Emergence of functional and structural properties of the head direction system by optimization of recurrent neural networks
Authors: Christopher J. Cueva
Peter Y. Wang
Matthew Chin
Xue-Xin Wei
Venue:  (2020)
Keywords: recurrent network
head direction system
neural circuits
neural coding

Abstract: Recent work suggests goal-driven training of neural networks can be used to model neural activity in the brain. While response properties of neurons in artificial neural networks bear similarities to those in the brain, the network architectures are often constrained to be different. Here we ask if a neural network can recover both neural representations and, if the architecture is unconstrained and optimized, also the anatomical properties of neural circuits. We demonstrate this in a system where the connectivity and the functional organization have been characterized, namely, the head direction circuit of the rodent and fruit fly. We trained recurrent neural networks (RNNs) to estimate head direction through integration of angular velocity. We found that the two distinct classes of neurons observed in the head direction system, the Compass neurons and the Shifter neurons, emerged naturally in artificial neural networks as a result of training. Furthermore, connectivity analysis and in-silico neurophysiology revealed structural and mechanistic similarities between artificial networks and the head direction system. Overall, our results show that optimization of RNNs in a goal-driven task can recapitulate the structure and function of biological circuits, suggesting that artificial neural networks can be used to study the brain at the level of both neural activity and anatomical organization.

Published as a conference paper at ICLR 2020
EMERGENCE OF FUNCTIONAL AND STRUCTURAL PROPERTIES OF THE HEAD DIRECTION SYSTEM BY OP- TIMIZATION OF RECURRENT NEURAL NETWORKS
Christopher J. C Peter Y. Wang, Matthew Chin; Xue-Xin Wei
Columbia University New York, NY 10027, USA
ABSTRACT
Recent work suggests goal-driven training of neural networks can be used to model neural activity in the brain. While response properties of neurons in artificial neural networks bear similarities to those in the brain, the network architectures are often constrained to be different. Here we ask if a neural network can recover both neural representations and, if the architecture is unconstrained and optimized, the anatomical properties of neural circuits. We demonstrate this in a system where the connectivity and the functional organization have been characterized, namely, the head direction circuits of the rodent and fruit fly. We trained recurrent neural networks (RNNs) estimate head direction through integration of angular velocity. We found that the two distinct classes of neurons observed in the head direction system, the Compass neurons and the Shifter neurons, emerged naturally in artificial neural networks as result of training. Furthermore, connectivity analysis and in-silico neurophysiology revealed structural and mechanistic similarities between artificial networks and the head direction system. Overall, our results show that optimization of RNNs in goal-driven task can recapitulate the structure and function of biological circuits, suggesting that artificial neural networks can be used to study the brain at the level both neural activity and anatomical organization.
INTRODUCTION
Artificial neural networks have been increasingly used to study biological neural circuits. In particular, recent work in vision demonstrated that convolutional neural networks (CNNs) trained to perform visual object classification provide state-of-the
models that match neural responses along various
stages of visual processing (Yamins et al. 2014) Sealigh-Razaay & Kriegeskorte] 2014]) Yamins &
DiCarlo 2016 Cadieu et al. [2014] Güçlü & van Gerven 20155] Kriegeskorte 2015). Recurrent
neural networks (RNNs) trained on cognitive tasks have also been used to account for neural response
characteristics various domains (Mante al. 2013 Sussillo al. 2013 Song al. 2016 Cueva &
Wei. 2018 Banino et al.) 2018 Remington et al., 2018 Wang et al. 2018 Orhan & Ma. 2019 Yang
al. 2019). 3. While these results provide important insights on how information processed neural
circuits, is unclear whether artificial neural networks have converged upon similar architectures as
the brain to perform either visual or cognitive tasks. Answering this question requires understanding the functional, structural, and mechanistic properties of artificial neural networks and of relevant neural circuits.
We address these challenges using the brain's 's internal compass the head direction system, system that has accumulated substantial amounts of functional and structural data over the past few decades in rodents and fruit flies (Taube et al. 1990a b Turner-Evee
et al. [017]] Green al. 2017: Seelig & Jayaraman] 2013) Stone et al., 2012 [ et al. 2013) Finkelstein et al., [015]] Wolff et al. 2013
Green & Maimon 2018). We trained RNNs to perform a simple angular velocity (AV) integration
task (Etienne & Jeffery] 2004) and asked whether the anatomical and functional features that have
emerged as a result of stochastic gradient descent bear similarities to biological networks sculpted
"equal contribution
'Correponddecc:: ccueva @ gmail.com, beixxxpu@@gmai..com

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by long evolutionary time. By leveraging existing knowledge of the biological head direction (HD) systems, we demonstrate that RNNs exhibit striking similarities in both structure and function. Our results suggest that goal-driven training of artificial neural networks provide framework to study neural systems at the level both neural activity and anatomical organization.
