Unsupervised learning of control signals and their encodings in Caenorhabditis elegans whole-brain recordings

Abstract

A major goal of computational neuroscience is to understand the relationship between synapse-level structure and network-level functionality. Caenorhabditis elegans is a model organism to probe this relationship due to the historic availability of the synaptic structure (connectome) and recent advances in whole brain calcium imaging techniques. Recent work has applied the concept of network controllability to neuronal networks, discovering some neurons that are able to drive the network to a certain state. However, previous work uses a linear model of the network dynamics, and it is unclear if the real neuronal network conforms to this assumption. Here, we propose a method to build a global, low-dimensional model of the dynamics, whereby an underlying global linear dynamical system is actuated by temporally sparse control signals. A key novelty of this method is discovering candidate control signals that the network uses to control itself. We analyse these control signals in two ways, showing they are interpretable and biologically plausible. First, these control signals are associated with transitions between behaviours, which were previously annotated via expert-generated features. Second, these signals can be predicted both from neurons previously implicated in behavioural transitions but also additional neurons previously unassociated with these behaviours. The proposed mathematical framework is generic and can be generalized to other neurosensory systems, potentially revealing transitions and their encodings in a completely unsupervised way.

1. Introduction

The nematode Caenorhabditis elegans (C. elegans) is an ideal model organism for probing the relationship between structure and function in neuronal networks as it is comprised of only 302 sensory, motor and inter-neurons whose stereotyped synaptic connections (i.e. its connectome) are known from serial section electron microscopy [1–3]. Indeed, C. elegans is perhaps the simplest organism to display many of the hallmark features of high-dimensional networked biological systems, including the manifestation of low-dimensional patterns of activity associated with functional behavioural responses [4]. Thus, the nervous system must reduce the high-dimensional representation of environmental stimuli into much lower-dimensional representations of motor commands [4–10]. Low-dimensional representations have been separately considered in posture (behavioural) analysis [11,12] as well as in previous analysis of calcium imaging data [4,13]. These representations can be used to characterize the evolution of both postures [12] and neuron population dynamics [14,15]. In this work, we exploit emerging whole-brain imaging recordings to posit a data-driven model of neurosensory integration in C. elegans, showing that a global linear framework, with the addition of internally generated control signals, explains and reproduces much of the activity of the network.

It has long been observed that C. elegans produces a small number of stable discrete behaviours (e.g. forward and backward motion, and turns), and that these behaviours change both spontaneously [5] and very quickly in response to external stimuli [16–18] or stimulation of even a single neuron [19–21]. A potential dynamical systems explanation for this observation is that of discrete behaviours as fixed points on an underlying manifold with some transition signals that move the system between them [22–24]. A purely linear dynamical system of the form x_k+1 = Ax_k cannot produce the observed multiple fixed points, where x_k are the data at time point k and the matrix A maps the state one step into the future. However, piecewise methods, like switching (hybrid) linear dynamical systems [15,25–27], circumvent this by segmenting the dynamics into patches with different linear dynamics (and thus different, though unique, fixed points) in each patch. An alternative method uses different phase loops and the phase along them to predict behaviour, producing conserved nonlinear dynamics in a special phase space [14]. Recent efforts have also attempted to explicitly model nonlinear neuronal and synaptic dynamics to approximate biophysical models of the nervous system [9,28–33], but this has currently been limited to subsets of neurons or single behaviours. This work instead focuses on how a single, global, linear dynamical system model with simple and interpretable additions can capture the nonlinear dynamics via appropriate framing as a control problem.

We use the recently developed data-driven method of dynamic mode decomposition with control (DMDc) [7,34], demonstrating that nonlinearities are not needed to describe many of the interactions in the system. However, we also extend the classic DMDc algorithm to handle unsupervised learning of control signals (Methods section), discovering that the system produces control signals intrinsically even in the absence of external stimulation. Finally, we study the neurons where these control signals are encoded in the network using sparse variable selection methods and the novel ‘elimination pathway’ of the encoding (see Methods). This novel method is used to remove experimentally known neurons from the sparse selection set, and we find that previously unknown neurons still successfully encode the control signals.

We provide code written in MATLAB [35] for a full analysis pipeline that uses raw data and, if available, external behavioural labels to discover the intrinsic dynamics, the effects of control on the state of the system and the encoding of the control signals.

2. Methods

Two whole-brain imaging datasets are used in this paper, one with no sensory stimulus [4] and one with oxygen modulation [13]. Our analysis relies on two established mathematical methods: DMDc and sparse optimization. A brief summary of each is given below.

