Natural image synthesis for the retina with variational information bottleneck representation
Anonymous Author(s)
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Abstract
Visual prostheses provide artificial vision to the blind by encoding the naturalvisual scene onto electrical signals that stimulate the undamaged neurons in theretina.
Encoding the high-dimensional visual input onto the limited number ofelectrodes of the prosthesis results in unnatural perceptions that often induces usersto avoid using the device due to the very high cognitive burden.
But, even in theunimpaired natural visual system, is all the visual scene information required toexplain the neuronal responses?
In this work, we search for answers to this questionby developing a joint model of the natural visual input and neuronal responses usingthe Information Bottleneck (IB) framework that is able to represent features of theinput data into a few latent variables that play a role in the prediction of the outputs.
The correlations between data samples, acquired from published experiments onex-vivo retinas, is accounted for in the model by a Gaussian Process (GP) prior.
Theproposed IB-GP model performs competitive to the state-of-the-art feedforwardconvolutional networks in prediction of spike responses to natural stimuli.
Weobserve that reconstructed stimuli from latent variables show robustness in spikeprediction across models.
Finally, the IB-GP model is used in a closed loop iterativeprocess to obtain reduced-complexity inputs that elicit responses as those elicitedby the original stimuli.
Introduction
A fundamental challenge in neuroprosthetics is finding the proper input to a sensory or motor systemthat yields a desired functional output.
This is achieved naturally by the underlying physiologicalcircuit in an unimpaired system.
For example, in the vision system, when an image is formed ontothe photoreceptors, the electrical neural activity is processed by several layers of neurons within theretina and a train of electrical neural spikes is sent to the visual cortex via the optic nerve producinga perception of the visual scene.
Retinal degenerative diseases may break this flow of information atone point while the rest of the circuit remains intact [24] [22].
In this case, prostheses could be usedto artificially stimulate neural cells in the retina to restore the lost functionality [38] [20] [16].
Thecurrent technology in the visual prosthesis consists of an array of stimulating electrodes.
The numberof stimulating electrodes is ∼ 4 orders of magnitude smaller than the number of photoreceptors [15].
Given this fact, one can ask: even in the natural vision system, is it possible to reduce the complexityof the input stimuli and still obtain the same functional behavior?
In other words, what are the reducedspatiotemporal stimuli that elicit the same response as that of the higher-dimensional original input?
Nirenberg et al.
[26] showed that a good encoder of the retina responses dramatically increases theprosthesis performance.
In this work, we focus on the natural vision system and search for answers tothese questions.
Our system consists of the spatiotemporal stimuli impinging on the photoreceptors inthe retina up to the spiking responses emitted by the Retinal Ganglion Cells (RGCs).
Linear-nonlinear (LN) [11] and generalized linear models (GLMs) [28] [29] were proposed for encoding of the retinalneural codes.
Although successful for artificial stimuli such as white noise, they could not faithfullydescribe RGCs responses to natural stimuli [17].
The convolutional neural network (CNN) modelof McIntosh et al.
[25] was the first model capable of such computation.
This was followed byother efforts such as employing recurrent neural networks [5] or variants of CNNs [23].
Althoughhigh-dimensional in nature, neural activity can be well explained by low-dimensional representations[12] [14] [31].
This opens up the possibility of using models that learn a latent representation of data,such as VAEs [18] and ICAs [21].
Accordingly, latent dynamical methods that inspect sources oflow-dimensional structure in the neural response have recently gained growing interest [19] [37].
Inspired by the Deep Variational Information Bottleneck (DVIB) [2] [9], we introduce the Information Bottleneck Gaussian Process (IB-GP), a latent space model that can extract the principal featuresof the joint stimuli-responses distribution that are sufficient for predicting the retina responses.
Theschematic of the IB-GP is depicted in Fig.
This model is a variational approximation to the original Information Bottleneck framework of Tishby et al.
The objective of IB is to obtain a reducedrepresentation of the input source that preserves maximum information about the output response.
DVIB assumes the IB’s latent codes are i.i.d.
This is a poor assumption for the retina dataset.
Thereason for this is two-fold.
First, while observing a scene, images captured by the retina are naturallyvery similar.
Second, during fixation, the eyes undergo dynamic movements that shift the gaze’scenter.
