Download terceraparte - Dante R. Chialvo

Ghost, phantoms and illusions are
the neural basis of tonal music
Dante R. Chialvo
Northwestern University.
d-chialvo@northwestern.edu
Chicago, IL.
ChialvoLab.Northwestern.Edu
Resumen corto:
Si el ojo
puede ser
considerado
una cámara
fotográfica,
Short Summary:
If the eye can
be seen as a
camera,
el oido
seguro que No es
un microfono
the ear
surely is NOT a
microphone
We look at mathematical models to determine how the auditory periphery
(the 1st neuron) could solve complex problems usually attributed to
central processors.
First, we solve a problem first investigated by Pythagoras, later by Ohm
and von Helmholtz: how the brain determines the pitch of a complex
sound. We will make three points:
1)
The output of even a single noisy neuron driven by multiple frequencies
are spikes spaced at the inverse of the “ghost” frequency we call
“pitch”.
2)
We find an expression to predict the frequency of the “ghost”
resonance for some arbitrary inputs.
3)
The ghost resonance suggests the presence of phantoms and other
sensory illusions which will be discussed ten years from now.
Then, we look at the neural mechanism of consonance, the basis of tonal
music, finding that can be mapped almost trivially to the same pitch
problem we just solved.
Last, investigating ways to falsify the theory we found new results applying
to another classic unsolved problem, the octave enlargement.
The problem: How do we perceive pitch of a complex tone ?
The big big picture
The general tendency in the field is:
More complex acoustic attributes => much higher cortical process.
Notice that this chart starts at the brainstem…
Instead we look at generic dynamical properties of
neurons
Algo asi como que puede hacer una neurona por si sola?
Tres cosas preliminares:
1) Miremos como luce el input
2) Que sabemos acerca de como responden las neuronas
- a señales periodicas sin ruido (deterministic)
3) Que sabemos acerca de como responden las neuronas
- a señales periodicas ruidosos (stochastic)
The “fundamental” is not so fundamental!
Spectrum
x(t)
x(t)
x(t)
x(t)
Time
x(t) = (cos f1 t + cos f2 t +cos f3t+ … cos fnt) /n
fn= q f0
12345678
q
Frequency
The “fundamental” is not so fundamental
(Regardless of phase or harmonicity)
Spectrum
Equal phase
x(t)
Equal phase
shifted
x(t)
Df
Random phase
x(t)
Random phase
shifted
x(t)
Time
x(t) = (cos f1 t + cos f2 t +cos f3t+ … cos fnt) /n
fn= q f0 + Df
Df
12345678
q
Frequency
Patterns of spikes are predicted by Farey’s series*
n:m
N:M
1:0
1:1
2:1
n+N:m+M
…
4:1
3:1
5:2
3:2
5:3
4:3
5:3
continue
3:2
2:1
3:1
*Chialvo, Nature 330, 1987
The devil staircase and the Farey series
The average pattern of spikes are period adding
(N:1) sequences
1:1
1:0
<1:1>
<2:1>
<3:1>
2:1
3:1
4:1
etc:1
no
no
no
no
Inter-spike interval
How we perceive pitch of a complex tone ?
Cual es el proceso neuronal mas simple que
produce spike trains conteniendo el codigo del
pitch de un tono complejo?
Respuesta corta:
The non dynamical threshold-crossing model
If x(t) > U th
the system emits a ‘‘spike’’
f2 =
f1
3*f0
= 2*f0
“Ghost Stochastic Resonance” = noise intensity at the largest
proportion of spikes with intervals ~ 1/ f0
The neuron respond to a frequency not present in the input (“ghost”)
Ghost resonance for mistuned two-frequencies 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
Df
plotted as a function of
f1
The neuron response to mistuned complex tones can be
generalized for even or odd number of tones
Two freq.
For inputs composed
frequencies:
kf0  Df ,
of
Three freq.
N
sinusoidal
signals
of
(k  1) f 0  Df , ... (k  N  1) f 0  Df
the resonance occur at frequencies:
Data, Theory and Numerics overimposed (no fitting)
How it compares with In Vivo Experiments (Cariani)
Our theory
300
n=2
Est. Pitch (Hz)
n=3
250
200
150
100
475
n=4
n=5
n=6
625
775
Center Frequency f2 (Hz)
Cariani P.A. and Delgutte B., J. Neurophysiol. 76, 1698-1716, 1996
Ghost resonance is a general phenomenon
Short list of recent Ghost Stochastic Resonance work:
• Molecular motors (ratchets) driven by 2F ( Fabio Marchesoni and
colleagues).
