Download Cosyne14_Kastner (pdf, 2 MiB) - Infoscience

Second Order Phase Transition Describes
Maximally Informative Encoding in the Retina
David B.
0.25
0
Input
S
p(x)
1
n
O
Kastner & Baccus 2011
0.3
0.2
0.1
-2
-0.2
P (1) P ( spike )
0.2
µ x
P ( 0 ) = 1 P (1)
0.5
-2
i
p(ri x)log 2 p(ri x)dx
i
H(r|x)
0.2
0
-0.04 0 0.04
Spacing (m)
0.4
0.4
0.2
0
0 0.005
h/
( - c)/
c
Retinal populations remain poised at a critical point
1
0.3
0.2
0.1
1
0
-0.5
0
( - c)/
The differences in the model due to
different average rates (as occurs
across contrasts), can be normalized,
allowing for a comparison across
different experimental conditions
1
0.5
0
-0.5
0
( - c)/
Data, placed onto the normalized model, and fit by
+ B ( h ) . Grey points indicate
equation m = A
c
the spinodal line.
0.35
0.3
0.2
0.3
Contrast ( )
n=2
n=2
h
P (11) , P (10 ) , P ( 01) , P ( 00 )
0µ 2
Rate limitation enforces
sparse responses
p(x)
p ( ri ) log 2 p ( ri )
0.6
h 0.34
m
Mapping between maximally informative solutions in neural
circuits and the Ising model of phase transitions in physics
2
Four responses patterns
for two neurons
0.4
1
0.2
0.5
0
-2
0
2
0
Input (x)
Placement of response
probabilities that maximizes
information given noise and rate.
r = P1 (1) + P2 (1)
Relative information
p(spike|x)
1
1
0.3
0.9
0.2
0.8
-1
0
Contrast
I ( x,r ) = H (r) H ( r x )
1
3
0.1
c
Optimal dynamic range placement in the retina
p(spike|x)
1
-
0.2
The spacing that provides the maximal
information for two response functions
with the different slopes ( ) across many
different slopes and average rates.
Response probability
0
0
2
P ( spike ) = p ( spike x ) p ( x ) dx
0
Mutual information
p( x)
Normalized µ
w
o
l
O
Normalized
um
i
ed
M
O
0
1
Spacing (m)
Information as a function of the spacing
between two response functions that have
slopes and a rate constraint taken from two
cells recorded simultaneously. Black dots
indicate the spacing of the data. Curves for
the different contrasts are offset for clarity.
Relative information
O
t
s
a
F
0
( - c)/
( - c)/
0
Data
<r>
ff
ff
ff
Response slope
introduces noise
H (r x) =
Estimated
Model
µ2 - µ1
Fraction of cells
0
range
Fast Off adapting
Fast Off sensitizing
c
0.1
0
-0.1
0.5
2
Input (x)
H (r ) =
0.35
1
0.4
0.1
0.4
-0.1 0
200
<r>
Steady state nonlinearities for
the two populations measured
across multiple contrasts
Input distribution
0.4
0.2
0
0.01
0.45
Fast On adapting
The response function with less noise
should have the lower threshold
Neurons modeled as having binary outputs
1+ exp
-0.5 0 0.5
2
Spacing (m)
Population Encoding Model
p ( spike x ) =
0
-0.1
c
Spacing (m)
0.5
20
0
0.47
m
Spacing (m)
0.4
Susceptibility
0.3
Example response functions,
recorded simultaneously, and
average for many cells plotted
relative to c, the point where
the optimal coding strategy
changes between redundant
and coordinated encoding.
m
0.3
-2
0.93
Data
0.35
Nonlinearities
All divergences and power-law scaling within the model are
consistent with a second order phase transition
Normalized
m
Model
The information provided by two
response functions with the same slope
( ) as the spacing ( 2 – 1) between the
functions vary. Black line indicate the
maximal information at a given .
Only off cell types, in
salamanders , split the
encoding between distinct
cell types. On cells do not
split into multiple cell types.
2
Sharpee
Specific heat
Two ganglion cell populations encode similar visual features,
but have distinct sensitivity and plasticity
Adaptation
Sensitization
Tatyana O.
Second order phase transition occurs with
increasing response function noise
Rate (Hz)
Neurons have limited dynamic ranges. Theoretical
studies have derived how a single neuron should place
its dynamic range to optimize the information it
transmits about the input. However, neural circuits
encode with populations of neurons.
Receptive Fields
Temporal
Spatial
Stephen A.
1
Baccus ,
of Neurobiology, Stanford University 2Computational Neurobiology Laboratory, The Salk Institute
3Present address: Laboratory of Computational Neuroscience, EPFL
Information (bits)
1Department
1,2,3
Kastner ,
1
0.9
0.8
0.2
0.3
Contrast ( )
Average fraction of
the maximal
information captured
by 7 pairs of
simultaneously
recorded adapting
and sensitizing cells
across different
contrasts
Conclusions
• Coordinated fast-Off populations provide the optimal amount of
information about their inputs given noise and an energy constraint
• The absence of multiple types of On cells in salamanders is
predicted by optimal information transmission
• Maximal information transmission with populations of neurons has a
direct parallel with second order phase transitions from physics
• fast-Off ganglion cells reside near a critical point