In this study, We propose a EEG analysis model using a nonlinear oscillator with one degree of freedom. It doesn’t have a random term. our study method identifies six model parameters experimentally. Here is the detail: https://kenyu-life.com/2018/11/03/modeling_of_eeg/ Created by Kenyu Uehara

1. Experimental Identification of Model Parameters and the Statistical
Processing Using a Nonlinear Oscillator applied to EEG Analysis
◯Kenyu UEHARA, Takashi SAITO
Graduate School of Science and Technology for Innovation,
Yamaguchi University, Japan
Proceedings of the ASME 2018 International Mechanical
Engineering Congress & Exposition IMECE2018
November 9 - 15, Pittsburgh, Pennsylvania, USA
１．Brief overview of EEG characteristics
２．Modeling EEG and Analysis way
EPIA(Eeperimental Parameter Identification Analysis)
３．Experimental
Relax state vs. Stress state
４．Result and Discussion

2. Introduction
EEG: ElectroEncephaloGraphy
±10 ~ 100 μV
Recordings brain activity
Investigating Sleep Cycle
Diagnosing Epilepsy Brain Activity and Mental Activity
Epilepsy…?
Christine Lowe Having an Epileptic Seizure (Graphic)
delta
theta
alpha
beta
Rem
or
non Rem

3. How we analyze EEG to see the mental state
＜FFT : Estimation human mental state＞
Real time FFT analysis
Frequency
4~ 8~ 13~ 16~ 30~ (Hz)
θ α Low-β High-β γ
θ α γδ
delta
theta
alpha
beta
Time
＜More investigating EEG : nonlinear analysis＞
・EEG has a CHAOS
・For example:
To see the quality of stress
「concentrating with joy」
「concentrating while being impatience」
etc.
EEG・・・Bio signals from the scalp
（It has biological information, therefore, human mental state can be estimated）
β
4 8 13 30 (Hz)
Known in EEG band power
Alpha waves associate with an increase level of
relaxation.
In stress state, the beta power increases.

4. 1．EEG signal・・・Chaos : “Bounded” + “Aperiodic”
（Unstable fluctuation, but different from self-destructive state）
What is the characteristics of EEG signal
・Bounded regularity : the frequency band increases with stress.
・Irregularity : the same state is not repeated.
→ this wave form is not repeated anymore
EEG
EEG signal
Irregularity
Thousand of fluctuation
NonlinearityLinearity
Regularity
Bounded
Periodic Aperiodic
<EEG signal is chaos>
・Long-term unpredictability
→ short-term forecasting is possible．
・sensitivity to initial state
→ fluctuation is deterministic
Bounded regularity

5. 1．EEG signal・・・Chaos : “Bounded” + “Aperiodic”
（Unstable fluctuation, but different from self-destructive state）
What is the characteristics of EEG signal
・Bounded regularity : the frequency band increases with stress.
・Irregularity : the same state is not repeated.
→ this wave form is not repeated anymore
EEG
EEG signal
Irregularity
Thousand of fluctuation
NonlinearityLinearity
Regularity
Bounded
Periodic Aperiodic
<EEG signal is chaos>
・Long-term unpredictability
→ short-term forecasting is possible．
・sensitivity to initial state
→ fluctuation is deterministic
Bounded regularity
Chaotic EEG is difficult to analyze, but the quantitative analysis can be
performed if the time-series data is below a certain critical time
because it fluctuates according to deterministic dynamics.