Therefore, incorporating the temporal correlations among data samples should provide abetter model.
Accordingly, IB-GP uses a Gaussian Process prior to modeling the retina dataset.
Ourcontributions in this work are:
• A probabilistic model : We propose IB-GP, a new model for the RGCs spike train in response tothe complex natural visual stimuli using the IB framework.
The model integrates a GP prior intothe IB latent code that can learn the temporal dynamics of the data in a lower dimension space.
Thismodel shows favorable performance compared to the state-of-the-art feedforward CNN models ofthe retina tested on real-world experimental data.• Latent dynamics analysis : We analyze the latent space of the proposed model and extract lowdimensional dynamics consistent with the high-dimensional data dynamics.
We show that theextracted dynamics are more predictive of the neural responses compared to dynamics extracted bya model with no temporal prior constraint.
• Closed loop image synthesis : Utilizing the IB-GP model, which only allows principal features of the input stimuli to pass the bottleneck, the original input stimuli are pruned to contain the minimum information required for producing the RGC responses.
The resulting reduced-dimension optimized stimuli could be used for the next round of measurements in an iterative process.
In the following, we start by reviewing the related work in section 2, the general setting of the problemand our model are presented in section 3, and the closed-loop stimuli optimization procedure is outlined in section 4.
Experiments and results are presented in sections 5 and 6 1 .
Finally, the resultsare discussed in section 7.
Related work
In the RGC modeling literature, our work is the closest to the work of McIntosh et al.
[25] in whicha CNN is used to model the responses of RGCs to a set of input stimuli.
Despite the simplicity ofthis approach, the CNN cannot be directly utilized to extract the principal features of the stimulicontributing to the neural responses.
On the other hand, neural latent models such as [19] [27] [32]are primarily applied to cortical recordings to learn sources of neural variability in the motor cortexdata.
Unlike our model, which directly relates the stimuli to the neural responses, the stimuli in thoselines of work are utilized as a guide star to disentangle the latent factors.
Therefore, these methodsare not directly applicable to modeling the RGC responses to input stimuli.
In the closed-loop physical system input optimization literature, our work is related to [30][33][36][4].
In particular, for neuroscience applications, authors in [4] control the activity of the individualneuronal sites in V4 by optimizing individual input stimulation, whereas in our work, thanks to thelearned latent code of the entire population of the targeted neurons, all stimuli are optimized together.
In the work of Shah et al.
[33], the RGCs responses to electrical stimulation were optimized byfirst developing a model for the electrical stimuli and the spiking probabilities and then using themodel for adaptive stimulation.
The model is obtained by parameterizing the spike amplitude andelectrical stimulation threshold by a few parameters and then maximizing an evident lower bound onthe spiking probabilities.
Our work does not assume any relevant relation between the input stimuliand the spiking probabilities.
Instead, the relevant parts of the input for predicting the output arediscovered automatically.
Methods3.1 IB-GP model
Notation and problem formulation The input stimuli dataset X ∈ R M × T consists of T consecutive spatial inputs of size M , i.e. X =
[ x M, 1 , x M, 2 , · · · , x M,T ] , that are projected in the retina at aconstant rate to elicit count responses of the form Y ∈ N N × T , where Y =
[ y N, 1 , y N, 2 , · · · , y N,T ]and N denotes the number of RGCs.
With the above assumptions, we formulate the problem asfinding the latent variables Z ∈ R L ×T that have maximal mutual information
I ( Z , Y ) with targets
Y .
We note that Z is comprised of T consecutive data points, i.e. Z =
[ z L, 1 , z L, 2 , · · · , z L, T ] where
T ≤ T .
Z is simultaneously constrained to have minimal mutual information with X .
Therefore, theconstraint optimization problem can be written as
[Equation]
where
[Equation]
ζ designates the parameters of the model (notation henceforth dropped for brevity) and β is a variableto adjust the amount of reduced and preserved information in Z .
To obtain these latent variables, weapproximate their posterior distribution with a parametric stochastic encoder p ϕ
( Z | X ) .
Due to thespatio-temporal nature of the data, assumption of data independence along the time dimension is notvalid.
The same is true in the latent space.
To account for this fact, we assume a posterior of the form:
[Equation]
where z l denotes the l -th latent vector of size T .
