real EEGsignal and the predicted signal experiences a sudden upwarddeflection For instance in the case of subject 1 the value ofthe Hurst exponent is decreasing in the time span of 119905 = 1 sto about 119905 = 1118 s and 119905 = 1 s to about 119905 = 1127 srespectively for real EEG signal and the predicted signal afterthat a sudden upward deflection can be seen which standsfor experiencing the visual stimulus and again the trendshows the same behavior The overall decreasing behaviorstands for the phenomenon that when a longer time span isconsidered the less the human brain ldquoremembersrdquo its initialstateThe same behavior can be seen in other Hurst exponentplots

By analyzing the behaviors which have been seen inFigures 7 and 8 it can be said that on one hand the uncertaintyof the prediction was low and on the other hand the accuracyof the prediction was very good as the predicted signalresembles the real EEG signal

All the analyses which have been done in this researchshow that EEG signals can be modeled by the solutionof fractional partial differential equations and thus thebehavior of system modeled by means of such equations canin principle not only be predicted but also quantifies

7 Conclusion

In this paperwe introduced a newmathematicalmodel whichquantifies the human brain response to external stimuli Wedeveloped this model by applying the fractional diffusionequation to human EEG signals The model generates amultifractal time series which shows a quantitative con-currence with the real EEG signals Using this model wesuccessfully predicted the EEG signals of different subjectsupon receiving a visual stimulus This model shall be furtherapplied in case of different external stimuli where the resultscan be verified against the real EEG signal which means theprediction of the human behavior by forecasting the EEG


10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6

Figure 7 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

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6

Figure 8 The grand average of the Hurst exponent variations for the recorded EEG signals (black solid line) and the grand average of theHurst exponent variations for the predicted signals (red dashed line) for 1 second after stimulation in the case of the visual stimulus subject1 to subject 6


Table 2 Comparison between the real and the predicted signals

Subject The initiation time for the response (ms) Response duration (s) Peak to peak voltage (120583V)Real Predicted Real Predicted Real Predicted

1 P118 P127 0052 0047 1397 13072 P120 P120 0054 0067 1286 13753 P110 P120 0058 0060 1524 15244 P100 P120 0078 0057 1628 15505 P112 P121 0043 0041 1250 11816 P120 P130 0050 0044 1385 1550Average 113 123 0055 0052 1411 1414

signal On the other hand this model also can be employedin order to predict different abnormal brain activities suchas epileptic seizure by at least some seconds before the timeof occurrence If so a seizure warning and the expected timeof this epilepsy occurrence can be generated leading to thefuture monitoring of this disease

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Kaneoke T Urakawa and R Kakigi ldquoVisual motion direc-tion is represented in population-level neural response asmeasured by magnetoencephalographyrdquo Neuroscience vol 160no 3 pp 676ndash687 2009

[2] B Lee Y Kaneoke R Kakigi and Y Sakai ldquoHuman brainresponse to visual stimulus between lowerupper visual fieldsand cerebral hemispheresrdquo International Journal of Psychophys-iology vol 74 no 2 pp 81ndash87 2009

[3] A Sorrentino L Parkkonen M Piana et al ldquoModulation ofbrain and behavioural responses to cognitive visual stimuli withvarying signal-to-noise ratiosrdquo Clinical Neurophysiology vol117 no 5 pp 1098ndash1105 2006

[4] U Will and E Berg ldquoBrain wave synchronization and entrain-ment to periodic acoustic stimulirdquo Neuroscience Letters vol424 no 1 pp 55ndash60 2007

[5] G Haiman H Pratt and A Miller ldquoBrain responses to verbalstimuli among multiple sclerosis patients with pseudobulbaraffectrdquo Journal of the Neurological Sciences vol 271 no 1-2 pp137ndash147 2008

[6] H Eswaran R Draganova and H Preissl ldquoAuditory evokedresponses a tool to assess the fetal neurological activityrdquoAppliedAcoustics vol 68 no 3 pp 270ndash280 2007

[7] K Sutani S Iwaki M Yamaguchi K Uchida and M TonoikeldquoNeuromagnetic brain responses evoked by pleasant andunpleasant olfactory stimulirdquo International Congress Series vol1300 pp 391ndash394 2007