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Figure 1: Overview of head direction system in rodents, fruit flies, and the RNN model. a, e) tuning
curve of an example head direction (HD) cell in rodents (a, adapted from Taube (1955)) and fruit flies (c, adapted from Turner-Evan et al. (217)). b, d) Tuning curve of an example angular velocity (AV) selective cell in rodents (b, adapted from Sharp et al. (200T) and fruit flies (d, adapted from Turner-Evans et al. (2017)). e) The brain structures the fly central complex that are crucial for maintaining and updating heading direction, including the protocerebral bridge (PB) and the ellipsoid body (EB). f) The RNN model. All connections within the RNN are randomly initialized. g) After
training. the output of the RNN accurately tracks the current head direction.
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MODEL
2.1
MODEL STRUCTURE
trained our networks to estimate the agent's current head direction by integrating angular velocity over time (Fig. [h. Our network model consists of a set of recurrently connected units (N = 100), which are initialized to be randomly connected, with no self-connection allowed during training. The dynamics of each unit in the network (t()) is governed by the standard continuous-tim RNN equation:
dr,(1) (t((t) + Σ (t) dt
Σ Wi + b, (c)
for i = 1,..., The firing rate of each unit rt(t), is related to its total input (t) through rectified tanh nonlinearity, rt(t) max(0, tanh(x;(t)), Every unit in the RNN receives input from all other units through the recurrent weight matrix Wree and also receives external input, [()), through the weight matrix Win These weight matrices are randomly initialized so no structure priori introduced into the network. Each unit has associated bias, b, which is learned and an associated noise term, (t). sampled at every timestep from a Gaussian with zero mean and constant variance. The network was simulated using the Euler method for T 500 timesteps of duration 7/10 (T set to be 250ms throughout the paper).
Let 0 be the current head direction. Input to the RNN is composed of three terms: two inputs encode the initial head direction in the form sin and cos(fp). and a scalar input encodes both clockwise (CW, negative) and counterclockwise, (CCW, positive) angular velocity every timestep. The RNN
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is connected to two linear readout neurons, y1 (t) and (z()), which are trained to track current head direction in the form of sin(0) and cos( The activities of y1 (t) and (2(() are given by:
yz(t) Σ Ware (t(()
(2)
2.2 INPUT STATISTICS
Velocity at every timestep (assumed be 25 ms) is sampled from zero-inflated Gaussian distribution (see Fig. 5 Momentum is incorporated for smooth movement trajectories, consistent with the observed animal behavior flies and rodents. More specifically, we updated the angular velocity as o + momentumm&AVC 1), where X is zero mean Gaussian random variable with standard deviation of one. In the Main condition, we set a = 0.03 radians/timestep and the momentum to be 0.8, corresponding to mean absolute AV of ~ ~10 deg/s. These parameters are set to roughly match the angular velocity distribution of the rat and fly (Stackman & Taube| 1998 Sharp et al.) 200T Bender & Dickinson 2006 Raudies & Hasselmo 2012). In section [
manipulate the magnitude of AV by changing a to see how the trained RNN may solve the integration task differently.
2.3 TRAINING
We optimized the network parameters Wree, Win b and Wout to minimize the mean-squared error in equation (3) between the target head direction and the network outputs generated according to equation plus metabolic cost for large firing rates (L2 regularization on r).
E =
> (t) f.j
yj
target (t))²
(3)
Parameters were updated with the Hessian-free algorithm (Martens & Sutskever 2011). Similar results were also obtained using Adam (Kingma & Ba 2015
3 FUNCTIONAL AND STRUCTURAL PROPERTIES EMERGED IN THE NETWORK
We found that the trained network could accurately track the angular velocity (Fig. Lg). We first examined the functional and structural properties of model units in the trained RNN and compared them to the experimental data from the head direction system in rodents and flies.
3.1 EMERGENCE OF HD CELLS (COMPASS UNITS) AND HD AV CELLS (SIIFTERS)
Emergence of different classes of neurons with distinct tuning properties
We first plotted the neural activity of each unit as a function of HD and AV (Fig. 2p) This revealed two distinct classes of units based on the strength of their HD and AV tuning (see Appendix Fig. b.,bbc). Units with essentially zero activity are excluded from further analyses. The first class of neurons exhibited HD tuning with minimal AV tuning (Fig. 2b. The second class of neurons were tuned to both HD and AV and can be further subdivided into two populations one with high firing rate when animal performs CCW rotation (positive AV), the other favoring CW rotation (negative AV) (CW tuned cell shown in Fig. 2g).. Moreover, the preferred head direction of each uub-population of neurons tile the complete angular space (Fig. (p). Embedding the model units into 3D space using t-SNE reveals clear compass-likk like structure, with the three classes of units being separated (Fig. 2
Mapping the functional architecture RNN to neurophysiology
Neurons with HD tuning but not AV tuning have been widely reported in rodents (Taube et al. 1990a
Blair & Sharp 1995 Stackman & Taube 1998), although the HD* AV tuning profiles neurons are rarely shown (but see Lozano et al. (0017). By re-analyzing the data from Peyrache et al. (2015). we find that neurons in the anterodorsal thalamic nucleus (ADN) of the rat brain are selectively tuned to HD but not AV (Fig. 2H, also see Lozano al. (2017h). with HD & AV tuning profile similar to what our model predicts. Preliminary evidence suggests that this might also be true for ellipsoid body (EB) compass neurons in the fruit fly HD system (Green et al. 2017 Murner-Evan al. 2017).