6. ２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Linear term
Nonlinear oscillator：linear term ＋ Duffing types nonlinear term
Nonlinear Periodic excitation
＜Modeling＞ Modeling of EEG, one DOF, no random term
EEG x (mV)
Internal dynamics
of EEG
Influence of
external
Our study method identifies six model parameters experimentally.
A, B, C, P1, P2, ω
We have studied this model
with abnormal epileptic EEG
abnormal EEGnormal EEG

7. Previous study : model parameter identified based on the least squares method
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Error =
N
∑
i=0
(··xi + A ·xi + Bxi + Cx3
i − P1cosωti − P2sinωti)
“Study on Detection of Epileptic Discharges Based on a Duffing Oscillator Model”, Takahiro M, Yasumi U, Masami F, Michiyasu S and Takashi S,
Proceedings of the ASME 2014 IMECE, Vol.3, No. IMECE2014-38107, pp.003T03A083.
model parameter identification based on the LSM
nonlinear parameter C has been greatly reduced in abnormal
abnormal EEGnormal EEG
abnormal EEGnormal EEG
Periodic Spike waveComplex fluctuation
→ large C → Small C
Abnormalities occur in a living body,
all nonlinearity decrease.
This is in agreement with our result

8. Previous study : model parameter identified based on the least squares method
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Error =
N
∑
i=0
(··xi + A ·xi + Bxi + Cx3
i − P1cosωti − P2sinωti)
“Study on Detection of Epileptic Discharges Based on a Duffing Oscillator Model”, Takahiro M, Yasumi U, Masami F, Michiyasu S and Takashi S,
Proceedings of the ASME 2014 IMECE, Vol.3, No. IMECE2014-38107, pp.003T03A083.
model parameter identification based on the LSM
nonlinear parameter C has been greatly reduced in abnormal
abnormal EEGnormal EEG
abnormal EEGnormal EEG
Periodic Spike waveComplex fluctuation
→ large C → Small C
Abnormalities occur in a living body,
all nonlinearity decrease.
This is in agreement with our result
This result revealed the relationship between the abnormalities of EEG
and the model parameters.
However, specific indexes of model parameters and physiological
characteristics in healthy condition have not been investigated.

9. Objective
we aim to investigate the characteristic of model
parameters for EEG analysis.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Linear term
Nonlinear oscillator：linear term ＋ Duffing types nonlinear term
Nonlinear Periodic excitation
EEG x (mV)
1. Parameter identification
(EEG data : relax state and stress state)
2. Comparison between two states.

10. This method identify model parameters for time sliding window.
Identified parameters evaluate human mental state.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Experimental EEG time series data
A
B
C
P1
P2
ω
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Objective function
Simulation model
The objective function is the average
of squared difference between
experimental and simulation value.
Record
Sim
Exp
Window
２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
Linear term Nonlinear Periodic excitation

11. This method identify model parameters for time sliding window.
Identified parameters evaluate human mental state.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Experimental EEG time series data
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Objective function
Simulation model
The objective function is the average
of squared difference between
experimental and simulation value.
Record
２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
Linear term Nonlinear Periodic excitation
Sim
Exp
A
B
C
P1
P2
ω
Window

12. This method identify model parameters for time sliding window.
Identified parameters evaluate human mental state.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Experimental EEG time series data
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Objective function
Simulation model
The objective function is the average
of squared difference between
experimental and simulation value.
Record
２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
Linear term Nonlinear Periodic excitation
A
B
C
P1
P2
ω
Sim
Exp
Window

13. This method identify model parameters for time sliding window.
Identified parameters evaluate human mental state.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Experimental EEG time series data
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Objective function
Simulation model
The objective function is the average
of squared difference between
experimental and simulation value.
Record
２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
Linear term Nonlinear Periodic excitation
A
B
C
P1
P2
ω
．．．
．．．
．．．
．．．
．．．
．．．
Sim
Exp
Window・・・・・・・・・

14. This method identify model parameters for time sliding window.
Identified parameters evaluate human mental state.
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Experimental EEG time series data
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Objective function
Simulation model
The objective function is the average
of squared difference between
experimental and simulation value.
Record
２．Modeling of EEG fluctuation (Short-term forecasting, deterministic)
・・・Modeling → Identification model parameters experimentally
Linear term Nonlinear Periodic excitation
A
B
C
P1
P2
ω
．．．
．．．
．．．
．．．
．．．
．．．
Sim
Exp
Window・・・・・・・・・
A
B
C
P1
P2
ω
．．．
．．．
．．．
．．．
．．．
．．．
Evaluate the human mental state (relax and stress)
Investigate the characteristic of model parameters