[8] W G Sannita ldquoStimulus-specific oscillatory responses of thebrain a timefrequency related coding processrdquo Clinical Neu-rophysiology vol 111 no 4 pp 565ndash583 2000

[9] A Miyanari Y Kaneoke Y Noguchi et al ldquoHuman brainactivation in response to olfactory stimulation by intravenousadministration of odorantsrdquo Neuroscience Letters vol 423 no1 pp 6ndash11 2007

[10] K Matz M Junghofer T Elbert K Weber C Wienbruchand B Rockstroh ldquoAdverse experiences in childhood influ-ence brain responses to emotional stimuli in adult psychiatricpatientsrdquo International Journal of Psychophysiology vol 75 no3 pp 277ndash286 2010

[11] K L Phan D A Fitzgerald K Gao G J Moore M E Tancerand S Posse ldquoReal-time fMRI of cortico-limbic brain activityduring emotional processingrdquo NeuroReport vol 15 no 3 pp527ndash532 2004

[12] P G Mullins L M Rowland R E Jung and W L Sibbitt JrldquoA novel technique to study the brainrsquos response to pain protonmagnetic resonance spectroscopyrdquo NeuroImage vol 26 no 2pp 642ndash646 2005

[13] D Simon K D Craig W H R Miltner and P Rainville ldquoBrainresponses to dynamic facial expressions of painrdquo Pain vol 126no 1ndash3 pp 309ndash318 2006

[14] W Freeman Predictions on Neocortical Dynamics Derived fromStudies in Paleocortex Birkhauser Boston Boston Mass USA1991

[15] W J Freeman ldquoSimulation of chaotic EEG patterns with adynamic model of the olfactory systemrdquo Biological Cyberneticsvol 56 no 2-3 pp 139ndash150 1987

[16] K Seetharaman H Namazi and V V Kulsih ldquoPhase laggingmodel of brain response to external stimuli-modeling of singleaction potentialrdquo Computers in Biology and Medicine vol 42no 8 pp 857ndash862 2012

[17] D T J Liley P J Cadusch and J JWright ldquoA continuum theoryof electro-cortical activityrdquoNeurocomputing vol 26-27 pp 795ndash800 1999

[18] D A Steyn-Ross M L Steyn-Ross L C Wilcocks and JW Sleigh ldquoToward a theory of the general-anesthetic-inducedphase transition of the cerebral cortex II Numerical simula-tions spectral entropy and correlation timesrdquo Physical ReviewEmdashStatistical Nonlinear and Soft Matter Physics vol 64 no 1p 1918 2001

[19] M L Steyn-Ross D A Steyn-Ross J W Sleigh and L CWilcocks ldquoToward a theory of the general-anesthetic-inducedphase transition of the cerebral cortex I A thermodynamicsanalogyrdquo Physical Review E vol 64 no 1 Article ID 011917 2001

[20] M L Steyn-Ross D A Steyn-Ross J W Sleigh and D TJ Liley ldquoTheoretical electroencephalogram stationary spec-trum for a white-noise-driven cortex evidence for a generalanesthetic-induced phase transitionrdquo Physical Review E Sta-tistical Physics Plasmas Fluids and Related InterdisciplinaryTopics vol 60 no 6 pp 7299ndash7311 1999

[21] M L Steyn-Ross D A Steyn-Ross J W Sleigh and D RWhiting ldquoTheoretical predictions for spatial covariance of the


electroencephalographic signal during the anesthetic-inducedphase transition increased correlation length and emergence ofspatial self-organizationrdquo Physical Review EmdashStatistical Non-linear and Soft Matter Physics vol 68 no 2 18 pages 2003

[22] M A Kramer H E Kirsch and A J Szeri ldquoPathological pat-tern formation and cortical propagation of epileptic seizuresrdquoJournal of the Royal Society Interface vol 2 no 2 pp 113ndash1272005

[23] V Kulish andW Chan ldquoModeling of brain response to externalstimulirdquo in Proceedings of the 7th NTU-SGH Symposium Singa-pore 2005

[24] AA Fingelkurts AA Fingelkurts andC FHNeves ldquoNaturalworld physical brain operational andmind phenomenal space-timerdquo Physics of Life Reviews vol 7 no 2 pp 195ndash249 2010

[25] V K Jirsa and H Haken ldquoField theory of electromagnetic brainactivityrdquo Physical Review Letters vol 77 no 5 pp 960ndash9631996

