Dante R. Chialvo

Department of Physiology

Northwestern University

Chicago, IL.

d-chialvo@northwestern.edu Papers: www.chialvo.net

Ghost Resonances: How Brains Can See What Isn't Out There

We work out a problem first investigated by Pythagoras: how the brain determine the pitch of a complex sound?

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, 2003. (others in www.chialvo.net)

Pythagoras in the blacksmith’s shop

Pythagoras experimenting with the “proportions of music”.

Que determina el pitch de un sonido complejoEs un problema clasico

Important historical references:Important historical references:

G. OhmG. Ohm. Ann. Phys. Chem. 59, . Ann. Phys. Chem. 59, 513-565, 1843. 513-565, 1843.

A. Seebeck.A. Seebeck. Ann. Phys. Chem. Ann. Phys. Chem. 60, 449-481, 1843. 60, 449-481, 1843. (Experimental results dismissed (Experimental results dismissed by Helmholtz )by Helmholtz )

H.L.F. von HelmholtzH.L.F. von Helmholtz “On the “On the sensations of tone as a sensations of tone as a physiological basis for the physiological basis for the theory of music” (now we know theory of music” (now we know he was wrong on this)he was wrong on this)

Pitch: “Atributo de un tono por el cual se lo puede Pitch: “Atributo de un tono por el cual se lo puede ordenar en una escala de bajo a alto”ordenar en una escala de bajo a alto”Example:(High versus Low pitch voice, etc)Example:(High versus Low pitch voice, etc)

Para un tono simple (diapason):Para un tono simple (diapason): Pitch: Pitch: es la es la frequenciafrequencia del tono. del tono.

Para un sonido complejo:Para un sonido complejo: Pitch: Pitch: Es un atributo subjective.Es un atributo subjective. No hay un calibre objectivoNo hay un calibre objectivo one can use to judge the pitch of one can use to judge the pitch of a complex tone.a complex tone.

La ilusion de la “missing fundamental” :La ilusion de la “missing fundamental” : Escuchamos fEscuchamos f0 0 , que NO existe…, que NO existe…

a+b

a

b

tiempo

512 Hz f1= 2f0

f2= 3f0

Un tono complejo Un tono complejo El tiempo entre los picos de interferencia es ~ 1/fEl tiempo entre los picos de interferencia es ~ 1/f00

Un tono simpleUn tono simple

Un tono simpleUn tono simple

768 Hz

F0 256 Hz

512+768 Hz

Tres observaciones:Tres observaciones:

1. El pitch percibido 1. El pitch percibido “shifts” se corre“shifts” se corre

2.2. Linias puntos ~ 1/n Linias puntos ~ 1/n peropero los los

Datos ~ 1/(n+.. algo)Datos ~ 1/(n+.. algo)

3. Ambiguidad 3. Ambiguidad (multiples (multiples perceptiones para perceptiones para cada f )cada f )

Experimental results from three listeners; Schouten et al. Experimental results from three listeners; Schouten et al. J. Acoust . Soc. Am. 34 (8) 1962. “Pitch of the residue”J. Acoust . Soc. Am. 34 (8) 1962. “Pitch of the residue”

Los experimentos de Schouten’s de pitch shift (1962)Los experimentos de Schouten’s de pitch shift (1962)

p p ff

0.2

1.0

1.4

0.6

1.8

2.2

Freq

uenc

y (k

Hz)

Amplitude

ff

pp

Que mecanismo neural mechanism reproduce Que mecanismo neural mechanism reproduce quantitativamentequantitativamente los datos de Schouten’s? los datos de Schouten’s?

The bottom line:The bottom line:

and:and: explique de una los otros asectos explique de una los otros asectos relevante derelevante de la fenomenologia del pitch la fenomenologia del pitch

Sospechamos de dos processes envueltos en la percepcion de la altura de tonos complejos :

uno lineal y otro no lineal

1) 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 ?

Podria ser que todo lo que Podria ser que todo lo que necesitamos es resonancia necesitamos es resonancia

estocastica de la interferencia?estocastica de la interferencia?

Sidetrack I: Que es resonancia Sidetrack I: Que es resonancia estocastica?estocastica?

Intensidad del Ruido

Outp

ut

Sig

nal/N

ois

e

Sistema Lineal

NoLinear

SR is characterized by an improvement of the OUTPUT signal/noise by an increase in the INPUT noise….

There is an optimum There is an optimum noise intensity for which noise intensity for which the neuron fires at the the neuron fires at the rate of the missing rate of the missing fundamental (i.e., fundamental (i.e., ffo o ))

Stochastic Resonance para 2 frequencias (harmonicas)Stochastic Resonance para 2 frequencias (harmonicas)

Neuron response to two-frequencies tones for increasing noise. 1/f0 are the most probable inter-spike intervals

x(t)= A (cos f1 t + cos f2 t ) +

f1 = qfo f2 = (q+1)fo

A noisy threshold easily A noisy threshold easily detects the peaksdetects the peaks

Ghost resonance para two-frequencies (inharmonic) Ghost resonance para two-frequencies (inharmonic) signalssignals

x(t)= A (cos f1 t + cos f2 t ) +

f1 = qfo + f

f2 = (q+1)fo + f

Neuron response to two-frequencies tones for increasing f plotted as a function of f1

with

k=2 k=3 k=4k=5

k=6 fp ~ 1/(k+1/2)

