Revealing computational mechanisms of retinal prediction via model reduction
Abstract
Recently, deep feedforward neural networks have achieved considerable success in modeling biological sensory processing, in terms of reproducing the input-output map of sensory neurons.
However, such models raise profound questions about the very nature of explanation in neuroscience.
Are we simply replacing one complex system (a biological circuit) with another (a deep network), without understanding either?
Moreover, beyond neural representations, are the deep network’s computational mechanisms for generating neural responses the same as those in the brain?
Without a systematic approach to extracting and understanding computational mechanisms from deep neural network models, it can be difﬁcult both to assess the degree of utility of deep learning approaches in neuroscience, and to extract experimentally testable hypotheses from deep networks.
We develop such a systematic approach by combining dimensionality reduction and modern attribution methods for determining the relative importance of interneurons for speciﬁc visual computations.
We apply this approach to deep network models of the retina, revealing a conceptual understanding of how the retina acts as a predictive feature extractor that signals deviations from expectations for diverse spatiotemporal stimuli.
For each stimulus, our extracted computational mechanisms are consistent with prior scientiﬁc literature, and in one case yields a new mechanistic hypothesis.
Thus overall, this work not only yields insights into the computational mechanisms underlying the striking predictive capabilities of the retina, but also places the framework of deep networks as neuroscientiﬁc models on ﬁrmer theoretical foundations, by providing a new roadmap to go beyond comparing neural representations to extracting and understand computational mechanisms.
Introduction
Deep convolutional neural networks (CNNs) have emerged as state of the art models of a variety of visual brain regions in sensory neuroscience, including the retina [1, 2] and deeper visual areas.
Their success has so far been primarily evaluated by their ability to explain reasonably large fractions of variance in biological neural responses across diverse visual stimuli.
However, fraction of variance explained is not of course the same thing as scientiﬁc explanation, as we may simply be replacing one inscrutable black box (the brain), with another (a potentially overparameterized deep network).
Indeed, any successful scientiﬁc model of a biological circuit should succeed along three fundamental axes, each of which goes above and beyond the simple metric of mimicking the circuit’s inputoutput map.
First, the intermediate computational mechanisms used by the hidden layers to generate
Δ t x y Δ t x
Motion reversal t [s]
Schwartz et al.
R a t e [ H z ] x
Training input Natural scene movie
Training output Ganglion cells’ response
Experimental data Omitted stimulus response
Model output
Schwartz et al.
s
R a t e [ H z ] t [s]
Training procedure: natural scenes
Testing procedure: structured stimulus (A) (
[Figure]
[Figure] (A) Training procedure: We analyzed a threelayer convolutional neural network (CNN) model of the retina which takes as input a spatiotemporal natural scene movie and outputs a nonnegative ﬁring rate, corresponding to a retinal ganglion cell response.
The ﬁrst layer consists of eight spatiotemporal convolutional ﬁlters (i.e., cell types), the second layer of eight convolutional ﬁlters, and the fully connected layer predicting the ganglion cells’ response.
The deep learning model reproduces [2] (B) an omitted stimulus response [5], and (C) the motion reversal response [6].
responses should ideally match the intermediate computations in the brain. Second, we should be able to extract conceptual insight into how the neural circuit generates nontrivial responses to interesting stimuli.
And third, such insights should suggest new experimentally testable hypotheses that can drive the next generation of neuroscience experiments.
However, it has been traditionally difﬁcult to algorithmically extract computational mechanisms, and consequently conceptual insights, from deep CNN models due to their considerable complexity.
Here we provide a method to do so based on the idea of model reduction, whose goal is to systematically extract a simple, reduced, minimal subnetwork that is most important in generating a complex
CNN’s response to any given stimulus.
Such subnetworks then both summarize computational mechanisms and yield conceptual insights.
We build on ideas from interpretable machine learning, notably methods of input attribution that can decompose a neural response into a sum of contributions either from individual pixels [3] or hidden neurons [4].
To achieve considerable model reduction for responses to spatiotemporal stimuli, we augment and combine such input attribution methods along with dimensionality reduction.
From deep CNNs to neural mechanisms through model reduction
To extract understandable reduced models from the millions of parameters comprising the deep
CNN in Fig.
1A and [2], we ﬁrst carve out important sub-circuits using modern attribution methods [3, 4], and then reduce dimensionality by exploiting spatial invariances present in the artiﬁcial stimuli carefully designed to speciﬁcally probe retinal physiology (Fig.1B-C).
We proceed in 3 steps:
Step (1): Quantify the importance of a model unit.
Since one main goal here is model reduction, we consider attributing the ganglion cell response back to the ﬁrst layer of hidden units to quantify their importance.
We denote by z [1] cxy ( t ) =
W [1] cxy (cid:126) s ( t )
+ b cxy the pre-nonlinearity activation of the ﬁrst layer (denoted by [1] ) hidden units, where W cxy and b cxy are the convolutional ﬁlters and biases of a unit in channel c ( c = 1 , . .
. , 8 ) at convolutional position ( x, y )
(with x, y = 1 , . . . , 36 ), and s ( t ) is the spatiotemporal input stimulus.