In Eq. 2, µ lϕ ∈ R T and Σ lϕ ∈ R T ×T are the meanand covariance matrix of a multi-variate normal distribution that are functions of the inputs.
Wechoose the structure of the covariance of the multivariate normal distribution in Eq.
2 to reflect the time correlations in the data.
Hence, similar to previous work [3] [7] [13], we construct the Σ lϕ in ourmodel by the product of bidiagonal matrices:
[Equation]
We note that the low-rank GP kernel assumption makes the computation required for drawing samplesfrom
p ϕ linear in time [3].
Expanding the IB objective in Eq.
1 (details in the Appendix 1), we have:
[Equation]
Although all terms in the RHS of Eq. 4 are fully defined, computing marginal distributions p ( Z ) andp ( Y | Z ) may be intractable.
We use a variational approximation for p ( Y | Z ) that is parameterized withθ .
On the other hand, the prior on the latent variables, i.e. p ( Z ) , is modeled using GPs.
In particular,we assume a GP prior on Z defined as a multivariate normal:
[Equation]
where ρ is the variational approximation for the prior and K is the covariance function that modelsthe temporal correlations in the latent space.
In more details, the covariance between the τ -th and theτ ′ -th
samples is computed as K ττ ′ =
K ( τ, τ ′ ) where K is the kernel function.
We used the Cauchykernel:
[Equation]
where σ and l are the magnitude and temporal scales of the kernel function, respectively.
Cauchykernels have proved successful in modeling of the time-series systems with multi-scale dynamics asshown before [13].
Substituting the variational approximations of the intractable marginals in Eq.
4and using the fact that the Kullback Leibler (KL) divergence is always positive, we obtain a lowerbound for the IB objective:
[Equation]
where τ ( t ) is to represent time dependence up to time t .
See Appendix 1 for more details.
2 Other models considered
Maximum likelihood feedforward CNN is used as a comparison baseline.
We use the same CNNnetwork that was shown in the previous work to obtain state-of-the-art results on a larger version ofthe Natural dataset [25].
The predicted outputs of the network are the averaged maximum likelihoodestimation of the retina responses given the input stimuli.
Artificial noise is injected into the modelduring optimization to account for the variability in retinal spiking.
IB-Disjoint optimizes the objective from Eq.
7 using the same training procedure as that of theoriginal method.
The difference lies in that the IB-Disjoint assumes latent samples are temporallyindependent.
Accordingly, an isometric Gaussian distribution is employed as the prior in Eq.
Thismodel is akin to the standard Variational Auto-encoder [18], or vanilla DVIB [2] in which samples inthe data and latent space are assumed to be independent (no latent GP).
Closed loop stimulation
In this section, we take advantage of the model developed in the previous sections to devise analgorithm that uses the prior recorded data to optimize the stimulation by iteration in subsequentmeasurements in a closed loop.
Specifically, we assume the latent variables of the learned model,henceforth referred to as the forward model , have captured the principal rules governing the underlyingbiological system.
We intend to find a transformed version of the original input stimuli that yields a set of latent variables that produce the most correlated responses with the original target responsesin the subsequent round of measurements.
Of course, the original stimuli themselves are one set ofsolutions.
However, this is not a useful transformation.
Instead, we are interested in obtaining the besttransformation subject to a constraint on their complexity.
Hence, we define a parametric functionf ξ : R M × T → R M × T that maps the complex original stimuli X onto the transformed stimuli X ∗ .
Passing X ∗ to the forward model, the parameter of the mapping functions, i.e. ξ , are optimized sothat the forward model’s output responses Y ∗ are the most correlated with the original responses
Y .
Denoting the original stimuli and responses as X orig := X and Y orig :
=
Y , the objective functionreads as:
[Equation]
where:
[Equation]
Therein, p ϕ and q θ are the encoder and decoder of the forward model trained on the prior data.
D 1is a measure to ensure the closeness of the original target outputs and the responses of the forwardmodel to X ∗ .
D 2 is a similarity measure between the original and synthesized stimuli X orig and X ∗which encourages the image synthesizer to focus on finding the essential features in the stimuli ratherthan synthesizing entirely new solutions.
Finally, D 3 is used to constrain the synthesized stimuli
X ∗to be smooth.
Contributions of D 2 and D 3 to the total loss in Eq.