2.1. Dynamic mode decomposition with control

Our data-driven strategy is based upon the dynamic mode decomposition (DMD). DMD provides a linear model for the dynamics of the state space x_j = x(t_j). Specifically, it finds the best-fit linear model:

$${\mathbf{X}}^{\prime }=\mathbf{A}\mathbf{X},$$ 2.1

where $\mathbf{X}=[{\mathbf{x}}_1\hspace{0.17em}{\mathbf{x}}_2\hspace{0.17em}\dots \hspace{0.17em}{\mathbf{x}}_{m-1}]$ and ${\mathbf{X}}^{\prime }=[{\mathbf{x}}_2\hspace{0.17em}{\mathbf{x}}_3\hspace{0.17em}\dots \hspace{0.17em}{\mathbf{x}}_m]$ are temporal snapshots of the system that are offset by one time step. There are a number of variants for computing A [7], with the exact DMD simply positing $\mathbf{A}={\mathbf{X}}^{\prime }{\mathbf{X}}^{†}$ where $†$ denotes the Moore–Penrose pseudo-inverse.

DMDc [34] capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and actuation signal u_j = u(t_j). For a matrix of input signals $\mathbf{U}=[{\mathbf{u}}_1\hspace{0.17em}{\mathbf{u}}_2\hspace{0.17em}\dots \hspace{0.17em}{\mathbf{u}}_{m-1}]$, DMDc regresses instead to the linear control system

$${\mathbf{X}}^{\prime }=\mathbf{A}\mathbf{X}+\mathbf{B}\mathbf{U}.$$ 2.2

Note that DMDc uses only snapshots in time of the state space and control input, making it compelling for systems whose governing equations are unknown. The DMDc equation is graphically represented in figure 1. The governing matrices (A and B) along with the control signal (U) produce a predictive model, such that the state of the system far in the future can be predicted. For instance, the third time step can be estimated from the first via

$${\mathbf{x}}_3=\mathbf{A}(\mathbf{A}{\mathbf{x}}_1+\mathbf{B}{\mathbf{u}}_1)+\mathbf{B}{\mathbf{u}}_2.$$ 2.3

Figure 1.

A 3-step framework for modelling neurosensory integration. Top: (1) Transition signals are learned from data with an assumption of linear dynamics. (2) A DMDc model is learned which uses dynamics, transition signals and actuation. These are global models, and are capable of reconstructing much of the data dynamically from an initial state. (3) Where and at what time scales control signals are encoded in the neural activity are studied using sparse linear models, where sparsity is now in neuron space. Bottom: the sensory–computation–behaviour pathway, a graphical description of the above mathematical steps. Each term in the above equations can be freely translated into this biological process. Transitions (green) actuate neurons via their own connectivity (yellow). Neuron traces (blue) evolve according to intrinsic dynamics (red), and also encode (light blue) the transition signals (green).

2.2. Variable selection via sparse linear models

If internally generated control signals are present, then there are two possibilities: they are random and fundamentally unpredictable, or they are encoded in the network. We explore the degree to which these signals are encoded using sparse variable selection algorithms and time-delay embedding, where data from further in the past are used. Note that sparsity here is in neuron space. Mathematically, each time step of the control signal is written as

$${\mathbf{u}}_{k+1}={\mathbf{K}}_1{\mathbf{x}}_k+{\mathbf{K}}_2{\mathbf{x}}_{k-1}+\cdots .$$ 2.4

There are multiple methods that are often used to perform this variable selection task [36]. However, these methods may make mistakes in their selections [37], and in general it is unclear how unique the selection is. The behaviours of C. elegans have been well studied, and each onset is associated with well-known neurons. Variable selection methods will almost certainly discover these well-known neurons, but by exploring further in the ‘elimination path’, less obvious encodings can be discovered. Algorithmically, this is the sequential removal or simulated ablation of the most important neuron for all time delays, and then a re-fitting of the sparse model in this way:

$${\mathbf{u}}_{k+1}={\tilde{\mathbf{K}}}_1{\tilde{\mathbf{x}}}_k+{\tilde{\mathbf{K}}}_2{\tilde{\mathbf{x}}}_{k-1}+\cdots ,$$ 2.5

where $\tilde{\mathbf{x}}$ and $\tilde{\mathbf{K}}$ have a subset of the original rows, but U is the same. The quality of signal reconstruction is quantified here as the number of false positives and false negatives in the reconstructed signal. Event detection is defined as a minimum number of frames above a hard threshold, as shown in figure 5 and discussed in the electronic supplementary material. Note that this algorithm does not produce optimal solutions, but rather explores the space of sparse solutions which may have different experimental interpretations.

Figure 5.

The control signals are encoded in the neuron population data. (a,b) Using equation (2.4), control signals (U) reconstructed via linear encoding (K_i) of the data (X_i) including time delays, with all neurons (a) or four neurons removed (b). The removed neurons are: SMDDR, 81, SMDVL and SMDVR as shown on the x-axis of (c). Event detection is determined via a hard threshold for each signal (dotted line). See electronic supplementary material, section III, for more discussion of this threshold. (c) Neurons are eliminated in order of the largest magnitude given to them by the linear model. The number of false detections increases significantly only after eight neurons have been removed. (d,e) The weights given to the top 10 most important neurons for different iterations.

