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Attributes of a neural code for perceptual/cognitive computation
Presented Feb 12, 2003Mathematical Biosciences Institute
Ohio State University
David W. Arathorn
Copyright(c)2003 David W. Arathorn
Unless otherwise indicated all diagrams from
“Map-Seeking Circuits in Visual Cognition”
David W. ArathornStanford University Press
M ap-S eek in g C ircu its In V isu al C ogn ition
David W . A ra thorn
from Stanford University Press
A neural code must be able to support a
sufficiently rich computational repertoire
• It must encode a sufficient range of values with sufficient accuracy and discriminability
• It must close under a sufficient set of operations implementable with neural mechanics
Two alternative approaches to problem, not mutually exclusive
• Try to deduce the code from inputs
• Try to deduce the code, or at least specific characteristics of the code, from computational requirements
Starting with the second approach
• What are the operations necessary to support a realistically complex computation?
– combine, – match, – map, inverse map, – compete, – attenuate, – scale – a nonlinearity (if not inherent)
A circuit composed of the following operations…
is capable of a variety of visual functions, computing inverse kinematics, and other cognitive computations
s s
2
f
w1
g1
g2
g3
q1
q2
q3
b
b
w
m w11
m w22
forward backward
r
f
d1( )r d2( )r d3( )r
i ig d b r
map, attenuate, combine
s s
2
f
w1
g1
g2
g3
q1
q2
q3
b
b
w
m w11
m w22
forward backward
r
f
d1( )r d2( )r d3( )r
k km f b w
k k k
k
m f b w b w w�� attenuate, combine
match, non-linearity
s s
2
f
w1
g1
g2
g3
q1
q2
q3
b
b
w
m w11
m w22
forward backward
r
f
d1( )r d2( )r d3( )r
i iq d b r�
comp , g g q
match, map
competition
Mapping module for multilayer circuits
s
d 1( )r
g 1g 2
g 3
q 1
q 2
q 3
b b
forw ard backw ard
r
s
d 1 ( )b-1
r
d 2 ( )b-1 d 3 ( )b
-1
d 2( )r d 3( )r
FORWARD BACKWARD
input image
layer 1
layer 2
memory
algebraic versions of the basic operations
i ig d b r
k km f b w
k k k k
k k
m f b w b w w��
i iq d b r�
comp , g g q
1) map, attenuate, combine: b forward, b and r backward superpositions
2) match, non-linearity: memory response and forward/backward match
3) competition: mapping coefficient competition
1i ig d r b
��
Role of the Ordering Property
In two places there is a match between a superposition and a pattern (which itself may be a superposition from another layer in multilayer circuits.)
si S s are components of superposition si
r S r is not component of superposition si
Ordering property is
P(sj si > r si ) >> P(sj si < r si )
for sj S
Depends on sparsity of si and number of components of si
s1
s2
sr
Edge-filtered images are sparse high dimensional vectors
During convergence only a small excess needed: sj si > r si
r is composed of parts of various si and is dismantled during convergence
FORWARD BACKWARD
input image
layer 1
layer 2
memory
First, search concurrently for locating patterns anywhere…
Then, search seqentially for confirming feature patternsin locations predicted by head model….
