Modelling neuronal behaviour with time series regression: Recurrent Neural Networks on synthetic C. elegans data
Authors: Gonçalo Leote Cardoso Mestre
Ruxandra Barbulescu
Arlindo L. Oliveira
L. Miguel Silveira
Venue: ICLR 2022 Submitted (2022)
Keywords: Data-driven models
RNNs
LSTMs
GRUs
C. Elegans
Time series regression
black-box

Abstract: Given the inner complexity of the human nervous system, insight into the dynamics of brain activity can be gained from understanding smaller and simpler organisms, such as the nematode C. elegans. The behavioural and structural biology of these organisms is well-known, making them prime candidates for benchmarking modelling and simulation techniques. In these complex neuronal collections, classical white-box modelling techniques based on intrinsic structural or behavioural information are either unable to capture the profound nonlinearities of the neuronal response to different stimuli or generate extremely complex models, which are computationally intractable. In this paper we investigate whether it is possible to generate lower complexity black-box models that can capture the system dynamics with low error using only measured or simulated input-output information. We show how the nervous system of C. elegans can be modelled and simulated with data-driven models using different neural network architectures. Specifically, we target the use of state of the art recurrent neural networks architectures such as LSTMs and GRUs and compare these architectures in terms of their properties and their RMSE, as well as the complexity of the resulting models. 
We show that GRU models with a hidden layer size of 4 units are able to accurately reproduce the system's response to very different stimuli.

Under review as a conference paper at ICLR 2022
MODELLING NEURONAL BEHAVIOUR WITH TIME SE- RIES REGRESSION: RECURRENT NEURAL NETWORKS ON SYNTHETIC C. elegans DATA
Anonymous authors Paper under double-blind review
ABSTRACT
Given the inner complexity of the human nervous system, insight into the dynamics of brain activity can be gained from understanding smaller and simpler organisms, such as the nematode C. elegans. The behavioural and structural biology of these organisms is well-known, making them prime candidates for benchmarking mod- elling and simulation techniques. In these complex neuronal collections, classical white-box modelling techniques based on intrinsic structural or behavioural infor- mation are either unable to capture the profound nonlinearities of the neuronal response to different stimuli or generate extremely complex models, which are computationally intractable. In this paper we investigate whether it is possible to generate lower complexity black-box models that capture the system dynamics with low error using only measured or simulated input-output information. We show how the nervous system of C. elegans can be modelled and simulated with data-driven models using different neural network architectures. Specifically, we target the use of state of the art recurrent neural networks architectures such LSTMs and GRUs and compare these architectures in terms of their properties and their RMSE, as well as the complexity of the resulting models. We show that GRU models with a hidden layer size of 4 units are able to accurately reproduce the system's 's response to very different stimuli.
INTRODUCTION
The study of the human brain is probably one of the greatest challenges in the field of neuroscience. Recent developments in experimental neuroscience have considerably increased the availability of novel recordings and reconstructions shedding further light into the structure and function of the brain as well as many other systems. But understanding the complexities behind the relations between structure and function well as the behaviour of such systems across multiple scales in these neuronal collections is constrained by the methods available to study them. This challenge has raised interest in many related fields, such electrophysiological analysis, imaging techniques, brain-related medicine, computational modelling and simulation. model reduction. Many of these efforts, while not directly providing specific information regarding structural or functional dynamics. do supply large volumes of recordings, measurements or simulations of observable input-output behaviour. The availability of these large datasets raises the question whether low complexity, data-driven, black-box models can be used to model such input-output relations with low error, avoiding the excessive inner detail which may not be known or available.
To determine whether such an approach can be used for large, complex systems, one research direction the study of smaller and simpler nervous systems, for which the underlying principles of network organization and information processing are easier to postulate. These organisms can become useful models to gain insight into the fundaments of neuronal dynamics and whole brain organization, develop and test modelling methods, simulation instruments and model
to validate hypotheses, reduction techniques. The hope that the knowledge gained from these analyses and the techniques developed for these simpler organisms can later be used model more complex systems.