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Figure 2: Emergence of different functional cell types in the trained RNN. a) Joint HD* AV tuning plots for individual units the RNN. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown function of head direction (x-axis) and angular velocity (y-axis). For each unit, AV ranges from -550 to 550 deg/s, and HD ranges from to 360 deg. b) Preferred HD) of each unit within each functional type. Approximately uniform tiling of preferred HD for each functional type of model ncurons is observed. c) 3D embedding of model neurons using t-SNE, with the distance between two units defined a compass-like structure from one view angle (left) and are segregated according to AV another view angle (right). Each dot represents neurons in the rat anterodorsal thalamic nucleus, or ADN (plotted based on data from Peyrache et al. (2015), downloaded from CRCNS website). White indicates high firing rate. e) Joint AV* tuning plots for two example neurons from the PB of of the fly central complex, adapted from Turner- al. (2017). Red indicates high firing rate. (f.g.h,i) Detailed tuning properties of model neurons match neural data. f) HD tuning curves for model Compass units exhibit shifted peaks at high CW (green) and CCW rotations (blue). g) HD tuning eurves for model Shifters exhibit peak shift and gain changes when comparing CW (green) and CCW (blue) rotations. h) HD tuning curves for CW (solid) and CCW (dashed) conditions for two example neurons in the postsubiculum of rats, adapted from Stackman & Taube 1998). i) HD tuning curves for CW (solid) and CCW (dashed) conditions for two example neurons in the lateral mammillary nuclei rats, adapted from Stackman Taube (1998).
minus their firing rate correlation, exhibits
unit. d) Joint HD* AV tuning plots for two example HD
Neurons tuned to both HD and AV tuning have also been reported previously in rodents and fruit flies (Sharp et al. 2001 Stackman Taube 1998, Bassett Taube 2001], although the joint HD* AV tuning profiles of neurons have only been documented anecdotally with few cells Turner-EEvans et al. (0077). In addition, in the fruit fly heading system, neurons the two sides of the protocerebral bridge (PB) (Pfeiffer & Homberg] 2014) are also tuned to CW and CCW rotation, respectively. and tile the complete angular space, much like what has been observed in trained network (Green et al. 2017
rodents, certain cells are also observed to display HD and AV tuning (Fig. 22).
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Turner- Evane---Eaas---- et al., 2017). These observations collectively suggest that neurons that are HD but not AV selective in our model can be tentatively mapped to "Compass" units in the EB, and the two sub-populations of neurons tuned to both HD and AV map to "Shifter" neurons right PB, respectively. We will correspondingly refer to our model neurons as either "Compass' units "CW/CCW Shifters' (Further justification of the terminology will be given in sections 3.2 3.3
the left PB and
Tuning properties of model neurons match experimental data
We next sought to examine the tuning properties of both Compass units and Shifters of our network in greater detail. First, we observe that for both Compass units and Shifters, the HD tuning curve varies function of AV (see example Compass unit in Fig. 2 and example Shifter in Fig. 2g).
Population summary statistics concerning the amount of tuning shift are shown in Appendix Fig.
7a. The preferred HD tuning is biased towards more CW angle CW angular velocities, and vice
versa for CCW angular velocities. Consistent with this observation, the HD tuning curves in rodents were also dependent upon AV (see example neurons in Fig. Ph.i) (Blair & Sharp 1995 Stackmann & Taube, 1998; Taube & Muller, 1998 Blair et al. 1997 1998). Second, the AV tuning curves for the Shifters exhibit graded response profiles, consistent with the measured AV tuning curves flies and rodents (see Fig. [b.d). Across neurons, the angular velocity tuning curves show substantial diversity
(see Appendix Fig. 6p). also consistent with experimental reports (Turner-Evans et al. 2017).
In summary, the majority of units in the trained RNN could be mapped onto the biological head direction system in both general functional architecture and also in detailed tuning properties. Our model unifies diverse set of experimental observations, suggesting that these neural response properties are the consequence of a network solving an angular integration task optimally.
