Understanding the functional and structural differences across excitatory and inhibitory neurons
Authors: Sun Minni
Li Ji-An
Theodore Moskovitz
Grace Lindsay
Kenneth Miller
Mario Dipoppa
Guangyu Robert Yang
Venue:  (2020)
Keywords: Neuroscience

Abstract: One of the most fundamental organizational principles of the brain is the separation of excitatory (E) and inhibitory (I) neurons. In addition to their opposing effects on post-synaptic neurons, E and I cells tend to differ in their selectivity and connectivity. Although many such differences have been characterized experimentally, it is not clear why they exist in the first place. We studied this question in deep networks equipped with E and I cells. We found that salient distinctions between E and I neurons emerge across various deep convolutional recurrent networks trained to perform standard object classification tasks. We explored the necessary conditions for the networks to develop distinct selectivity and connectivity across cell types. We found that neurons that project to higher-order areas will have greater stimulus selectivity, regardless of whether they are excitatory or not. Sparser connectivity is required for higher selectivity, but only when the recurrent connections are excitatory. These findings demonstrate that the functional and structural differences observed across E and I neurons are not independent, and can be explained using a smaller number of factors.

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UNDERSTANDING THE FUNCTIONAL AND STRUC- TURAL DIFFERENCES ACROSS EXCITATORY AND IN- HIBITORY NEURONS
Anonymous authors Paper under double-blind review
ABSTRACT
One of the most fundamental organizational principles of the brain is the sepa- ration of excitatory (E) and inhibitory (I) neurons. In addition to their opposing effects on post-synaptic neurons, and I cells tend differ in their selectivity and connectivity. Although many such differences have been characterized exper- imentally, is not clear why they exist in the first place. We studied this question in deep networks equipped with E and I cells. We found that salient distinctions between E and neurons emerge across various deep convolutional recurrent net- works trained perform standard object classification tasks, and explored the necessary conditions for the networks to develop distinct selectivity and connec- tivity across cell types. We found that neurons that project to higher-order areas will have greater stimulus selectivity, regardless of whether they are excitatory or not. Sparser connectivity required for higher selectivity, but only when the re- current connections are excitatory. These findings demonstrate that the functional and structural differences observed across E and neurons are not independent, and can be explained using smaller number of factors.
Deep neural networks have become powerful tools to model the brain (Yamins & DiCarlo 2016 Kriegeskorte| 2015). They have been used to successfully model various aspects of the sensory Yaamins et al. 2014 Kell et al. [218)) cogmitive (Mante et al. 2013 Wang et al.) 2018 Yang et al., 2019). and motor system (Sussillo et al. 2015). Deep networks have been particularly effective at capturing neural representations in higher-order visual areas that remain challenging to model with
other methods (Yamins et al. 2014 Khaligh-Razavi Kriegeskorte 2014).
Demonstrating that these models can mimic the brain in some respects is only starting point in the process of advancing neuroscience with deep learning. Once some relationship between artificial and biological networks is established, the model can be dissected to better understand how certain biological computations are mechanically implemented (Sussillo & Barak 2013). Although notori- ously difficult to interpret, deep networks are still much more accessible than the brain itself. This is how deep networks can help address the "how" question. On top that, deep networks can be used in normative approach to answer the "wwy"" question. Suppose found that certain architecture, objective function (dataset), and training algorithm could together lead to a neural network matching some features of the brain. Then we can ask why the brain evolved such features by testing which el- ement of the architecture, dataset, and training is essential for such features to evolve in the artificial networks. For example. [Lindsey et al. 2019) showed early layers of convolutional neural network can recapitulate the center- surround receptive field observed in retina, but only when followed by an information bottleneck similar the one that exists from retina to cortex. This finding demonstrates how realistic biological constraints can be used to explain the emergence of known properties of the brain.
Despite the many successes of applying deep networks to model the brain, standard architectures deviate from the brain in many important ways. Recently, neural networks that incorporate funda- mental structures of the brain are becoming increasingly common (Kar al. 2019 Miconi et al.) 2018 Song et al. 2016). One fundamental structural principle of the brain is the abundance re- current connections within any cortical area. Cortical neurons receive substantial proportion of their inputs from other neurons in the same area (Harris & Shepherd (015).. Convolutional recur- rent networks trained object classification tasks perform comparably purely feedforward

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networks with similar numbers of parameters, while being able to better explain temporal responses in higher visual areas (Nayebi et al. 2018 Kar et al. 2019).
One of the most fundamental organizational principles of the cortex is the separation of excitatory (E) and inhibitory (I) neurons (Dalc) 1935 Eccles et al. 1954). Dale's law states that each neuron expresses a fixed set of neurotransmitters, which results in either excitatory inhibitory downstream effects. In addition to the difference in their immediate impacts on post-synaptic neurons, E and I neurons differ in several other important ways. There are several times (4-10x) more E neurons than neurons (Hendry et al, 1987). Neurons that project to other areas, the so-called "principal neurons", are all excitatory in the cortex (Bear et al., 2007). In the mammalian sensory cortex, neurons are overall more selective to stimuli than I neurons in the same area as extensively reported in mice (Kerlin et al., 2010: Znamenskiy et al. 2018) and to a lesser extent other species (Wilson et al. 2017). Finally, E neurons are more sparsely connected with each other, with connection probability of around 10% (Harris & Shepherd 2015), compared to neurons, which target almost all neighboring E neurons (Pleffer al. 2013).
Although many of these differences across E and neurons are well-known and routinely incorpo- rated in computational models (Somers et al. 1995, McLaughlin et al.] 2000), it remains unclear whether they can emerge in deep neural networks from training. is also unknown whether these differences are all independent properties, each contributing to the computation, if some differ- ences are natural results of the others. To answer these questions, we first trained many variants classical object classification tasks. Across
of deep convolutional recurrent neural networks all variants, three structural principles are built in: Dale's law, the abundance of excitatory neu- rons, and the role of excitatory units as principal neurons. However, other observed properties of biological E-I circuits are not hardwired, and therefore could only evolve under the pressure of performing the task. We found that functional and structural differences across and I neu- rons emerged through training. The development of these characteristics allows us to address the "why" question by removing the built-in differences across and neurons and monitoring whether specific functional and structural properties still emerge. Our anonymized code is available at
MULTI-CELL CONVOLUTIONAL RECURRENT NETWORKS
The networks we use model the visual cortex consist of 2 layers of purely feedforward, convo- lutional processing, followed by 2 or 3 layers of recurrent processing (Figure [ The feedforward layers correspond loosely to retina and thalamus, while the recurrent layers correspond to cortex. Each recurrent layer consists of excitatory and inhibitory neurons (channels).
