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338 ISBN 9781538643426
Proceedings of IEEE International Conference on Applied System Innovation 2018
IEEE ICASI 2018 Meen, Prior & Lam (Eds)
1
Fractional Order PID System for Suppressing Epileptic Activities
Ahmed Soltan1, Lijuan Xia1, Andrew Jackson2, Graeme Chester1, and Patrick Degenaar1
School of Engineering, Newcastle University, Newcastle upon Tyne, UK
Institude of Neuroscience, Newcastle University, Newcastle upon Tyne, UK
Email: (ahmed.abdelaal, l.xia , andrew.jackson , graeme.chester , @ncl.ac.uk, patrick.degenaar)@newcastle.ac.uk
Abstract
Epilepsy is a dynamic disorder of the brain at the system level
which is due to abnormal activity of the brain cells. A closed
loop control system is designed in this work to detect the
epileptic seizures and hence to suppress it through stimulating
the brain cells. ProportionalIntegralDerivative (PID) is the
most extensively used closed loop controller because of its
simple implementation and robust performance. Although
some efforts have been done to use PID controller to suppress
the abnormal brain activities, the previously proposed systems
were limited for specific cases or model parameters due to the
stability constraints. This work is proposing to use the
fractional order PID which is the generalization of the
traditional PID system to suppress the epileptic seizures. By
using the fractional PID, different stability domains are created
based on the fractional orders and hence more degree of
freedom for the system parameters. In this work, the Neural
Mass Model (NMM) is used as a test platform for the controller
for suppressing the brain activity. A graphical technique for
stability contours is illustrated in this work to make the
parameters determination easy for different stability conditions.
MATLAB simulations are conducted to verify the controller
performance, and the simulation results show the ability of the
controller to suppress the focal epilepsy seizures at different
scenarios.
Key words: Fractional calculus, PID, Epilepsy suppression,
closedloop control, feedback.
I. Introduction
Brain neurological disorders such as epilepsy are dynamical
disorders due to abnormal activities of the brain cell [1]. Indeed,
many electrodes/optrodes have been proposed in the last few
years to interface with brain cells [2, 3, 4]. It is impossible to
cure such disorders using feedforward stimulation techniques
due to it dynamical and unpredictable behaviour. Hence,
closedloop controllers are used to detect the abnormal
activities and then through the control algorithm the controller
decides to suppress these abnormal activities [5, 6]. Although
many algorithms have been proposed to be used for closed
loop control systems, most of these techniques have not been
implemented in an implantable device. This is because the
parameter tuning was done manually or the parameters value
is very large. Moreover, previously proposed closedloop
controllers for brain interface have limited number of
parameters (for example for PI controller, there are two
parameter) while the cell culture models such as Neural
Mass Model (NMM) [6, 7] has many variables change to
model the cell response. As a result, it is very difficult to match
the change in these large number of parameters system (9
parameters for NMM) with only two parameters (for PI as an
example).
On the other hand, a significant amount of research has been
done on using fractional calculus in different areas of research
such as circuit design [8], biomedical modelling [9] and natural
phenomena [10]. This is because the additional parameters
introduced by using fractional calculus which introduce
additional degree of freedom and flexibility in the optimization
process. Fractional calculus is the general form of the integer
order and one common definition of the fractional calculus is
Caputo definition of fractional order derivative which is given
in (1) [11]:
(1)
Where is the initial conditions, is the required time of
calculation and is the fractional order and it is a real number.
Hence, in this work a generalized form of the Proportional
IntegralDerivative (PID) will used to control the epileptic
seizures. The proposed PID consists of fractional order
derivative of order and fractional order integration of order
rather the traditional PID. Indeed, the fractional PID has
been established previously in the control theory applications
and it showed a great flexibility in the system design and tuning
[11]. Moreover, the NMM model is used to model the neural
behavior and to generate artificial epileptic signal to test the
controller system.
