Sensorimotor Transformations
Maurice J. Chacron and Kathleen E. Cullen
Outline
• Lecture 1:
- Introduction to sensorimotor
transformations
- The case of “linear” sensorimotor transformations:
refuge tracking in electric fish
- introduction to linear systems identification
techniques
- Example of sensorimotor transformations:
Vestibular processing, the vestibulo-occular reflex
(VOR).
Outline
• Lecture 2:
- Nonlinear sensorimotor transformations
- Static nonlinearities
- Dynamic nonlinearities
Lecture 1
Sensorimotor transformation:
if we denote the sensory input as a vector S
and the motor command as M, a
sensorimotor transformation is a mapping
from S to M :
M =f(S)
Where f is typically a nonlinear function
Examples of sensorimotor
transformations
- Vestibulo-occular reflex
- Reaching towards a visual target, etc…
Example: Refuge tracking in weakly
electric fish
Refuge tracking
Refuge tracking
Sensory input
Motor output
Error
Results
- Tracking performance is best
when the refuge moves slowly
- Tracking performance degrades when
the refuge moves at higher speeds
- There is a linear relationship between
sensory input and motor output
(Cowan and Fortune, 2007)
Linear systems identification
techniques
Linear functions
• What is a linear function?
F(ax1 + bx 2 ) = aF(x1) + bF(x2 )
• So, a linear system must obey the following
definition:
Out = F(aIN1 + bIN2 ) = aF(IN1) + bF(IN2 )
Linear functions (continued)
• This implies the following:
a stimulus at frequency f1 can only cause a
response at frequency f1
Linear transformations
OUT(t )  [T  IN](t )   (t )
assume output is a convolution
of the input with a kernel T(t) with
additive noise. We’ll also assume that all
terms are zero mean.
- Convolution is the most general linear
transformation that can be done to a signal
+¥
[T * IN ](t) = ò T (t - t ')IN(t ')dt '
-¥
An example of linear coding:
• Rate modulated Poisson process
time dependent firing rate
time
Linear Coding:
Example:
Recording from a P-type
Electroreceptor afferent.
There is a linear relationship between
Input and output
Gussin et al. 2007 J. Neurophysiol.
Instantaneous input-output transfer function:
Fourier decomposition and transfer
functions
- Fourier Theorem: Any “smooth” signal
can be decomposed as a sum of sinewaves
- Since we are dealing with linear transformations,
it is sufficient to understand the nature of linear
transformations for a sinewave
Linear transformations of a sinewave
• Scaling (i.e. multiplying by a non-zero
constant)
• Shifting in time (i.e. adding a phase)
f
sin(wt + f )®
sin(wt
) ®G
sin(wt
+ f+ f+ )y )
f (Ax + By) = G
Asin(w
) + Bsin(w t + f + y )
t 1++fy
[ Asin(w
1t +1f
1 ) + Bsin(w2 t2 + f 2 2]
= AG
Af (x)
+ Bf1t (y)
sin(w
+ f1 ) + BG sin(w2 t + f2 )
= Af (x) + Bf (y)
Cross-Correlation Function
C (t , )  X (t ) S (t   )    X (t )  S (t   ) 
For stationary processes:
In general,
C (t , )  C ( )
C ( )  C ( )
Cross-Spectrum
• Fourier Transform of the Cross-correlation function
• Complex number in general
a+ib
a: real part
b: imaginary part
i = -1
~
~
~
~*
~*
C ( f )  X ( f ) S ( f )    X ( f )  S ( f ) 
Representing the cross-spectrum:
~
~
i
C ( f ) | C ( f ) | e
| C( f ) |= real[C( f )] + imag[C( f )]
2
~
 imag [C ( f )] 

  arctan 
~

real
[
C
(
f
)]


2
: phase
: amplitude
Transfer functions (Linear Systems
Identification)
OUT(t )  [T  IN](t )   (t )
assume output is a convolution
of the input with a kernel T(t) with
additive noise. We’ll also assume that all
terms are zero mean.
~
~
~
~
OUT ( f )  T ( f ) I N ( f )   ( f )
Transfer function
Calculating the transfer function
~
~
~ *
~ *
~
~
~ *
I N ( f ) OUT ( f )  I N ( f )T ( f ) I N ( f )  I N ( f ) ( f )
multiply by:
~ *
IN ( f )
and average over noise realizations
~*
~
~
~ *
~
~ *
C ( f )  T ( f ) I N ( f ) I N ( f )  I N ( f ) ( f )
~*
C (f)
~
T(f ) ~
PIN ( f )
=0
Gain and phase:
gain = T( f )
~
 imag [T ( f )] 

  arctan 
~

real
[
T
(
f
)]


Sinusoidal stimulation
at different frequencies
Stimulus
20 msec
Response
Gain
Combining transfer functions
input
T1 ( f )
T2 ( f ) ...
... TN ( f )
output
N
OUT ( f ) = IN( f )Õ Ti ( f )
i=1
Where transfer functions fail…
Vestibular system
Cullen and Sadeghi, 2008
Example: vestibular afferents
CV=0.044
CV=0.35
Regular afferent
paffn67301b.mat: hhv - fr - - 1 -
120
120
100
100
1
Firing rate
(spk/s)
Head velocity
(deg/s)
80
80
60
60
40
40
20
20
0
0
`
-20
-20
-40
-40
4.691
6.691
8.691
10.691
12.691
14.691
16.691
18.691
20.691
22.691
24.691
Irregular afferent
paffn70501b.mat: hhv - fr - - 1 -
160
160
140
140
1
120
120
100
100
80
Firing rate
(spk/s)
60
60
40
40
20
Head velocity
(deg/s)
80
0
`
20
0
-20
-20
-40
-40
29.992
34.992
39.992
44.992
49.992
54.992
Signal-to-noise Ratio:
SNR( f ) 
Presponse ( f )
Pnoise ( f )
noise  response   response 
Borst and Theunissen, 1999
Using transfer functions to
characterize and model refuge
tracking in weakly electric fish
Sensory input
Motor output
Error
Characterizing the sensorimotor
transformation
1st order
2nd order
1
H (s) =
as + b
1
H (s) = 2
as + bs + c
s = i 2p f
Modeling refuge tracking using
transfer functions
sensory
input
sensory
processing
motor
output
motor
processing
Modeling refuge tracking using
transfer functions
sensory
input
sensory
processing
motor
output
1
Ms 2
Newton
Simulink demos
Mechanics constrain neural processing
Summary
• Some sensorimotor transformations can be
described by linear systems identification
techniques.
• These techniques have limits (i.e. they do not
take variability into account) on top of
assuming linearity.