# Stochastic resonance

AI and Robotics

Nov 15, 2013 (4 years and 6 months ago)

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Stochastic Resonance

a tutorial

Kang
-
Hun Ahn

Dept. of Physics,

Chungnam Nat’nl Univ.

http://neurodyn.umsl.edu/sr/

40

120

400

Noise

Level

Noise helps !

Paddlefish catch planktons better with electrical noise

Reflex system of human brain functions better with bloody pressure noise

Output: heartbeat

Stochastic Resonance ?

A model : a driven particle interacting with
environment

Basic equation: Langevin equation

Basic equation:

Langevin Equation

;
0
)
(
2
2

x
V
dt
x
d
m
)
(
)
(
2
2
t
F
x
V
dt
x
d
m
env

Consider a system of inertial mass
m

that interacts with its

environment through a conservative potential
V
(x)

and in addition through a complex interaction term

characterized both by friction and noise.

Without friction the dynamic equation is Newton’s equation

Friction and noise in the system is due to the interaction of the

mass m with a large number of degress of freedom in the environ
-

ment. It can be included by adding a time
-
dependent environmelntal

force term to Newton’s equation

Paul Langevin

(1872
-
1946)



dt
t
F
F
T
k
N
N
B
)
(
)
0
(
2

0
)
(
);
(
)
(
2
2

t
F
t
F
dt
dx
x
V
dt
x
d
m
N
N

In many dissipative systems the environmental force can be separated into a

dissipation

(or loss) term proportional to the ensemble average velocity and

a
noise

term due to a random force

Equations of this form are known as
Langevin equations
.

The dissipative term in the Langevin equation causes energy to be transferred

from the system to the environment.

Thermal equilibrium in a system controlled by the Langevin equation is achieved

through the second moment of the noise force, which must satisfy:

Dissipation and Noise is due to the Environment


d
e
t
F
t
F
S
i
N
N



)
(
)
(
2
1
)
(


T
k
d
e
T
k
S
B
i
B



)
(
2
2
1
)
(
)
'
(
2
)
'
(
)
(
t
t
T
k
t
F
t
F
B
N
N

Fundamental Relation between Environmental Noise,
Dissipation and Temperature (Einstein 1905, age 26)

If we assume the noise force is uncorrelated for time scales over which the

harmonic oscillator responds, we have so called
white noise
, and

Noise

Dissipation

Temperature

We can define a spectral density for the (noise) force
-
force correlation function as

For white noise the
spectral density

is constant (independent of frequency):

A particle in a double
-
well:

A simple model for SR

Assume overdamping;

No oscillation

Assume white noise

The mean passage time from

c to c

; C. Gardiner, Handbook of Stochastic Methods (Berlin,Springer)

Significant contribution from

c
z
y

,
0
If

V
Expanding the potential up to second order

Kramer

s formula

Transition rate

The passage times are exponentially distributed with the mean value

W
/
1


Consider
small periodic modulation

V
V

1

The driving force do not perform deterministic transition

Then

for
smaller than intrawall relaxation rate

s

Stochastic resonance; The mechanism

When the transition time is about one half cycle of the periodic modulation,

The response is optimally enhanced.

A numerical result

2
.
0
/
1

V
V
s
s
100
/
2

The power spectrum of the dynamical process

Signal to noise ratio;

Quantifying the stochastic resonance

Sampling time

)
/
(
log
10
)
(
0
10
P
P
dB
P

54
.
0

s
t
10000
max

Average over

20 samples

The power spectrum

)
(
log
10
)
(
10
SNR
dB
SNR

Caution: SNR is not insensitive to the sampling time.

Stochastic Resonance in sterocilla?

Maybe
««

with

Prof. S.Park

x

y(t)

Proposal for a biomimetic nanoelectromechanical sensor

With Prof. Y.D. Park

Stochastic resonance is just a phenomenon among many
phenomena in biological systems

There are many interesting effects in nonlinear systems which
might be useful to biomimetics,…stochastic synchronization,
mode locking, solitonic wave, etc.

Demonstration of complex biomimetic systems utilizing such a
various phenomena which truly mimic living organs ( in the
sense of their functions) will be a great challenge.