II. Neural Mass Model
Neural Mass Model (NMM) is one of the common
computational models used to model a complex neurological
phenomena. The NMM is based on a biologically plausible
parametrization of the of the layered neocortex dynamic
behavior
Fig. 1 (a) schematic diagram shows the relation between different
neural subpopulation in the NMM model, (b) block diagram of the
PID controller to suppress the epileptic seizures.
2
In this model, a cortical sheet is represented by three
interacting populations; the main subpopulation, the excitatory
subpopulation and the inhibitory subpopulation as shown in
Fig. 1. These subpopulations are connected and interacted with
each other via a connectivity constants which represents the
average synaptic constants. The transfer function of the NMM
model is given in (1).
(2)
Where
is the subpopulation transfer function,
is the average excitatory synaptic gain, is the average
time constant for the excitatory synaptic. And
,
and are the average gain and time constant inhibitory
synaptic respectively. And are the connectivity
constants that represent the interaction between subpopulations.
Indeed, NMM is able to describe and generate general cortical
electrical activities such as electroencephalogram and epileptic
seizures. Hence, NMM is used in this work to generate
epileptic seizures to demonstrate how the proposed control
system will suppress these seizures. To simplify the analysis of
the control system, the transfer function of the NMM will be
represented by (3). This also, will make the parameters of the
control system independent on the parameters of the NMM
model. (3)
Where and are the real and imaginary parts of
respectively. From (2), the amplitude of the activity
signal depends on many parameters which shows a wide
variation on the amplitude.
Fig. 2 Change of the magnitude of the NMM which represents the
change in the neural response due to the change on the C1 and C2
parameters.
Table I Summary of the NMM model parameters used in the analysis
Parameter
Value
10
3.25
17
0.02
0.0108
2.5
0.56
6
For example, the change of the magnitude due to the change on
the parameters is illustrated in Fig. 2. Moreover, the
impact of the parameters on the amplitude is nonlinear and
nonpredictable as shown in Fig. 2. Hence, a flexible system is
required to compensate for these changes during the runtime.
The analysis illustrated in Fig. 2 and the following analysis is
done using the parameters value summarized in Table I.
III. Control System
A. Fractional order PID
Proportional Integral – Derivative (PID) controller is the
most used closed –loop in the field of control theory because
of its simple implementation and robust performance. Hence,
a modified version of the traditional PID will be used in this
work to control the neural activities. The modified PID is based
on using fractional order elements to achieve the fractional
integration and derivative. Then, the polynomial of the
fractional PID (FPID) is given in (4).
(4)
The system (3) consists of a fractional integral of order and
integration constant while, the derivative is of order and
derivative constant, the propagation constant is. From
(4) the major advantage of using the FPID is the additional
degree of freedom introduced by have two more design
parameters. Due to the vast number of variables in the
NMM, the value of the constants and are very large
for the integer order PID. This in turn makes the control system
very slow to compensate for the changes. Moreover, these
large values make the implementation process very
challenging as it results in large values for the components.
Hence, this is not suitable for the medical applications as it
require very small circuit size and hence very small
components.
On the other hand, the fractional orders in case of FPID can
be used to compensate for the large values of the components
and limit the transfer function parameters as will be shown in
the following analysis. From Fig. 1(b), the transfer function of
the system in case of detecting a seizure is given by (5).
(5)
From (4), the characteristic equation of the controller is
given as follows:
(5)
From (4, 5), the FPID controller response is function also of
the fractional orders which increases the design degree of
freedom and flexibility.