We examined many variants for the recurrent layer. For brevity, here we focus on two representative architectures. The first one, which for convenience refer to the StandardEl model, consists of recurrently connected E and I channels:
Currg Wx+ X, + WE * E, - WI- * 1--1 + bg
Curry = Wx X2 + W g- / E W * 4--1 + by
E, fE E,- + (1 fE) (CurTE)
4 = f.=frola
- fi) (Curry)
X, is the input from the previous layer, E, and I are the activity of the excitatory and inhibitory neurons. Currg and Curry are the imput currents for E and I neurons respectively. Batch normaliza- tion's is applied to Curr and we tested variants where batch normalization is also applied to Curry- fE and fi are forget gates, implemented as o (f) where is the sigmoid function and f is trainable 1-d tensor with the same size as the number of channels. Dale's law is implemented by setting all relevant weight matrices WE-+., Wi,, to be non-negative using an absolute function, W = WW] where the trainable variable is the non-siin-constrranee weight matrix W. bg and by are bias terms. We found that task performance drops substantially if rectified linear function is used to impose the sign constraint instead. Only excitatory neurons in each recurrent layer makes long-range connec- tions to the next layer, making them the principal neurons (PN). In comparison, inhibitory neurons all interneurons (Int), making within-layer connections.
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PN
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Convolutional recurrent network with multiple cell types. The Standard model for CIFARIO 10 starts with two layers of regular convolutional processing, followed by 2 layers of re- current processing. Each recurrent layer consists of three types of cells. The excitatory principal neurons (PN) output non-negative connection weights and target the next area. Output-gatie (OG) and input- (IG) neurons are inhibitory and only target neurons within the same area.
Figure 1:
In the brain, E and neurons be further divided into many subtypes that differ in their inputs, output targets, and gene expression (Markram et al. 2004). Two major types of inhibitory neurons account for 60% of all inhibitory neurons in the cortex (Rudy et al. 2011). The first type expresses the molecule Somatostatin (SST) and inhibits dendrites of E neurons, the structure that receives inputs from other neurons. The second type expresses the protein Parvalbumin (PV) and inhibits soma of E neurons, where outputs are generated. Experimental evidence suggested that these input- and output-controlling inhibition can function like subtractive or multiplicative gates (Lee et al. 2012 Wilson et al. 2012). As was previously pointed out (Costa et al. 2017), this motif in which principal neurons those that can project other areas-are recurrently controlled by multiple gates reminiscent of the design of common recurrent units in machine learning, including Long Short-
Term Memory (LSTM) (Hochreiter Schmidhuber 1997) and Gated Recurrent Unit (GRU) (Cho al. 2014) networks.
Motivated by these findings, we introduced a convolutional recurrent layer with two distinct types of inhibitory interneurons (Int), which we refer to the Standard model. These two types of neurons control/gate the input and output of the excitatory neurons respectively, and are referred to as the input-gating (IG) and output-gating (OG) neurons. Taken together, Standard model is deseribed by
f. o Tf(Wxxx X₁ + WPN-S * PN, +b)),
IG, PWWxI10 X, WPN-IG PN,-1 biG),
4. (WIG-PN * IG, + bb),
OG, (W Xx + WPN- PN,-1 + boG),
0, (WoG-NP OG; + bo),
C, f1 C, + (1 i,) a X X X, WPN-PX PN be),
PN, -C,
Here IG,, OG,, PN, are the activity the input-gating, output-gating and principal neurons. C is the pre-activation values (or membrane potential) of the principal neurons, corresponding to the cell state in LSTM. Batch normalization is applied C.. IG and OG neurons inhibit PNs through inhibitory currents i, and o,, fi stands for the forget gate, similar to LSTM. All neurons are rectified linear units (ReLU) (see Appendix for justification), a. = max(x, J 0). The nonlinearity of the gate variable is the sigmoid function for multiplicative and forget gates (Ø,, oy), and ReLU for subtractive gates ...). * stands for convolution, while stands for element-wise multiplication. The Standard model is closely related LSTM in its of forget, input, and output gates. However, in LSTM, there is only a single type of neurons, the principal neurons, from which the input/output gates are directly generated.
REPRODUCING FUNCTIONAL AND STRUCTURAL DIFFERENCES ACROSS CELL TYPES
In this section, we will show that several experimentalll-ooseeve functional and structural dif- ferences across excitatory (PN) and inhibitory (Int) neurons can naturally emerge in E-I networks
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Figure 2: Model excitatory neurons have higher orientation selectivity and natural image se- lectivity than inhibitory neurons. (a,b) The average gOSI (a), orientation skewness (b), and image skewness (c) for each type of neurons in the Standard networks. Error bar is the 95% confidence interval computed with bootstrapping across 5 networks.
trained on image classification tasks. These results will be illustrated mainly in the Standard model, however, they hold for all variants E-I networks tested, including the StandardE/ model.
We trained the Standard model the image classification datasets CIFARIO (Krizhevsky & Hinton 2009) and ImageNet (Deng et al. 2009). We used common training techniques including momen- tum (Polyak) 1964) and L.2 regularization. For CIFARIO 10, regularization coefficient is 0.0002, the initial learning rate is 0.1 and decays 10-fold at epochs 100, 150, and 200, in all 250 epochs. The network consists of two convolutional feedforward layers of 16, and 32 channels each, followed by two recurrent layers. The first recurrent layer contains 64 PN, 16 IG, and 16 OG channels. For each type of neurons, the number of channels doubled in the second recurrent layer. The network is un- folded for 4 time steps, and the classification output is read-out from the final time step using fully connected linear layer. For all conditions, we trained networks with different random seeds. The network reaches approximately 85% test accuracy CIFARIO. 10, comparable to convolutional net- works similar depth (Hinton et al. 2012). See Appendix A for more details and hyperparameters for ImageNet.
We ask whether training leads to qualitative differences between excitatory and inhibitory neurons in the network, observed in the brain. In mouse cortex, inhibitory neurons are functionally less selective than excitatory neurons (Kerlin et al., 2010; Znamenskiy et al.] 2018), meaning that they tend to respond to different stimuli with similar values. We first measured the selectivity static oriented gratings for excitatory and inhibitory neurons (Figure Rp). The selectivity is quantified in two ways. First, computed the global Orientation Selectivity Index (gOSI) for each neuron (Wörgötter & Eysel 1987).
M Σ
M
gOSl,
(e 20
Σ
(0h(
(3)
where ry (8x) is the j-th neuron's 's response to the k-th grating stimulus at orientation 6g. gOSI is close to when the neuron highly selective to orientation. For each type of neurons, we report the average gOSIs across all channels in the center of the convolutional layer. The second measure of selectivity the skewness of the distribution of responses to various stimuli (Samonds et al.) 2014). The skewness 7/ for neuron j is defined as
EM (ryA 73)3
M Dn (Ti -,,
7
where ry.k is the j-th neuron's response to the k-th stimulus, and F, is the average response. The skewness higher if a neuron is strongly selective to a small number of stimuli. For both gOST (Figure 2 and orientation skewness (Figure2p). the selectivity is substantially higher for excitatory neurons (PN) compared to inhibitory neurons (IG, OG), in both recurrent layers (areas 3 and 4). Next, measured the selectivity natural images. We computed the skewness of responses to all images in the test set of CIFAR10. Again, excitatory neurons both areas and have higher selectivity to natural images than inhibitory neurons (Figure 2p).
