1. Ghost Resonances: How Brains Can See What Isn't Out There
Dante R. Chialvo
We work out a problem first investigated by Pythagoras: how the brain determine the pitch of a complex sound?
Department of Physiology
Northwestern University
Chicago, IL.
Papers:

2. We discuss a case in which the NON LINEAR STOCHASTIC ASPECTS OF SENSORY TRANSDUCTION ARE ESSENTIALS AND CAN NOT BE REPLACED BY LINEAR ONES.
Not always what we perceive is “out there”. How is that and why we care?
We will make three points:
1)The response of a single noisy neuron to inputs composed by multiple frequencies is a preferred resonant “ghost” frequency
2) We find an algebraic expression predicting the frequency at which this “ghost” resonance occur for any arbitrary inputs.
3) The resonant frequency coincide with the perceived “pitch”.
Papers: Physical Review E, 2002; Chaos, (others in

3. Que determina el pitch de un sonido complejo Es un problema clasico
Important historical references:
G. Ohm. Ann. Phys. Chem. 59, , 1843.
A. Seebeck. Ann. Phys. Chem. 60, , (Experimental results dismissed by Helmholtz )
H.L.F. von Helmholtz “On the sensations of tone as a physiological basis for the theory of music” (now we know he was wrong on this)
Pythagoras in the blacksmith’s shop
Pythagoras experimenting with the “proportions of music”.

4. Example:(High versus Low pitch voice, etc)
Pitch: “Atributo de un tono por el cual se lo puede ordenar en una escala de bajo a alto”
Example:(High versus Low pitch voice, etc)
Para un tono simple (diapason):
Pitch:
es la frequencia del tono.
Para un sonido complejo:
Es un atributo subjective.
No hay un calibre objectivo
one can use to judge the pitch of
a complex tone.

5. La ilusion de la “missing fundamental” :
Escuchamos f0 , que NO existe…
El tiempo entre los picos de interferencia es ~ 1/f0
Un tono complejo
a+b Hz
Un tono simple
512 Hz a
f1= 2f0 b
768 Hz
f2= 3f0
tiempo
Un tono simple
F0 256 Hz

6. Los experimentos de Schouten’s de pitch shift (1962)
Tres observaciones:
1. El pitch percibido “shifts” se corre
Linias puntos ~ 1/n pero los
Datos ~ 1/(n+.. algo)
3. Ambiguidad (multiples perceptiones para cada f )
Dp  Df Dp Df
Experimental results from three listeners; Schouten et al. J. Acoust . Soc. Am. 34 (8) “Pitch of the residue”

7. The bottom line:
Que mecanismo neural mechanism reproduce quantitativamente los datos de Schouten’s?
and:
explique de una los otros asectos relevante de la fenomenologia del pitch

8. Sospechamos de dos processes envueltos en la percepcion de la altura de tonos complejos :
uno lineal y otro no lineal
Lineal: tonos Complejos vistos como interferencia constructive .
2) Nolineal: un proceso “seleccionando” los maximos de la interferencia
Son los intervalos entre los picos de interferencia equivalente al perceived pitch? ..como haria una neurona para detectar esos picos ?

9. Sistema Lineal NoLinear Sidetrack I: Que es resonancia estocastica?
Podria ser que todo lo que necesitamos es resonancia estocastica de la interferencia?
Intensidad del Ruido
Output Signal/Noise
Sistema Lineal
NoLinear
SR is characterized by an improvement of the OUTPUT signal/noise by an increase in the
INPUT noise….

10. Stochastic Resonance para 2 frequencias (harmonicas)
A noisy threshold easily detects the peaks
x(t)= A (cos f1 t + cos f2 t ) + e
f1 = qfo
f2 = (q+1)fo
There is an optimum noise intensity for which the neuron fires at the rate of the missing fundamental (i.e., fo )
Neuron response to two-frequencies tones for increasing noise. 1/f0 are the most probable inter-spike intervals

11. Ghost resonance para two-frequencies (inharmonic) signals
k=2
k=3
k=4
k=5
k=6
fp ~ 1/(k+1/2)
x(t)= A (cos f1 t + cos f2 t ) + e
with
f1 = qfo + Df
f2 = (q+1)fo + Df
Neuron response to two-frequencies tones for increasing D f plotted as a function of f1

12. Ghost resonance for mistuned three-frequencies signals
k=2
k=3
k=4
k=5
k=6
fp ~ 1/(k+1)
x(t)= A (cos f1 t + cos f2 t + cos f3 t) + e
with
f1 = qfo + Df
f2 = (q+1)fo + Df
Neuron response to three-frequencies tones for increasing Df plotted as a function of f1
f3 = (q+2)fo + Df