B. Stability analysis
Basically, PID controller is a feedback loop system and hence
Proceedings of IEEE International Conference on Applied System Innovation 2018
IEEE ICASI 2018 Meen, Prior & Lam (Eds)
339
ISBN 9781538643426
1
Fractional Order PID System for Suppressing Epileptic Activities
Ahmed Soltan1, Lijuan Xia1, Andrew Jackson2, Graeme Chester1, and Patrick Degenaar1
School of Engineering, Newcastle University, Newcastle upon Tyne, UK
Institude of Neuroscience, Newcastle University, Newcastle upon Tyne, UK
Email: (ahmed.abdelaal, l.xia , andrew.jackson , graeme.chester , @ncl.ac.uk, patrick.degenaar)@newcastle.ac.uk
Abstract
Epilepsy is a dynamic disorder of the brain at the system level
which is due to abnormal activity of the brain cells. A closed
loop control system is designed in this work to detect the
epileptic seizures and hence to suppress it through stimulating
the brain cells. ProportionalIntegralDerivative (PID) is the
most extensively used closed loop controller because of its
simple implementation and robust performance. Although
some efforts have been done to use PID controller to suppress
the abnormal brain activities, the previously proposed systems
were limited for specific cases or model parameters due to the
stability constraints. This work is proposing to use the
fractional order PID which is the generalization of the
traditional PID system to suppress the epileptic seizures. By
using the fractional PID, different stability domains are created
based on the fractional orders and hence more degree of
freedom for the system parameters. In this work, the Neural
Mass Model (NMM) is used as a test platform for the controller
for suppressing the brain activity. A graphical technique for
stability contours is illustrated in this work to make the
parameters determination easy for different stability conditions.
MATLAB simulations are conducted to verify the controller
performance, and the simulation results show the ability of the
controller to suppress the focal epilepsy seizures at different
scenarios.
Key words: Fractional calculus, PID, Epilepsy suppression,
closedloop control, feedback.
I. Introduction
Brain neurological disorders such as epilepsy are dynamical
disorders due to abnormal activities of the brain cell [1]. Indeed,
many electrodes/optrodes have been proposed in the last few
years to interface with brain cells [2, 3, 4]. It is impossible to
cure such disorders using feedforward stimulation techniques
due to it dynamical and unpredictable behaviour. Hence,
closedloop controllers are used to detect the abnormal
activities and then through the control algorithm the controller
decides to suppress these abnormal activities [5, 6]. Although
many algorithms have been proposed to be used for closed
loop control systems, most of these techniques have not been
implemented in an implantable device. This is because the
parameter tuning was done manually or the parameters value
is very large. Moreover, previously proposed closedloop
controllers for brain interface have limited number of
parameters (for example for PI controller, there are two
parameter) while the cell culture models such as Neural
Mass Model (NMM) [6, 7] has many variables change to
model the cell response. As a result, it is very difficult to match
the change in these large number of parameters system (9
parameters for NMM) with only two parameters (for PI as an
example).
On the other hand, a significant amount of research has been
done on using fractional calculus in different areas of research
such as circuit design [8], biomedical modelling [9] and natural
phenomena [10]. This is because the additional parameters
introduced by using fractional calculus which introduce
additional degree of freedom and flexibility in the optimization
process. Fractional calculus is the general form of the integer
order and one common definition of the fractional calculus is
Caputo definition of fractional order derivative which is given
in (1) [11]:
(1)
Where is the initial conditions, is the required time of
calculation and is the fractional order and it is a real number.
Hence, in this work a generalized form of the Proportional
IntegralDerivative (PID) will used to control the epileptic
seizures. The proposed PID consists of fractional order
derivative of order and fractional order integration of order
rather the traditional PID. Indeed, the fractional PID has
been established previously in the control theory applications
and it showed a great flexibility in the system design and tuning
[11]. Moreover, the NMM model is used to model the neural
behavior and to generate artificial epileptic signal to test the
controller system.
II. Neural Mass Model
Neural Mass Model (NMM) is one of the common
computational models used to model a complex neurological
phenomena. The NMM is based on a biologically plausible
parametrization of the of the layered neocortex dynamic
behavior
Fig. 1 (a) schematic diagram shows the relation between different
neural subpopulation in the NMM model, (b) block diagram of the
PID controller to suppress the epileptic seizures.
