adaptive estimation of firing patterns of hindmarsh-rose ......series data. hindmarsh-rose model (hr...
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Adaptive Estimation of Firing Patterns of Hindmarsh-Rose
Neurons and Synchronization Detection with Instantaneous
Lyapunov Exponents
Ryuta Ito Yusuke Totoki
Haruo Suemitsu Takami Matsuo
Oita University, Japan
Outline
• Background and Objectives
• Hindmarsh-Rose model
• Adaptive observer for LTV-MIMO systems by Zhang
• Synchronization measure
• Simulation results
• Concluding remarks
Background and Objectives
• We propose an adaptive observer that allows us to estimate the parameters and the input signal simultaneously using the time-varying adaptive observer proposed by Zhang.
• We present an synchronization measure by using that is a real-time decay rate of time series data.
Hindmarsh-Rose model (HR model)
(Belykh[1,2005])
Dynamics of Hindmarsh-Rose model
• It is known that the HR neuronal model generates various firing patterns depending on the argument value of the differential equation.(P. Arena et al., Chaos, Soliton and Fractals,2006)
• When fixing to I=0.05(Single neuron): – When a=[1.8,2.85] ,tonic bursting(TB) : IBN – When a >= 2.9,tonic spiking(TS) :ISN
• When fixing to a = 2.8 (Coupled neurons): – When I = [0,0.18] ,tonic bursting – When I = [0.2,5] ,tonic spiking
IBN(Intrinsic Bursting Neuron) ISN(Intrinsic Spiking Neuron)
6
The parameters a and I are key parameters that
determine the firing pattern. We assume that the
membrane potential x is available, but the others
are immeasurable. In this case, we consider
following problems:
(1) Estimate y and z using the available signal x.
(2) Estimate the parameter a or I to distinguish
the firing patterns by using early-time
dynamic behaviors.
IBN(intrinsic bursting neuron) Single neuron
0 200 400 600 800 1000-1.5
-1
-0.5
0
0.5
1
1.5
2
time
x
0 200 400 600 800 1000-1.1
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
time
z
0 200 400 600 800 10000
1
2
3
4
5
6
7
time
y
x
y
z
-10
12
3
0
5
10-1.5
-1
-0.5
0
xy
zz
y x
ISN(intrinsic spiking neuron)
Single neuron
0 200 400 600 800 1000-4
-2
0
2
4
6
8
10
12
time
x
0 200 400 600 800 1000-8
-7
-6
-5
-4
-3
-2
-1
0
time
z
0 200 400 600 800 10000
50
100
150
200
250
time
y
-50
510
15
0
100
200
300-8
-6
-4
-2
0
xy
z
x
y
z
z
x y
Adaptive observer
for LTV-MIMO systems by Zhang
Zhang proposes a new approach to the design of an adaptive observer for the following linear time-varying MIMO system of the form
The system is a global exponentially stable adaptive observer for the MIMO system of the form
Adaptive observer
• Application to HR neuron
We rewrite the single HR neuron as
a vectorized form
Only membrane potential x can be measured.
• Adaptive observer with HR neuron
Using Zhang’s adaptive observer, we have the following
Definition of instantaneous Lyapunov exponent(ILE)
We define instanteous Lyapunov exponents
with respect to a decay rate function φ(t) as :
Moving average of instantaneous Lyapunov exponent
Synchronization measure
The ILE is sensitive to dynamical
noises. To reduce the effect of
noises, we introduce the moving
average of the ILE as another
measure of the ILE as another
measure of the decay rate as :
Estimation of a single neuron
• Observer by Zhang
When IBN
When ISN
0 200 400 600 800 1000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time
x;x
1
0 200 400 600 800 10000
1
2
3
4
5
6
7
time
y;y
1
0 200 400 600 800 1000-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
time
z;z
1 blue lines : states
red lines : estimates
xx ˆ, yy ˆ, zz ˆ,
16
• Observer by Zhang
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
I
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
a
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time
d a+®
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
c ¹b
1
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
time
c¹c
1
Estimation of parameter θ
a I
a+α μb μc
Model with coupled neurons
Dynamical Equations (I.Belykh et al.[1])
We call the neurons the neurons respectively thi iii zyx ,,
Synaptically coupled
Observer by Zhang (IBN-IBN-Weak coupling)(1)
First neuron: IBN neuron Second neuron: IBN neuron
8.21 a8.22 a
05.0sg
Weak coupling
1st neuron
2nd neuron
observer
1I2I
1x
Two IBN neurons synchronize as bursting neurons in the
coupling of two same IBN neurons with the coupling strength
05.0sg
Observer by Zhang (IBN-IBN-Weak coupling)(2)
0 200 400 600 800 1000-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time
x;x
1
0 200 400 600 800 10000
1
2
3
4
5
6
7
time
y;y
1
0 200 400 600 800 1000-1
-0.8
-0.6
-0.4
-0.2
0
time
z;z
1
xx ˆ, yy ˆ,
zz ˆ,)(z
n
(blue lines: states ) (red lines: estimates)
with I = 0.3
)(zn
Observer by Zhang (IBN-IBN-Weak coupling)(3)
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
a
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
I
1
0 200 400 600 800 10000
1
2
3
4
5
time
d a+®
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
c ¹b
1
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
time
c¹c
1
I:timevarying
a I
a+α μb μc
)(In
The states (red lines) and its estimates (blue lines)
)(In
Concluding remarks • The adaptive observer's composition
– Simultaneous estimation by observer of Zhang
1. It is possible to estimate five simultaneous parameters by measuring only the membrane potential.
2. It is not easy to estimate a time-varying parameter.
• ILE
– Stability criterion
• Stability that is weaker than exponent stability
– Synchronous detection
• Synchronization measure of two signals
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