Caenorhabditis Elegans (C. elegans) belongs to this category of organisms and is quickly becoming one of the benchmarks in whole brain organization studies. C. elegans is nematode (roundworm) of

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about mm in length with a compact nervous system consisting of less than 1000 cells across all sexes and around 15000 connections (Cook et al., 2019). This rather small nervous system allows the worm to solve basic problems such as feeding, predator avoidance and mate-finding The relative simplicity of C. elegans allowed for its almost complete deseription from different perspectives and scales, from its genetics and genomics the molecular biology, structural anatomy, neuronal function, circuits and behaviour. This information is available in comprehensive databases of genetics and genomics (Hunt-Newbury al., 2007), electron micrographs and associated data, online books and atlases of the neurobiology, structural and behavioural anatomy (Jackson et al.] 2014). Creating a realistic model that encapsulates all this information is not a trivial task. Open- source databases of digitally reconstructed neurons (Gleeson et al. (211)), computational models (Szigeti et al. 2014) and collaborative solutions (Cantarelli et al., 2018) are opening the door for more flexible, multi-scale and multi-algorithm simulation environments for C. elegans and other complex biological systems.
The underlying models are based on the connectome, the map of the neuronal connections in the brain. Usually described neuronal network, the connectome is a graph where the nodes are the neurons and the edges represent the synapses. The complete connectome of C. elegans contains 302 neurons for the adult hermaphrodite (Varshney et al., 2011) and 385 neurons for the male (Cook et al. 2019), but for the latter the respective 3D reconstructions are not yet published. Digital reconstructions for the male are only available for the posterior nervous system of 144 neurons (Jarrell et al. 2012).
The more complex the organism. the more complicated the resulting model, needing more computa- tionally demanding and potentially intractable simulations of its dynamic behaviour. This increased complexity stems from the detailed modelling of the internal structure. However, in many cases, especially of highly complex systems, this detail not available since the internal mechanisms may not be well known or mapped or it may be simply impossible to examine and record. For that reason, frequently one is really only interested in the peripheral, or input-output behaviour. This motivates our efforts not only to place the focus more on observable onput- data, as well as to try and generate reduced models that avoid extraneous detail not necessary to explain these peripheral relations.
In this work we propose methodology for generating a reduced order model of the neuronal behaviour of organisms using only peripheral information. We use C. elegans as a proxy for our study, Realistic models of C. elegans, which take into account spatial distribution and biophysical properties of neuronal compartments have been reported in the literature (Gleeson et al., 2018). We start with similar model created ho-house. Our model comprises the complete connectome of the adult hermaphrodite of C. elegans, with 302 multi-compartmentt neurons and 6702 synapses (Varshney al. 2011). The model was validated (Anonymous) 202T against four scenarios described in related literature (Kim et al. 2019): Forward Crawling Motion (FCM) for the full network, Ablation of AVB interneurons FCM, Ablation of AVA interneurons FCM and the Nictation behaviour. We reproduce here the FCM scenario, in which we apply stimulus on the touch sensitive sensory neurons and the interneurons known to be part of the forward movement circuit and we check the activity of the motor neurons associated with forward locomotion. Since we find strong activity in most of these neurons, we conclude that the worm moves forward.
Based on synthetic data extracted from this high-fidelity model, we create a completely equation cruet data-driven model assuming no prior knowledge of the original system's 's structure and equations, using neural networks trained on datasets representing the system's 's response to different input signals. The ultimate goal is to generate a reduced model to replace the original, detailed one. This reduced model should be able to reproduce with reasonably low RMSE the behaviour of the realistic model while having fewer degrees of freedom. In this work we focus on the issue of reduced RMSE, which we equate fidelity in reproducing the system dynamics, showing that we can produce sufficiently accurate models for amalysing the behaviour of the C. elegans nervous system using neural networks.
RELATED WORK AND CONTEXT
The connectome-based models mentioned above are often termed white-box models, as they are based on direct knowledge and access to the internal structure and parameters' values of the modelled system. These are distributed models, where each neuron has a 3D description and position in space and the synapses associated with neuronal sections. Such models enable highly accurate simulation of the dynamic behaviour of organisms, but easily become extremely complex as they incorporate detailed structural and functional information of the system.
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While the white-box approach ensures access and evaluation of inner parameters during simulation, it has been shown that the activity of complex networks of neurons can often be described by relatively few distinct patterns, which evolve on wwwdimensional subspaces (Karasözen 2020). This knowledge, together with the ever-present need to avoid potential numerical intractability in large-scale networks with many degrees of freedom, has generated renewed interest in applying model reduction, often also referred to as model compression, to these neuronal networks, including techniques such as Dynamic Mode Decomposition (DMD) (Brunton et al. 2016), Proper Orthogonal Decomposition (POD) (Kellems et al., 2009) and Discrete Empirical Interpolation (DEIM) (Lehtimäki et al. (019). Depending on the level of morphological accuracy of the underlying models, reduction techniques can have any shade of grey from white-box to black-box, the latter assuming no preliminary knowledge of the system structure and building the model solely out of knowledge of its input-output behaviour.
