Transcript Neural Network Models in Vision

Neural Network Models
in Vision
Peter Andras
[email protected]
The Visual System
R
LGN
V1
V3
V2
Lower
V5
V4
Higher
Neurons
Rod
Horizontal
Bipolar
Amacrine
Ganglion
Neuron Models
The McCullogh-Pitts model
x1
w1
x2
Inputs
xn-1
xn
z   wi xi ; y  H ( z )
w2
x3
…
n
i 1
..
w3
.
wn-1
wn
Output
y
Neuron Models
The Hodgkin-Huxley Model
K+ Na+
+
+
Na
+
Na Na
K+
Na+
K+
K+
Na+
dV
3
4
C  I 0  cNa x1 x2 (V  E Na )  cK x3 (V  EK )  cL (V  EL )
dt
dxi
  i (1  xi )   i xi ; i  1,2,3
dt
Na+
K+
K+
K+
Na+
Na+ K+
Modelling Methodology
Physiological measurements
Electrode
Response
Stimulus
Other methods: EEG, MRI, PET, MEG, optical recording,
metabolic recording
Modelling Methodology
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18
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8
6
4
2
0
Response characterisation in
terms of stimulus properties
20
18
16
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10
8
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2
0
Spike count
5
5
19
17
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13
9
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7
5
Spike count
Temporal frequency
Stimulus
9.
8.
5
7.
5
6.
5
5
5.
4.
5
20
18
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10
8
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0
3
180
Spatial frequency
1
Degrees
160
140
120
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60
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20
0
Spike count
3.
5
2.
5
1.
0.
5
4
Modelling Methodology
Models:
A. Statistical models: large number of neurons, with a
few well-defined properties, the response is analysed at
the population level;
Modelling Methodology
Models:
B. Macro-neural models: simplified model
neurons organised in relatively simple
networks, the overall input-output relationship
of the full network is analysed;
Modelling Methodology
Models:
C. Micro-neural models: the neurons are
modelled with many details and models of
individual neurons or networks of few
detailed neurons are analysed.
Modelling Methodology
Physiological measurements
Response characterisation
Model selection
OBJECTIVE 1: match the measured response properties
by the response properties of the model.
OBJECTIVE 2: test the theories, generate predictions.
Neural Network Models
Retina: ON and OFF centre ganglion cells
Bipolar cells
+1
-1
ON
OFF
Preferred stimulus
Neural Network Models
Retina: ON and OFF centre ganglion cells
Measured response of an ON cell
The response of a model ON cell
20
18
20
18
16
14
16
14
12
10
12
10
Spike rate
8
6
8
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2
0
4
2
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Spike rate
Neural Network Models
V1: Orientation selective cells
LGN cells
Preferred
stimulus
Neural Network Models
V1: Orientation selective cells
Measurement
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0
Model
0
0
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0
Degrees
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0
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0
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10
80
60
Spike count
40
0
20
0
0
Degrees
18
0
16
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14
0
12
10
80
60
Spike count
40
0
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0
Neural Network Models
V1: Ocular dominance patterns and orientation maps
Neural Network Models
V1: Ocular dominance patterns and orientation maps
Neuron = Feature vector:
• orientation preference; • eye preference;
• spatial frequency;
• temporal frequency;
Training principles:
• the neuron fires maximally when the stimulus
matches its preferences set by the feature vector;
• the neuron fires if its neighbours fire;
• when the neuron fires it adapts its feature vector
to the received stimulus.
Neural Network Models
V1: Ocular dominance patterns and orientation maps
Mathematically:
Neurons: (wi , ci); wi – feature vector; ci – position vector;
Training set: xt , training vectors, they have the same
dimensionality as the feature vectors;
Training:
i* = index of the neuron for which d(wi*, xt) < d(wi, xt),
for every i  i*;
wi = (1-) wi +  xt , for all neurons with index i, for
which d(ci, ci*) < .
Neural Network Models
V1: Contour detection
Stimulus
Neural Network Models
V1: Contour detection
Neural interactions: specified by interconnection weights.
Mechanism: constraint satisfaction by mutual modification
of the firing rates.
Result: the neurons corresponding to the contour position
remain active and the rest of the neurons become silent.
Neural Network Models
V5: Motion direction selective cells
Orientation
selective cells
delay effect
-1
+1
Preferred stimulus
Neural Network Models
Visual object detection
Object
Features:
• colour;
Invariant
combination of
features
• texture;
• edge distribution;
• contrast distribution;
• etc.
Object
detection
Neural Network Models
Visual object detection
Method 1: Hierarchical binary binding of features
Colour
Texture
Edges
Contrast
This method leads to
combinatorial explosion.
Neural Network Models
Visual object detection
Method 2: Non-linear segmentation of the feature space.
Colour
Texture
Edges
Contrast
Learning by back-propagation
of the error signal and
modification of connection
weights.
Neural Network Models
Visual object detection
Method 3: Feature binding by synchronization.
Critical Evaluation
• Neural network models typically explain certain selected
behavioural features of the modelled neural system, and they ignore
most of the other aspects of neural activity.
• These models can be used to test theoretical assumptions about
the functional organization of the neurons and of the nervous
system. They provide predictions with which we can determine the
extent of the validity of the model assumptions.
• One common error related to such models is to invert the causal
relationship between the assumptions and consequences: i.e., the
fact that a model produces the same behavior as the modelled, does
not necessarily mean that the modelled has exactly the same
structure as the model.
Revised View of the Neural
Network Models
Revised interpretation:
• neurons = anatomical / functional modules (e.g., cortical
columns or cortical areas);
• connections = causal relationships (e.g., activation of bits
of LGN causes activation of bits of V1);
• activity function of a neuron = conditional distribution of
module responses, conditioned by the incoming stimuli;
Revised View of the Neural
Network Models
Neural network model
x1
x2
x3
x4
Bayesian network model
f1(x1)
P(y1|x1)
y1
f2(x2) y
2
f(y1, y2, y3, y4)
P(x1, x2, x3, x4)
y
y
f3(x3) 3
f4(x4)
y1
x1
y4 yi = fi(xi)
y = f(y1, y2, y3, y4)
x2
x3
P(y2|x2)y
2
P(y | y1, y2, y3, y4)
y
y3
P(y3|x3)
x4
y4
P(y4|x4)
P(x1, x2, x3, x4)
P(yi | xi)
P(y | y1, y2, y3, y4)
Revised View of the Neural
Network Models
Advantages of the Bayesian interpretation:
• relaxes structural restrictions;
• makes the models conceptually open-ended;
• allows easy upgrade of the model;
• allows relaxed analytical search for minimal
complexity models on the basis of data;
• allows statistically sound testing;
Conclusions
• Neuron and neural network models can capture important
aspects of the functioning of the nervous system. They allow us to
test the extent of validity of the assumptions on which the models
are based.
• A common mistake related to neural network models is to invert
the causal relationship between assumptions and consequences.
This can lead to far reaching conclusions about the organization of
the nervous system on the basis of natural-like functioning of the
neural network models that are invalid.
• The Bayesian reinterpretation of neural network models relaxes
many constraints of such models, makes their upgrade and
evaluation easier , and prevents to some extent incorrect
interpretations.
Seminar Papers
1. PNAS, 93, 623-627, Jan. 1996
2. PNAS, 96, 10530-10535, Aug. 1999