Ghost resonance for mistuned Ghost resonance for mistuned three-frequenciesthree-frequencies signalssignals

x(t)= A (cos f1 t + cos f2 t + cos f3 t) +

f1 = qfo + f

f2 = (q+1)fo + f

f3 = (q+2)fo + fNeuron response to three-frequencies tones for

increasing f plotted as a function of f1

with

k=2 k=3 k=4k=5

k=6 fp ~ 1/(k+1)

La solucion: la respuesta de la neuroan a tonos complejos compuestos de pares o impares de frecuencias

Para estimulos con N sinusoides de frecuencias:

ffNkffkfkf 000 )1(...,)1(,

la resonancia ocurre a frequencias:

N=2 N=3

Ghost resonance for 2, 3, 4, or 5 frequencies Ghost resonance for 2, 3, 4, or 5 frequencies signalssignals

k=2

k=7

Theory, Numerical and Experimental (Schouten)

Pitch ~ 1/(k+1)three-frequenciesthree-frequencies

Spike trains from cat auditory primary afferent after Cariani P.A. and Delgutte B., J. Neurophysiol. 76, 1698-1716, 1996

Center Frequency f2 (Hz)625

100

200

300

150

250Es

t. P

itch

(Hz)

475 775

n=2

n=3

n=4n=5

n=6

How it compares with In Vivo Experiments (Cariani)

Our theory

Y entonces ?

Moral #0Supersocion Lineal de tonos puros

Ocurre “afuera nuestro”

Moral #1Fundamental is not so fundamental

Time 1 2 3 4 5 6 7 8

The “fundamental” is not so fundamental!

x(t)

x(t)

x(t)

x(t)

x(t)= (cos f1 t+ cos f2t+cos f3t+ … cos fnt) /n

fn= q f0

Frequency

q

Spectrum

The “fundamental” is not so fundamental (Regardless of phase or harmonicity)

x(t)

x(t)

x(t)

x(t)

shifted

shifted

FrequencyTime

x(t)= (cos f1 t+cos f2t+cos f3t+ … cos fnt) /nfn= q f0 +f

f

fRandom phase

Random phase

Equal phase

Equal phase

1 2 3 4 5 6 7 8

q

Spectrum

Both signals produce equal pitch percept of ~ 100 Hz

Amplitude modulated white noise

Complex harmonic tone

Time Frequency

The fundamental is not so fundamental at all !

Moral #2Same phenomena across different systems

Replicated experimentally in semiconductor lasers with optical feedback

UPC (Barcelona): Javier Martin-Buldu , Jordi Garcia Ojalvo, Carme

Torrent and UIB (Mallorca): Claudio Mirasso

Ghost Resonance in a Semiconductor Laser with Optical Feedback Europhys. Letters, 64(2), 2003.

Missing Fundamental

Similar phenomenon already reported in vision*

*K. Fujii et al, Psychological Research (2000) 64:149-154.

The “shift visual experiment” is awaiting!!

“ “Ghost” Stochastic Resonance in more Ghost” Stochastic Resonance in more elaborated numerical modelselaborated numerical models

FHN neuron modelFHN neuron modelsimplest modelsimplest model

Replicated experimentally in the FitzHugh-FitzHugh-NagumoNagumo analog circuit.

Frequency of f 1 (Hz)

Sp

ike I

nst.

Fre

qu

en

cy (

1/I

SI

, H

z)two three

four five

k=2

k=7

Analog “FHN on a chip”

“Newsweek”

Ultrasound 1

Using the same principle for “Target delivery” of audio?

Ultrasound 2

American Tech. San Diego

Blah-Blah-logy:Blah-Blah-logy:The problem of the pitch perception is formalized The problem of the pitch perception is formalized as a as a linear interferencelinear interference of individual tones being of individual tones being nonlinearly detectednonlinearly detected by a by a noisy thresholdnoisy threshold. .

Thus, the perceived pitch is associated with Thus, the perceived pitch is associated with generic dynamic found in excitable systems which generic dynamic found in excitable systems which we call “ghost Stochastic Resonance”.we call “ghost Stochastic Resonance”.

Two general expressions relating mistuning and Two general expressions relating mistuning and pitch shift agree extremely well with the available pitch shift agree extremely well with the available experimental data.experimental data.

Results from psychophysics agrees well with the Results from psychophysics agrees well with the numerics and the theory.numerics and the theory.

Meta Blah-Blah-logy:Meta Blah-Blah-logy:Forget adding sine waves! I am looking at pulse Forget adding sine waves! I am looking at pulse trains and action potentialstrains and action potentials. Interference is even . Interference is even more robust. The same analysis shed light into the more robust. The same analysis shed light into the issue of coincidence detection of spike trains.issue of coincidence detection of spike trains.

Binaural pitch:Binaural pitch: A third neuron reads the virtual A third neuron reads the virtual pitch of two incoming spike trains. (Upcoming paper)pitch of two incoming spike trains. (Upcoming paper)

Why the dominance region of virtual pitch is where Why the dominance region of virtual pitch is where it is? it is? Numerical results shows thatNumerical results shows that the existence the existence region can be predicted from the same interference region can be predicted from the same interference problem.problem.

Looking to test in auditory nerve…Looking to test in auditory nerve…

Cochlear Implants…Cochlear Implants…

Other senses, theory of music (consonances) Other senses, theory of music (consonances) more…more…

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) 050902(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) 1226-1230 (2003).

Theoretical plumber:in biology be carefull

what you ask for