Black-box approaches are often built upon data-driven models, sometimes Barning-based, which have the ability to grasp more naturally and more efficiently the complexity induced by the profound nonlinearities in the neuronal transmission of information. Machine-learning techniques are used to extract data-driven reduced order models for systems arising from differential equations describing the intrinsic dynamies (Regazzoni et al., 2019) and even extract the governing equations of the estimated model (Sun et al. 2020). It is therefore quite natural to consider using state of the art learning methods for developing reduced models of neuronal behaviour using data obtained from available recordings or even simulations obtained with more complex models.
Especially designed to capture temporal dynamic behaviour, Recurrent Neural Networks (RNNs), in their various architectures such as Long Short-Term Memory (LSTMs) and Gated Recurrent Units (GRUs), have been extensively and successfully used for forecasting or detecting faults in multivariate time series data (Massaoudi et al. 2019), (Gallicchio et al. 2). (Yuan et al., 2020). Fiilonoy
ct al. 2016). Bidirectional LSTMs were used to model genome data by [Tavakoli (2019), whereas a combination of CNNs and LSTMs generates a model for epileptic seizure recogmition using EEG signal analysis in Xu et al. (2020). An attempt to model the human brain activity based on fMRI using RNNs (LSTMss and GRUs) is reported in Güçlü & van Gerven (017)) In recent years, deep network approaches were used to model realistic neural activity data (Molano-Mazoo et al. 2018). (Bellec et al., 2021). 2021). (Karampatziakis 2010). Few studies examined the behavioural output of network models of C. elegans using machine-learning techniques. RNNs are generated in a grey-box manner to study the chemotaxis behaviour (Xu et al. (2100) or predict the synaptic polarities (Lanza et al.
2021 of C. elegans, yet these models only include a subset of the connectome.
3 METHODS
Given that the starting point is in fact represented by time series data obtained from simulations of the realistic connectome-based model, the modelling task is akin sequence sequence conversion for which the most suitable neural network models are sequential ones.
In this work analyze the suitability of three of the most commonly used architectures for recurrent neural networks. We start with the least complex unit, the simple RNN, originally proposed in the 1980's to model sequence data (Rumelhart et al. 1986), (Werbos 1988). (Elman) 1990). The second model used for the recurrent layer is the LSTM unit, developed by Hochreiter & Schmidhuber (997) and later improved with the introduction of the forget gate to adaptively release internal resources when necessary (Gers et al. 1999). Finally we analyze its sibling, the GRU (Cho et al. 2014).
3.1 RECURRENT NEURAL NETWORKS
RNNs |Rumelhart et al. 1986). Werbos 1988, Elman 1990) are a family of neural networks used for processing sequential data, particularly adept to processing a sequence of values x(1) ..., x(*), and in most cases capable to process sequences of variable length. RNNs appear from the relaxation of
the condition on Feedforward Neural Networks (FFNNs) that neurons in a given layer do not form connections among themselves.
Although RNNs which are trained using Backpropagation Through Time (BPTT) (Weroos 1990). seem to be good model for sequential tasks, they are known to suffer mainly from two issues, vanishing and exploding gradients (Bengio et al.) 1994). Exploding gradients (Bengio et al. 1994) refer to a large increase the norm of the gradient during training, which appears due to the explosion
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Ct-1
C2 h
h
by-3 tanh hy
bynh
h,
Figure 1: Comparison between the three different units, RNN, LSTM and GRU, respectively.
of long term components that can grow exponentially more than short term ones. This the less common of the two problems and there known solutions to handle it, such the clipping gradient technique (Pascanu et al., 2012). A harder to solve problem is the vanishing gradient issue (Bengio et al. 1994). which refers when long term components go exponentially fast norm O, making impossible for the model to learn the correlation between temporally distant events.
In order to faithfully reproduce the dynamics of our system the simulations used for generating the datasets require the use of fine time steps, leading to long data sequences. This in turn implies that the response at given time will depend on values which are far back the sequence. This situation, however unavoidable, may lead the RNN to experience difficulties in learning our data resulting in model with unacceptable RMSE.