3.2 CONNECTIVITY STRUCTURE OF THE NETWORK
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Figure 3: Connectivity of the trained network is structured and exhibits similarities with the con- nectivity in the fly central complex. a) Pixels represent connections from the units in each column to the units in each row. Excitatory connections are in red, and inhibitory connections are in blue. Units are first sorted by functional classes, and then are further sorted by their preferred HD within each class. The black box highlights recurrent connections to the Compass units from Compass units, from CCW Shifters, and from CW Shifters. b) Ensemble connectivity from each functional cell type the Compass units highlighted in a), in relation to the architecture of the PB & EB in the fly central complex. Plots show the average connectivity (shaded area indicates one s.d.) as function of the difference between the preferred HD of the cell and the Compass unit it is connecting to. Compass units connect strongly to units with similar HD tuning and inhibit units with dissimilar HD tuning. CCW Shifters connect strongly to Compass units with preferred head directions that slightly CCW-shifted to its own, and CW Shifters connect strongly to Compass units with preferred head directions that slightly CW-shifted its own. Refer to Appendix Fig. 8p for the full set of ensemble connectivity between different classes.
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Previous experiments have detailed subset of connections between EB and PB neurons in the fruit fly. We next analyzed the connectivity of Compass units and Shifters in the trained RNN to ask whether it recapitulates these connectivity patterns test which has never been done to our knowledge in any system between artificial and biological neural networks (see Fig. 3
Compass units exhibit local excitation and long-range inhibition
We ordered Compass units, CCW Shifters, and CW Shifters by their preferred head direction tuning and plotted their connection strengths (Fig. [a). This revealed highly structured connectivity patterns within and between each class of units. We first focused on the connections between individual Compass units and observed pattern of local excitation and global inhibition. Neurons that have similar preferred head directions are connected through positive weights and neurons whose preferred head directions are anti-phase are connected through negative weights (Fig. Bb) This pattern is consistent with the connectivity patterns inferred in recent work based on detailed calcium imaging and optogenetic perturbation experiments (Kim et al., (017)) with one caveat that the connectivity pattern inferred in this study is based on the effective connectivity rather than anatomical connectivity, We conjecture that Compass units in the trained RNN serve to maintain a stable activity bump in the absence of inputs (see section 3.3
as proposed in previous theoretical models [Turing 1952 Amari
1977: Zhang 1996).
Asymmetric connectivity from Shifters to Compass units
We then analyzed the connectivity between Compass units and Shifters. We found that CW Shifters excite Compass units with preferred head directions that are clockwise to its own, and inhibit Compass units with preferred head directions counterclockwise to its own (Fig. 3p)) The opposite pattern is observed for CCW Shifters. Such asymmetric connections from Shifters the Compass units are consistent with the connectivity pattern observed between the PB and the EB in the fruit fly central complex (Lin et al. 2013) Green et al.] 2017 (Tumer-Evans et al. 2017). and also in agreement with previously proposed mechanisms of angular integration (Skaggs et al. 1995] Green et al., 2017 TurnerEEvan et al. 2017 Zhang 1996) (Fig. 3b). We note that while the connectivity between PB
Shifters and EB Compass units are one-to-one (Lin et al. 2013 Wolff et al. 2013 Green et al. (017). the connectivity profile in our model is broad, with a single CW Shifter exciting multiple Compass
units with preferred HDs that are clockwise to its own, and vice versa for CCW Shifters.
In summary, the RNN developed several anatomical features that are consistent with structures reported or hypothesized in previous experimental results. A few novel predictions are worth mentioning. First, our model the connectivity between CW and CCW Shifters exhibit specific recurrent connectivity (Fig. [). Second, the connections from Shifters to Compass units exhibit not only excitation in the direction of heading motion, but also inhibition that is lagging in the opposite direction. This inhibitory connection has not been observed in experiments yet but may facilitate the rotation of the neural bump in the Compass units during turning ( (Wolff et al. [2013] Franconville] et al. 2018 Green et al.) 2017 Green & Maimon 2018). In the future, EM reconstructions together with functional imaging and optogenetics should allow direct tests of these predictions.
3.3 PROBING THE COMPUTATION IN THE NETWORK
We have segregated neurons into Compass and Shifter populations according to their HD and AV tuning, and have shown that they exhibit different connectivity patterns that are suggestive of different functions. Compass units putatively maintain the current heading direction and Shifter units putatively rotate activity on the compass according to the direction of angular velocity. To substantiate these functional properties, we performed series of perturbation experiments by lesioning specific subsets of connections.