13. La solucion: la respuesta de la neuroan a tonos complejos compuestos de pares o impares de frecuencias
N=2
N=3
Para estimulos con N sinusoides de frecuencias:
la resonancia ocurre a frequencias:

14. Ghost resonance for 2, 3, 4, or 5 frequencies signals
k=2
k=7

15. Theory, Numerical and Experimental (Schouten)
Pitch ~ 1/(k+1)
three-frequencies

16. How it compares with In Vivo Experiments (Cariani)
Our theory
Spike trains from cat auditory primary afferent after Cariani P.A. and Delgutte B., J. Neurophysiol. 76, , 1996

17. Y entonces ?

18. Supersocion Lineal de tonos puros Ocurre “afuera nuestro”
Moral #0
Supersocion Lineal de tonos puros
Ocurre “afuera nuestro”

19. Fundamental is not so fundamental
Moral #1
Fundamental is not so fundamental

20. The “fundamental” is not so fundamental!
Frequency q
Spectrum
x(t)
x(t)
x(t)
x(t)
Time
x(t)= (cos f1 t+ cos f2t+cos f3t+ … cos fnt) /n
fn= q f0

21. The “fundamental” is not so fundamental
(Regardless of phase or harmonicity)
Spectrum
Equal phase
x(t)
Equal phase
shifted Df
x(t)
Random phase
x(t)
Insert here perception of the noise modulated
shifted Df
Random phase
x(t)
Time q
x(t)= (cos f1 t+cos f2t+cos f3t+ … cos fnt) /n
fn= q f0 +Df
Frequency

22. The fundamental is not so fundamental at all !
Complex harmonic tone
Amplitude modulated white noise
Time
Frequency
Both signals produce equal pitch percept of ~ 100 Hz

23. Same phenomena across different systems
Moral #2
Same phenomena across different systems

24. Replicated experimentally in semiconductor lasers with optical feedback
Ghost Resonance in a Semiconductor Laser with Optical Feedback
Europhys. Letters, 64(2), 2003.
UPC (Barcelona): Javier Martin-Buldu , Jordi Garcia Ojalvo, Carme Torrent and UIB (Mallorca): Claudio Mirasso

25. Similar phenomenon already reported in vision*
Missing Fundamental
The “shift visual experiment” is awaiting!!
*K. Fujii et al, Psychological Research (2000) 64:

26. “Ghost” Stochastic Resonance in more elaborated numerical models
simplest model
FHN neuron model

27. Replicated experimentally in the FitzHugh-Nagumo analog circuit.
two
three
four
five
k=2
k=7
Spike Inst. Frequency (1/ISI , Hz)
Analog “FHN on a chip”
Frequency of f 1 (Hz)

28. “Newsweek” Using the same principle for “Target delivery” of audio?
Ultrasound 1
Ultrasound 2
American Tech. San Diego

29. Blah-Blah-logy:
The problem of the pitch perception is formalized as a linear interference of individual tones being nonlinearly detected by a noisy threshold.
Thus, the perceived pitch is associated with generic dynamic found in excitable systems which we call “ghost Stochastic Resonance”.
Two general expressions relating mistuning and pitch shift agree extremely well with the available experimental data.
Results from psychophysics agrees well with the numerics and the theory.

30. Meta Blah-Blah-logy:
Forget adding sine waves! I am looking at pulse trains and action potentials. Interference is even more robust. The same analysis shed light into the issue of coincidence detection of spike trains.
Binaural pitch: A third neuron reads the virtual pitch of two incoming spike trains. (Upcoming paper)
Why the dominance region of virtual pitch is where it is? Numerical results shows that the existence region can be predicted from the same interference problem.
Looking to test in auditory nerve…
Cochlear Implants…
Other senses, theory of music (consonances) more…

31. Recent work (www.chialvo.net):
Chialvo DR, Calvo O, Gonzalez DL, Piro O, Savino, GV: Subharmonic resonance and synchronization in neuronal systems. Physical Review E, 65(5) (R) (2002).
Calvo O, Chialvo DR: Ghost resonance in an electronic circuit. IJBC in press (2005).
Buldu M, Chialvo DR, Mirasso C, Torrent C, Garcia Ojalvo J: Ghost Resonance in a Semiconductor Laser with Optical Feedback. Europhys. Letters 64(2)2003.
Chialvo DR: Illusions and ghost resonances: How we could see what isn’t out there. (Unsolved Problems of Noise, IAP Proceedings, 2002).
Chialvo DR: A neural mechanism for the missing fundamental phenomenon. Chaos, 13(4) (2003).

32. Theoretical plumber: in biology be carefull what you ask for