3.2 LONG SHORT-TERM MEMORY
The Long Short-Term Memory unit (Hochreiter & Schmidhuber, 1997) appeared solution the vanishing gradient problem, later improved with the inclusion of the forget gate (Gers et al. 1999). LSTM unit consists of three main gates, the input gate TT that controls whether the cell state updated, the forget gate (2) that defines how the previous memory cell affects the current one and the output gate (3). which controls how the hidden state updated. Note that LSTM units exhibit major difference from RNN simple units, since besides the hidden state they also output cell state to the The LSTM mechanism is described by the following:
next LSTM unit, as it can seen Figure [
(Wix Uph + bi)
ē o W. + Uchr + bo)
f. = (WPx: + Uph + by)
(2)
Ct = o
(5)
O (Wox + Uoh,11 b.)
(3) h, o o((c).
(6)
where W,, U, Wr. Ur. W.. Uo. We and Uc, are the weights and b; by. b, and b the biases. All these 12 parameters are usually learned by the model, while (....) and ((...) are the logistic sigmoid and the hyperbolic tangent activation functions, respectively. The outputs of the LSTM unit, the hidden state and the cell state, are computed using (5) and (6), respectively. The computation of the cell state requires the candidate cell state, obtained through
3.3 GATED RECURRENT UNITS
The of LSTM units in recurrent neural networks already produced models that were able to learn very distant dependencies (Gers et al. 2000). but these units are complex structures composed of three gates. For that reason, in 2014 new type of unit, the GRU (Cho al. 2014), was suggested, described as follows:
2, o Wax + U.h.11 + b,) W, + U,h, br)
(7) hg o( W(Wn + Un(re o h + by)
(9)
(8) h, (1 2x) he + zt fif.
(10)
In and 9.. the weights, Wz. Uz. W U.. Wn. Un and the biases by, b bn the parameters that the model should learn. o(...) ) and o(...) are, again, the logistic sigmoid and the hyperbolic tangent activation functions, respectively.
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0.0
05
0.5
1.
Figure 2: Example input (top row) and output (bottom row) time sequences.
The GRU (Figure is only composed of two gates, the update gate 7 and the reset gate 8 The GRU only outputs the hidden state (TO), computed based on the candidate hidden state (9) The update gate controls how much of the past information needs to be passed along to the future, while the reset gate used to decide how much information the model should forget.
EXPERIMENTAL SETTING
4.1
DATA
The starting model based the complete connectome of the adult hermaphrodite of C. elegans, with 302 nulti-compartmental neurons and 6702 synapses. The neurons are described by 3D geometrical information extracted from NeuroML and LEMS files (Gleeson al. 2019) for C.elegass. We added membrane biophysical properties and connectivity data (chemical synapses and gap junctions) from Gleeson al. (2018) for the complete connectome. This is high-fidelity model, since due the level of detail taken into account we assume it reproduces with fidelity the real output of C. elegans neurons. The simulations reproduce the Forward Crawling Motion scenario, by applying varying input currents to two sensory neurons ("PLMI.", "PLMR"") and two interneurons ("AVBL.", "AVBR") and record the responses of four neurons known to have strong activity during forward locomotion ("DB1", "LUAL", "PVR" and "VB1") 1"). The resulting system described in Python and simulated in NEURON (Carnevale Hines 2006), one of the traditional neural simulators.
The Python code invokes NEURON to generate the neuronal network, to simulate its behaviour with respect to certain imput signals (currents) and to save the responses the four specific neurons (voltages) previously identified important for test case. We simulate the high-fidelit model for 500 ms with two time steps - 0.5 ms and 0.1 ms and 40 different shapes for the input currents. The input-output variations extracted into datasets of 40 snapshot files each, which are further used to train the neural nets. These files are available in the Supplementary material.
To train and tune the hyperparameters, learning rate and batch size, the data divided into three sets: training, validation and test. The separation of data is done as follows: training set 50% of the data, validation
25% and for test the remaining 25%. The separation is done by hand, so that validation and test sets are diverse and demanding as possible. Three examples of the diverse set of inputs and outputs are shown Figure
4.2 MODELLING
The models are developed in Python VVn Rossum & Drake 2009), using the libraries Keras (Chollet eet al. 2013) and Tensorflow al. 2015). Details on the code and dependencies to run the experiments listed in Readme file available together with the code in the Supplementary material.
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0.7
0.5
0.5
(70) dhoenio
06
0.2
6.1
011
Figure 3: Average training and validation RMSE of ten simulations, with recurrent layers of size 16 and (left) and 64 and 32 (right).
RNN-16
LSTM-8
GRU-8
RNN-64
LSTM-32
GRU-32
Figure 4: Experiment 1: real (red & blue) and predicted (green & black) sequences for DB1 and LUAL (1st and 3rd rows) and PVR and VB1 (2nd and 4th rows) for two sequences of the test set (one selected simulation out of ten).
