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Title: A Unified Theory of Early Visual Representations from Retina to Cortex through Anatomically Constrained Deep CNNs

Paper content:
The vertebrate visual system is hierarchically organized to process visual information in successive stages.
Neural representations vary drastically across the first stages of visual processing: at the output of the retina, ganglion cell receptive fields (RFs) exhibit a clear antagonistic center-surround structure, whereas in the primary visual cortex (V1), typical RFs are sharply tuned to a precise orientation.
There is currently no unified theory explaining these differences in representations across layers.
Here, using a deep convolutional neural network trained on image recognition as a model of the visual system, we show that such differences in representation can emerge as a direct consequence of different neural resource constraints on the retinal and cortical networks, and for the first time we find a single model from which both geometries spontaneously emerge at the appropriate stages of visual processing.
The key constraint is a reduced number of neurons at the retinal output, consistent with the anatomy of the optic nerve as a stringent bottleneck.
Second, we find that, for simple downstream cortical networks, visual representations at the retinal output emerge as nonlinear and lossy feature detectors, whereas they emerge as linear and faithful encoders of the visual scene for more complex cortical networks.
This result predicts that the retinas of small vertebrates (e.g. salamander, frog) should perform sophisticated nonlinear computations, extracting features directly relevant to behavior, whereas retinas of large animals such as primates should mostly encode the visual scene linearly and respond to a much broader range of stimuli.
These predictions could reconcile the two seemingly incompatible views of the retina as either performing feature extraction or efficient coding of natural scenes, by suggesting that all vertebrates lie on a spectrum between these two objectives, depending on the degree of neural resources allocated to their visual system.
Why did natural selection shape our visual representations to be the way they are?
Traditionally, the properties of the early visual system have been explained with theories of efficient coding, which are based on the premise that the neural representations are optimal at preserving information about the visual scene, under a set of metabolic constraints such as total firing rate or total number of synapses.
These theories can successfully account for the antagonistic center-surround structure of receptive fields (RFs) found in the retina BID0 BID18 BID35 BID21 BID13 , as well as for the oriented structure of RFs found in the primary visual cortex V1 BID30 BID3 .However
, a number of properties of the early visual system remain unexplained. First,
it is unclear why RF geometries would be so different in the retina and V1. A study
BID36 has proposed that both representations are optimal at preserving visual information under different metabolic constraints: a constraint on total number of synapses for the retina, and one on total firing rate in V1. However
, it is unclear why the two systems would be optimized for these two different objectives. Second
, there is a great diversity of ganglion cell types at the output the retina BID17 , with each cell type tiling the entire visual field and performing a specific computation. Interestingly
, some of these types perform a highly nonlinear computation, extracting specific, behaviorally-relevant cues from the visual scene (e.g. direction-selective cells, objectmotion-selective cells), whereas other types are better approximated by a quasi-linear model, and respond to a broad range of stimuli (e.g. midget cells in the primate BID32 and quasi-linear pixel-encoders in the mouse BID20 ). Intriguingly
, although quasi-linear and more nonlinear types exist in species of all sizes (e.g. primate parasol cells are nonlinear BID10 ), the proportion of cells performing a rather linear encoding versus a nonlinear feature detection seems to vary across species. For example
, the most common ganglion cell type in the primate retina is fairly well approximated by a quasi-linear pixel-encoder (midget cells, 50% of all cells and >95% in the central retina BID32 BID11 ), whereas the most common cell type in mouse acts as a specific feature detector, thought to serve as an alarm system for overhead predators (W3 cells, 13% of all ganglion cells BID38 ). Again, theories
of efficient coding have not been able to account for this diversity of computations found across cell types and across species.The limitations of current efficient coding theories might reside in the simplistic assumption that the objective is to simply relay indiscriminately all visual information to the next stages of processing. Indeed, the ultimate
goal of the visual system is to extract meaningful features from the visual scene in order to produce an adequate behavioral response, not necessarily to faithfully encode it. A recent line of work
has proposed using the information bottleneck framework as a way to move beyond the simplistic objective of information preservation towards more realistic objectives BID5 . Another study has shown
that by changing the objective from efficiently encoding the present to efficiently encoding the future (predictive coding), one could better account for the spatio-temporal RFs of V1 cells BID34 . Although promising, these
approaches were limited to the study of a single layer of neurons, and they did not answer the aforementioned questions about cross-layer or cross-species differences. On the other hand, deep convolutional
networks have proven to be accurate models of the visual system, whether they are trained directly on reproducing neural activity BID28 BID4 , or on a behaviorally relevant task BID37 BID14 BID4 ), but they have not yet been used to study the visual system through the lens of efficient coding theories.In this study, we trained deep convolutional neural networks on image recognition (CIFAR-10, BID23 ) and varied their architectures to explore the sets of constraints that could have shaped vertebrates' early visual representations through natural selection. We modeled the visual system with a series
of two convolutional networks, one corresponding to the retina and one downstream network corresponding to the ventral visual system in the brain. By varying the architecture of these networks
, we first found that a reduction in the number of neurons at the retinal output -corresponding to a realistic physical constraint on the number of fibers in the optic nerve -accounted simultaneously for the emergence of center-surround RFs in our model of the retina, and for the emergence of oriented receptive fields in the primary visual relay of the brain. Second, we found that the degree of neural resources
allocated to visual cortices in our model drastically reshaped retinal representations. Given a deep visual cortex, the retinal processing emerged
as quasi-linear and retained substantial information about the visual scene. In contrast, for a shallow cortex, the retinal processing
emerged as nonlinear and more information-lossy, but was better at extracting features relevant to the object classification task. These observations make testable predictions on the qualitative
differences that should be found in retinal representations across species, and could reconcile the seemingly incompatible theories of retinal processing as either performing efficient encoding or feature detection.