Another major qualitative difference between excitatory and inhibitory neurons in the brain is the higher density of I-to-E connections (i.e., Int-to-PN) compared E-to-E (i.e., PN-to-PN). In
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network, found that the distribution of all recurrent connection weights is multi-modal (Figure B)), with a group of strong connections and many much weaker connections. To assess the connec- tion density, chose threshold exp(-10). and quantified the proportion of connection weights that exceed this threshold. This threshold is chosen to separate the strongest mode from the weaker modes in the distribution of all weights (Figure 3p.. The distribution of PN-to-PN connections is spread out (Figure 3p), leaving a substantial proportion of connections below the threshold. In con- trast, the distributions of both IG-PN and OG-PN connection weights mostly concentrated above the threshold (Figure (), leading to higher connection density (Figure Be). Therefore, sparser excitatory connectivity emerged in the network after training, in agreement with biological observa- tions.
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Figure 3: Model excitatory neurons are more sparsely connected. (a-d) The distribution of connection weights for all recurrent connections (a), and for the PN-to-PN (b), OG-to-PN (c), and IG-PN connections (d) in area 3 of one standard network. Blue line: the threshold used to compute the proportion of strong connections, namely those that exceed the threshold. (c) The connection density (proportion of strong connections) for three types of excitatory and inhibitory connections across area 3 and 4.
The findings above are summarized in Figure 4 which also includes results from the StandardEl model, and from networks trained on ImageNet. The emergent differences across E and I neurons are in qualitative agreement with experimental estimates. To test the generality of these results, trained networks with subtractive multiplicative gates (Figure Dh), networks that structurally interpolate between the Standard and StandardEl models (Figure 9)), and networks with or without batch normalization or dropout (Figure De [7p. In all variations of networks that reached accuracy above 80%, the excitatory neurons are more selective and less connected.
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Figure 4: The E-I differences emerge across network variations and datasets. The E-I dif- ferences are summarized using the difference in image skewness between principal neurons (PN) and interneurons (Int), and the difference in connection density between Int-to-PN and PN-to-PN connections. Light circles: individual networks. Dark circles: average. Experimental estimates are derived from Znamenskiy al. (018). [Song al. (2005), and Packer & Yuste (201T) (Appendix
By monitoring the orientation selectivity, image selectivity, and connection density throughout the training process, we found that the differences across excitatory and inhibitory cells emerged early on (Figure 5). Even though excitatory and inhibitory neurons started out with similar selectivity
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Figure 5: The E-I differences emerge early in training. The orientation skewness (a), image skewness (b), and connection density (c) for areas (top) and 4 (bottom) throughout training.
(close to 0), and similar connectivity (close to 1), the differences across cell types become substantial after 5-30 epochs. These functional and structural differences remain stable throughout training, even as the training performance continues to improve (Figure TOP. Together, these results argue that there exists strong optimization pressure differentiate the selectivity and connectivity of excitatory and inhibitory neurons.
3 WHY DO EXCITATORY AND INHIBITORY NEURONS HAVE DISTINCT SELECTIVITY AND CONNECTIVITY?
We have shown that neural networks with excitatory and inhibitory neurons develop different selec- tivity and connectivity, qualitatively reproducing long-standing findings in the brain. Now we ask why these differences emerge.
The built-in structural asymmetry between model E and neurons must be critical for their emergent differences. We have incorporated three major forms of asymmetry that exist in the brain. First, there is an asymmetry in numbers. Excitatory neurons are 4 to 10 times more abundant than inhibitory neurons in the brain. In all models tested so far, there are more excitatory than inhibitory neurons in each area. Second, there is asymmetry in projection targets. In the cortex, all principal neurons are excitatory. Meanwhile, all inhibitory neurons are interneurons, meaning that they only connect other neurons within the same area. Third, there is by definition an asymmetry in action. Excitatory neurons excite other neurons while inhibitory neurons inhibit. When the activation func- tion of neuron is rectified and non-saturating (for example, ReLU), excitatory inputs the neuron can move it into responsive regime, where inhibitory inputs can make neuron responsvee In this section, we will remove individual asymmetry, and test which one led to the observed dif- ferences in selectivity and connectivity. In the Standard model, there is an additional asymmetry because inhibitory neurons use multiplicative gates to control inputs and outputs of excitatory neu- rons. This asymmetry is not presented in the StandardE/ model. Therefore, we will present results based on variations of both the Standard and the StandardEl model trained on CIFARIO.
Asymmetry in numbers In our Standard model, the ratio between the number of OGs(IGs) and PNs is 1:4. It is conceivable that excitatory neurons can afford to be more selective to orientations and images because there are more of them. To test this hypothesis, we independently varied the numbers of E and I channels in the recurrent layers. The number of E or channels ranges from to 256 in area 3. Area 4 always has the doubled number of channels. We found that the number of E, but not I, channels have strong influence on both image selectivity (Figure 6b. left) and connection density (Figure left) of E neurons. In contrast, the inhibitory selectivity is consistently low, regardless of the number of E or channels (Figure (4, right). The inhibitory connection density depends moderately on the number of and channels, decreasing with a larger number of I channels (Figure 6b, right).

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Figure 6: Selectivity and connectivity in networks with various numbers of excitatory and inhibitory channels. (a) The image skewness of principal neurons (PN, left) and interneurons (Int, right) for networks of various numbers of excitatory and inhibitory channels. (b) The connection density of PN-PN connections and Int-PN connections for various networks. Results from areas and 4, and from two types of interneurons are combined. All networks shown have accuracy above 60%.
Even though excitatory neurons become more selective and less connected when there are more of them, the asymmetry in numbers alone does explain the emergent E-I differences. When the number of excitatory and inhibitory channels are the same and both high, excitatory neurons still have higher selectivity and sparser connectivity. Similar selectivity and connectivity, task accuracy depends mainly on the number of excitatory channels (Figure IT. meaning that it is not necessary to have a high number of inhibitory channels in these networks. Without any inhibitory channels, task accuracy only drops moderately in our Standard and StandardE/ models. However, without batch normalization, accuracy drops to chance level when recurrent layers consist of only excitatory neurons (see Figure 17 and Appendix D.. This potentially explains why there are 4-8 times less inhibitory neurons than excitatory neurons the brain.
Asymmetry in projection In both the cortex and all networks tested so far, principal neurons are exclusively excitatory. Here, we decoupled this relationship by training InhPN networks, where long-range connections originate from inhibitory, instead of excitatory, neurons. We do so by flip- ping the sign of all local recurrent connections. The connections from principal neurons to the next layer are kept excitatory, because we had difficulty training networks with inhibitory long-range pro- jections past 80% accuracy. InhPN networks can still achieve accuracy similar to that of the standard network. However, the now-inhibitory principal neurons (PN) remain more selective orientation and natural images compared to the now-excitatory interneurons (Int) (Figure 7 green). This result argues that higher selectivity is not property inherent to excitatory neurons. Whichever type of neurons serves as the principal neurons would demand higher selectivity, presumably because the principal neurons need to carry detailed stimulus information the next layer.