8 are tuned by hyperparameter α .
Two major mechanisms help to prevent f ξ from being simply an identity mapping.
One, during thetraining of the f ξ , the encoder’s latent variables Z ∗ are masked by zeroing out all the latent variablesexcept for the most informative one.
This way, X orig is no longer the optimal solution that minimizesthe first term in the objective function.
Two, f ξ is implemented using an Autoencoder architecturewith a limited latent capacity.
Finally, the optimized stimuli for subsequent rounds of measurementscan be obtained by the following procedure: (1) map the original stimuli to the optimized stimuliusing mapping f ξ , (2) encode the new input X ∗ to the posterior Z ∗ , (3) decode the latent variables,(4) construct the metric D and backpropagate the error to fit f ξ , (5) once converged, use the mappingfunction to obtain the optimized stimuli X ∗ (6) send new inputs X ∗ to the true biological system (orits model) and observe the new true system outputs, (7) to repeat the procedure, use the true system’snew inputs and outputs to fine-tune the forward model and (8) repeat.
Experiments
5.1 Dataset
We focus on applying our model to the time-series dataset containing the 2D train of input stimuli(images) entering the vision circuit of an example subject and the corresponding elicited countresponses of a group of neurons.
Specifically, we use two real-world experimental datasets.
The firstone, Natural dataset, contains natural scene images as input stimuli and spike trains from nine RGCsin Salamander [25] 2 .
The second dataset, Brownian-movement , is a public dataset which containsimages with disc-like shapes having Brownian movements and the spike trains elicited in 91 Rat’s
RGCs [8].
5.2 Forward modeling
Training specifications We used the models outlined in section 3 to fit the neural activities andinput stimuli.
The Natural dataset contains 359000 input-output samples in the training set, of which20% was randomly chosen for validation.
The test set contains 5000 averaged repeated trials ofnovel stimuli.
We note that the output responses were first binned at 10 ms and then smoothed usinga Gaussian filter with a standard deviation equal to the bin size.
The Brownian-movement datasetcontains 129600 pairs of input images and output spikes binned at 12.5 ms in the training set, ofwhich 20% was randomly chosen for validation.
The test set contains 1800 novel stimuli.
We variedthe number of neurons in the Brownian dataset for which we tested our model.
We report the resultsfor nine neurons in the main paper and 15, 21, 27, and 36 neurons in Appendix 6.
Models and evaluation metrics To account for the discrete nature of the output responses (counttime series), we employ a Poisson regression model for both IB-GP and IB-Disjoint decoder aswell as the feedforward CNN loss term, i.e. q θ ( y t | Z ) =
Po ( y t | f θ ( Z )) where f θ is the parametricnetwork.
We sweep the number of latent factors and the prior pressure constraint β for both IB-GPand IB-Disjoint.
For the Natural and Brownian datasets, the number of latents is swept between 1to 15 and 1 to 50 variables, respectively.
Except for the latent space of the IB-GP, which requiresmore variables to model its covariance matrix, IB-GP and IB-Disjoint are implemented using thesame architectures.
All hyperparameters in the training of the two models are fixed to be the same.
The architectures of the networks and the training setup are explained in more detail in Appendix 3.
We perform a three-fold fitting of the models for the spike prediction results.
We report the models’
Poisson negative log-likelihoods on the ground truth data.
Moreover, we use the Pearson Correlation[6] between the ground truth data and the networks’ prediction as another metric for evaluating theperformance of the models.
Traversal analysis We perform traversal analysis on the latent space of IB-GP and IB-Disjointmodels to study the amount of information picked by each latent variable.
To this end, the encoderp ϕ ( Z | X ) is fed with a batch of input stimuli to initialize the latents.
In the IB-GP model, the l -thlatent is sampled from N ( 0 , m K ) where m is a multiplier that is traversed in the range [ − 3 , 3] whilekeeping all the other latents to their inferred values.
The latent variables in the IB-Disjoint model aresampled similarly but from the normal univariate distribution N ( 0 , m I ) .
The latents are then passedto the decoder that reconstructs the output responses.
The output responses of the traversed model arethen compared against the non-traversed responses.
5.3 Closed-loop image synthesis
We use both the IB-GP and IB-Disjoint with a latent size of 4.
The prior constraint parameter β is setto 0.05.