[26] A Einstein ldquoUber die von der molekularkinetischen The-orie der Warme geforderte Bewegung von in ruhendenFlussigkeiten suspendierten Teilchenrdquo Annalen der Physik vol322 no 8 pp 549ndash560 1905 Reprinted in A Einstein ldquoInves-tigations on the theory of the Brownian movementrdquo editedwith notes by R Furth translated by A D Cowper LondonMethuen 1926

[27] V V Kulish Partial Differential Equations Pearson Singapore2nd edition 2010

[28] Y Demirel Nonequilibrium Thermodynamics Transport andRate Processes in Physical Chemical and Biological SystemsElsevier Amsterdam The Netherlands 2007

[29] C H Bennett P Gacs M Li P M Vitanyi and W H ZurekldquoInformation distancerdquo IEEE Transactions on Information The-ory vol 44 no 4 pp 1407ndash1423 1998

[30] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[31] H E Hurst ldquoThe long-term storage capacity of reservoirsrdquoTransactions of the American Society of Civil Engineers vol 116pp 770ndash799 1951

[32] B BMandelbrot and J RWallis ldquoNoah Joseph and operationalhydrologyrdquoWater Resources Research vol 4 no 5 pp 909ndash9181968

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Journal of

ObesityJournal of



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OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


the phase-lagging model of single action potential [16]When the microscopic models are extended to a macrolevelthen different methods are employed Many of these modelsassume the cortical region to be a continuum Liley et aldeveloped a set of nonlinear continuumfield equationswhichdescribed the macroscopic dynamics of neural activity in thecortical region [17] These equations were used by Steyn-Ross et al who introduced noise terms into them to give aset of stochastic partial differential equations (SPDEs) Theyalso converted the equations governed by Liley et al intolinearized ODEs This model could predict the substantialincrease in low frequency power at the critical points ofinduction and emergence They later used this model tostudy the electrical activity of an anaesthetized cortex [18ndash21] Kramer et al started with the equations given by Steyn-Ross and coworkers and neglected the spatial variation andthe stochastic input They believed that this gave rise to a setof ordinary differential equations (ODEs) for the modelingof the cortical activity They showed that the results obtainedfrom the SPDE model agree with clinical data in an approx-imate way [22] but they also stated that the spatial samplingof the cortex was poor because of inherent shortcomingsin the equipment used Kulish and Chan have suggested anovel method for the modeling of the brain response usingfundamental laws of nature like energy conservation and theleast action principle The model equation obtained has beensolved and the results show a good agreement with real EEGs[23]

Despite rapid advances in the studies related to theanalysis of the human brain response there has been lessprogress in the mathematical modeling of the human brainactivity due to external stimuli Yet it seems that the con-temporary level of developments in physics and mathematicsmakes establishing quantitative correlations between externalphysical stimuli and the brain responses to those stimuli

This paper attempts to introduce a new mathematicalmodel of the human brain response to external stimuli basedon the fractional diffusion equation At first we talk aboutthe macroscopic level of brain organization and EEG signalas an alert state of human behavior monitoring Then byintroducing fractional diffusion equations and consideringthe EEG signal as a fractal time series we model the EEGsignal using mathematical equations This model is thensolved and after discussing different parameters in the modelwe provide some results and discuss about the modelrsquossolution in details Some concluding remarks are provided atthe end

2 Macroscopic Level of Brain Organizationand the EEG Signal

In order to study the human neural activity one can considerdifferent levels of brain organization at different scales intime and space from a single neuron (microscopic level) tothe whole brain organization (macroscopic level) In fact thewhole brain activity cannot be observed by measuring theactivity of just a single neuron as far it is informative as itcontributes to study the entire population of which it is amember

40

30

20

10

0

minus10

minus20

minus301 11 12 13 14 15 16 17 18 19 2

Time (s)

Volta

ge (120583

V)

Figure 1 EEG signal of a subject in response to an external stimulus

In this research we focus on the macroscopic level ofbrain organization Macroscopic level of brain organizationrefers to the level of neural assembliesrsquo population in whicheach neural assembly interacts with other neural assem-blies in close and distant cortical areas exhibits spatial-temporal behaviour and paints the human behaviour [24]The functional behaviour of the brain is encoded in thesespatial-temporal structures and can be extracted from themacroscopic quantities dynamics observed by EEG signalmostly [25]

