Transcript No Slide Title

Auditory Perception:
from Neuron
to Cognition and Behaviour
John van Opstal
Institute for Neuroscience
Dept. of Biophysics
Radboud University Nijmegen
Detailed section through the cochlea:
Outer
Hair cells
Organ of Corti
Inner
Hair cell
Auditory nerve
Basilar Membrane
Tonotopic
organization of
the Cochlea:
Originates from:
• Instantaneous coupling
of the BM to the
intra-cochlear fluid
• Location-dependent
stiffness of the BM
(Von Békésy)
The cochlea:
• decomposes sounds into their frequency components
• represents sounds tonotopically
• has a relation to the sound’s identity
• has NO direct relation to the sound’s location
Problems:
Sound localization results from the
neural processing of implicit acoustic cues
in the tonotopic input.
Sound segregation and identification is
ill-posed.
The total sound field is
the linear superposition of an unknown number of
sound sources, in which each sound source by itself
is a linear superposition of an unknown number of tones:
your
t e st s
t a
rts n o
w
NSources
S(t) 

high
 tones
Source  1 freq  low
The auditory system needs to
reconstruct the original sound
sources from this superimposed sound field.

Encoding formats of sensory input
Visuomotor System:
Auditory System:
retinotopic code
V
tonotopic code
AMP
(dB)
ITD’s
ILD’s + HRTF’s
R
E = +20

F
.5
1
2
4
8
16
Freq (kHz)
E = - 10
E = - 40
Auditory Scene Analysis:
Ill-posed problem:
How does the auditory
system parse the superposition of sounds into
the original acoustical
objects?
To the auditory system, the incoming sound is
an a-priori unknown superposition of unknown sound
sources (‘auditory objects’)!
General Organization of the Mammalian Auditory System
II
subcortical
pathways
ACOUSTIC
I
Signals en Systems:
To understand why the auditory system represents
sounds in the way it does, we need to cover some
elementary background of signal analysis.
Mathematical concepts for today:
- sine and cosine: relations and properties
- integration and summation (the very basics only…)
- delta functions: relations and properties
- time domain and frequency domain
- this is all related to Fourier analysis (= cochlea!)
A signal can be described by a mathematical
function in which time is the independent variable:
x = x(t)
One can characterize signals in different types
and into different ways:
• Periodic vs. Non-Periodic
• Deterministic vs. Stochastic
• Continuous vs. Discrete
• Periodic vs. Transient
Example of deterministic signals:
the harmonic and the damped harmonic
x ( t )  A  cos( 2  f 0  t )

• harmonic
• periodic
x(t)  A  e
 t /
 cos( 2  f 0  t )
• transient /
non-periodic

The most elementary periodic auditory signal:
the pure tone
x ( t )  A  sin( 2  f  t   )
• has an infinite duration
• is periodic
• is known as the “harmonic function”
x ( t )  A  sin(   t   )
 = 2f is the angular frequency (in rad/s)
The most elementary transient signal:
the Dirac deltapulse, or ‘click’


x ( t )   (t )
0
voor t=0
elders
• has an infinitely short duration
• is non-periodic
• the surface (integral) is 1, by definition
• called the “Dirac deltapulse, or impulse function”


  ( t )  dt  1




f ( t )   (t  t 0 )  dt  f (t 0 )

n.b. This last property is crucial
for linear systems analysis!
A simple approximation of the Dirac-impulse function
is a rectangular pulse, e.g. at time tk:
1
T
k


tk tk+∆T

 ( t  t k )  lim
T  0
k
EVERY SIGNAL CAN BE COMPOSED
OF ELEMENTARY DIRAC PULSES:
For example:
A step signal can be described by a sequence of
pulses with the same height:
Step = sum of pulses (of width ∆t ) with weight 1

Mathematically: s( t ) 
  (t  t
k 0
k
)
Any arbitrary signal can be approximated by a
sequence of pulses with variable height (weight):
k
x ( t k ) k


Sum of
weighted pulses

x (t) 
 x (t
k  
k
)  k
Finally, an exact description of an arbitrary
signal x(t) is done with Dirac-impulses
(lim ∆T->0):

x(t) 

x ( )   ( t   )  d 

This is called a representation of signal x in the
time domein, by impulse functions.
This description is generally true. It works for

all signals, even if we don’t have an explicit
analytical expression for x(t). However, it is not always
the most convenient description of a signal.
Alternative:
For EACH periodic function, x(t), with period T0, holds:
Signal x(t) can be written as a sum of elementary harmonic
functions, i.e. sines and cosines, with each sine/cosine its
own frequency, which is an integer multiple of the signal’s
groundfrequency, and its own amplitude:

x (t)  a0 
a

n
 cos( 2  nf 0  t ) 
n 1
b
n
 sin( 2  nf 0  t )
n 1
an en bn are the so-called Fourier coefficients
n=1,2,3,4,...... is the spectral number, and
fo is de ground frequency of the signal
f0 
1
T0
T0 is the signal’s period
Fourier-series written slightly differently:

x(t) 
c
n
 cos( 2  nf 0  t   n )
n 0
amplitude
phase
frequency
EXAMPLE:
Quantitatively:
for the Fourier coefficients it can be proven that:
an 


bn 
2
T0
2
T0
T0

x ( t )  cos(
T0
0
)  dt
T0
0

2  nt
x ( t )  sin(
2  nt
)  dt
T0
With this in mind, we can now compute the Fourier
coefficients ourselves for some simple examples.
Fourier-analysis on periodic signals:
A discrete spectrum:
f = n•f0, met n=1,2,3,4,...
short pulse
long pulse
broad
bandwidth
narrow
bandwidth
shortest
slightly longer
broad
less
broad
narrower
yet longer
narrowest
longest
T   f  2
Also non-periodic signals have a Fourier
spectrum.
In this case, the Fourier series becomes a
Fourier integral, for which the period is
infinitely long, and the ground frequency
(and the frequency interval), df, goes to zero:

x (t)  a0 
 a( f )  cos( 2  f

 t )  df 
0
 b( f )  sin( 2  f
 t )  df
0

x(t) 
 c ( f )  cos( 2  f
 t   ( f ))  df
0
c(f) is now a continuous amplitude spectrum, and
(f) is the continuous phase spectrum of x(t)
End first lecture
End first lecture
End first lecture
End first lecture