A unified theoretical account for the structural differences between the receptive field shapes of retinal neurons and V1 neurons has until now been beyond the reach of efficient coding theories.
BID21 found that efficient encoding of images with added noise and a cost on firing rate produce center-surround RFs, whereas the same task without noise produces edge detectors.
However, this observation (as they note) does not explain the discrepancy between retinal and cortical representations.
BID36 propose a different set of constraints for the retina and V1, in which the retina optimizes for a metabolic constraint on total number of synapses, whereas V1 optimizes for a constraint on total firing rate.
It is not clear why each of these constraints would predominate in each respective system.
Here we show that these two representations can emerge from the requirement to perform a biologically relevant task (extracting object identity from an image) with a bottleneck constraint on the dimensionality of the retinal output.
Interestingly, this constraint differs from the ones used previously to account for center-surround RFs (number of synapses or total firing rate).
It is worth noting that we unsuccessfully tried to reproduce the result of BID21 in our network, by adding noise to the image and applying an L1 regularization to the retina-net activations.
In our framework (different than the one of BID21 in many ways), the receptive fields of the retina-net without bottleneck remained oriented across the full range of orders of magnitude of noise and L1 regularization that permitted successful task performance.There is a long-standing debate on whether the role of the retina is to extract relevant features from the environment BID25 BID17 BID32 , or to efficiently encode all visual information indistinctly BID2 BID0 BID18 .
In this work, we show that our model of the visual system, trained on the same task and with the same input statistics, can exhibit different retinal representations depending on the degree of neural resources allocated to downstream processing by the ventral visual stream.
These results suggest the hypothesis that, despite its conserved structure across evolution, the retina could prioritize different computations in different species.
In species with fewer brain resources devoted to visual processing, the retina should nonlinearly extract relevant features from the environment for object recognition, and in species with a more complex ventral visual stream, the retina should prioritize a linear and efficient transmission of visual information for further processing by the brain.
Although all species contain a mix of quasi-linear and nonlinear cell types, the proportion of quasi-linear cells seems to vary across species.
In the mouse, the most numerous cell type is a two-stage nonlinear feature detector, thought to detect overhead predators BID38 .
In contrast, the most common ganglion cell type in the primate retina is fairly well approximated by a linear filter (midget cells, 50% of all cells and >95% in the central retina BID32 BID11 ).
Note however that two-stage nonlinear models are also present in larger species, such as cat Y-type cells and primate parasol cells BID10 , making it difficult to make definitive statements about inter-species differences in retinal coding.
To gain a better understanding of these differences, it would be useful to collect a dataset consisting of recordings of complete populations of ganglion cells of different species in response to a common bank of natural scenes.A related question is the role of the parcellation of visual information in many ganglion cell types at the retinal output.
A recent theory of efficient coding has shown that properties of midget and parasol cells in the primate retina can emerge from the objective of faithfully encoding natural movies with a cost on the total firing rate traversing the optic nerve BID29 .
On the other hand, many cell types seem exquisitely sensitive to behaviorally relevant features, such as potential prey or predators BID17 .
For example, some cell types in the frog are tuned to detect moving flies or looming predators BID25 .
It is an intriguing possibility that different cell types could subserve different functions within a single species, namely efficient coding of natural scenes for some types and extraction of behaviorally-relevant features for others.