2
In this model, a cortical sheet is represented by three
interacting populations; the main subpopulation, the excitatory
subpopulation and the inhibitory subpopulation as shown in
Fig. 1. These subpopulations are connected and interacted with
each other via a connectivity constants which represents the
average synaptic constants. The transfer function of the NMM
model is given in (1).
(2)
Where
is the subpopulation transfer function,
is the average excitatory synaptic gain, is the average
time constant for the excitatory synaptic. And
,
and are the average gain and time constant inhibitory
synaptic respectively. And are the connectivity
constants that represent the interaction between subpopulations.
Indeed, NMM is able to describe and generate general cortical
electrical activities such as electroencephalogram and epileptic
seizures. Hence, NMM is used in this work to generate
epileptic seizures to demonstrate how the proposed control
system will suppress these seizures. To simplify the analysis of
the control system, the transfer function of the NMM will be
represented by (3). This also, will make the parameters of the
control system independent on the parameters of the NMM
model. (3)
Where and are the real and imaginary parts of
respectively. From (2), the amplitude of the activity
signal depends on many parameters which shows a wide
variation on the amplitude.
Fig. 2 Change of the magnitude of the NMM which represents the
change in the neural response due to the change on the C1 and C2
parameters.
Table I Summary of the NMM model parameters used in the analysis
Parameter
Value
10
3.25
17
0.02
0.0108
2.5
0.56
6
For example, the change of the magnitude due to the change on
the parameters is illustrated in Fig. 2. Moreover, the
impact of the parameters on the amplitude is nonlinear and
nonpredictable as shown in Fig. 2. Hence, a flexible system is
required to compensate for these changes during the runtime.
The analysis illustrated in Fig. 2 and the following analysis is
done using the parameters value summarized in Table I.
III. Control System
A. Fractional order PID
Proportional Integral – Derivative (PID) controller is the
most used closed –loop in the field of control theory because
of its simple implementation and robust performance. Hence,
a modified version of the traditional PID will be used in this
work to control the neural activities. The modified PID is based
on using fractional order elements to achieve the fractional
integration and derivative. Then, the polynomial of the
fractional PID (FPID) is given in (4).
(4)
The system (3) consists of a fractional integral of order and
integration constant while, the derivative is of order and
derivative constant, the propagation constant is. From
(4) the major advantage of using the FPID is the additional
degree of freedom introduced by have two more design
parameters. Due to the vast number of variables in the
NMM, the value of the constants and are very large
for the integer order PID. This in turn makes the control system
very slow to compensate for the changes. Moreover, these
large values make the implementation process very
challenging as it results in large values for the components.
Hence, this is not suitable for the medical applications as it
require very small circuit size and hence very small
components.
On the other hand, the fractional orders in case of FPID can
be used to compensate for the large values of the components
and limit the transfer function parameters as will be shown in
the following analysis. From Fig. 1(b), the transfer function of
the system in case of detecting a seizure is given by (5).
(5)
From (4), the characteristic equation of the controller is
given as follows:
(5)
From (4, 5), the FPID controller response is function also of
the fractional orders which increases the design degree of
freedom and flexibility.
B. Stability analysis
Basically, PID controller is a feedback loop system and hence
340 ISBN 9781538643426
Proceedings of IEEE International Conference on Applied System Innovation 2018
IEEE ICASI 2018 Meen, Prior & Lam (Eds)
3
stability represents one of the main constrains of the system.
Indeed, a graphical method for stability analysis of the integer
order PID is proposed in [5].
Fig. 3 Illustrative diagram to show different stability domains for
different fractional orders
Yet, the values of the transfer function parameters that ensure
stable system are very large using the technique proposed in
[12]. This affects the PID delay performance and the
implementation.
On the other hand, fractional calculus gives more flexibilities
in stability analysis. This is because, every fractional order
system can be expressed in a different Riemann sheet as shown
in the illustrative diagram of Fig. 3.