Each model tested consists of one recurrent layer described in Section 3 followed by a dense layer. The dense layer performs a simple linear transformation for each sequence point to convert the output of the recurrent layer, size "hidden size", into the four outputs.
For consistent comparison, fixed the optimizer to Adam (Kingma & Ba 2015) and the loss function the root mean squared error. The other hyperparameters, learning rate and batch size, were tuned through experimentation (further details available in the Supplementary material). The experiments on each the two hyperparameters were conducted separately for the three different
Table 1: The average RMSE of ten simulations, for RNN with 16 and 64 hidden units and for LSTM and GRU with and hidden units, for the iteration with the smallest validation loss.
RNN-16
LSTM-8
GRU-8
RNN-64
LSTM-32
GRU-32
Training
1.0444-001
9.9371e-03
7.7912e-03
6.2831e-02
7.6058e-03
6.7129e-03
Validation
1.2986e-01
3.6109c-02
3.4760e-02
8.8764e-02
3.6437e-02
3.5098c-02
Test
1.3406e-01
1.4904e-02
1.0005e-02
9.5476e-02
1.5230e-02
1.2384e-02

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Table 2: The average RMSE of ten simulations obtained with the GRU model, for different sizes of the recurrent layer.
Units
Units
Units
Units
32 Units 64 Units
Training
4.2999c-02
1.0525e-02
7.7912e-03
6.7886c-03
6.7128e-03
5,8068c-03
Validation
5.3732e-02
3.4800c-02
3.4760e-02
3.4961c-02
3.5098c-02
3.6230e-02
Test
6.0047e-02
1.1771e-02
1.0005e-02
9.3625e-03
1.2384e-02
1.6841e-02
2
4
8
16
32
64
Figure 5: Experiment 2: real (red & blue) and predicted (green & black) sequences for DB1 and LUAL (1st and 3rd rows) and PVR and VB1 (2nd and 4tht rows) for two sequences of the test set (one selected simulation of ten).
recurrent layers with a fixed hidden size of 16 units, and the other hyperparameter fixed. Each model was trained for 1000 epochs, with the final model chosen as the best iteration the validation set. After this experimentation we chose to fix the batch size at 32 for all three models, the learning rate at 0.001 for the RNN and 0.05 for the LSTM and the GRU as these values seemed provide the best results. Different activation functions were also experimented, but did not end up having positive impact on the results, therefore we kept the default ones.
EXPERIMENTS AND RESULTS
In Experiment (Section 5.1) we compare the performance of the three types of layers, RNN, LSTM and GRU. Experiment (Section 5.2) carries comparison between different sizes of the GRU recurrent layer in order to determine the optimal size under some RMSE constraints. Finally, Experiment 3 (Section (.3) is investigation upon the ability of the models reproduce data resulting from simulations with finer time step, therefore involving longer sequences with more data points. For all the experiments, the loss is computed as the average RMSE of simulations.
5.1 EXPERIMENT
In this experiment we compare the performances of the three types of units the dataset correspond- ing to the coarser time step (0.5 ms). In the interest of faimess we use layers with comparable number of parameters: RNN with 16 hidden units (404 total parameters) against a LSTM and a GRU, both with 8 units (452 and 348 parameters, respectively); and a RNN with 64 units (4676 parameters) against a LSTM and GRU with 32 units (4868 and 3684 parameters, respectively).
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00
0.
1210 dom
0.45
6.7
(80) pieeii
04
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050 soaning
Drandd -Tainng
0.25
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68
0.5
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0.1
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10
117
Figure 6: Average training and validation RMSE of ten simulations. Left: for 6 different hidden sizes of the GRU-based recurrent layer, for the iteration with the smallest validation loss. Right: for the two datasets with two different hidden sizes with GRU-based recurrent layer.
Table 3: The average RMSE of ten simulations obtained with the GRU model, for the two datasets and two sizes of the recurrent layer (4 and 8 hidden units).