• Models of climate changes (Holger Braun and colleagues).
• Lasers (Martin Buldu, Jordi Garcia-Ojalvo and colleagues).
• Electric fish sensory system (Longtin and colleagues)
Ghost resonance in vision*
Missing Fundamental
*K. Fujii et al, Psychological Research (2000) 64:149-154.
From Ghosts to Phantoms
Another twist:
As seen, the intervals between spikes is determined by the
“shape” in time of the complex tone,
Thus, what will be perceived for amplitude modulated white
noise?
Shhhh…Shhhh…Shhhh…Shhhh…Shhhh…
Input Signal
1/fo
The input signal frequency spectrum
noise
Notice that the input signal spectrum is flat
White noise is perceived not as noise but as a periodic
signal with 1/fo frequency
The input spectrum is flat
The ouput spikes at preferred
intervals ~ 1/f0
Obviamente si el oido fuese un microfono, debieramos
percibir ruido....
White noise is perceived not as noise but as a periodic
signal with 1/fo frequency
A short list of what we have learned:
1) The output of even a single noisy neuron driven
by multiple frequencies are spikes spaced at
the inverse of the “pitch”.
2) A simple expression predicts the pitch for
many arbitrary inputs (which perfectly agrees
with both psychophysics and neural data).
3) The mechanism (“ghost stochastic resonance”)
suggests the presence of phantoms and other
sensory illusions.
4) The perception of ghosts and phantoms clearly
show that the ear is not a microphone.
Consonance
The problem:
What is the neural mechanism of consonance
The first law of nature ruled by arithmetic of
integers:
Pythagoras discovered that two similar strings under the same tension sounded
together sound pleasant if the length of the strings are in the ratio of two small
integers …. (1:1, 2:1, 3:2… n:m )
A tendency is to think on terms of frequency
More coincident
harmonics imply
better consonance
1:1 > 2:1 > 3:2 > 4:3 > 45:32
Inconsistent!
According with this picture
pure tones’ consonance is
left undefined
La solucion: Tonos puros que suenan consonante tienen la
misma altura y la neurona descarga a intervals de
1/altura
Unison
Octave
Galileo (and independently Mersenne) explained consonance as the regularity
of the intervals “such that the eardrum is not kept at perpetual torment”
La solucion: Tonos complejos que suenan consonante tienen
la misma altura y la neurona descarga a intervals de
1/altura
Unison
Octave
Interval is more telling than 1/Frequency
Conclusion 1:Two tones (simple or complex)
sound consonance when they have the
same pitch
Conclusion 2: Altua y consonance son la
misma cosa para una neuroan
Conclusion 3:The model objectively judges
both pitch and consonance (algo asi como
un pichichometro o consonometro)
Conclusion 4: Dynamical properties of the
auditory periphery can do highly nontrivial processing including the extraction
of the pitch of arbitrarily complex tones
and the judgment of consonance.
What else?
The problem: to be consonant higher octaves need a few more %
Psycho-acoustic Data
Spike Trains Data
So called “enlargement of the octave”
The solution:
Enlargement of the octave can be explained again on
the basis of dynamical properties of neurons, (i.e.,
incomplete recovery from the previous spike) but I
am out of time…
Time to summarize:
BlahBlahlogy
By looking at the dynamics of toy neuron models we work out a long
unsolved problem:
The brain (can) determine(s) the pitch of a complex sound by the
combination of a linear superposition plus a nonlinear noisy
threshold crossing process.
We find the expression predicting the frequency of the “ghost”
resonance for some arbitrary inputs.
Phantoms and other illusions can be created in similar ways and
surely are hiding new surprises.
Consonance can be judged even by a single neuron in exactly the
same way than pitch.
The ear is not a microphone
Papers: www.chialvo.net
Papers: www.chialvo.net
How we perceive pitch of a complex tone ?
Labeling the time of each spike as tspike, average the
forcing for t=tspike-Dt,tspike+Dt.
What do you see for the case of SR?
Answer: The input periodic forcing
What do you see for the case of Ghost SR?
Answer: The ghost
What do you see for the case of AM white noise?
Answer: Nothing.
Replicated experimentally in semiconductor lasers
with optical feedback
Ghost Resonance in a Semiconductor Laser with Optical Feedback
UPC (Barcelona): Javier Martin-Buldu , Jordi Garcia Ojalvo, Carme Torrent and
UIB (Mallorca): Claudio Mirasso