Moreover, the system can be designed for a specific
stability parameters as given in [12]. From (6), the transfer
function parameters and which satisfy the stability
condition are given by:
(7.a)
(7.b)
(7.c)
Where are given by:
(8.a)
(8.b)
From (7), the PID controller parameters are function of the
fractional orders. Hence, stability contours are function of
the fractional orders. Then, the stability contour can be
designed for specific value of the PID controllers by changing
the fractional orders only. For traditional PI controller (
and), the stability contours will be given by (9)
which is the same as the contour relations given in [6]. This
confirms that the analysis given in [6] is a special case of the
analysis introduced in this work.
(9.a)
(9.b)
Different stability contours for the traditional PI of (9) are
shown in the surface diagram of Fig. 4(a). The value of has
a very broad range of change ranges from 0 to based on
the frequency range.
(a)
(b)
Fig. 4 Analysis for the PID system using the parameters of Table I and
(a) Surface diagram of the
stability contours for the integer order PID system which show large
values of versus that achieve stability condition, (b) Stability
contours for the fractional PID system with which shows the
value of decreased by one decade.
Furthermore, the stable region decrease with the increase of
and hence the system will require long time interval to
stabilize.
On the other hand, by regenerating the analysis of Fig. 4(a)
with only changing the fractional order from 1 to 0.4, the
value of scales down by one order of magnitude as depicted
in Fig. 4(b). Hence, the fractional order can be used to
compensate for the large value of the controller parameters.
Moreover, the value of change broadly with the change in
the magnitude change of the NMM model as shown in Fig. 5
while the magnitude change represented by the change in .
Yet, the fractional order can be used to tune the parameter
in order to compensate for the change in the NMM amplitude
while keeping the value of smaller than the traditional case.
Fig. 5 Change of with respect to the fractional order and
using the parameters of Table I and
.
4
IV. Case study
To verify the analysis introduced in the previous sections, a
time domain analysis for the whole system of Fig.1 before and
after including the PID response. The system is simulated using
MATLAB for 16 second; the first 8 seconds represents only
the NMM response in the time domain while the second 8
seconds after including the impact of the PID controller in the
loop. In this work, the decision about the existence of epileptic
seizures is based on a threshold value for simplifying the
analysis.
The MATLAB simulation is illustrated in Fig. 6 for two
scenarios; the first one is for the traditional PID system (=
1,=0). The second case is for a fractional PID with =
0.5 and =0. The simulation has been done for zero input
error. The output from the system is almost same as shown in
Fig. 6 although PID parameters are different for both cases. For
the integer order PID =125,=1.54,=0. On the
other hand, for fractional PID the parameters are =
0.5,=125,=200,=0. This shows the impact of
using the fractional order calculus to compensate for the large
values of the integer order PID. Hence, fractional order PID
make the system implementation more suitable for biomedical
implantable devices. Especially, it has been shown that the
fractional operator can be implemented on FPGA using less
than 10 Flipflops [13].
Fig. 6 Simulation of the NMM system with both integer order and
fractional order PID using same parameters for the NMM mentioned
in Table I
V. Conclusion
A fractional order PID system is proposed in this work to
control epileptic seizures. The fractional order is able to
compensate for the large values of the system parameters
which make the system suitable for the implantable devices. A
surface diagram for different stability contours has been
introduced for both the integer and fractional order PID system.
Finally a case study is presented to verify the analysis and a
very good matching between the outputs of the fractional and
integer order PID is found.
Acknowledgement
The authors would also like to thank the Wellcome Trust
(102037/Z/13/Z) and the Engineering and Physical Sciences
Research Council (NS/A000026/1) for funding the CANDO
(www.cando.ac.uk) project.
References
[1]
Junwen Luo, et al, "Optogenetics in Silicon: A neural processor
for predicting optically active neural networks," IEEE
transactions on biomedical circuits and systems, vol. 11, no. 1,
pp. 1527, 2017.