Dataset1 4
Dataset2 4
Datasetl
Dataset2
Training
1.0525c-02
1.0878c-02
7.7912c-03
8.1769c-03
Validation
3,4800c-02
3.4554c-02
3.4760c-02
3.5440c-02
Test
1.1771e-02
1.2107e-02
1.0005e-02
1.0434e-02
Figure 3 shows the evolution of the training and validation losses during the training process. The simple RNN unit tends to take more time learn, being also slightly less stable during the end of the training process. Although this is not a good indicator, it is not as alarming as the behaviour shown in Figure where it clear that the simple RNN unit is not able reproduce the outputs with the desired reduced RMSE, while the LSTM and the GRU perform well. A summary of this experiment's results is shown Table Since the simple RNN unit did not perform sufficiently well by not being able to reproduce the output with minimal RMSE, we are left with the LSTMs and GRUs units. Given that the GRU is the less complex unit of the two, we consider the main option and keep the LSTM as an alternative architecture.
5.2 EXPERIMENT 2
The GRU, due to its low RMSE and relative simplicity, therefore emerged as the prime candidate unit for our modelling purposes. However, we now want to determine how small the models can be without compromising the overall error. The focus of this second experiment is therefore to test different sizes of the recurrent layer and determine the smallest size that still able to generate a model with sufficiently low RMSE. We test both the LSTM and GRU units using the dataset with the coarser time step of 0.5 ms. Since the LSTM does not produce noticeable improvement over the GRU with a similar number of parameters, we only report here the results obtained with the GRU for six different sizes of the recurrent layer: 2, 4. 8, 16, 32, 64 (Table 2. Figure 6 (left) illustrates the evolution of the training and validation losses during the learning process, where one can see that even for size 8 the model reaches low and stable loss. In fact, from Figure and Table 2 we can state that GRU with a size of hidden units optimal reproduce the outputs with low RMSE.
5.3 EXPERIMENT 3
In this experiment we explore the models' behaviour in the same simulation interval using data sampled with different time steps as this leads to sequences of varying lengths. From a methodology standpoint this is important since even though time-wise the dynamics do not change, the temporal dependencies that the model has to leam are farther back in the sequence, which increases the difficulty of the learning process. We run the model for two datasets, one with the coarser (0.5 ms) time step and one with a finer (0.1 ms) one. The experiment is done only for the GRU, with 4
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Datasetl-
Dataset2
Datasetl 88
Dataset2 8
Figure 7: Experiment 3: real (red & blue) and predicted (green & black) sequences for DB1 and LUAL (1st and 3rd rows) and PVR and VB1 (2nd and 4th rows) for two sequences of the test set (one selected simulation of ten).
and 8 units. Even though the model takes more time to converge for the finer time step. ends up stabilizing with loss of the same order in both cases and the model fits the test data well, as shown in 6 (right) and Table The plots in Figure further strengthen this idea. We can conclude that GRU with 4 hidden units is consistently able reproduce the outputs of the original model with reasonably low RMSE, for various inputs, for data sampled with coarser and finer time steps.
6 CONCLUSION
In this paper we create models for the C. elegans nervous system with three different recurrent neural networks architectures: simple RNNs, LSTMs and GRUs The objective to further generate low-order description to replace the original, detailed model in the NEURON simulator. To achieve this goal we seek a model as simple as possible and therefore the ideal unit would appear to be the simple RNN. However, this unit does perform sufficiently well compared to the other two architectures. The LSTT and GRU give comparable results in terms of overall fidelity, measured through RMSE, for different sizes of the recurrent layer. Due its simplicity, GRU is preferable and with a hidden size of 4 units, is able to reproduce with high fidelity, i.c. low RMSE, the original model's responses to different types of stimuli. Furthermore, from computational standpoint, explicitly inferring the response of the GRU model to such stimuli will vastly outperform the cost associated with simulating the high-fidelity model within NEURON, which has solve the set of nonlinear equations implicit in the connectome network. However, to quantify the potential advantage and to provide a fairer comparison would require including our model into NEURON and perform an identical simulation.
Further work will concentrate on improving the automation in choosing appropriate stimuli for the training, validation and test sets as well as optimal parameter selection. This will require systematic analysis of compression possibilities of the learning-bast based models with error control. These results nonetheless show that it is feasible to develop recurrent neural network models able to infer input-output behaviours of real biological systems, enabling researchers to advance their understanding of these systems even in the absence of detailed level of connectivity.