In this study we allowed only a limited number of cell types (i.e. convolutional channels) at the retinal output (1 to 4), in order to have a dimensionality expansion between the retinal representation and the representation in the ventral visual stream (32 channels), an important condition to see the retinal center-surround representation emerge.
By using larger networks with more channels in the retina-net and the VVS-net, we could study the emergence of a greater diversity of neuron types in our retina-net and compare their properties to real retinal cell types.
It would also be interesting to extend our model to natural movies.
Indeed, most feature detectors identified to date seem to process some form of image motion: wide-field, local or differential BID32 .
Adding a temporal dimension to the model would be necessary to study their emergence.In conclusion, by studying emergent representations learned by a deep network trained on a biologically relevant task, we found that striking differences in retinal and cortical representations of visual information could be a consequence of the anatomical constraint of transmitting visual information through a low-dimensional communication channel, the optic nerve.
Moreover, our computational explorations suggest that the rich diversity of retinal representations found across species could have adaptively co-evolved with the varying sophistication of subsequent processing performed by the ventral visual stream.
These insights illustrate how deep neural networks, whose creation was once inspired by the visual system, can now be used to shed light on the constraints and objectives that have driven the evolution of our visual system.
The following analysis corroborates our qualitative observation that a dimensionality bottleneck in the retina-net yields center-surround retinal receptive fields and oriented, edge-detecting receptive fields in the first layer of the VVS-net (V1).
For a given receptive field, we quantified its orientedness as follows: we displayed rectangular bar stimuli of all possible combinations of width, orientations and spatial translations that fit in the input image window.
Among all these combinations, we selected the bar stimulus width, orientation, and translation that yielded the strongest response from the RF.
Bars with the same width as the best stimuli were presented at all orientations and translations, and for each orientation, we select the strongest response it produced (across all translations).
In this manner we obtained a measure of the strength of a receptive field's preference for all orientations.We measured the strength of each RF preference (maximum strength of response) for its preferred orientation and for the orthogonal orientation, and computed the ratio of these strengths.
Completely isotropic filters would be expected to give a ratio of 1, while oriented filters should give higher ratios.
Note however that some deviation from 1 may indicate noise in the filter rather than true orientedness.
For each network layer, we averaged this ratio across filters (for convolutional layers with multiple layers) and trials (re-training of the same neural network architecture with different random initializations).
We found that the average ratios were 1.56(±0.22) for the retinal output, 3.05(±0.30) for the first VVS-net layer, and 2.57(±0.27) for the second VVS-net layer, where error margins given are 95% confidence intervals.
To help assess whether retinal RFs were more isotropic than expected by chance, we compared them to receptive fields composed of random Gaussian noise as a baseline.
These give an average ratio (as computed above) of 1.97(±0.08), significantly higher than that for retinal RFs.
Furthermore, the standard deviation of RF preference across orientations was significantly lower for the retinal RFs (0.118 ± 0.036) than for random RFs (0.177 ± 0.007), also indicating that retinal RFs were more isotropic than expected by chance.We also plot the average RF preference for different orientations at each layer to more comprehensively assess the isotropy of RFs at each network layer.
To aggregate results across multiple trials and filters, we rotated the coordinates of each receptive field such that its preferred orientation was vertical, and averaged our results across filters and trials.
(See FIG4 .The results confirm our qualitative observations that (1)
RFs in the second layer of a vanilla network (N BN = 32) are highly oriented ( FIG4 ) (2) RFs in the second layer (retina output) of a bottleneck network (N BN = 1) are much more isotropic, consistent with center-surround RFs ( FIG4 top), and (3) RFs in the layer immediately following the retina-net in the bottleneck network are oriented ( FIG4 .We
also quantitatively corroborate our observation that oriented receptive fields in the V1 layer pool input from oriented arrays of center-surround filters in the retina-net output layer. We
apply our method of isotropy quantification described above to the weight matrix for each input-output filter combination in the V1 convolutional layer. We
find that this weight matrix itself exhibits orientedness across filters and trials, confirming our observation ( FIG4 ). To
investigate whether neurons in our model's early layers more closely resembled simple or complex cells, we performed the following analysis. As
before, we obtained local linear approximations of receptive fields by computing the gradient in input space with respect to the response of a given neuron. Rather
than beginning with a blank input, we ran multiple trials with different randomly initialized inputs. A purely
linear cell would give the same result no matter the initialization; a somewhat nonlinear but still "simple" cell is expected to give similar results across initializations.

assistant: text: We reproduced neural representations found in biological visual systems by simulating their neural resource constraints in a deep convolutional model.