1. INTRODUCTION

A neuron’s receptive field succinctly describes neural feature selectivity and quantitatively predicts responses to synthetic and natural stimuli. Simple receptive field models that account for responses to some stimuli often require modification to account for responses to others. This is often described as modulation by input statistics, i.e., context. Characterizing these modulatory effects remains a challenge. We address this problem in primary visual cortex, where such modulatory effects are considered to have substantial functional significance. We analyzed how receptive fields changed between two stimulus sets - formed with two-dimensional Hermite functions - that were matched in luminance, contrast, spatial extent, and spatial frequency compositions, but differed in their two-dimensional organization:

3. COMPARISON BETWEEN MODELS

5. PROCRUSTES ANALYSIS

7. SUMMARY

Tatyana O. Sharpee and Jonathan D. Victor Sloan-Swartz Center for Theoretical Neurobiology and Dept. of Physiology, UCSF, San Francisco, CA, 94143;

Weill Medical College of Cornell University, New York, NY 10021.

Spatial Symmetry and Stability of V1 Receptive Fields Revealed with Two-Dimensional Hermite Functions†*

*†

6. SYMMETRY AND LABILITY OF RECEPTIVE FIELD COMPONENTS

4. CONTEXT-DEPENDENCE

rank01234567

Cartesian stimulus set polar stimulus set

cosine

x

y

sine

2. METHODS

We computed V1 receptive fields (n=51, 34 from cat V1 and 17 from macaque V1) using two complementary approaches:

I. In the two-pathway model, the neural response is a sum of three elements: - stimulus filtered with the linear part of receptive field (L filter) contributes linearly to the predicted firing rate. - stimulus filters with the even part of receptive field (E filters) contributes to the predicted firing rate through full-wave rectification (absolute value of the stimulus convolved with E filter). - mean firing rate is added.

II. We compute receptive fields of the linear-nonlinear model as most informative dimensions (MID) (Sharpee, Rust, & Bialek 2004). We will denote these receptive fields as

linear-nonlinear model

<0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

5

10

15

20

25

num

ber

of c

ells

0.5<0 0.0 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.00

5

10

15

20

25

num

ber

of ce

lls

c34 site 2 experiment 1 unit 6 cell #27

-0.1

0

0.1

c30 site 3 experiment 3 unit 6 cell #4

Lcart

Mcart

Mpolar

Lpolar

Lcart

Mcart

MpolarLpolar

-0.1

0

0.1Examples of receptive fields with no context-dependent changes.

c30 s ite 3 experiment 3 unit 3 cell #3

-0.1

0

0.1

Lcart Mcart

MpolarLpolar

c33 s ite 1 experiment 1 unit 1 cell #11

-0.1

0

0.1

Lcart Mcart

MpolarLpolar

original RF

Mpolar

Lpola

r

0 1 2 3 4 5 6 70123

4

5

6

7

Lpola

r

Mpolar

01 2 3 4 5 6 70123

4

5

6

7

Lca

rt

Mcart

01 2 3 4 5 6 70123

4

5

6

7

Lca

rt

Mcart

01 2 3 4 5 6 70123

4

5

6

7

01 2 3 4 5 6 70123

4

5

6

7

Mcart

Mpola

r

0 1 2 3 4 5 6 70123

4

5

6

7

Lcart

Lpola

r

0 1 2 3 4 5 6 70123

4

5

6

7

Mcart

Mpola

r

Lcart

Lpola

r

0 1 2 3 4 5 6 70123

4

5

6

7

rms of diagonal elements for thebest rotation matrix

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

M filtersL filters

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

M filtersL filters

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6

rms of receptive fieldcomponents

Lcart

Mcart

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6

with symmetry argument01 2 3 4 5 6 7

012

3

4

5

6

7

01 2 3 4 5 6 7012

3

4

5

6

7

Vertical flip of the

coordinate system

Receptive field components that are anti-symmetric with respect to 180 degree spatial rotation (odd-rank components) are more susceptible to context dependent changes that the symmetric components (even-rank components).

Procrustes analysis allows for a non-parametric characterization of context-dependent changes in receptive fields.