lynx image courtesy U.S Fish and Wildlife Service
iter 1
iter 25
(next slide)
iter 1 iter 20
iter 40
iter 60 iter 80 iters 20+40+60+80
pattern x-loc y-loc orient yscale xscaleeyes 110 69 7.500 0.850 0.850 ------snout 109 48 7.500 0.850 0.800------left ear 83 108 12.500 1.000 0.900------right ear 144 99 27.500 0.900 0.850
Pattern location – related to tracking
antelope image courtesy U.S Fish and Wildlife Service
Training model, 3D normals
Signal encoding has to “interoperate” with plausible synaptic encoding... e.g. encoding of 3D normals
1
2
3
41
2
3
4
00.250.5
0.75
1
1
2
3
4
0
50
100
1500
50
100
150
0
0.5
1
0
50
100
150φ
ψ
k - encode( <ψ,φ>) encode(<90,90>)
16 synapse weights (wi = e -| v-ci | 2 )
per <x,y,z> encode spherical orientation of normal
determine if viewpoint orientation perpendicular to model normal
Input image
0 50 100 150 200
0
50
100
150
200
3d models courtesy www.3Dcafe.com
0 50 100 150 200
0
50
100
150
200
3D projection
rotation in plane
scale/aspect
translation
iter 1 iter 25
25, 0.01, 0.15, 10, F, -10.0, -5.0, 2.5, 1.0, 1.0, 0.05, 1.0, 1.0, 0.05, 0.7, 1.4, 0.1, 1.3, 1.3, 0.1, 1.4, 1.4, 0.1, -75.0, -15.0, 5.0, 10.0, 45.0, 5.0 /1, 1, 25 pig.dat1, 50 / testsamples, interval pig1dis4_h.datpig1dis4_v.datpig1dis4_l.datpig1dis4_r.dat
----------------- 40000 25667929 1 17708 108 111 1.000 2 17709 109 111 0.533
------------------
1 1 25667929 1 2 -5.000 1.000 1.000 1.000 2 1 -7.500 1.000 1.000 0.363------------------
1 1 25667929 1 3 1.000 1.300 1.400 1.000 2 4 1.100 1.300 1.400 0.911 3 2 0.900 1.300 1.400 0.409 4 5 1.200 1.300 1.400 0.283------------------
13 8 25667929 1 67 -35.000 25.000 1.000 2 66 -35.000 20.000 0.823 3 59 -40.000 25.000 0.730 4 65 -35.000 15.000 0.289 5 58 -40.000 20.000 0.137 6 75 -30.000 25.000 0.065
translation
rotation in plane
scaling/aspect
3D projection
another view
25, 0.01, 0.15, 10, F, -30.0, 0.0, 3.0, 1.0, 1.0, 0.05, 1.0, 1.0, 0.05, 0.7, 1.4, 0.1, 1.3, 1.3, 0.1, 1.4, 1.4, 0.1, -75.0, -15.0, 5.0, 10.0, 45.0, 5.0 /1, 1, 25 ------
1 1 25661176 1 8 -6.000 1.000 1.000 1.000 2 7 -9.000 1.000 1.000 0.902 3 9 -3.000 1.000 1.000 0.681 4 10 0.000 1.000 1.000 0.335 5 6 -12.000 1.000 1.000 0.195------
1 1 25661176 1 5 1.200 1.300 1.400 1.000------
13 8 25661176 1 17 -65.000 15.000 1.000 2 27 -60.000 25.000 0.591 3 16 -65.000 10.000 0.526 4 26 -60.000 20.000 0.294
rotation in plane
scaling/aspect
3D projection
iter 1
iter 25
layer 1 layer 2 layer 3
translation rotation scaling/perspect
mapping convergence
iter 1
iter 25
Recharacterization: mappings as data
-10 -5 0 5 10-15
-10
-5
0
5
10
15
-10 -5 0 5 10-15
-10
-5
0
5
10
15
memory pattern 2 components input pattern
locations of activemappings at endof convergence
Concurrently active translational mappings pop out figure defined by repeated pattern.
Another composed inverse mapping problem:inverse kinematics
1
2
3
p0
p3
seg 1
seg 2
seg 3
constr3(F2)
constr2(F1)
Convergence to inverse kinematic solution
0 20 40 60 800
20
40
60
80
0 20 40 60 800
20
40
60
80
0 20 40 60 800
20
40
60
80
10 20 30 40 50 60 70 80
10
20
30
40
50
60
70
80
Layer 3/Segment 3 Layer 2/Segment 2 Layer 1/Segment 1
iter 1
iter 2
iter 5
0 20 40 60 800
20
40
60
80
activate end effector targetlocation
Can the basic operations be implemented with neural or other analog mechanisms?
