1

FitzHugh-Nagumo equation can describe frequency responses

in human vision

Atsushi Osa, Masaru Suzuki, Koki Otaka,

Shoichi Kai, Hidetoshi Miike

Yamaguchi University

Motion smear/ Sharpening

• The visual system integrates signals over time.

Motion blur Motion blur Motion sharpening Motion sharpening

Motion sharpening

• When objects move fast they look sharper than when they are stationary. – Ramachandran et al.(1974), Bex et al. (1995), Hammett & Bex (1996)・・・

Brightness (stationary)

Perceived shape (moving)

Moving

A typical impulse response function can show the motion sharpening

(Pääkkönen & Morgan 2001)

An impulse response function of the human vision systemAn impulse response function of the human vision system

Steep slope

gradual slope

moving edge

brightness

position

moving edge

brightness

position

motion Sharpening motion Sharpening

motion smear motion smear

From frequency response of vision to impulse response

Sensitively to sin-wave flicker Sensitively to sin-wave flicker

Frequency (Hz)

Sen

siti

vely

Impulse response Impulse response

IFT IFT

model D. H. Kelly (1971)

Impulse responses depending on the average illuminance

• D. H. Kelly (1971)

frequency response frequency response

Impulse responses Impulse responses

Average illuminance

● 9300 td 〇 850 td △ 77 td ◇ 7.1 td □ 0.65 td ■ 0.06 td

Frequency (Hz)

Sen

siti

vely

retina model retina model

2

Frequency response of vision

Average illuminance

● 9300 td 〇 850 td △ 77 td ◇ 7.1 td □ 0.65 td ■ 0.06 td

Sen

siti

vely

Frequency (Hz)

low-pass low-pass

band-pass band-pass

dark stimulus dark stimulus

light stimulus light stimulus

In liner system

• RLC circuit

ftACdt

dqR

dt

qdL 2cos

12

2

The resonant frequency doesn’t depend on the inputThe resonant frequency doesn’t depend on the input

LC

10

Purpose

Liner Transfer function I O

Nonlinear transfer function

using FHN eq.

I O

FitzHugh-Nagumo equation

Spike generations in a neuron

u: activator v: inhibitor

abvudt

dv

IvuuCdt

duext

3

depolarization

hyperpolarization

u

v

Using equation

abvudt

dv

vuuCdt

du 3

CIaubbuCdt

dubuC

dt

ud 113 32

2

2Input Input

one variable

0113 32

2

2

aubbuCdt

dubuC

dt

ud

FHN eq. FHN eq.

Input : infinitesimal amplitude

• Input: I=I0＋Aeit, I0: constant (average illuminant), A: infinitesimal amplitude

• Output: , u0: fixed point

• Power spectrum

bbuCCCubi

ACu

133

~2

0

2

0

2

22

0

2222

0

224

22*

13132

~~

bbuCuCCb

CAuuP

tuuu i

0 )e(~+=

3

Power spectrum

•

– D>=0 : Low-pass

– D<0: Band-pass

• p: peak frequency

22

0

22 132 uCCbD

2/Dp

22

0

2222

0

224

22*

13132

~~

bbuCuCCb

CAuuP

low-pass low-pass

band-pass band-pass

Pp)

2224

22

22 pp

p

bCC

CAP

b=0.8

C=5

Numerical analysis

abvudt

dv

vuuCdt

du 3

ftCACIaubbuCdt

dubuC

dt

ud2cos113 0

32

2

2

Input: I

Input: I0: average illuminance、 A: amplitude of stimulus

ftCACI 2cos0

one variable

0113 32

2

2

aubbuCdt

dubuC

dt

ud

Impulse f=0.1Hz f=0.25Hz

f=0.5Hz f=0.75Hz f=1.0Hz

f=2.5Hz f=5.0Hz f=7.5Hz

Output in various frequency u v

a=0.8

b=0.8

C=5

A=1

sec sec sec

sec sec sec

sec sec sec

A=0.01

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

0.01 0.1 1

Pow

er

Frequency (Hz)

0

0.22

0.44

0.49

0.53

Pp) Pp)

Analysis Analysis Numerical analysis Numerical analysis I I0

Light stimulus Light stimulus

dark stimulus dark stimulus

Difference

Variation range of the peak frequency p

narrow narrow

wide wide

FHN model Human vision

Bigger amplitude

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.05 0.5

Po

wer

Frequency (Hz)

0 0.22

0.44 0.49

0.53

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.05 0.5

Po

we

r

Frequency (Hz)

0 0.22

0.44 0.49

0.53 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.05 0.5

Po

we

r

Frequency (Hz)

0 0.22

0.44 0.49

0.53

Input: I

Input: I0: average illuminance、 A: amplitude of stimulus

ftCACI 2cos0

I I0 I I0

I I0

A=0.01 A=0.01 A=0.1 A=0.1 A=0.4 A=0.4

4

0.0001

0.001

0.01

0.1

1

0.05 0.5

Pow

er

Frequency (Hz)

0 0.1

0.16 0.26

0.34 0.44

0.49 0.53

I I0

A=0.5 A=0.5

Frequency response Frequency response of vision

Frequency response Frequency response of FHN eq.

FHN with noise

avtbudt

dv

vuuCdt

du

)(

3

＋

abvudt

dv

vuuCdt

du 3

noise

Effect of multiplicative noise

0.05 0.5

0 0.1

0.16 0.26

0.38 0.44

0.53

0.05 0.5

0 0.1

0.16 0.18

0.26 0.38

0.44 0.53 0.001

0.01

0.1

1

0.05 0.5

Pow

er

0 0.1 0.16 0.26 0.3 0.34 0.38 0.44 0.53

=0 =0 ：0 ~ 0.4 (uniform distribution) ：0 ~ 0.4 (uniform distribution)

：0 ~ 1 (uniform distribution) ：0 ~ 1 (uniform distribution)

I I0 I I0 I I0

The most similar result

0.001

0.01

0.1

1

0.05 0.5

Pow

er

Frequency (Hz)

0 0.1

0.16 0.18

0.26 0.38

0.44

avtbudt

dv

vuuCdt

du

)(

3

＋

a=0.8

b=0.8

C=5

：0 ~ 0.4 (uniform distribution)

I I0

Conclusions

• FitzHugh-Nagumo equation can describe frequency responses in human vision.

– Dynamical noise stabilizes the response.

FHN eq. I O

avtbudt

dv

vuuCdt

du

)(

3

＋0.001

0.01

0.1

1

0.05 0.5

Po

we

r

Frequency (Hz)

0 0.1 0.16 0.18 0.26 0.38 0.44

Future work1

• More accurate simulation of human vision.

• D.H.Kelly’s retina model + Our model

0.001

0.01

0.1

1

0.05 0.5

Po

we

r

Frequency (Hz)

0 0.1 0.16 0.18 0.26

5

Future work 2

• Simulation of motion sharpening

FHN eq.

avtbudt

dv

vuuCdt

du

)(

3

＋

Motion sharpening Motion sharpening

Future work 3

• Decrease in the flicker fusion rate relates to mental fatigue. • Flicker fusion rate : the frequency at which

an flicker light stimulus appears to be completely steady

• The mechanism has not been revealed yet. • The multiplicative noise in our model works to

decrease the sensitively of flicker. Dynamical noise in brainDynamical noise in brain

Mental fatigue Mental fatigue

relationship ? relationship ?

Thank you for your kind attention