[2]
Hung Cao, et al, "An integrated $\mu$LED optrode for
optogenetic stimulation and electrical recording," Biomedical
Engineering, IEEE Transactions on, vol. 60, no. 1, pp. 225229,
2013.
[3]
R. Ramezani, et al, "OnProbe Neural Interface ASIC for
Combined Electrical Recording and Optogenetic Stimulation,"
IEEE Transactions on Biomedical Circuits and Systems, vol. In
press, 2018.
[4]
Hubin Zhao, et al, "A Scalable Optoelectronic Neural Probe
Architecture With SelfDiagnostic Capability," IEEE
Transactions on Circuits and Systems I: Regular Papers, p. In
Press, 2018.
[5]
M. T. Salam , et al,"Seizure Suppression Efficacy of Closed
Loop Versus OpenLoop Deep Brain Stimulation in a Rodent
Model of Epilepsy," IEEE Transactions on Neural Systems and
Rehabilitation Engineering, vol. 24, no. 6, pp. 710719, 2016.
[6]
Junsong Wang , et al, "Suppressing epileptic activity in a neural
mass model using a closedloop proportionalintegral
controller," Scientific reports, vol. 6, p. 27344, 2016.
[7]
Ben H Jansen, et al, "Electroencephalogram and visual evoked
potential generation in a mathematical model of coupled
cortical columns," Biological cybernetics, vol. 73, no. 4, pp.
357366, 1995.
[8]
A. Soltan Ali, et al, "Fractional order Butterworth filter: active
and passive realizations," IEEE Journal on emerging and
selected topics in circuits and systems, vol. 3, no. 3, pp. 346
354, 2013.
[9]
T. J. Freeborn, et al, "Cole impedance extractions from the step
response of a current excited fruit sample," Computers and
electronics in agriculture, vol. 98, pp. 100108, 2013.
[10]
Dipanjan Saha, et al, "Effect of initialization on a class of
fractional order systems: experimental verification and
dependence on nature of past history and system parameters,"
Circuits, Systems, and Signal Processing, vol. 32, no. 4, pp.
15011522, 2013.
[11]
R. Caponetto, Fractional order systems: modeling and control
applications, World Scientific, 2010.
[12]
A. Soltan, et al,"Fractional order sallenKey and KHN filters:
stability and poles allocation," Circuits, Systems, and Signal
Processing, vol. 34, no. 5, pp. 14611480, 2014.
[13]
M. F. Tolba , et al,"FPGA realization of Caputo and Grunwald
Letnikov operators," in 2017 6th International Conference on
Modern Circuits and Systems Technologies (MOCAST), 2017.
Proceedings of IEEE International Conference on Applied System Innovation 2018
IEEE ICASI 2018 Meen, Prior & Lam (Eds)
341
ISBN 9781538643426
3
stability represents one of the main constrains of the system.
Indeed, a graphical method for stability analysis of the integer
order PID is proposed in [5].
Fig. 3 Illustrative diagram to show different stability domains for
different fractional orders
Yet, the values of the transfer function parameters that ensure
stable system are very large using the technique proposed in
[12]. This affects the PID delay performance and the
implementation.
On the other hand, fractional calculus gives more flexibilities
in stability analysis. This is because, every fractional order
system can be expressed in a different Riemann sheet as shown
in the illustrative diagram of Fig. 3.
Moreover, the system can be designed for a specific
stability parameters as given in [12]. From (6), the transfer
function parameters and which satisfy the stability
condition are given by:
(7.a)
(7.b)
(7.c)
Where are given by:
(8.a)
(8.b)
From (7), the PID controller parameters are function of the
fractional orders. Hence, stability contours are function of
the fractional orders. Then, the stability contour can be
designed for specific value of the PID controllers by changing
the fractional orders only. For traditional PI controller (
and), the stability contours will be given by (9)
which is the same as the contour relations given in [6]. This
confirms that the analysis given in [6] is a special case of the
analysis introduced in this work.
