Transcript Integrative Vision

Local Stability Analysis
Step One: find stationary point(s)
Step Two: linearize around all stationary points (using Taylor expansion),
the Eigenvalues of the linearized problem determine nature of stationary point:
Real parts:
positive: growth of fluctuations, instability
negative: decay of fluctuations, stability
Imaginary parts:
if present, solutions are oscillatory (spiraling)
spiraling inward or outward if non-zero real parts
Overall: point (asymptotically) stable if all real parts negative
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Examples of nonlinear activation functions (transfer functions):
a. “half-wave rectification”
F L   GL  L0 
a.
b.
c.
L : L  0

 0 : else
b. “sigmoidal function”
F L  
rmax
1  expg1 L1/ 2  L 
c. rectified hyperbolic tangent
F L  rmax tanhg2 L  L0 
Note: we will typically consider the
activation function as a fixed property
of our model neurons but real neurons
can change their intrinsic properties.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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The Naka-Rushton function
A good fit for the steady state firing rate of neurons in several visual areas (LGN,
V1, middle temporal) in response to a visual stimulus of contrast P is given by:
 rmax P N
:P0
 N
N
F P    P1/ 2  P

0 : else

P ½ , the “semi-saturation”, is the stimulus contrast (intensity) that produces half of
the maximum firing rate rmax. N determines the slope of the non-linearity at P ½ .
Albrecht and
Hamilton (1982)
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Interaction of Excitatory and Inhibitory
Neuronal Populations
Motivations:
• understand the emergence of oscillations in excitatory-inhibitory networks
• learn about local stability analysis
Consider 2 populations of excitatory and inhibitory neurons with firing rates v:
MEE
vE
MIE
MEI
vI
Dale’s law: every neuron is either excitatory or inhibitory, never both
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Mathematical formulation:
[ ]+
Parameters: MEE = 1.25, MEI = -1, gammaE = -10Hz, tauE = 10ms
MII = 0, MIE = 1, gammaI = 10 Hz, tauI = varying
Stationary point:
v  26.67, v  16.67
0
E
0
I
vE
MIE
vI
MEE
MEI
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Phase Portrait
*nullclines, zero-isoclines
stationary point
*
*
A: Stationary point is intersection of the nullclines. Arrows indicate direction
of flow in different area of the phase space (state space).
B: real and imaginary part of Eigenvalue as a function of tauI .
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Linearization around stationary point gives
the following matrix A with these Eigenvalues:
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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For tauI below critical value of 40ms, Eigenvalues have negative real
parts: we see damped oscillations. Trajectory spirals to stable fixed point
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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When tauI grows beyond critical value of 40ms, a Hopf bifurcation occurs
(here tauI=50ms): stable fixed point → unstable fixed point + limit cycle
Here, the amplitude of the oscillation grows until the non-linearity “clips” it.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Neural Oscillations
•
interaction of excitatory and inhibitory neuron populations can lead to
oscillations
•
very important in, e.g. locomotion: rhythmic walking and swimming motions:
Central Pattern Generators (CPGs)
•
also very important in olfactory system (selective amplification)
•
also oscillations in visual system: functional role hotly debated. Proposed as
solution to binding problem:
• Idea: neural populations that represent features of the same object
synchronize their firing
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Binding Problem
•
•
what and where (how) pathways in visual system
how do you know what is where?
Synchronization
no
yes
circle
triangle
up
visual field
down
neural
representation
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
spike trains
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Competition and Decisions
Motivation: ability to decide between alternatives is fundamental
Idea: inhibitory interaction between neuronal populations representing different
alternatives is plausible candidate mechanism
The most simple system:
e1 
e2 
K1
K2
1

1

 e1  S K1  3e2 
 e2  S K 2  3e1 
Winner-take-all
(WTA) network
 100x 2

S x   1202  x 2 : x  0
 0
:x0
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Stationary States and Stability
e1 
1

 e1  S K1  3e2  ,
e2 
1

 e2  S K 2  3e1 
 Mx 2

S  x    2  x 2 : x  0
 0
:x0
dS
2M 2 x

dx  2  x 2 2
The stationary states for K1=K2=120:
• e1 = 50, e2 = 0
• e2 = 50, e1 = 0
• e1 = e2 = 20

Linear stability analysis:
1) for e1 = 50, e2 = 0 :
  1
A  
 0
0 
 , with   1 / 
1
 

→ “stable node”
2) for e1 = e2 = 20 : (τ=20ms)
  1
A   8
  5
 58 
 , with 1  0.13, 2  0.03 → “unstable saddle”
1 
 
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Matlab Simulation
Behavior for strong identical input: K1=K2=K=120
Plase Plane
60
50
45
50
40
40
30
E2
E1 (red) & E2 (blue)
35
25
30
20
20
15
10
10
5
0
0
0
50
100
150
200
Time (ms)
250
300
350
400
0
10
20
30
E1
40
50
60
one unit wins the competition and completely suppresses the other
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Continuous Neural Fields
So far: individual units, with specific connectivity patterns
Idea: abstract from individual neurons to continuous fields of neurons, where
synaptic weights patterns become homogeneous interaction kernels
Variant 1:
continuous labeling of input or
output domain
Variant 2:
continuous labeling of twodimensional cortical space
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Recurrent Simple Cell Model
Question: how is orientation selectivity achieved? (feedforward vs. recurrent accounts)
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Classic Hubel and Wiesel Model
simple cell sums input from
geniculate On and Off cells in
particular constellation
complex cell sums inputs from
simple cells with same orientation
but different phase preference
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Recurrent Model
 / 2 d '
dv( )