During many years scientists have studied the humanbehavior by recording and analysis of the EEG signals fromdifferent areas of the brain The EEG signal is the com-position of different frequency bands (oscillatory activities-Alpha Beta ) which are structured coordinately (spatiallytemporally) In fact this signal has different characteristicsthat can be used in order to study the human brain responseto external stimuli in our research For instance Figure 1shows the EEG signal of a subject who received an externalstimulus at 119905 = 1 s In this figure the brain response to externalstimulus can be seen as a sudden upward deflection at about119905 = 125 s after the application of stimulus at 119905 = 1 s

3 Fractional Diffusion Equation

Here in order to develop our model we start with a simpleequation

120597119865

120597119905

= 119863nabla

2119865

(1)

where

nabla

2119865 =

120597

2119865

120597119909

2

(2)

Equation (1) is the well-known diffusion equation wherethe coefficient 119863 is the diffusivity of the medium for theproperty 119865 In fact this equation arises in many descriptionsof biological and physical phenomena including Brownianmotion [26] gradient driven chemical diffusion (with Fickrsquoslaw) and heat transfer (heat diffusion with Fourierrsquos Law) Itis noteworthy that (1) is written in the Euclidean space

On the other hand the diffusion process can also beenstudied for fractals in fractal space Fractals (such as randomwalk) may be defined as self-similar geometric objects whosescaling exponent (dimension) satisfies the Szpilrajn inequal-ity

alefsym ge 119863

119879 (3)






120597

2119867119865

120597119905

2119867= 119862

2(2119867minus1)119863


2119865

(4)

where nabla2119865 = 120597

2119865120597119909



2119865120597119905

2= 119862

minus2119863

2nabla



2nabla

2119865





120597

2119867119881

120597119905

2119867= 119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119881

120597120578

2 (5)









120591 =

119863eff119862

2 (6)



0 le 120578 lt infin (7)





119905 ge 0 (8)

Also regarding (5)

119881 (120578 0) = 119881

0 (9)


lim120578rarr+infin

119881 (120578 119905) = const lt infin (10)




0




119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119884

120597120578

2minus 119904

2119867119884 = 0 (11)



119884 (120578 119904) = 119860

1(119904) 119890


+ 119860

2(119904) 119890

120578119904119867[119862(2119867minus1)119863eff(1minus119867)]

(12)

where1198601(119904) and119860





119884 (120578 119904) = 119860

1(119904) 119890


(13)


119889119884 (120578 119904)

119889120578

= minus

119860

1(119904) 119904

119867119890


[119862

(2119867minus1)119863eff(1minus119867)

]

(14)



119884 (120578 119904) = minus119904

minus119867119862

(2119867minus1)119863eff(1minus119867)

119889119884 (120578 119904)

119889120578

(15)


119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 120597

minus119867

120597119905

minus119867(

120597119881

120597120578

) (16)



120597

120572119891

120597119905

120572=

1

Γ (minus120572)

int

119905

0

119891 (120585) 119889120585

(119905 minus 120585)

120572+1 Re (120572) lt 0 (17)


minus

120597119881

120597120578

=

120593

119863eff (18)


119881 (120578 119905) = 119881

0+ 119862

(2119867minus1)119863eff(minus119867) 1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (19)






120593 (119909 119905) = 120593

0(119909 119905) exp[minus

(119905 minus 119905

lowast)

2

120590

2] (20)











119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(21)



119881 (119909 119905) = 119881

0minus (

119863

120591

)

12

int

119905

0

120597119881

120597119909

T0(

119905 minus 120585

2120591


2120591

) 119889120585

(22)


0(119911) is the




119881

0minus

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

= 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(23)



0from both


radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

minus 120591


1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867= 0

(24)

Then we can write

int

119905

0

radic

119863

120591

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905 minus 120585)

1minus119867119889120585 = 0

(25)


radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867= 0

(26)

Thus

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

= 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867

(27)


119863eff119863

= [2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(28)



lim119911rarrinfin

119863eff119863


[2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(29)

lim119911rarrinfin


lim119911rarrinfin

T0(119911) =

exp (119911)radic2120587119911



lim119911rarrinfin

119863eff119863


[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] (31)