Perturbation while holding constant head direction
We first lesioned connections when there is zero angular velocity input. Normally, the network maintains stable bump of activity within each class of neurons, i.e., Compass units, CW Shifters, and CCW Shifters (see Fig. b... We first lesioned connections from Compass units to all units and found that the activity bumps in all three classes disappeared and were replaced by diffuse activity in a large proportion of units. As a consequence, the network could not report an accurate estimate of its current heading direction. Furthermore, when the connections restored, bump formed

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Storing constant HD
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Figure 4: Probing the functional role of different classes model neurons. a-j) Perturbation analysis in the case of maintaining constant HD. a) Under normal conditions, the RNN output matches the target output. b) Population activity for the trial shown a), sorted by the preferred HD and the class of each unit, i.c., Compass units, CCW Shifters, CW Shifters. c-j) RNN output and population activity when a specific set of connections are set to zero during the period indicated by blue arrows in and j). k-v) Perturbation analysis in the case of shifting HD. For each manipulation, CW rotation and CCW rotation are tested, resulting in two trials. k-n) Normal case without any perturbation. o-v) RNN output and population activity when connections from CW Shifters (o-r) and CCW Shifters (s-v) are set to zero. Refer to the main text for the interpretation of the results.
again without any external input (Fig. 4dd, suggesting the network can spontancously generate an activity bump through recurrent connections mediated by Compass units.
We then lesioned connections from CW Shifters to all units and found that all three bumps exhibit CCW rotation, and the read-out units correspondingly reported CCW rotation of heading direction (Fig. F.. Analogous results were obtained with lesions of CCW Shifters, which resulted in a CW drifting bump of activity (Fig. 4g.h). These results are consistent with the hypothesis that CW and CCW Shifters simultaneously activate the compass, with mutually cancelling signals, even when the heading direction is stationary. When connections are lesioned from both CW and CCW Shifters to all units, we observe that Compass units are still capable of holding a stable HD activity bump (Fig. 4 .). consistent with the predictions that while CW/CCW Shifters are necessary for updating heading during motion, Compass units are responsible for maintaining heading.
Perturbation while integrating constant angular velocity
We next lesioned connections during either constant CW or CCW angular velocity. Normally, the network can integrate AV accurately (Fig. k-). As expected, during CCW rotation, we observe a corresponding rotation of the activity bump in Compass units and in CCW Shifters, but CW Shifters display low levels of activity. The converse is true during CW rotation. We first lesioned connections from CW Shifters to all units, and found that significantly impaired rotation in the CW direction,

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and also increased the rotation speed in the CCW direction. Lesioning of CCW Shifters to all units had the opposite effect, significantly impairing rotation in the CCW direction. These results are consistent with the hypothesis that CW/CCW Shifters are responsible for shifting the bump in CW and CCW direction, respectively, and are consistent with the data in Green et al. (2017), which shows that inhibition of Shifter units in the PB of the fruit fly heading system impairs the integration of HD. Our lesion experiments further support the segregation of units into modular components that function separately maintain and update heading during angular motion.
Medium condition)
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Figure 5: Representations in the trained RNN vary the input statistics change. a) The AV distribution used to train the RNN in the low angular velocity condition. b) A histogram of the slopes of the AV tuning curves for individual units in the low AV condition. e) Heatmaps of the joint AV and HD tuning for each unit in the low AV condition. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown as a function of head direction (x-axis) and angular velocity (y-axis). d,e.f) Same convention a-c, but for the main condition (f is the same as Fig. 22). g.h,i) Same convention as a-c, but for the high angular velocity condition.
4 ADAPTATION OF NETWORK PROPERTIES TO INPUT STATISTICS
Optimal computation requires the system to adapt to the statistical structure the inputs (Barlow 1961 Attneave 1954). In order understand how the statistical properties of the input trajectories affect how a network solves the task, we trained RNNs to integrate inputs generated from low and high AV distributions.
When networks are trained with small angular velocities, we observe the presence of more units with strong head direction tuning but minimal angular velocity tuning. Conversely, when networks trained with large AV inputs, fewer Compass units emerge and more units become Shifter-like and exhibit both HD and AV tuning (Fig. (c.f.i.. We sought to quantify the overall AV tuning under each velocity regime by computing the slope of each neuron's 's AV tuning curve at its preferred HD angle. We found that by increasing the magnitude of AV inputs, more neurons developed strong AV tuning (Fig. hh.. In summary, with slowly changing head direction trajectory. it is advantageous to allocate more resources to hold stable activity bump, and this requires more Compass units. In contrast, with quickly changing inputs, the system must rapidly update the activity bump integrate head direction, requiring more Shifter units. This prediction may be relevant for understanding the diversity of the HD systems overall head turning behavior depending on the ecological demand (Stone et al. 2017 Seelig & Heinze 2017, Finkelstein et al., 2018).