8. ACKNOWLEDGMENTS

We thank F. Mechler, M.A. Repucci, and K.P. Purpura for help with experiments. This research was supported by the Swartz Foundation, and NIH grants 5K25MH068904 to TOS and 2RO1EY009314 to JDV.

stimulus

mean rate

sum to compute

the firing rate

L filter

E filter

Xstimulus filtered by RF

firing r

ate

two-pathway quasi-linear model

M filter

Correlation coefficient between receptive fields of the two-pathway and the linear -nonlinear model

l51 site 6 experiment 1 unit 1 cell #47

Lcart Mcart

MpolarLpolar

-0.1

0

0.1

-0.1

0

0.1

c30 s ite 7 experiment 1 unit 1 cell #8

Lcart Mcart

MpolarLpolar

c34 s ite 1 experiment 1 unit 1 cell #22

-0.05

0

0.05

Lcart Mcart

MpolarLpolar

l51 site 6 experiment 1 unit 2 cell #48

Lcart Mcart

MpolarLpolar

-0.1

0

0.1

num

ber

of ce

lls

<0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

5

10

15

20

25

30

35

Cartesianstimuli

p<0.01

p>0.050.01<p<0.05

num

ber

of ce

lls

<0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

5

10

15

20

25

30

35

Polar stimuli

Receptive fields computed using the two models were consistent with each other for most neurons, in 44/51 for Cartesian stimuli and 42/51 for polar stimuli. L filters of the two-pathway quasi-linear model and M filters of the linear-nonlinear model were compared.

Even for correlation coefficients that are far from 1, there still appears to be a strong resemblance between the L and M filters. --> The debiasing appears to be conservative in the sense that the correlation coefficients appear to underestimate the similarity of the estimated receptive fields.

Correlation coefficients were debiased to compensate for finite data effects using the jackknife method (Efron and Tibshirani, 1998). This can lead to negative correlation coefficients (even though only shapes of receptive fields are compared, and not their polarity.

Comparison of the nonlinear firing rate function in the two models.

-1.0 -0.5 0 0.5-1.0

-0.5

0

0.5

asymmetry index (M filters)

asym

met

ry index

(L

and E

filt

ers)

Cartesian (r=0.97)

polar (r=0.97)

For the two-pathway quasi-linear model, we choose asymmetry index as:

It is based on the relative powerand of the L and E filters.

A LE =|L | − |E ||L | + |E |

AM =f 0 − f −f + − f 0

|L | =il2i

|E | =ie2i

For the linear-nonlinear model, we measurethe asymmetry of the firing rate function as:

It is based on the average firing rates F. , f , and f_for all stimuli with positive, zero, or negativecorrelations with the receptive field.

0+

Correlation coefficient between receptive fields derived from Cartesian and polar stimulus sets.

p<0.01

p>0.050.01<p<0.05

two-pathway model linear-nonlinear model

L filtersrms of jackknife components

M filt

ers

rms

of ja

ckkn

ife

com

ponen

ts

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2polarCartesian

Receptive field estimates have lower variability in the two-pathwayquasi-linear model than of the linear-nonlinear model.

In approximately half of the cells, receptive fields showed significant differences with context, i.e. between Cartesian and polar stimulus sets.

Examples of changes in receptive fields with context:

Any apparent linear relationship between the receptive field coefficients along the inverting and non-inverting patterns must be due to chance. To ensure that symmetry-violating coefficients are zero, we add receptive fields in the mirror coordinate systems.

c33 s ite 1 experiment 1 unit 2 cell #12

-0.1

0

0.1

Lcart Mcart

MpolarLpolar

c30 s ite 3 experiment 3 unit 2 cell #2

-0.1

0

0.1

Lcart Mcart

MpolarLpolar

chan

ge

in o

rien

tation

chan

ge

in s

pat

ial

freq

uen

cy

To non-parametrically characterize any systematic differences in receptive fields at the population level, we compute the best linear transformation between two sets of receptive fields. This “Procrustes” matrix represents a rotation in 36-component space (dimensionality of receptive fields.

Reducing the number of free parameters with symmetry arguments:

-

-

Significant changes with context were more common for receptive fields computed with the two-pathway model (39/51) than with the linear-nonlinear model (20/51), as is the case for the first two of the examples above. Nevertheless, context-dependent changes were always qualitatively similar in both models, suggesting that lack of significance in the case of the linear-nonlinear model was due to the higher variability in these receptive field estimates.

transformation ~ unit matrix

Deviations from the unit matrix characterize context-dependent changes.Higher rank are more susceptible to context-dependent changes than the lower-rank

Reducing the number of free parameters with symmetry arguments,makes it apparent that odd-rank components are more labile than the even-rank ones:

Effect not observed in sets of random vectors.

The difference in lability between even and odd rank components is not due to a differences in magnitude of these components, which decrease with rank monotonically, or their variability.

Higher-rank components show greater context-dependent changes than the more localized, lower rank components.

rms of receptive fieldcomponents

rankrank

Lpolar

Mpolar

rms of diagonal elements for thebest rotation matrix