Interestingly, in the InhPN networks, the connectivity among inhibitory principal neurons (PN) become dense despite maintaining high selectivity (Figure green). This result is in stark contrast with the sparser connectivity needed for highly selective excitatory neurons in the Standard model (Figure (). We speculate that these seemingly contradictory results could potentially be understood in the context of recurrently connected ReI.U neurons. When ReLU neuron is highly selective, typically, strongly activated by only small subset of stimuli. If this selectivity supported by recurrent excitatory connections, then neuron only needs to receive inputs from small set of excitatory neurons with similar preferred stimuli. Meanwhile, the selectivity is supported by recurrent inhibition, then neuron needs to receive inputs from nearly all other neurons, except for the small subset of neurons with similar preferences. Therefore, recurrent connections between highly selective inhibitory neurons should be dense, instead of sparse.
Asymmetry in action The namesake difference between excitatory and inhibitory neurons is the sign of their connection weight values. All connections stemming from excitatory (inhibitory) neuron are constrained to be non-negative (non (non-positive). We can release this constraint on the sign of connection weights, while keeping other E-I asymmetries. Such NoConstraint networks

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Figure 7: Removing asymmetry between excitatory and inhibitory neurons. Difference in nection density against difference in image selectivity for areas (a) and 4 (b) for the original networks (blue), their InhPN variants (green), and NoConstraint variants (orange). Light symbols: individual networks; dark symbols: average. Circle: Standard model and its variants; triangle: StandardEl and its variants.
achieve slightly better accuracy compared the Dale's law-obeying networks. In the NoConstraint networks, the formerly-excitaton principal neurons (PN) developed higher selectivity compared to the formerly-inhibitory interneurons (Int) (Figure 7 orange), consistent with our previous finding that principal neurons have higher selectivity regardless of the sign of their outputs.
Again consistent with previous results, in the NoConstraint networks, the principal neurons no longer have sparser connectivity, instead, they are almost fully connected (Figures 7 14p. This sult again indicates that the link between connectivity and selectivity depends crucially on whether Dale's law is applied. In network without Dale's law, neuron receive inputs from all other neurons and remain selective, as long as the inputs from most neurons cancel out, leaving this neuron effectively driven by small of inputs.
Impact of training and dataset The emergent differences between model E and I neurons is not only a result of the built-in E-I asymmetry, but also a result of the training process. Before training, model E and neurons have the same selectivity and connectivity (Figure 5 During the training process, neurons rapidly gained high selectivity, even before the network approaches high task ac- curacy (Figure TOp. These results suggest that the emergence of high selectivity (and possibly sparse connectivity) is not necessarily dependent on the network performing object classification tasks well. To test this hypothesis, we trained networks classify set of randomly labeled CIFARIO images (see Appendix on the training (chance level on the validation set, expected). Higher selectivity and sparser connectivity among excitatory neurons still emerge in such networks (Figure (6), although less con- sistently across areas or random seeds. Understanding the minimal task necessary for the emergence of E-I differences would be interesting for future work.
K (Zhang et al. 2016). After training, networks can achieve 20- - 50% accuracy
DISCUSSION
We have shown that recurrent neural networks equipped with excitatory and inhibitory cells are capable of capturing several important features of the brain, including higher selectivity and sparser connectivity among excitatory neurons. These qualitative features emerged from the pressure to perform the task, suggesting that these qualities are indeed beneficial to task performance. This allows us to study what aspects of the network give rise to this distinction between excitatory and inhibitory neurons. We found that the higher selectivity of excitatory neurons is mainly driven by their role as the principal neurons that transmit information to upper layers. When Dale's law obeyed, a higher selectivity necessitates sparser connectivity among excitatory neurons. Our
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findings predict that if brain area contains inhibitory principal neurons, these neurons should be more selective than intemeurons of the same area.
Our model omits many elements of biological neural circuits, such as synaptic dynamics, spiking, long-range feedback connections, and so on. Our work aims at exploring the necessary computa- tional conditions for several experimentally-ooseeve implication of our work that the biological features mentioned above are not required for such emer- gent properties. Rather than attempting implement as many biologically-releann network features in our models as we could, we include features that represent a step forward in biologically fidelity relative to most deep networks (e.g., excitatory and inhibitory neurons). but which are still feasible to implement on high-ding dinen-immnsional visual classification tasks. It is an interesting challenge for ongo- ing and future work further incorporate biological components, including those mentioned above, into deep neural networks.
properties emerge. It is, in fact, another
Optimization algorithms like stochastic gradient descent combined with large datasets are highly effective at tuning connection weights to perform tasks. However, it remains unlikely to observe a specific set of desired structural principles emerge naturally through training. standard networks to obey Dale's law. We have not explored whether Dale's law could emerge out of training. Such an emergence is theoretically possible, but has not been demonstrated, to the best of our knowledge. There are at least three possibilities: (1) the optimization algorithm is not strong enough to discover this solution because remains a exponentially small part of the solution space (1/2%) of the entire solution space, N being the number of connection weights). Similarly, it would be difficult for a ResNet structure (He et al., 2016) to develop from training a vanilla 100+ layer deep feedforward network. (2) The task we used is appropriate for the emergence of Dale's law. It is conceivable that Dale's law is beneficial to some tasks that have yet been identified. (3) Finally, it is possible that Dale's law is result of compromise that the brain has to strike due to its biological nature, and is irrelevant to general computing machines. Understanding the nature of the computational benefit of Dale's law (if any) would be major achievement computational neuroscience, and may shed light on better designs of artificial neural networks.
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REFERENCES
Mark Bear, Barry W Connors, and Michael A Paradiso. Newroscience, volume Lippineott Williams Wilkins, 2007.
Kyunghyun Cho, Bart Van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Hol- ger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv: 1rXivv1406.10788 2014.
François Chollet. Xception: Deep learning with depthwise separable convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1251-1258, 2017.
Rui Costa, Ioannis Alexandros Assael, Brendan Shillingford, Nando de Freitas, and Tim Vogels. Cortical microcireuits gated-recurrent neural networks. In Advances Neural Information Processing Systems, pp. 272-2822222 2017.
Henry Dale. Pharmacology and nerve-endings, 1935.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hi- erarchical image database. In 2009 IEEE conference computer vision and pattern recognition, pp. 248-255 Ieee, 2009.
John C Eccles, P Fatt, and K Koketsu. Cholinergic and inhibitory synapses in a pathway from motor-axon collaterals motoneurones. The Journal of physiology, 126(3):524 562, 1954.
Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pp. 315-323, 2011.
Kenneth D Harris and Gordon MG Shepherd. The neocortical circuit: themes and variations. Nature neuroscience, 18(2):170, 2015.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
Stewart H Hendry, HD Schwark, EG Jones, and J Yan. Numbers and proportions of gaba- immunoreactive neurons in different areas of monkey cerebral cortex. Journal of Neuroscience, 7(5):1003-151, 1987.
Geoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, llya Sutskever, and Ruslan R Salakhutdi- nov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1007.0580, 2012.
Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735-1780, 1997.
Kohitij Kar, Jonas Kubilius, Kailyn Schmidt, Elias Issa, and James J DiCarlo. Evidence that recurrent circuits are critical to the ventral streams execution of core object recognition behavior. Nature neuroscience, pp. 2019.