different animal species, different species exhibit different
Jayaraman 2015
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5 DISCUSSION
Previous work in the sensory systems have mainly focused on obtaining an optimal representation
[Barlow] [961] Laughlin| 1981; Linsker 1988 Olshausen & Field 1996 SSionncllli & Olshausen
Yamins et al., 2014 Khaligh-Razavi & Kriegeskorte| 2014)) 2014) with feedforward models. Several
recent studies have probed the importance of recurrent connections in understanding neural computa- tion by training RNNs to perform tasks (e.g., Mante et al. (2013); Sussillo et al. (2015): Cueva &
Wei (2018), ) but the relation of these trained networks to the anatomy and function of brain circuits are not mapped. Using the head direction system, we demonstrate that goal-driven optimization of recurrent neural networks can be used to understand the functional, structural and mechanistic properties of neural circuits. While we have mainly used perturbation analysis to reveal the dynamics of the trained RNN, other methods could also be applied to analyze the network. For example, in Appendix Fig. TO using fixed point analysis (Sussillo & Barak 2013; Maheswaranathan et al. 2019). we found evidence consistent with attractor dynamics. Due to the limited amount of experimental data available, comparisons regarding tuning properties and connectivity are largely qualitative. In the future, studies of the relevant brain areas using Neuropixel probes (Jun et al. (017)) and calcium imaging (Denk et al. 1990) will provide a more in-depth characterization of the properties of HD circuits, and will facilitate a more quantitative comparison between model and experiment.
In the current work, we did not impose any additional structural constraint on the RNNs during training, asides from prohibiting self-connections. We have chosen to do so in order to see what structural properties would emerge consequence of optimizing the network solve the task. It is interesting to consider how additional structural constraints affect the representation and computation in the trained RNNs. One possibility would be to have the input or output units only connect to subset of the RNN units. Another possibility would be to freeze a subset of connections during training. Future work should systematically explore these issues.
Recent work suggests it is possible to obtain tuning properties in RNNS with random connec- tions (Sederberg & Nemenman 2019). We found that training was necessary for the joint HD*AV tuning (see Appendix Fig. 9 to emerge. While Sederbbre & Nemenman (2019) consider simple binary classification task, our integration task is computationally more complicated. Stable HD tuming requires the system to keep track of HD by accurate integration of AV, and to stably store these values over time. This computation might be difficult for a random network to perform and, more generally, completely random networks may lead to different memory representations than the attractor geometry we observe in Fig. 10 (Cueva et al. 2019).
Our approach contrasts with previous network models for the HD system, which are based on hand-crafted connectivity (Zhang) 1996 Skaggs et al. 1995 Xie et al., 2002 Green et al. 2017]
Kim et al., 2017, Knierim & Zhang 2012, Song & Wang 20055] Kakaria & de Bivort 2017 Stone
et al. 2017). Our modeling approach optimizes for task performance through stochastic gradient descent. We found that different input statisties lead to different heading representations in an RNN, suggesting that the optimal architecture of a neural network varies depending on the task demand- an insight that would be difficult to obtain using the traditional approach of hand-crafting network solutions. Although we have focused on simple integration task, this framework should be of general relevance to other neural systems as well, providing a new approach to understand neural computation at multiple levels.
Our model may be used as a building block for Al systems to perform general navigation (Pei et al., 2019). In order to effectively navigate in complex environments, the agent would need to construct a cogmitive map of the surrounding environment and update its own position during motion. A cireuit that performs heading integration will likely be combined with another circuit integrate the magnitude of motion (speed) to perform dead reckoning. Training RNNs to perform more challenging navigation tasks such as these, along with multiple sources of inputs, i.e., vestibular, visual, auditory, will be useful for building robust navigational systems and for improving our understanding of the computational mechanisms of navigation in the brain (Cueva & Weil 2018 Banino et al., 2018.
ACKNOWLEDGMENNTS
Research supported by NSF NeuroNex Award DBI-1707399 and the Gatsby Charitable Foundation. We would like to thank Kenneth Kay for careful reading of an earlier version of the paper, and Rong Zhu for help preparing panel d in Figure 2

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A
APPENDIX
Tuning for head direction
Tuning for angular velocity
VIVV
AAA
Keevviy of unitt 10
A
AA A
Activity of unit t0 27
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A
Head direction (D to 360 dog!
Angular (-350 350 deg/s)
Shifters
Strength of turing for (dlope)
Nof gctive
Compass
Strength of tuning for HD (max- - min response)
Figure 6: Tuning properties and unit classification. a) HD tuning curves for all 100 units in the RNN based on the Main condition. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown bottom were not active, did not contribute to the network, and were not included in the analyses. All 88 units with some activity were included in the analyses. b) Similar to a), but for AV tuning curves. c) Classification into different populations using HD and AV tuning strength.
function of head direction. The 12 units at the
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b
Population summary of tuning shift
Population summary of gain change
CCW Shifters median 00.22
CW Shittess medion 13.5"
Shittrs, median Compass 5.04e.5
23 Compass units median 14"
6
Number of units
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Peak shift HD tuning when comparing CW and CCW rotations (deg)
Gain change HD tuning when comparing CW and CCW rotations
Figure 7: Population summary tuning shift and gain change when comparing the CW and CCW rotations. a) Population summary of tuning shift. b) Population summary of change of the peak firing rate HD tnning curves.