These hyperparameters are chosen as a trade-off between the latent space compression and
RGCs neural activity fitting accuracy.
The image synthesizer f ξ is implemented using an Autoencoderarchitecture [35] with a latent size of 15.
The rationale for this choice of architecture is explained in
Appendix 3.
To allow for multi-phase iterative optimization, we use a neural network as the proxyfor the true biological system trained on the Natural dataset.
This model is trained once and keptfixed.
We use the feedforward CNN network of McIntosh et al.
[25] as the true model in order torule out the possibility of obtaining correlated responses due to the similarity of the IB-GP modeland the true model.
We optimize the stimuli in a three-phase closed-loop experiment.
We used the
Poisson regression loss as the D 1 measure in the objective function 8.
The similarity measure D 2between the synthesized and original stimuli is computed using a Mean Squared Error (MSE).
Total
Variation (TV) metric [10] is used as the D 3 measure.
We analyze the contributions of all measuresto the quality of the synthesized stimuli by sweeping the hyperparameter α .
Results
6.1 Forward modeling
RGC neural activity fitting Performances of the IB-GP, IB-Disjoint, and Feedforward CNN infittings to neural activities versus the number of latents for both Natural and Brownian datasets areplotted in Figure 2.
Figure 2a-d shows that IB-GP outperforms IB-Disjoint almost in all cases.
For afixed β value, the performance of the IB-Disjoint gets closer to that of the IB-GP with the increase inthe number of latents.
This indicates that IB-GP learns neural activities with a fewer number of latents.
Similar is the case with β value decreasing to the extent where IB-Disjoint’s performance reachesthat of the IB-GP.
For less constrained models, IB-Disjoint surpasses IB-GP (refer to Appendix 5).
Interestingly, IB-GP achieves good neural prediction performance for the Natural dataset even withtwo latents.
For the Brownian dataset, IB-GP keeps its performance gap with IB-Disjoint even for alarge number of latents.
This is due to the strong correlation between the Brownian stimuli dataset(refer to Appendix 4 to observe some examples of stimuli).
Both IB-GP and IB-Disjoint outperformthe Feedforward CNN when a sufficient number of latents are used (4 latents and β < 0 .
for
Natural and (27 latents and β < 0 .
01 ) for the Brownian dataset.
Comparison of the IB-GP and IB-Disjoint learned dynamics In Figure 3, we analyze the learneddynamics of the IB-GP and IB-Disjoint trained under the same conditions on the Natural dataset [Figure] the same as (a,b) but for the Brownian dataset. (results for the Brownian dataset are in Appendix 7).
For β = 0 .
and 15 latents, performancesof the IB-GP and IB-Disjoint in prediction of RGCs responses are similar (correlation ∼ 0 .
45 vs.∼ 0 .
43 , respectively).
Nevertheless, the amount of information picked by latents in each model variessignificantly.
To show this, we perform a traversal analysis of the latents and report the variability inthe output responses for each traversed latent in Figure 3(a,d).
The KL divergence for each latentis also plotted.
As observed in the figure, IB-GP lumps most of the information required to explainthe neural responses in one factor.
In comparison, this information is spread over five factors in
IB-Disjoint.
Analyzing the dynamics of the latents in each model can provide further insight.
Figure3(b,e) visualizes the inferred dynamics of the most informative latent in each model after reducingthe dimension to 2 with T-SNE.
As observed in the figure, the dynamics of the most informativelatent in the IB-GP have 10 clusters.
Strikingly, these ten clusters correspond to the ten unique stimuliin the dataset.
We note that each batch of data in the Natural dataset contains 1000 consecutiveimages. Every 100 images are in fact, one unique image that is jittered and displayed on the retina.
Auto-Correlation (AC) plots of the latents corroborate these findings.
As evident from Figure 3(c,f), AC of the most informative latent in the IB-GP closely follows the statistics of the true and predictedresponses of the model (averaged over all neurons).
While the AC of the IB-Disjoint’s predictedresponses is similar to that of the true responses, the AC of its latents is very different.
To demonstratethe efficacy of the IB-GP model in fitting the RGCs responses, neural activity predictions of bothmodels are shown in Figure 3 (g,h).
Refer to Appendix 8 for a more detailed interpretation of IB-GP’slatent dynamics.