Alexander JE Kell, Daniel LK Yamins, Erica N Shook, Sam Norman-Haignere, and Josh H Mc- Dermott. A task-optimized neural network replicates human auditory behavior, predicts brain responses, and reveals a cortical processing hierarchy. Neuron, 08(3):630-644 2018
Aaron M Kerlin, Mark Andermann, Vladimir Berezovskii, and R Clay Reid. Broadly tuned response properties of diverse inhibitory neuron subtypes in mouse visual cortex. Neuron, 67(5): 858 18,-871, 2010.
Seyed-Mahdi Khaligh-Razze Razivgi and Nikolaus Kriegeskorte. Deep supervised, but not unsupervised, models may explain it cortical representation. PLoS computational biology, 10011):e003311, 2014.
Nikolaus Kriegeskorte. Deep neural networks: a new framework for modeling biological vision and brain information processing. Annual review vision science, 1:417-446, 2015.
10

---

Under review as a conference paper at ICLR 2020
Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Tech- nical report, Citeseer, 2009.
Seung- Seung-Hee Lee, Alex Kwan, Siyu Zhang. Victoria Phoumthipphavong, John G Flannery, Sotiris C Masmanidis, Hiroki Taniguchi, Z Josh Huang, Feng Zhang. Edward Boyden, al. Activation of specific interneurons improves v1 feature selectivity and visual perception. Nature, 488(7411): 379, 2012.
Jack Lindsey, Samuel A Ocko, Surya Ganguli, and Stephane Deny. A unified theory of early visual representations from retina cortex through anatomically constrained deep cnns. arXiv preprint arXiv:1901.00945, 2019.
Valerio Mante, David Sussillo, Krishna V Shenoy, and William T Newsome. Context-dependent computation by recurrent dynamics in prefrontal cortex. nature, 503(7474):78,, 2013.
Henry Markram, Maria Toledo-Rodriguez, Yun Wang, Anirudh Gupta, Gilad Silberberg, and Caizhi Wu. Interneurons of the neocortical inhibitory system. Nature reviews neuroscience, 5(10):793, 2004.
David McLaughlin, Robert Shapley, Michael Shelley, and Dingeman J Wielaard. A neuronal net- work model of macaque primary visual cortex (v1): Orientation selectivity and dynamics in the input layer 4ca. Proceedings of the National Academy of Sciences, 97 97(14):8087-8022 2000.
Thomas Miconi, Jeff Clune, and Kenneth O Stanley. Differentiable plasticity: training plastic neural networks with backpropagation. arXiv preprint 184.:v8804.044000 2018.
Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10), pp. 807-814, 2010.
Aran Nayebi, Daniel Bear, Jonas Kubilius, Kohitij Kar, Surya Ganguli, David Sussillo, James DiCarlo, and Daniel L Yamins. Task-driven convolutional recurrent models of the visual system. In Advances in Neural Information Processing Systems, pp. 5290-5301, 2018.
Adam M Packer and Rafael Yuste. Dense, unspecific connectivity of neocortical parvalbumin- positive interneurons: canonical microcircuit for inhibition? Journal of Neuroscience, 317))): 13260-13271, 2011.
Carsten K Pfeffer, Mingshan Xue, Miao He, 7. Josh Huang, and Massimo Scanziani. Inhibition of inhibition in visual cortex: the logic of connections between molecularly distinct interneurons. Nature neuroscience, 1668):1068, 2013.
Boris Polyak. Some methods of speeding up the convergence of iteration methods. USSR Com- putational Mathematics and Mathematical Physics, 4(5):1-17, 1964.
Rishi Rajalingham, Elias B Issa, Pouya Bashivan, Kohitij Kar, Kailyn Schmidt, and James J DiCarlo. Large-scale, high-resolution comparison of the core visual object recognition behavior of humans, monkeys, and seate-of-the-ae deep artificial neural networks. Journal of Neuroscience, 38(33): 7255-7269, 2018
Daniel B Rubin, Stephen D Van Hooser, and Kenneth D Miller. The stabilized supralinear network: unifying circuit motif underlying multi-input integration in sensory cortex. Neuron, 85(2):402- 417, 2015.
Bernardo Rudy, Gordon Fishell, SooHyun Lee, and Jens Hjerling-Lefflee Three groups of interneu- rons account for nearly 100% of neocortical gabaergic neurons. Developmental neurobiology, 71 (1):45-61, 2011.
Jason M Samonds, Brian R Potetz, and Tai Sing Lee. Sample skewness as a statistical measurement of neuronal tuning sharpness. Neural computation, 26(5):860 906. 2014.
David C Somers, Sacha B Nelson, and Mriganka Sur. An emergent model of orientation selectivity in cat visual cortical simple cells. Journal of Neuroscience, 15(8)55448-546, 1995.

---

Under review as a conference paper at ICLR 2020
H Francis Song, Guangyu R Yang, and Xiao-Jing Wang. Training -niiattrr-inhiiiitory recurrent neural networks for cognitive tasks: a simple and flexible framework. PLoS computational biol- ogy, 12(2)=e1004792. 2016.
Sen Song, Per Jesper Sjöström, Markus Reigl, Sacha Nelson, and Dmitri B Chklovskii. Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS biology, 3(3)e66, 2005.
David Sussillo and Omri Barak. Opening the black box: low-dimensional dynamics in high- dimensional recurrent neural networks. Neural computation, 25(3):626-649, 2013.
David Sussillo, Mark M Churchland, Matthew T' Kaufman, and Krishna V Shenoy. A neural network that finds a naturalistic solution for the production of muscle activity. Nature neuroscience, 18(7): 1025, 2015.
Jing Wang, Devika Narain, Eghbal A Hosseini, and Mchrdad Jazayeri. Flexible timing by temporal scaling of cortical responses. Nature neuroscience, 21(1): 2018.
Daniel E Wilson, Gordon B Smith, Amanda I. Jacob, Theo Walker, Jordane Dimidschstein, Gord Fishell, and David Fitzpatrick. Gabaergic neurons in ferret visual cortex participate in functionally specific networks. Neuron, 93(5):1058-10633 2017.
Nathan R Wilson, Caroline A Runyan, Forea L Wang, and Mriganka Sur. Division and subtraction by distinct cortical inhibitory networks in vivo. Nature, (188(7411):443 2012.
Kong-Fatt Wong and Xiao-Jing Wang. A recurrent network mechanism of time integration in per- ceptual decisions. Journal of Neuroscience, 2614):131 2006.
Wörgötter and U Th Eysel. Quantitative determination of orientational and directional components in the response of visual cortical cells to moving stimuli. Biological cybernetics, 5766)3449-355, 1987.
Daniel LK Yamins and James DiCarlo. Using goal-driven deep learning models to understand sensory cortex. Nature neuroscience, 19(3):356, 2016.
Daniel LK Yamins, Ha Hong, Charles F Cadieu, Ethan Solomon, Darren Seibert, and James hicrarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619 8-24,, 2014.
DiCarlo. Performance-optimee
Guangyu Robert Yang, Madhura R Joglekar, Francis Song, William Newsome, and Xiao-Jing Wang. Task representations in neural networks trained to perform many cognitive tasks. Nature neuroscience, pp. 1, 2019.