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SUPPLEMENTARY MATERIAL
The Supplementary Material containing the datasets, the models, the source code and the instructions run them is available the following anonymous GitHub repository:
12


Reviews:
[{"review_id": "bgeT2Y-AWMb", "reviewer": "Reviewer_mRji", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "\nThe paper uses a connectome-based in-house model of the worm, utilizing compartmental neuron models and chemical and electrical synapses to generate the ground truth data. In general, I found that insufficient details are provided about how this model was built & validated (how were the biophysical properties chosen, what synapse models were used, etc). The authors state that the simulator is \"high-fidelity\" because they \"assume it reproduces with fidelity the real output of C. elegans neurons.\". The model might be high-complexity, but as far as I could see, fidelity was not demonstrated anywhere. In fact, the only test case, also used for all subsequent modeling work with neural networks, is \"forward locomotion\". In it, 2 sensory neurons and 2 interneurons are stimulated to generate activity in 4 neurons known to be active in locomotion. It's possible that additional information is available in the supplementary materials which I was not able to access (\"ERRORS.The value of \"offset\" is out of range. It must be >= 0 && <= 17825792. Received 17825795\"). Even if that's the case, more of this information should be in the main text.\n\nI found section 3.1 with detailed discussion of GRU, LSTM, and problems related to training RNNs a bit too verbose. These are standard and well-known architectures at this point, and could just be briefly introduced with references to the original papers. The reader would benefit much more from learning the details of the C. elegans simulation.\n\nThe paper does a good job of systematically testing various variants of recurrent neural networks (LSTM, GRU, vanilla RNN) in a simple setup involving a single recurrent layer followed by a dense layer with 4 outputs. The authors focus on GRU, and conclude that a tiny neural network with just 4 GRU units is sufficient.\n\nThat a GRU network works for modeling time-series data is not particularly surprising. I found the experimental work presented in the paper far too limited to yield useful insights into using RNNs as a black-box model of real brain activity. Ideally, the experiments should cover more neurons and behaviors, and variations in synapse strengths (see doi:10.1038/s41586-021-03778-8) and the biophysical parameters. What is missing is also a comparison of the computational complexity of the simulation and the proposed reduced order model. One could also ask whether the model used in the simulation is actually necessary to produce this \"forward motion\" scenario -- could some neurons be excluded, or simpler neuron models used instead?\n\nSpecific comments:\n* The text mentions that the separation of the simulation results into train/test/validation was done manually. Please provide more details on the criteria you used to ensure diversity within each group.\n* In experiment 3, you state that \"[..] data sampled with different time steps as this leads to longer sequences\". This should probably be rephrased to highlight that the same physical time is simulated.\n* What exactly is the input to the network? Only the 2 currents on the sensory neurons or also on the interneurons?\n* Why is it necessary to provide stimulus on interneurons instead of just sensory neurons?\n\n\n", "summary_of_the_paper": "The paper shows that a small recurrent neural network can be used to predict the activity of 4 neurons in a C. elegans simulation with good accuracy as measured by RMSE.\n", "summary_of_the_review": "The paper effectively shows that a tiny GRU network can reproduce a few time-series generated from a larger simulation. The general research direction of building reduced-order models for such simulations is interesting, but the very limited empirical results presented here and lack of details about the simulation make it impossible for me to recommend acceptance.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "7Jwu4GWLeTY", "reviewer": "Reviewer_okcY", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "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": "## Strenghts\n- The paper is very well written and is an enjoyable read. It explains\n  well the problem it is trying to address and why it's important, and\n  it introduces all concepts and techniques it uses in an accessible\n  way. I think this is particularly important for this paper, as its\n  main audience would presumably be computational neuroscientists, so\n  it seems a good idea not to take for granted a deep knowledge of\n  (say) recurrent network architectures.\n- Addressing the problem of model reduction using off-the-shelf\n  machine learning approaches is a timely choice, given the very rapid\n  increase of experimental data available for modeling in this field.\n- The main thesis of the paper is well supported by the evidence,\n  which is laid out in a logical and clear progression.\n\n## Weaknesses\n- Mirroring what I wrote above in one of the \"strenghts\" points: the\n  fact that paper is a rather simple application of off-the-shelf RNN\n  architectures to a well-defined modeling problem means that the\n  paper is not hugely ambitious, and that it doesn't bring much in the\n  way of theoretical or conceptual insight; however, in my opinion\n  this does not detract from the quality of the work.\n\n## Minor suggestions:\n- When discussing related literature, I invite the authors to consider\n  the inclusion of a broader selection of papers that in recent years\n  applied deep network approaches to modeling neural activity data, as\n  the works currently cited in the paper investigate only fMRI data\n  (which is only distantly related to actual neural activity), or are\n  specific to C Elegans. Two examples that come to mind are\n  Molano-Mazon et al, ICLR 2018, *Synthesizing realistic neural\n  population activity patterns using generative adversarial networks*,\n  and Bellec et al, NeurIPS 2021, *Fitting summary statistics of neural\n  data with a differentiable spiking network simulator*.