Campass units
CCW Shifters
CW Shifters
Compass units
CCW Shifters
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CCW Shifters
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Figure 8: Connectivity of the trained network. a) Pixels represent connections from the units in each column to the units in each row. Units are first sorted by functional classes, and then are further sorted by their preferred HD within each class. Excitatory connections in red, and inhibitory connections are in blue. b) Ensemble connectivity from each functional cell type. Plots show the average connectivity (shaded area indicates standard deviation) function of the difference in preferred HD.
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b
Angulor velodity (-550 to 550 deg/s)
Anguler veloddty (-550 550
direction
Head
Figure 9: Joint HD AV tuning of the initial, randomly connected network and the final trained network. a) Before training, the 100 units the network do not have pronounced joint HD x AV tuming. The color scale is different for each unit (blue minimum activity, yellow maximum activity) to maximally highlight any potential variation in the untrained network. b) After training, the units tuned to HD x AV, with the exception of 12 units (shown at the bottom) which not active and do not influence the network.
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b
deg
90 deg
180
deg
Head direction
180
Target
RNN output
1000
Timestep
PC1
d
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180
PC2
Target RNN output
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Figure 10: Attractor structure of the trained network. To store angular information the neural activity settles into compass-lin like geometry, with the position along the compass determined by the angle. a) The RNN is able to store angular information in the presence of noise. In this example sequence an initial input the RNN specifies the initial heading direction 270 degrees. The angular velocity input to the RNN is zero and so the network must continue to store this value of 270 degrees throughout the sequence. b) The activity of all units in the RNN are shown for 180 sequences, similar to a), after projecting onto the two principal components capturing most of the variance (91%). Fach line shows the neural trajectory over time for a single sequence. For example, the neural activity that produces the output shown in a) is highlighted with black dots (1 dot for each timestep). The activity of the RNN is initialized to be the same for all sequences, i.c. (0) in equation] is the same, so the activity for all sequences starts at the same location, namely, the origin of the figure. The activity quickly settles into a compass-li like geometry, with the location around the compass determined by the heading direction. (c,d) When the RNN is integrating nonzero angular velocity it appears to transition between the attractors shown in b), i.e. the attractors found when the network was only storing constant angle. Panel c) shows example sequence with nonzero angular velocity. In d) the neural trajectory from c) is superimposed over the attractor geometry from b).
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Reviews:
[{"review_id": "Hygbnylk5B", "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": "## Overview\nThis paper studies whether a recurrent neural network trained to solve a particular task (integration of angular velocity to generate head direction). Given trained recurrent networks that solve the task, the paper analyzes these using a number of different methods (visualizing tuning curves, perturbation experiments, and varying input statistics). Overall, I think this is a nice paper that shows the power of using artificial networks as a model system to answer neuroscientifically motivated questions. In particular, I found the perturbation analyses particularly illuminating. These kinds of experiments are only possible in these artificial networks, and can highlight/guide future biological experiments.\n\n## Major comments\n- \"Units with minimal tuning to both variables are discarded from further analysis\"  -- how many units end up being discarded? The reason I ask is because the number of discarded neurons affects this statement: \"Therefore, units in the trained RNN could be mapped on to the biological head direction system both in terms of general functional architecture and detailed tuning properties.\". If a large number of neurons are discarded, then the statement should be amended to say that \"a fraction of units in the trained RNN\" can be mapped onto the HD system. However, my understanding is that many biological neurons have tuning properties that are hard to classify, perhaps the discarded neurons could map on to these previously uncharacterized neurons?\n- The paper demonstrates a number of qualitiative similarities between artificial and biological networks (e.g. both contain neurons tuned for HD or AV). It would be even more compelling if, wherever possible, these comparisons were made to be quantitative.\n- \"We conjecture that Ring units in the trained RNN serve to maintain a stable activity bump in the absence of inputs\". I think a cool direct test of this idea would be to use the techniques in Barak & Sussillo 2013 (find fixed points of the recurrent network dynamics, and analyze the linearized system at those fixed points) to identify the ring attractor structure. In particular, numerical auto-differentiation can be used to automatically identify these points (c.f. the methods in https://arxiv.org/abs/1907.08549). Using these tools, can one find the ring attractor hidden in these networks?\n\n## Minor comments/questions\n- The layout in Fig 2a (and Fig 5c, 5f, and 5i) is a little misleading. If I understand correctly, the axes labels apply to each individual panel (which shows tuning for a particular neuron). However, since the labels extend across many panels, it looks as if the panels themselves are organized according to angular velocity and head direction, which doesn't make sense.