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.
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