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint 16 1.0153300, 2016.
Petr Znamenskiy, Mean-Hward Kim, Dylan Muir, Maria Florencia lacaruso, Sonja Hofer, and Thomas D Arsic-Flogel, Functional selectivity and specific connectivity of inhibitory neurons in primary visual cortex. bioRxiv, pp. 294835, 2018.
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A ADDITIONAL DETAILS OF THE MODELS
Our models consist of two feedforward layers and two recurrent layers for CIFARIO. Convolutions in the feedforward layers are regular convolutions. In the recurrent layers, we use depth-separable convolution, where a spatial convolution is applied on each channel separately and then followed by point-wise convolution that mixes channels (Chollet 2017). Depth-separable convolution is used because using regular convolution for recurrent connections leads to substantially lower (of- ten around chance-level) and less stable accuracy on ImageNet in our networks. When training on CIFARIO, networks with regular convolution can still reach around 85% accuracy, but we observed a greater variability across networks with different random seeds. Kernel size is 3x3 for all convo- lutions. To measure the connection density, the 4-d convolution matrix of size [3, 3, channels in, channels. [out] is recovered from the spatial and point-wise convolution kernels, and then averaged over the first two spatial dimensions. This would give channel-to-channee weight matrix, with a total of channels in xchannels _out connection weights. The results are similar when we do not average across space and instead analyze the 3x3xchannels in channells out connection weights. Density is quantified as the proportion of connection weights that exceed a chosen threshold. We picked a threshold exp(-10) that separates the strong mode from the weaker modes in the distri- bution of all connection weights. The precise value of the threshold does not impact our qualitative results. Max pooling applied on the output of each layer. Pooling stride is 1 for the first feedfor- ward layer, and 2 for other layers.
For ImageNet, the L2 weight regularization coefficient is 0.0001. The regularization strengths are taken directly from the official Tensorflow ResNet models for CIFARIO 10 and ImageNet. We found that, without weight regularization, the connection probability of both excitatory and inhibitory connections become close to 1, far from experimentally-obseee
values. The initial Iearning rate is and decays 10-fold at epochs 30, 60, 80 and 90, in all 100 epochs. Height and width of input images resized and cropped to 128 pixel. The network consists of two convolutional feedforward layers of 64, and 128 channels each, followed by three recurrent layers. The first recurrent layer contains 256 PN, 64 IG, and 64 OG channels. For each type of neurons, the number of channels is the same in the second recurrent layer and doubled in the third recurrent layer. Batch-normalizatioo is applied to the cell state C.. The network is unfolded for 5 time steps, and the classification output is read- out from the final time step using fully connected linear layer. For all conditions, we trained 5 networks with different random seeds. The network reaches approximately 55% test accuracy on ImageNet.
B BIOLOGICAL RELEVANCE OF RELU UNITS
Besides being the standard choices of activation function for modern deep convolutional neural networks Gloort et al. 2011 Nair & Hinton 2010; He et al. [0166), ReLU and its close variants (ELU, LeakyReLU, etc.) are widely used in deep neural network models for visual neuroscience (many of them being the same models used in computer vision) (Rajalingham et al. 2018 Nayebi
et al.
2018 Lindsey et al. 2019)
Admittedly, the ReLU activation function is not an accurate depiction of a biological f-I curve, however, it does capture the important rectification of activity at low input values. ReLU does not saturate at high input values, while neurons obviously do (> 200 sp/s). However, most cortical neu- rons do not operate in this high-activity saturating regime. Many rate-based computational models of neural circuits use activation functions that do not saturate at high values, e.g., Wong & Wang 2006 Rubin al. 2015).
C COMPUTING ORIENTATION SELECTIVITY
A static oriented grating a two-dimensional sinusoidal wave G(z y) satisfying:
G( v) cos(kl-(x -0o zinssnn (y yo cosd/ + Φ),
where 0 is the orientation, k is the spatial frequency, & is the phase and (æo. yo) is the center location. We generate a total of 432 gratings (Figure using 18 orientations (109 apart), 6 spatial frequencies
(1, 2, 3, 4, 6, 8) and 4 phases (90' apart).
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We present each grating image network, and each neuron's preferred orientation, spatial fre- quency and phase is chosen when the neuron has the maximal activity (averaged over time step). At its preferred spatial frequency and preferred phase, the global Orientation Selectivity Index (gOSI) computed based on all orientations. Unlike gOSI, the orientation skewness computed based on all 432 grating images.
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rons. (a) A total of 432 gratings, using 18 orientations (10" apart), 6 spatial frequencies (1, 2, 3. 4, 6, 8) and phases (909 apart). (b) The orientation tuning curves of example PN, OG, and IG neurons from area 3 of one Standard network. The number in each panel indicates the neuron's global Orientation Selectivity Index (gOSI).
D MODEL VARIANTS
We tested a number of model variants to test the generality of our findings. The results are summa- rized (Figure 9 in the same style as Figure
D.1 READOUT AND GATING
In our Standard model, the logits are read out from the last time step of the final recurrent layer (area 4). We also trained networks where the logits are read out from the final recurrent layer's activity summed across all time steps. In Figure Pa. We denote the former structure as "LastT", and the latter structure as "SumT".
Apart from multiplicative (Mul) input gates and subtractive (Sub) output gates in the standard net- work, we implemented other combinations of gating mechanisms. Figure a network is named as "Readout InputGateOupputaatt"" For example, our Standard model is named "Last' MulSub" this plot.
D.2 TRANSITIONING FROM Standard TO StandardEl
Figure 99 contains a set of models in transition from our Standard to StandardE/ model. NoOG ECurrBN removes the output gate of our Standard model and moves batch normaliza- tion from cell state Ct to the input current of PN neurons. Based NoOG ECurrBN_SSub by changing the multiplicative input gate into a simplified subtractive cur- rent. By furthering simplifying the forget gate structure to be the same as the StandardEl model, and adding recurrent structure for IG neurons, we get the NoOG ECurrBN SSub SFG Recurrl model, which is equivalent the StandardE/ model.