\n- Missing citation towards the end of pg. 3\n- Consider setting the panels in Fig 5a-h to have the same axes limits, for easier comparison.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "ByeXr3JAKH", "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 authors train RNNs to integrate motion cues, to determine the head direction of a simulated animal. They then investigate the tuning properties of the units in the RNN, and the connectivity between units, and make comparisons to the corresponding systems in the fly and rodent.\n\nSimilarities are quite strong: the RNN learns to reproduce the ring attractor seen in the fly, where head direction (HD) cells excite other HD cells with similar direction preference and inhibit those with dissimilar preference. Shifter cells are also observed, that integrate angular velocity cues and \"shift\" the representation of head direction: these show an intuitive asymmetric connectivity profile. \n\nI can't quite decide what to make of this paper. One the one hand, the recapitulation of the biological circuit in the RNN is fairly compelling. On the other hand, other models (admittedly with hand-tuned connectivity) yield quite similar findings. So my overall opinion comes down to the question of: how big a deal is it that the RNN learned this connectivity, instead of it being designed by a modeller like Xiao-Jing Wang?\n\nI have a few specific suggestions:\n\n1) It would be useful as a comparison to duplicate the analysis of unit tuning, connectivity, etc., in untrained networks. Recent work (arXiv:1909.10116 [q-bio.NC]) shows that, even in untrained randomly-connected RNNs, some units have (by chance) tuning, and that tuning depends on the connectivity between units in a manner that could somewhat resemble the attractor models.\n\n2) There were important details that I couldn't find. For example, how many of the RNN units receive the external inputs? Is it all of them? Is that realistic?\n\n3) I question the approach (first paragraph of Sec. 3.1) of only analyzing the units  with strong HD and/or AV selectivity. Presumably the mouse shows some units with weak tuning, and investigating the connectivity of the weakly-tuned units in the RNN (e.g., their connectivity to each other, and to HD and AV units) could be informative.\n\nOne analysis to consider: ablate the nonselective units after training, and run the RNN with no retraining. Does the head direction tracking function remain intact or degrade? Based on some previous work with feedforward nets (arXiv:1803.06959 [stat.ML]), I'd guess that the function could be seriously degraded by removing nonselective units. That modelling result could make a strong prediction for experiments: inactivation of nonselective neurons (e.g., using archaerhodopsin), would impair function.\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 thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "BkghycwRYr", "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 examines head direction representations in a RNN. The RNN was trained to report current head direction using initial head direction and angular velocity information as inputs. The authors compare the representations in the RNN to the representations found in the head direction systems of mice and flies. They find that the representations and connectivity matrices of the RNN recapitulate many aspects of the real brains’ head direction systems. These include the presence of cells tuned only for head direction (“ring cells”) and cells with combined head direction and angular velocity tuning (“shift cells”). These cells tile the space in a functional torus structure, as revealed by t-SNE, which resembles the structures seen in real brains. Similarly, the connectivity profiles and functional role of these cell types matches what is observed in the brains. Altogether, these results demonstrate that RNNs optimized for this task can capture both the functional and anatomical aspects of the systems found in animals. This validates the use of RNNs for studying these systems. In my opinion, it also suggests that information about the structure of brains could inform strong priors for ANNs.\n\nI think this is a great paper. It was a pleasure to read, the data was very clear, and the results very interesting. It should be accepted in my opinion. I have only a few minor notes for improvement.\n\n- Per my last note in the summary, it would be nice if the authors made the paper a little bit more relevant for machine learners by discussing how this information could potentially inform novel neural network architectures for navigation.\n\n- There’s a missing reference on page 3, second last paragraph.\n\n- Figure 5 a,d, and g would be easier to interpret if they all had the same x-range as g (that way, you could see the narrower ranges and wouldn’t have to notice that the scales were different).\n\n- Why have Figure S8 and Fig 3 separate? Aren’t they nearly identical? Fig S8 isn’t that much bigger, why not just make it Fig. 3?", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies properties that emerge in an RNN trained to report head direction, showing that several properties in natural neural circuits performing that function are detected. 
All reviewers agree that this is quite an interesting paper. While there are some reservations as to the value of letting a property of interest emerge as opposed to simply hand-coding it in, this approach is seen as powerful and valuable by many people, in that it suggests a higher plausibility that the emerging properties are actually useful when optimizing for that function -- a claim which hand-coding would not make possible. Reviewers have also provided valuable suggestions and requests for clarifications, and authors have responded by improving the presentation and providing more insights.
Overall, this is a solid contribution that will be of interest to the part of the ICLR audience that is interested in biological systems.
Decision: Accept (Spotlight)