Question: What is the optimal noise with regard to signal transmission?
Answer: There is a FINITE OPTIMAL level of noise at which the response of the system is maximum: STOCHASTIC RESONANCE (SR)
SR Mechanism
WEAK periodic signal
output
Gaussian white noise
Periodically forced Brownian particle in a bistable potential
zero noise: particle oscillates within one well finite noise: jumps between wells possible
Stochastic differential equation of motion:
x˙ =−U ' x Acos t D t
1 1 U x= x 4 − x 2 4 2
noise
+
A≪1 〈t 〉=0
X
variable
=
〈t t−〉=
1 1 U x , t = x 4− x 2− Acos t x 4 2
Time dependent assymetry of the potential
t=
2 t=
t=0
+
noise
3 t= 2
escape time ~
SYNCHRONIZED HOPPING
function of noise intensity (Kramers rate:
−
r K ∝e
V D
)
= ?
Time series at different noise intensities:
High noise
Synchronized hopping Intermediate noise
Low noise
SR history: recurrence of ice ages Benzi et al (1981, 1982) C. Nicolis (1982)
Does SR rule the periodically recurrent ice ages? (periodicity ~ 10000 years) Global climate: double well potential Small modulation of earth's orbital eccentricity: weak periodic forcing Short term climate fluctuations: Gaussian white noise
period ~ 10000 years !
MATHCING OF TWO ASTRONOMICAL TIME SCALES AT OPTIMAL NOISE
First experimental verification of SR Fauve and Heslot (1983) Scmitt trigger device output
Living organisms use noise for optimal detection Douglas et al Nature (1993)
PREDATOR (hungry fish)
hydrodynamically sensitive sensors
PREY (crayfish)
noise: water turbulence periodic force: water vibrations generated by fish tail
The crayfish detects the hungry fish easier on the background of water turbulence!
How does the brain analyze weak and noisy signals ? Simonotto et al PRL (1997)
=
The human visual system makes use of the noise induced enhancement in the information content
Coherence Resonance (CR)
Coherence Resonance
noise
Nonlinear system
output
“Stochastic Resonance without External Periodic Forcing” Gang et al PRL (1993)
SR: response of a bistable system to an external periodic forcing, noise present CR: coherent motion stimulated by the INTRINSIC dynamics of the system
NOISE INDUCED OSCILLATIONS SYSTEM BELOW BIFURCATION (Hopf, saddle node bif. on a limit cycle)
Prototype model for CR Fitz Hugh-Nagumo + noise
dx x3 =x− − y dt 3 dy =xaD t dt
∣a∣1
: stable fixed point
∣a∣1
: limit cycle
“Coherence Resonance in a Noise-Driven Excitable System” Pikovsky and Kurths PRL (1997)
Quantifying CR : measures of coherence Noise-induced limit cycle: 2 time scales !
correlation time : ∞
c =∫0 C 2 t dt autocorrelation function :
C =
〈 y t y t〉 , y = y−〈 y 〉 2 〈 y 〉
normalized fluctuations of pulse durations :
R p=
Var t p 〈t p 〉
low noise
intermediate noise
high noise
There is a maximum in the regularity of oscillations at optimal D
Applications of CR
● ● ● ● ●
Other neuronal and biological systems Chemical models Electronic circuits Semiconductor lasers .....
Conclusions
● ● ● ● ●
Noise enhanced signal transmission stimulated by periodic forcing (SR) Characteristic time scale matching Noise induced motion more regular at optimal noise intensity (CR) Excitability, timescale dependence on noise Constructive role of noise
Bibliography 1) 2) 3) 4)
Gammaitoni et al, Rev. Mod. Physics 70, 223 (1998) Anishenko et al, Physics-Uspekhi 42, 7 (1999) B. Lindner, PhD thesis (2002) H. Engel, Statistische Physik II (lecture notes)