NoOG_ECurin
we obtained
D.3 BATCH NORMALIZATION AND DROPOUT
In Figure De. we tested various networks with or without batch normalization or dropout. our Standard model, removing the batch normalization on cell states (NoPNCelIBN) replacing it with
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The E-I differences emerge across a range of network variations. (a-c) The E-I
differences are summarized using the difference in image skewness between principal neurons (PN) and interneurons (Int), and the difference in connection density between interneuron-to-Pr and PN- to-PN connections, in the same way Figure 4 (a) Models with different readout and gating mechanisms. The Standard model is denoted here as the MastT_MulSub model. (b) Models in transition from the Standard model to the StandardEl model. The latter is denoted here as the
NoOG ECurrBN SSub SFG Recurrl model. (c) Models with variations of batch normalization or dropout. All model variants develop the E-I differences. Results combined from areas and 4. Light circles: individual networks. Dark circles: average.
a dropout function (PNCellDrpt) (keep probability equals to 0.9) retains the performance above 80% and the E-I differences in both selectivity and density (Figure
For the Standard model with gating neurons, adding batch normalization for IG (AddIGCurrBN) or OG neurons (AddOGCurBBN) increases selectivity for the batch-normalized inhibitory neu- rons. Nevertheless, the inhibitory selectivity remains lower than that of PNs. The same trend is observed when batch normalization is added to inhibitory neurons in the StandardE/ model (EI CurrBN ICurrBN), or when dropout added to output-gating neurons in PNCelLDrpi (PN- CellDrplAAddOGGirpp))
E EXPERIMENTAL ESTIMATES OF E-I DIFFERENCES IN SELECTIVITY AND CONNECTIVITY
Here we intend to estimate the experimentaally-oeeeee differences in selectivity and connectivity across E and neurons. Znamenskiy et al. (2018) recorded from mouse V1 responses to gratings. They quantified the response skewness of PV+ and PV- neurons (their Fig. lb). PV, i.c. parval- bumin, is expressed only in a major subtype of inhibitory neurons. The difference between PV- and PV+ skewness is about 2.1. Song et al. (2055) measured the connection probability of pairs of excitatory neurons in rat VI. For pairs of cells with intersomatic distance lower than 100 microns, the connection probability is about 1 1% (their Fig. 2b). Packer & Yuste (2011) recorded from L2/3 of mouse barrel cortex (S1), and found that the connection probability from PV+ pyramidal cells with intersomatic distance less than 200 microns is around 42% (their Fig. 4f). The last two results obtained from different areas, species, and using different methodologies. so the quantitative value of the estimation is not to be trusted. However, the qualitative difference is robust.
F CLASSIFICATION PERFORMANCE DURING TRAINING
The classification performance the Standard model on CIFARIO increases rapidly in the first 10 epochs, and continues to improve after 150 epochs of training. The E-I difference in selectivity also increases rapidly in the first 10 epochs (Figure 5)),), while the difference in connection density plateaus at about epoch 50 (Figure 5t).
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Figure 11: Performance of networks with various numbers of excitatory (principal neuron, PN) and inhibitory (interneuron, Int) channels.
G PERFORMANCE OF NETWORKS WITH VARIOUS NUMBERS OF EXCITATORY AND INHIBITORY CHANNELS
The performance of networks with various numbers of excitatory and inhibitory channels is summa- rized (Figure IT same style as Figure The performance depends mainly on the number of excitatory (PN) channels, rather than the number of inhibitory (Int) channels. All networks shown in Figure and Figure have accuracy above 60%.
H PERFORMANCE OF THE InhPN AND NoConstraint NETWORKS
The performance of the standard, InhPN and NoConstraint networks is summarized in Figure I2 Such NoConstraint networks achieve slightly better accuracy compared the standard Dales law- obeying networks.
DETAILED COMPARISON BETWEEN THE STANDARD MODEL AND ITS VARIANTS
The Standard model is compared to its InhPN variants (Figures 13 and the NoConstraint variants
(Figure 14
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Performance CIFAR-10
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Figure 12: Performance of the JnhPN and NoConstraint networks compared with the standard models.
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Figure 13: Detailed comparison between the Standard model and InhPN network. Comparing gOSI (a), orientation skewness (b), image skewness (c), and connection density (d) for areas 3 (left)
and (right) between the Standard model and InhPN network.
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Figure 14: Detailed comparison between Standard model and NoConstraint network. Compar- ing gOSI (a), orientation skewness (b), image skewness (c), and connection density (d) for areas (left) and (right) between Standard model and the NoConstraint network. the NoConstraint model, the OG-PN connectivity is close to zero, suggesting that OGs are only weakly participating the computation.
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Figure 15: Detailed analysis of the StandardEl model.
DETAILED ANALYSIS OF THE StandardEl MODEL
The StandardEl model shows similar qualitative trends (Figure 15) the Standard model used in the main text (Figures
K TRAINING NETWORKS ON RANDOM LABELS
We trained networks CIFAR-10 with shuffled labels (Zhang al.) 2016). Our Standard network CIFARIO 10 contains 107K trainable parameters, compared used by Zhang al. (2016). therefore
1.6M parameters networks
used shuffled training sets 10X smaller than the original
training set. Unlike Zhang et al.] (2016), networks trained to random labels did not have regu- larizations turned off. If we turn off 1.2 regularization weights, then connection probability of excitatory and inhibitory neurons approach 1, similar to observed before (see Appendix
L ANALYZING THE IMPACT OF REMOVING INHIBITORY NEURONS
In the Standard networks, the classification accuracy appears to depend very weakly on the num- ber of inhibitory channels (Figure II] To text if inhibitory neurons are needed at all, we trained the Standard and StandardE/ networks with inhibitory channels. To our surprise, the accuracy dropped, but rather moderately (Figure [7]b (Standard, from 0.855 to 0.817; StandardEL, from 0.842 0.728). We sought understand what compensated for the loss of inhibitory neurons. We found
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Figure 16: Analysis of networks trained on shuffled labels. L2 is used to emphasize that 1.2 weight regularization intact networks trained on shuffled labels, unlike Zhang al. (2016). The S2/Et L2/EI Shuffled L2 networks achieve on average 43.9% accuracy the randomly- labeled training set.
that without Batch Normalization (BN) excitatory neurons, the loss of inhibitory neurons be- comes devastating (Figure [7p)), reducing the performance to chance level (Standard, from 0.837 to 0.107; StandardEl, from 0.711 0.122). Therefore, inhibitory neurons are particularly necessary in networks trained without BN.
To ensure that the main findings are independent of whether batch normalization is applied, we repeated our experiments of Figs. 6 and 7, but for Standard and StandardEl architectures with no BN. We were able to reproduce all major findings (Figure 17d).
We did notice that even though having inhibitory neurons is devastating, the accuracy can be rescued by having a single inhibitory channel (Figure [7p). which in convolutional network cor- responds to many inhibitory neurons with the same tuning. This is consistent with experimental findings that inhibitory neurons are weakly or non-selective.
We would also like to point out that, even with BN applied, inhibitory neurons are active and strongly connected with excitatory neurons, meaning that they participate in the proper functioning of the network.
To summarize, inhibitory neurons are always used, but only indispensable when BN is not applied. The main conclusions of the paper hold with or without BN.
