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