6.2 Closed loop image synthesis
Figure 4 reports the training results of the closed-loop experiment after the first round of measurements.
Both the IB-GP and IB-Disjoint are used as the forward model in the training of the image synthesizer.
We consider two scenarios when tuning the hyperparameters in the objective function of the imagesynthesizer.
Scenario one: the image synthesizer is trained only to reduce the complexity of theoriginal stimuli independently from the RGCs responses (equivalent to setting α = ∞ in theobjective function of the image synthesizer).
In this scenario, we also consider the case where the TV smoothness constraint is dropped.
These are important cases because they answer this validquestion: can a simple dimensionality-reduction technique be used for synthesizing images thatelicit responses correlated with the original ones?
Scenario Two: the image synthesizer is trained [Figure]
The variability in the accuracy of the model for traversed latent factors is depicted as blue shades in conjunction with RGCs responses ( α ̸ = 0 ).
Figure 4(a, b) plot the Pearson correlation andnegative log-likelihood of the RGCs responses predicted by the true model in response to the stimuliobtained by the image synthesizer network for the two scenarios.
The TV loss of the obtainedsynthesized stimuli in each case is plotted in Figure 4c .
As can be observed in the Figure 4d,dimensionality reduction of the stimuli without neural information results in non-smoothed images(column 4) or smoothed images (column 5) but poor RGC response accuracy (rightmost results in Fig.
4a-c).
On the contrary, incorporating RGC neural information obtains synthesized stimuli withexcellent smoothness and significant RGC response accuracy (Fig.
4a-c and column 2 in F4d).
Theseresults demonstrate the robustness of the proposed algorithm in finding synthesized images thatprovide accurate responses.
Strikingly, when IB-GP is used as the forward model in the closed-loopexperiment, complex curvature shapes appear in the synthesized images.
Expectedly, these shapesappear where the neurons’ receptive field is located.
The receptive fields (RFs) of the neurons,obtained by Spike-triggered Average (STA) analysis [1], are also shown in the figure.
Finally, wecontinue the optimization of the image synthesizer for the IB-GP model with hyperparameter α = 0 .
1for another two rounds.
The Pearson correlation of the synthesized stimuli in all three rounds readsas: 0.353, 0.366, and 0.361.
Details of the iterative optimization and evolution of the synthesizedimages in each phase are explained in Appendix 9.
The last row in the Figure 4d shows a failuremode of the image synthesizer algorithm.
We tested the image synthesizer (trained on the Naturaldataset) on the white noise data.
As seen from the figure, IB-Disjoint provides a superior synthesizedimage than other methods.
This is in line with our observations that the IB-GP model fails to fitneural data elicited by white noise.
More results are available in Appendix 10.
[Table]
k [Table]
li hood
[Table]
[Figure]
Discussion
This work presents the IB-GP model, a latent space variational model that learns a low-dimensionalrepresentation of the Natural stimuli and RGCs responses.
We observed that the IB formulation ofthe neural activity fitting allows learning of a low-dimensional representation of the stimuli and RGCresponses which is more predictive than the feedforward models.
We demonstrated that incorporatinga temporal prior on the latent space of the IB model factorizes the sources of neural variability intofewer latents.
The Latent analysis of the model revealed the superior performance of the IB-GP to IB-Disjoint, a variation of the proposed model with no temporal prior on the latents.
The proposedmodel was then used in a closed-loop experiment to synthesize input stimuli that, while of lowercomplexity, elicit neuronal responses as those elicited by the original complex stimuli.
Given thepromising results demonstrating the usefulness of the model, the IB-GP model has some limitationstoo.
First, it is unclear what type of temporal prior is sufficient for the best factorization of latents.
Although the Cauchy kernel has proved efficient in modeling multi-scale dynamics [13], it mightbe that other temporal kernels, a combination of several kernels, or even spatio -temporal kernels[9] could improve disentangling of the latents.
Second, we observed the failure mode of the IB-GPmodel in the fitting of the white noise stimuli due to the lack of temporal correlation.
Accordingly,the IB-GP model could be modified to work for all types of stimuli.
Third, computing the covariancesof the IB-GP prior is computationally intensive, especially for the long temporal datasets.
This mightalso be a possible negative societal impact of our work, resulting in a large carbon footprint.
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