r
 v( )  h( )  
 0  1 cos2   'v( ' )
 / 2 
dt


Stimulus with orientation
angle θ=0. A: amplitude, c:
contrast, ε: small
h( )  Ac1     cos2 
nonlinear amplification
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Superior Colliculus and Saccades
Representation of saccade
motor command in superior
colliculus: vector averaging
Yarbus
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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A Simple Model of Saccade
Target Selection
Question: how do you select the target of your next saccade?
Idea: competitive “blob” dynamics in 2 dimensional “neural field”
linear unit for global inhibition
layer of non-linear units
with local excitation
h(x, t )  h(x, t )  g (x)  S (h(x, t ))    S (h(x, t ))  I (x, t )   (x, t )
1 : h  0
g (x)  2 2 exp( x / 2 ) , S (h)  
, and
0 : h  0
1
2
2
  0 , is a const ant ,I : input,η : noise.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Stability Analysis of Saccade Model
h(x, t )  h(x, t )  g (x)  S (h(x, t ))    S (h(x, t ))  I (x, t )   (x, t )
1 : h  0
g (x)  2 2 exp( x / 2 ) , S (h)  
, and
0 : h  0
1
2
2
  0 , is a const ant ,I : input,η : noise.
Step 1: look for homogeneous stationary solutions
Step 2: find range of β for which homogeneous
stationary solution becomes unstable
Step 3: simulate system (Matlab), observe behavior
Step 4: estimate the size of the resulting blob as
a function of β
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
Reminder: Convolution


 
 
f ( x )  g ( x )   f ( x  x ' ) g ( x ' )dx '




f ( x)  g ( x)  g ( x)  f ( x)

 
f ( x )      f ( x ' ) dx '




F f ( x )  g ( x )  F f ( x )Fg ( x )
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Example Run
Initialization: 10 random spots of small activity, I=0, η small Gaussian iid noise
time
Result: a blob of activity forms at location determined by initial state and noise
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Results of Analysis
h(x, t )  h(x, t )  g (x)  S (h(x, t ))    S (h(x, t ))  I (x, t )   (x, t )
1 : h  0
g (x)  2 2 exp( x / 2 ) , S (h)  
, and
0 : h  0
1
2
2
  0 , is a const ant ,I : input,η : noise.
Step 1: look for homogeneous stationary solutions
• h0=0 works, β>1/A prevents fully active layer (A=area of layer)
Step 2: find range of β for which homogeneous stationary solution becomes unstable
• for small local fluctuation from h0=0 to grow, we need β<1/2πσ2
Step 3: simulate system (Matlab), observe behavior
• formation of single blob of activity suppressing all other activity in layer
Step 4: estimate the size of the resulting blob as a function of β, σ
• r  1 2 , for r  
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Matlab Code Fragments
% initialize layer
size = 50;
h = zeros(size,size);
for i=1:10
x = unidrnd(size);
y = unidrnd(size);
h(x,y)=h(x,y)+0.05;
end
% main loop
while(1)
active = (h>0);
I = conv2(active, g, 'same') - beta*(sum(sum(active)));
h = (1-alpha)*h + alpha*I + normrnd(0, noise, size, size);
% display plots, etc.
pause
end
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Discussion of Saccade Model
Positive:
• roughly consistent with anatomy/physiology
• explains how several close-by targets can win over strong but isolated target
• suggests why time to decision is longer in situations with several equally strong
targets
• similar models used in modeling human performance in visual search tasks
Limitations:
• only qualitative account
• in order to make precise quantitative predictions, it is typically necessary to take
more physiological details into account, which are mostly unknown:
• exact connectivity patterns
• non-linearities
• more than one area is involved
• what are all the inputs?
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Connection to Maximum
Likelihood Estimation
So far: purely bottom-up view: networks with this connectivity structure just happen
to exhibit this behavior and this may be analogous to what the brain does
New idea: use such dynamics to do Maximum Likelihood estimation
Want:
ˆ  arg max p(r | )


r: firing rate vector, Θ: stimulus parameter
New idea: blob dynamics + vector decoding works better than doing direct
vector decoding on the noisy inputs
Population vector decoding:
N
v pop  
ra
c
max a
a 1 a
r
where ca is the preferred
stimulus vector for unit a
1-d “blob” network with noisy input
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Binocular Rivalry, Bistable Percepts
r K1  e2 
2
 e1  e1 
(  a1 ) 2  K1  e2 
2
r K 2  e1 
2
 e2  e2 
(  a2 ) 2  K 2  e1 
2
 A a1   a1  e1 
 A a 2   a2  e2 
Idea:
extend WTA network by slow
adaptation mechanism.
Adaptation acts to increase semisaturation of Naka Rushton nonlinearity
ambiguous figure
binocular rivalry
L
R
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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Matlab Simulation
r K1  e2 
2
 e1  e1 
(  a1 )  K1  e
2
r K 2  e1 

2
2 
2
 e2  e2 
(  a2 )  K 2  e 
2
1 
2
,  A a1   a1  e1 
,  A a 2   a2  e2 
E1-A1 Projection of State Space
60
β=1.5
60
β=1.5
50
40
40
E1
E1 (red) & E2(blue)
50
30
30
20
20
10
10
0
0
1000
2000
3000
Time (ms)
4000
5000
6000
0
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
0
10
20
30
A1
40
50
60
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Discussion of Rivalry Model
Positive:
• roughly consistent with anatomy/physiology
• offers parsimonious mechanism for different perceptual switching phenomena, in a
sense it “unifies” different phenomena by explaining them with the same mechanism
Limitations:
• provides only qualitative account
• real switching behaviors are not so nice and regular and simple:
• cycles of different durations
• temporal asymmetries
• rivalry: competition likely takes place in hierarchical network rather than in just
one stage.
• spatial dimension was ignored
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch
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