(9.a)
(9.b)
Different stability contours for the traditional PI of (9) are
shown in the surface diagram of Fig. 4(a). The value of has
a very broad range of change ranges from 0 to based on
the frequency range.
(a)
(b)
Fig. 4 Analysis for the PID system using the parameters of Table I and
(a) Surface diagram of the
stability contours for the integer order PID system which show large
values of versus that achieve stability condition, (b) Stability
contours for the fractional PID system with which shows the
value of decreased by one decade.
Furthermore, the stable region decrease with the increase of
and hence the system will require long time interval to
stabilize.
On the other hand, by regenerating the analysis of Fig. 4(a)
with only changing the fractional order from 1 to 0.4, the
value of scales down by one order of magnitude as depicted
in Fig. 4(b). Hence, the fractional order can be used to
compensate for the large value of the controller parameters.
Moreover, the value of change broadly with the change in
the magnitude change of the NMM model as shown in Fig. 5
while the magnitude change represented by the change in .
Yet, the fractional order can be used to tune the parameter
in order to compensate for the change in the NMM amplitude
while keeping the value of smaller than the traditional case.
Fig. 5 Change of with respect to the fractional order and
using the parameters of Table I and
.
4
IV. Case study
To verify the analysis introduced in the previous sections, a
time domain analysis for the whole system of Fig.1 before and
after including the PID response. The system is simulated using
MATLAB for 16 second; the first 8 seconds represents only
the NMM response in the time domain while the second 8
seconds after including the impact of the PID controller in the
loop. In this work, the decision about the existence of epileptic
seizures is based on a threshold value for simplifying the
analysis.
The MATLAB simulation is illustrated in Fig. 6 for two
scenarios; the first one is for the traditional PID system (=
1,=0). The second case is for a fractional PID with =
0.5 and =0. The simulation has been done for zero input
error. The output from the system is almost same as shown in
Fig. 6 although PID parameters are different for both cases. For
the integer order PID =125,=1.54,=0. On the
other hand, for fractional PID the parameters are =
0.5,=125,=200,=0. This shows the impact of
using the fractional order calculus to compensate for the large
values of the integer order PID. Hence, fractional order PID
make the system implementation more suitable for biomedical
implantable devices. Especially, it has been shown that the
fractional operator can be implemented on FPGA using less
than 10 Flipflops [13].
Fig. 6 Simulation of the NMM system with both integer order and
fractional order PID using same parameters for the NMM mentioned
in Table I
V. Conclusion
A fractional order PID system is proposed in this work to
control epileptic seizures. The fractional order is able to
compensate for the large values of the system parameters
which make the system suitable for the implantable devices. A
surface diagram for different stability contours has been
introduced for both the integer and fractional order PID system.
Finally a case study is presented to verify the analysis and a
very good matching between the outputs of the fractional and
integer order PID is found.
Acknowledgement
The authors would also like to thank the Wellcome Trust
(102037/Z/13/Z) and the Engineering and Physical Sciences
Research Council (NS/A000026/1) for funding the CANDO
(www.cando.ac.uk) project.
References
[1]
Junwen Luo, et al, "Optogenetics in Silicon: A neural processor
for predicting optically active neural networks," IEEE
transactions on biomedical circuits and systems, vol. 11, no. 1,
pp. 1527, 2017.
[2]
Hung Cao, et al, "An integrated $\mu$LED optrode for
optogenetic
stimulation and electrical recording,"
Biomedical
Engineering, IEEE Transactions on,
vol. 60, no. 1, pp. 225
229,
2013.
[3]
R. Ramezani, et al, "OnProbe Neural Interface ASIC for
Combined Electrical Recording and Optogenetic Stimulation,"
IEEE Transactions on Biomedical Circuits and Systems, vol. In
press, 2018.
[4]
Hubin Zhao, et al, "A Scalable Optoelectronic Neural Probe
Architecture With Self
Diagnostic Capability,"
IEEE
Transactions on Circuits and Systems I: Regular Papers, p. In
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