Now computing the line integral F [ s ( t ; 1)]
=
[Equation]
[Equation] [Equation]
[1] ∂α over a straight path in spatiotemporal stimulus space from the zero stimulus to s ( t ) given by s ( t ; α ) = αs ( t ) where the path parameter α ranges from 0 to 1 yields
[Equation]
This represents an exact decomposition of the response r ( t ) into attributions A cxy from each subunit at the same time t (since all CNN ﬁlters beyond the ﬁrst layer are purely spatial).
This attribution further splits into a product of W [1] cxy (cid:126) s , reﬂecting the activity of that subunit originating from spatiotemporal ﬁltering of the preceding stimulus history, and an effective stimulus dependent weight
G cxy ( s ) from each subunit to the ganglion cell, reﬂecting how variations in subunit activity z [1] cxy [Equation] as the stimulus is turned on from 0 to s ( t ) yield a net impact on the response r ( t ) .
A positive (negative) effective weight indicates that increasing subunit activity along the stimulus path yields a net excitatory (inhibitory) effect on r ( t ) .
Step (2): Exploiting stimulus invariances to reduce dimensionality.
The attribution of the response r ( t ) to ﬁrst layer subunits in (1) still involves 8 × 36 × 36 = 10 , 368 attributions.
We can, however, leverage the spatial uniformity of artiﬁcial stimuli used in neurophysiology experiments to reduce this dimensionality.
For example, in the omitted stimulus response (OSR), stimuli is spatially uniform, implying W [1] c (cid:126) s ≡ W [1] cxy (cid:126) s is independent of spatial indices ( x, y ) .
Thus, we can reduce the number of attributions to the number of channels via
[Equation]
More generally for other stimuli with no obvious spatial invariances, one could still attempt to reduce dimensionality by performing PCA or other dimensionality reduction methods on the space of hidden unit pre-activations or attributions over time.
We leave this intriguing direction for future work.
Step (3): Building reduced models from important subunits.
Finally, we can construct minimal circuit models by ﬁrst identifying “important” units deﬁned as those with large magnitude attributions
A .
Second, we construct our model as a one hidden layer neural network composed of only the important hidden units, with effective connectivity from each hidden unit to the ganglion cell determined by the effective weights G in (1), or (2)
[Figure]
We apply our attribution method to identify the subcircuit responsible for the omitted stimulus response (OSR) phenomenon (Fig. 1B), in which a periodic sequence of full ﬁeld ﬂashes entrains a retinal ganglion cell to respond periodically, but when a single ﬂash is omitted, the ganglion cell produces an even larger response at the expected time of the response to the omitted ﬂash.
This
OSR phenomenon is observed across several species including salamander [5, 9].
Interestingly, for periodic ﬂashes in the range of 6-12Hz, the latency between the last ﬂash before the omitted one, and the burst peak in the response, is proportional to the period of the train of ﬂashes [5, 9], indicating the retina retains a short memory trace of this period.
Moreover, pharmacological experiments suggest
ON bipolar cells are required to produce the OSR [7, 9], which have been shown to correspond to the ﬁrst layer hidden units in the deep CNN [1, 2].
These phenomena raise two fundamental questions: what computational mechanism causes the large amplitude burst, and how is the timing of the peak sensitive to the period of the ﬂashes?
There are two theoretical models in the literature that aim to answer these questions.
One proposes that the bipolar cell activity responds to each individual ﬂash with an oscillatory response whose period adapts to the period of the ﬂash train [8].
However, recent direct recordings of bipolar cells suggest that such period adaptation is not present [10].
The other model claims that having dual pathways of
ON and OFF bipolar cells are enough to reproduce most of the aspects of the phenomena observed in experiments [7].
However, the model only reproduces the shift of the onset of the burst, and not a shift in the peak of the burst, which has the critical predictive latency [8].
Direct model reduction of the deep CNN in Fig.
1A using the methods of section 2 yields for the ﬁrst time an interpretable model that can reproduce all salient features of the OSR response, comprised of three important feedforward pathways that combine one OFF temporal ﬁlter with two ON temporal ﬁlters (Fig. 2(A-1)).
One ON ﬁlter has an earlier peak with a longer tail, while the other has a later peak with a shorter tail.
The ﬁlter with an earlier peak and longer tail yields stronger activity for higher frequency ﬂash trains, thereby explaining the shift of the burst response to earlier times for higher frequency trains (Fig.
2C).
Thus, our systematic model reduction approach yields a new model of the OSR that cures important inadequacies of prior models.
Moreover, it yields a new, experimentally testable scientiﬁc hypothesis that the OSR is an emergent property of three bipolar cell pathways with speciﬁc and diverse temporal ﬁltering properties.  Discussion
[Figure]
In summary, in the case of the retina, we have shown that complex CNN models obtained via machine learning can not only mimic sensory responses to rich natural scene stimuli, but also can serve as a powerful and automatic mechanism for generating valid scientiﬁc hypotheses about computational mechanisms in the brain, when combined with our proposed model reduction methods (Fig. 3).
Applying this approach to the retina yields conceptual insights into how a single model consisting of multiple nonlinear pathways with diverse spatiotemporal ﬁltering properties can explain decades of painstaking physiological studies of the retina.
Overall the success of this deep learning approach to scientiﬁc inquiry in the retina, which was itself not at all a priori obvious before this work, encourages future studies to explore this approach deeper in the brain