\n- If possible, it would be great to have an actual comparison of the\n  compute time/cost incurred in running the trained RNN (even just for\n  the final selected version of the architecture, the 4- or 8-units\n  GRU), versus running the ground-truth simulation with NEURON. I\n  apologise if this is given somewhere and I missed it.\n- Please make all figure labels bigger. Axes labels, tick labels and\n  legend contents are currently almost impossible to read!\n- on page 7, last line of section 5.1: \"*[...] we consider it the main\n  option and keepING the LSTM as an alternative architecture*\" → should\n  be \"keep\".\n\n", "summary_of_the_paper": "This paper investigates the use of recurrent neural networks as a\nmodel reduction tool in computational neuroscience. More specifically,\nthe authors consider the problem of predicting the activity of a set\nof four neurons in the C Elegans nematode worm, resulting from the\n(simulated) electrical stimulation of other neurons in the animal's\nconnectome. Three experiments are carried out, testing different\nnetwork architectures, network sizes, and the effect of increasing the\ntemporal resolution of the data. The results show that a small\nGRU-based network is sufficient for achieving excellent agreement with\nthe starting data, which was generated using computationally demanding\nsimulations of a network of multi-compartimental neurons. The paper\nalso includes an introduction on popular RNN architectures, and on the\nproblem of model reduction in computational neuroscience.\n", "summary_of_the_review": "This is a solid paper that addresses a question which will be of\ninterest to part of the ICLR community. The research is somewhat\nlimited in scope, but very well executed and written up, and overall a\nworthy contribution. It is also a good demonstration of the usefulness\nof open databases of neuroscience models and data.\n\nMy recommendation is to **accept**.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "3C_ISbzrQZ_", "reviewer": "Reviewer_Y3zd", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "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": "In this paper authors create models for the C. elegans nervous system with three different recurrent neural networks architectures: simple RNNs, LSTMs and GRUs. The objective is to further generate a low-order description to replace the original, detailed model in the NEURON simulator. \n\nAuthors should improve the automation in choosing appropriate stimuli for the training, validation and test sets as well as optimal parameter selection and perform a systematic analysis of compression possibilities of the learning-based models with error control.", "summary_of_the_paper": "Authors show how the nervous system of C. elegans can be modelled and simulated with data-driven models using different neural network architectures. Specifically, they target the use of state of the art recurrent neural networks architectures such as LSTMs and GRUs and compare these architectures in terms of their properties and their RMSE, as well as the complexity of the resulting models. Authors show that GRU models with a hidden layer size of 4 units are able to accurately reproduce the system’s response to very different stimuli.", "summary_of_the_review": "Overall its a weak accept. Authors should improve the automation in choosing appropriate stimuli for the training, validation and test sets as well as optimal parameter selection and perform a systematic analysis of compression possibilities of the learning-based models with error control.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper explores the use of recurrent neural networks to model neural activity time-series data. The hope is that computationally demanding biophysical models of neural circuits could be replaced by RNNs when the goal is simply to capture the right input-output functions. The authors show that they can fit RNNs to the behaviour of a complex, biophysical model of the C elegans nervous system, and they explore the space of hyperparameter and network choices that lead to the best fits.

The reviews for this paper were borderline, with scores of 3, 6, and 8. On the positive side, the reviewers agreed that the paper is very effective in demonstrating that the input-output behaviour of the biophysical model of C elegans can be replicated by RNNs. But, on the negative side there were concerns about the limited nature of the empirical results, lack of details about the simulation, too much emphasis in describing well-known RNN architectures, and lack of systematic strategy for applying this technique in other systems. The rebuttals did not change the borderline scores.

Thus, this is an instance where the AC must be a bit more involved in the decision. After reading the paper and reviews, the AC felt that this work was not sufficiently general in its application. Ultimately, using artificial neural networks to fit neural data is common practice nowadays, so really, this paper serves as a proof-of-concept for replacing a complex biophysical model with a simpler RNN. But, given that RNNs are quite good at modelling sequence data, it's not terribly surprising that this works. Moreover, though the authors do a very careful search over network design decisions, they don't provide a systematic strategy for others to employ if they so wished. Also, the authors do not provide much insight into what the RNNs learn that might help us to better understand the modelled neural circuits. And most importantly, this only demonstrates the effectiveness for systems where we have biophysical models with well-established accuracy, which is not the case for most neural circuits. Given these considerations, a reject decision was reached.
Decision: Reject