14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD


1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD



lim119911rarrinfin

119863eff119863


[1] = 1(32)


lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911


[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)




























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















120597

2119867119865

120597119905

2119867= 119862

2(2119867minus1)119863


2119865

(4)

where nabla2119865 = 120597

2119865120597119909



2119865120597119905

2= 119862

minus2119863

2nabla



2nabla

2119865





120597

2119867119881

120597119905

2119867= 119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119881

120597120578

2 (5)









120591 =

119863eff119862

2 (6)



0 le 120578 lt infin (7)





119905 ge 0 (8)

Also regarding (5)

119881 (120578 0) = 119881

0 (9)


lim120578rarr+infin

119881 (120578 119905) = const lt infin (10)




0




119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119884

120597120578

2minus 119904

2119867119884 = 0 (11)



119884 (120578 119904) = 119860

1(119904) 119890


+ 119860

2(119904) 119890

120578119904119867[119862(2119867minus1)119863eff(1minus119867)]

(12)

where1198601(119904) and119860





119884 (120578 119904) = 119860

1(119904) 119890


(13)


119889119884 (120578 119904)

119889120578

= minus

119860

1(119904) 119904

119867119890


[119862

(2119867minus1)119863eff(1minus119867)

]

(14)



119884 (120578 119904) = minus119904

minus119867119862

(2119867minus1)119863eff(1minus119867)

119889119884 (120578 119904)

119889120578

(15)


119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 120597

minus119867

120597119905

minus119867(

120597119881

120597120578

) (16)



120597

120572119891

120597119905

120572=

1

Γ (minus120572)

int

119905

0

119891 (120585) 119889120585

(119905 minus 120585)

120572+1 Re (120572) lt 0 (17)


minus

120597119881

120597120578

=

120593

119863eff (18)


119881 (120578 119905) = 119881

0+ 119862

(2119867minus1)119863eff(minus119867) 1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (19)






120593 (119909 119905) = 120593

0(119909 119905) exp[minus

(119905 minus 119905

lowast)

2

120590

2] (20)











119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(21)



119881 (119909 119905) = 119881

0minus (

119863

120591

)

12

int

119905

0

120597119881

120597119909

T0(

119905 minus 120585

2120591


2120591

) 119889120585

(22)


0(119911) is the




119881

0minus

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

= 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(23)



0from both


radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

minus 120591


1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867= 0

(24)

Then we can write

int

119905

0

radic

119863

120591

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905 minus 120585)

1minus119867119889120585 = 0

(25)


radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867= 0

(26)

Thus

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

= 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867

(27)


119863eff119863

= [2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(28)



lim119911rarrinfin

119863eff119863


[2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(29)

lim119911rarrinfin


lim119911rarrinfin

T0(119911) =

exp (119911)radic2120587119911



lim119911rarrinfin

119863eff119863


[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] (31)


14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD


1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD



lim119911rarrinfin

119863eff119863


[1] = 1(32)


lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911


[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)




























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119905 ge 0 (8)

Also regarding (5)

119881 (120578 0) = 119881

0 (9)


lim120578rarr+infin

119881 (120578 119905) = const lt infin (10)




0




119862

2(2119867minus1)119863eff2(1minus119867) 120597

2119884

120597120578

2minus 119904

2119867119884 = 0 (11)



119884 (120578 119904) = 119860

1(119904) 119890


+ 119860

2(119904) 119890

120578119904119867[119862(2119867minus1)119863eff(1minus119867)]

(12)

where1198601(119904) and119860





119884 (120578 119904) = 119860

1(119904) 119890


(13)


119889119884 (120578 119904)

119889120578

= minus

119860

1(119904) 119904

119867119890


[119862

(2119867minus1)119863eff(1minus119867)

]

(14)



119884 (120578 119904) = minus119904

minus119867119862

(2119867minus1)119863eff(1minus119867)

119889119884 (120578 119904)

119889120578

(15)


119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 120597

minus119867

120597119905

minus119867(

120597119881

120597120578

) (16)



120597

120572119891

120597119905

120572=

1

Γ (minus120572)

int

119905

0

119891 (120585) 119889120585

(119905 minus 120585)

120572+1 Re (120572) lt 0 (17)


minus

120597119881

120597120578

=

120593

119863eff (18)


119881 (120578 119905) = 119881

0+ 119862

(2119867minus1)119863eff(minus119867) 1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (19)