15. ３．Experimental (EEG of both Relax state and Stress state)
＜Experimental set-up＞
Relax state（non-stimulating landscape）
→ Stress state（１００-cell calculation）
・visual area O1
・Impedance 20Ω or less
・Shield room
・No caffeine
Internal 10-20 method
Relax
（landscape）
Stress
（100-cell calc.）
Meditation
- 60 0 120 240 Sec.
Measurement start
＜Result＞ We use alpha waves（8-13 Hz）in this study
State transitions at rest and concentration 40 seconds are not included in the analysis area
(Sampling 0.002 sec)
A
B
C
P1
P2
ω
．
．
．
．
．
．
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Parameters identification
for 1 second
Window =
1sec. 500 sample

16. ３．Analysis result・・・
··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
＜Error Value＞
Error(A, B, C, P1, P2, ω) =
1
N
N
∑
i=1
(xexp
i
− xsim
i )2
Average error
2.02
Average error
0.582
Whether the simulation using the
identified parameters is consistent
with the experiment result
・it is possible to simulate
complicated fluctuations of EEG
the identified parameter value evaluates the characteristics of the change in the
mental state of humans
Linear term Nonlinear Periodic excitation

17. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Parameters in Relax state Parameters in Stress state
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Statistical processing was performed on the model parameters
at both relax state and stress state
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
the
diagonal cell
the
diagonal cell

18. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
Comparison with relax state and stress state
・parameter C is 30.2 times smaller (decrease)
・parameter B is 1.37 times larger （increase）
・Parameter A is 1.10 times larger （increase）
• • • • • • • •
In the stress state …
in the stress state,
the waveform of the EEG will be simply
and linearly from chaotic complicated behavior.
parameter B and C could be
indicators of human mental state
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state

19. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal
External
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state
EEG x (mV)

20. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state

21. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state
・There was no correlation between the parameters A and B (both linear parameters)

22. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state
・There was no correlation between the parameters A and B (both linear parameters)
・Nonlinear parameter C and linear A have weak positive correlation

23. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state
・There was no correlation between the parameters A and B (both linear parameters)
・Nonlinear parameter C and linear A have weak positive correlation
・Nonlinear parameter C and linear B have strong negative correlation
・The results of these correlation indicate that the existence of the nonlinear
term is crucial in EEG analysis using the mathematical model.

24. ··x + A ·x + Bx + Cx3
= P1cosωt + P2sinωt
A
B
C
P1
P2
ω
A
B
C
P1
P2
ω
A B C P1 P2 ω A B C P1 P2 ω
Parameters in Relax state Parameters in Stress state
Internal Internal
External External
Linear term Nonlinear Periodic excitation
３．Analysis result・・・
Statistical processing was performed on the model parameters
at both relax state and stress state
・There was no correlation between the parameters A and B (both linear parameters)
・Nonlinear parameter C and linear A have weak positive correlation
・Nonlinear parameter C and linear B have strong negative correlation
These experimentally identified parameter maps can be useful tools as a way to visualize
the state of EEG in the future
・The results of these correlation indicate that the existence of the nonlinear
term is crucial in EEG analysis using the mathematical model.

25. Conclusion
we aim to investigate the characteristic of model
parameters for EEG analysis.
1. Parameter identification
(EEG data : relax state and stress state)
2. Comparison between two states.
・parameter C is 30.2 times smaller (decrease)
・parameter B is 1.37 times larger （increase）
・Parameter A is 1.10 times larger （increase）
• • • • • • • •
In the stress state …
・There was no correlation between the parameters A and B (both linear parameters)
・Nonlinear parameter C and linear A have weak positive correlation
・Nonlinear parameter C and linear B have strong negative correlation
Parameters relationship
・The results of these correlation indicate that the existence of the nonlinear
term is crucial in EEG analysis using the mathematical model.