– combine, – match, – map, inverse map, – compete (involves a max( ) operation) – attenuate, – scale – a nonlinearity (if not inherent)
Constraints imposed by neural mechanics
• non-linearity
• limited accuracy and reliability
• limited amplitude range
• signal warp– pulse spread in dendrite means signal
interactions at distance from synapse of origin cannot depend directly on temporal fine structure seen in the axonal signal
Neuronal circuit is constructed from cell pairs which implement reciprocal pathways with temporal or phase encoding
stage i Fi
stage i+1 Fi+1
Bi
Bi+1
inhibitorysynapse
excitatorysynapse
inhibitoryinter-cell
forward backward
V
t
V
t
DC 1.0 vs. DC 0.7
DC 1.0 vs. DC 0.5
• Vectors are self “normalizing• Amplitude encoding range is not limited to cell
dynamic range• Capacitance (pulse spread) is friendly
– Increases temporal differentiation– Decreases effect of spatial location of synapse
• Only monotonicity required in combination to implement addition-like operation
• Devil-take-the-hindmost competition implemented by inhibitory signal derived from signal front –keeps meaningful signals in narrow range
Some advantages of phase encoding
target 1
target 2
target 3
no targets
left
• Dendritic trees are non-homogeneous. i.e. local structure determines local signal interactions, global structure determines combination of local results
• All computation takes place in dendritic trees (in idealized cell).
• Signal interactions evolve primarily during rise time of dendritic pulse
• State can be held over between cycles to amplify signal differences
• Pulse length is long compared to propagation time: some location independence
Intra-dendritic computation using phase in real neurons
From SR Williams, GJ Stuart, 2002, “Dependence of EPSP Efficacy on Synapse Location in Neocortical Pyramidal Neurons”, Science 295:1907-10
signal propagation
sEPSPs
From SR Williams, GJ Stuart, 2002, “Dependence of EPSP Efficacy on Synapse Location in Neocortical Pyramidal Neurons”, Science 295:1907-10
amplitude - rise time relation
From SR Williams, GJ Stuart, 2002, “Dependence of EPSP Efficacy on Synapse Location in Neocortical Pyramidal Neurons”, Science 295:1907-10
phase difference – response amplitude function(also known as “coincidence detection”)
somaticsynapses
distal (620 um)synapses
distal (620 um)synapses
From SR Williams, GJ Stuart, 2002, “Dependence of EPSP Efficacy on Synapse Location in Neocortical Pyramidal Neurons”, Science 295:1907-10
bbcol
stage b
stage m
stage r
input field
rm = r-match ri = r-intersect
rbkwd rfwd
bfwd bbkwd
mbkwd mfwd
mapping competition
1 2
3 4
6 5
7 8
9
10
rmcli
rm ri
11 subtract
1r
1r�
q1
g1
1b
1b�
maxb�
max q
max iqq
(1)
(2)
(3)
(4)
(5)
(6)
(9)
(10)
(11)
Signal equivalentsin algorithmic circuit
(7)
(8)
1m
1m�
bbcol
stage r
input field
rbkwdrfwd
bfwd
bbkwd
mapping competition
1
2
6 5
9
10
rmcli
rm
ri
11 subtract
1r
1r�
q1
g1
1b
1b�
maxb�
max q
max iqq
(1)
(2)
(3)
(4)
(5)
(6)
(9)
(10)
(11)
Signal equivalentsin algorithmic circuit
3
4
Signal relationships in neuronal circuit
0 5u 10u 15u 20u 25u 30u 0 600m 1.2 1.8 2.4 3
3.6
rm ri
bbcol
0 5u 10u 15u 20u 25u 30u 0 600m 1.2 1.8 2.4 3
3.6 bbcol
rm ri
recognition state
non-recognition state
bbcol inhibits late rm signalsthus blocking paired ri signal
Is phase encoding the only way to satisfy the original requirements?
Encoding must support a sufficiently rich repertoire of computation– It must encode a sufficient range of values
with sufficient accuracy and discriminability– It must close under a sufficient set of
operations implementable with neural mechanics
Map-seeking circuits will work with any encoding that supports the operations listed earlier.
• Encodings must have started simple to have co-evolved with useful, robust mechanisms to implement operations on them.
An evolutionary caveat from engineering: complex and/or delicate systems generally do not work at first...but an evolutionary step has to work in “alpha release.”
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