120593 (119909 119905) = 120593

0(119909 119905) exp[minus

(119905 minus 119905

lowast)

2

120590

2] (20)











119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(21)



119881 (119909 119905) = 119881

0minus (

119863

120591

)

12

int

119905

0

120597119881

120597119909

T0(

119905 minus 120585

2120591


2120591

) 119889120585

(22)


0(119911) is the




119881

0minus

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

= 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(23)



0from both


radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

minus 120591


1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867= 0

(24)

Then we can write

int

119905

0

radic

119863

120591

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905 minus 120585)

1minus119867119889120585 = 0

(25)


radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867= 0

(26)

Thus

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

= 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867

(27)


119863eff119863

= [2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(28)



lim119911rarrinfin

119863eff119863


[2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(29)

lim119911rarrinfin


lim119911rarrinfin

T0(119911) =

exp (119911)radic2120587119911



lim119911rarrinfin

119863eff119863


[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] (31)


14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD


1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD



lim119911rarrinfin

119863eff119863


[1] = 1(32)


lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911


[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)




























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















119881 (120578 119905) = 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(21)



119881 (119909 119905) = 119881

0minus (

119863

120591

)

12

int

119905

0

120597119881

120597119909

T0(

119905 minus 120585

2120591


2120591

) 119889120585

(22)


0(119911) is the




119881

0minus

radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

= 119881

0minus 119862

(2119867minus1)119863eff(1minus119867) 1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867

(23)



0from both


radic

119863

120591

int

119905

0

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

) 119889120585

minus 120591


1

Γ (119867)

int

119905

0

120597119881

120597120578

119889120585

(119905 minus 120585)

1minus119867= 0

(24)

Then we can write

int

119905

0

radic

119863

120591

120597119881

120597120578

T0(

119905 minus 120585

2120591


2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905 minus 120585)

1minus119867119889120585 = 0

(25)


radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

minus 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867= 0

(26)

Thus

radic

119863

120591

120597119881

120597120578

T0(

119905

2120591

) exp(minus 119905

2120591

)

= 120591


1

Γ (119867)

120597119881

120597120578

1

(119905)

1minus119867

(27)


119863eff119863

= [2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(28)



lim119911rarrinfin

119863eff119863


[2

1minus119867Γ (119867) 119911

1minus119867T0(119911) exp (minus119911)]

2

(29)

lim119911rarrinfin


lim119911rarrinfin

T0(119911) =

exp (119911)radic2120587119911



lim119911rarrinfin

119863eff119863


[

2

1minus2119867Γ

2(119867)

120587

119911

1minus2119867] (31)


14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD


1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD



lim119911rarrinfin

119863eff119863


[1] = 1(32)


lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911


[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)




























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















14

12

1

08

06

04

02

0

0 2 4 6 8 10 12 14 16 18 20

z

DeffD


1

09

08

07

06

05

04

03

02

01

00 5 10 15 20

H = 1

H = 09H = 08

H = 07

H = 06

z

DeffD



lim119911rarrinfin

119863eff119863


[1] = 1(32)


lim119911rarrinfin

[

2

1minus2119867Γ

2(119867)

120587

119911


[119883 sdot

1

119911

2119867minus1] (33)

Thus

lim119911rarrinfin

119863eff119863

= 0(34)




























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






























119881 (120578 119905) = 119881

0+ 120591

12minus119867 1

radic119863eff

1

Γ (119867)

int

119905

0

120593 (120578 120585) 119889120585

(119905 minus 120585)

1minus119867 (35)








minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















minus4 m2s120593

0(119909 119905) 1 V sdotms

119905

lowast 0002 s120590 0001 s













7 Conclusion



10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(a) Subject 1

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)

V (120583V)

(b) Subject 2

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(c) Subject 3

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(d) Subject 4

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(e) Subject 5

10

5

0

minus5

minus10

100 110 120 130 140 150 160 170 180 190

t (s)V (120583V)

(f) Subject 6


0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(a) Subject 1

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(b) Subject 2

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(c) Subject 3

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(d) Subject 4

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(e) Subject 5

0980

0960

0940

0920

0900

0880

100 110 120 130 140 150 160 170 180 190

t (s)

H

(f) Subject 6









References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























References








































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article fractional diffusion based modelling and...

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