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Figure 17: Analyzing the impact of having no inhibitory neurons. (a,b) Accuracy in networks without inhibitory neurons. (a) Standard and StandardEl networks, (b) Similar architectures, but without batch normalization (NoNorm). (c) Performance of networks with varying numbers of ex- citatory and inhibitory channels. Maximum (left) and average (right) performance across 5 random seeds. (d) InhPN and NoConstraint variants of networks without batch normalization reproduced findings from Figure
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Reviews:
[{"review_id": "BylvOn4k5H", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The idea behind the paper is quite interesting and enticing. The authors attempt to emulate real-life human brain neural network structure with artificial neural networks similar in architecture to the visual cortex.\nThe authors claim that they observe similar patterns emerge in the Neural Networks when it comes to the interaction and topological properties of excitatory and inhibitory neurons.\n\nUnfortunately, the paper's claims seems to be undermined by several factors. First, the properties of the artificial neural networks that the authors build into the model (excitatory/inhibitory neurons relative abundance, inter-layer projection) are likely to be sufficient to explain the properties authors claim are informative in their model.\n\nIn addition to that, some architectural choices by the authors are highly disputable and counter-intuitive. For instance ReLU activation for Neurons would be contradicting any biological reality.\n\nBesides, to be biologically innovative, the authors' approach seems to miss a number of crucial factors - notably interaction between neurotransmitters, timing of discharges and synapses offsets, allowing single neurons to perform complex logical computations requiring artificial neural networks with simple scalar non-linear activation functions multi-layer sub-networks to learn.\n\nThe case the authors cite - similarity of architectures of convolutional neural networks and the pinwheel architecture of visual cortex V1/V2 areas, is one of the few examples of convergent architectures of real-world and artificial neural networks specializing in a task.\n\nFinally, the role of Inhibory neurons in the visual cortex seems to be\nwell understood, both biologically and mathematically (see for instance https://www.sciencedirect.com/science/article/pii/S0896627303003325 or https://www.sciencedirect.com/science/article/pii/S0896627303003325 or https://www.sciencedirect.com/science/article/pii/S092842570300072X)\n\nOverall, the quality of the article at least in its current state, does not seem to be ready for acceptance to ICLR, but I'm willing to adjust my opinion after reading the opinion of more qualified reviewers' in this area and the authors response.", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "Byl5iH3ptH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "[Update after rebuttal period]\n\nI would like to thank the authors for the detailed response and for addressing the concerns on such a tight schedule.\n\nOverall my two concerns appear to have been right: 1. The object classification task is not really relevant to elicit the observed behavior and 2. Inhibitory neurons are not essential (at least when training with batch norm). \n\nDo the conclusions of the paper about understanding of E/I cells in the brain still hold? I am not really sure. I would like to urge the authors to carefully assess their conclusions again in light of the new evidence and perhaps rephrase abstract, intro and discussion where necessary.\n\nHaving said that, I still like the paper and its approach. I think it provides valuable insights to the field (e.g. neuro-inspired architecture design, concerns regarding training objectives, etc.) and important steps in the right direction and could be a valuable contribution even if the original claims are not entirely substantiated.\n\n\n[Original review]\n\nThis paper seeks to understand why excitatory neurons in cortex are more stimulus selective than inhibitory cells and why excitatory cells are more sparsely connected to each other. The authors train a recurrent neural network model with a number of biological constraints on CIFAR-10. These constraints include Dale's law, an unequal number of E vs. I cells and only excitatory connections between layers (areas) of the network. I would summarize the main findings as follows:\n\n- Excitatory principal neurons are more selective and more sparsely connected to each other than local inhibitory neurons, consistent with biology\n\n- Stimulus selectivity and performance depend only on the number of excitatory cells, but neither on the E/I ratio nor the number of I cells\n\n- High selectivity appears to be a general property of principal cells, but does not depend on the sign of local connections, while sparse connectivity is mainly linked to the excitatory local connections\n\n\nStrengths:\n+ Architectural decisions are well motivated from a neuroscientific perspective\n+ Provides a hypothesis why certain connectivity and selectivity patterns emerge in the brain by incorporating biological knowledge into neural network models trained on a specific task\n+ Well written and clear, logic development of the arguments\n\nWeaknesses:\n- Unclear whether classification task is necessary to elicit the authors' observations\n- Interneurons seem unnecessary, raising concerns about relevance of results\n- Also trains on ImageNet, but only some results are shown; most from CIFAR\n\n\nOverall I like the paper from the perspective of a neuroscientist, as it provides a kind of normative account of why things in the brain might look the way they do. My enthusiasm is somewhat limited, however, because I am not convinced the classification task is actually what drives the authors' observations. I am willing to increase my score and argue more strongly for the paper if the authors can address this concern, detailed in the following:\n\nA few observations lead me to believe the classification task the networks are trained may not be important to elicit the observed behavior. \n\n1.) The emergent properties in stimulus selectivity show up almost immediately after training starts according to Fig. 5A+B. Performance, on the other hand, takes a few more epochs to pick up according to Fig. 10. \n\n2.) The connectivity preferences that emerge in Figure 5C appear at already 50 epochs where CIFAR-10 performance is at 50%. To put that into perspective, a linear SVM already achieves 40% on this task, and a one-layer multilayer perceptron with only 100 hidden neurons reaches that performance. \n\n3.) The results of Fig. 11 suggest that the number of inhibitory channels is not important. Did the authors try a stripped down version with only excitatory cells?\n\nI suggest the authors train on CIFAR-10 with shuffled labels on the training set [Zhang et al. 2016; https://arxiv.org/abs/1611.03530]. Although performance on the test set would be at chance (no object recognition capabilities), it would be interesting to see whether the selectivity and connectivity properties still emerge.\n\nFinally, (related to 3.) I wonder whether it makes sense to draw conclusions about differences between E and I cells from a model trained on a task where the number of inhibitory cells seems irrelevant. Wouldn't one need to have at least both types of neurons to be required for the task in order to draw such conclusions?\n\n\nMinor Comments:\n\n- I sometimes found the nomenclature of the multiple models you tried a bit confusing and hard to follow — especially in parts of the Appendix\n\n- The type of operations (convolution, element-wise) in equation 1, together with the meaning of nonlinearities like \\sigma_c are only defined one page after, making that section a bit hard to follow. ", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "B1g27loaKH", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper explores why the brain might have separate excitatory (E) and\ninhibitory (I) cells and how their different properties affect their\nnetwork connectivity.  The paper is novel and addresses an interesting\nquestion.\n\nThe paper follows Dale's law and has separate populations of E and I\ncells.  The cells differ in proportion (4 times more E cells) and in\nprojection patterns (E cells project to the next layer).  From these\nconstraints they found that some of the other observed differences\nbetween E and I cells developed, specifically greater selectivity of E\ncells and greater connectivity of I cells.\n\nBoth subtractive and divisive inhibition are modeled in the Standard model.\n\nThrough careful controls, they show that increased selectivity of E\nneurons depends on having a lot of E neurons (but there was not a\ncorresponding effect for I neurons).  Likewise the sparse connectivity\nof E neurons was due to a large number of E neurons.  They also showed\nthat being projection neurons was more important than being E cells\nfor strong selectivity.  Finally networks without Dale's law imposed\nperform slightly better.\n\nThese are all interesting findings that will benefit the field of\nneuroscience but may also lead to insights for the next architecture\nboost to improve deep learning.  I am in favor of acceptance.\n\n\n\n", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper explores the role of excitatory and inhibitory neurons, and how their properties might differ based on simulations.  A few issues were raised during the review period, and I commend the authors for stepping up to address these comments and run additional experiments.  It seems, though, that the reviewer's worries were born out in the results of the additional experiments: "1. The object classification task is not really relevant to elicit the observed behavior and 2. Inhibitory neurons are not essential (at least when training with batch norm)."  I hope the authors can make improvements in light of these observations, and discuss their implications in a future version of this paper. 
Decision: Reject