2.3. Learning control signals via sparse optimization

The DMDc algorithm requires knowledge of the control signals, U. Expert-identified state labels and an example neuron that displays strong state-dependent behaviour are shown in figure 2. However, these are only available because of the decades of C. elegans experimental work identifying (1) discrete behavioural states and (2) the command neurons for each activity. For new organisms, and in order to generate hypotheses about potential new states in C. elegans, the unsupervised problem, i.e. learning the signal directly from data, is of critical interest. In the related problem of compressed sensing, there are no dynamics (A = 0) and a sparse signal U can be recovered if certain conditions are satisfied [38]. However, these data contain plateaus and slow exponential decays, which signals that are sparse in time would not recover without dynamics.

Figure 2.

Transition signals in C. elegans. Top: a calcium imaging trace of a neuron connected with the discrete reversal behaviour. Behavioural labels and neuron identities are determined by experimentalists, as described in [4]. Green, forward; yellow, reversal; dark blue, ventral turn; light blue, dorsal turn. Below: these labels can be reframed as ‘onset’ signals, and are characteristically sparse in time. See electronic supplementary material, 3.6, for details on how these are generated from behavioural labels.

DMDc (2.2) can be thought of as an error minimization problem over the dynamics and actuation matrices, A and B. If the control signal is unknown, the minimization must be extended to the control signal U itself. However, there is now a trivial solution where the control signal dominates the model: X₂ = BU with A = 0. For this reason, an assumption must be made about the control signals. In this case, the statement that these signals are sparse is directly biologically interpretable, and means that the transitions between states should be rare as a percentage of frames. This ‘sparsity constraint’ can be expressed in a mathematically precise way using the ℓ₀ norm:

$$\underset{\mathbf{A},\mathbf{B},\mathbf{U}}{min}[||\mathbf{A}{\mathbf{X}}_1+\mathbf{B}\mathbf{U}-{\mathbf{X}}_2|{|}_2+\lambda ||\mathbf{U}|{|}_0].$$ 2.6

Directly solving this optimization problem is extremely difficult, although there are efficient algorithms in certain cases [39]. More recently, a convex relaxation of the ℓ₀ to an ℓ₁ norm is often solved [40], though this has been recently shown to lead to errors in its selection pathway [37]. We use a different approximation, the sequential least-squares thresholding algorithm as described in [41], which has been shown to converge to the minima of the original ℓ₀ problem [42,43]. The code is outlined in algorithm 1 and uses a modified form of Akaike information criteria (AIC) [44] to choose the best iteration of the algorithm. This modification reduces the importance of each element of the matrix U, because each entry is not a ‘global parameter’ as in the original derivation of AIC; more information and derivation is given in the electronic supplementary material. The matrix U in this algorithm is additionally constrained to be positive, for better interpretability as ‘on’ transition signals. Note that the elements of B can be negative, and thus the control signals may inhibit or excite affected neurons.

3. Results

3.1. Known transitions are discovered and characterized

Experimentalists have separated neural activity behaviour into discrete categories via thresholding the activity and derivatives of individual neurons [4], and the data this work is based on [4,13] posit a separation into four fundamental non-quiescent behaviours with subtypes: forward, reversals, and dorsal and ventral turns. However, open questions remain about the number of behaviours that exist and how discrete they are. Some works have posited up to six forward motion states and three reversal states, multiple turning subtypes, or even a continuum of behaviours [45,46]. As figure 3 shows, using unsupervised optimization three behavioural onsets can be discovered: reversal, and dorsal and ventral turns. Specifically, algorithm 1 recovers a time series that is highly correlated with the first few frames of experimentalist-labelled behaviours (see electronic supplementary material 1.1 for more details). Figure 3d,e demonstrates consistency across individuals and lends evidence from a dynamics perspective to support the view that these are the dominant behaviours at the neural activity level.

Figure 3.

Control signals (rows of U) can be learned from data via algorithm 1. (a–c) The onset of well-known states as determined by experts (above) and as learned (below) across 5 min (same window as figure 2). All signals are normalized to have a maximum of 1.0 and are thus arbitrary units (arb. units). (d,e) Correlation between expert and learned signals across 15 individual C. elegans, both with (e) and without (d) external stimulus. Oxygen stimulation protocol described in [13]. This stimulation paradigm causes a large number of sequential reversals that are less well separated than non-stimulated datasets, possibly contributing to the decreased reconstruction accuracy. Reversals (Rev), dorsal (DT) and ventral turns (VT) are consistently learned, but forward state (Fwd) onsets are never significant, as discussed in section II of the electronic supplementary material.

However, in no individuals could a signal correlated to the onset of forward motion be discovered. If this were a trivial state that displays no activity, a simple decay to a fixed point following a turn state would be sufficient to achieve a good reconstruction of the trajectory, even without an onset signal. However, fast-scale behaviours are known experimentally to occur within this state [47]. Thus, although there is activity, it falls outside the dynamical specification and sparsity assumptions of equation (2.6). Our results imply that for the onset of these behaviours, forward motion is more complex than reversals, meaning that it cannot be described as a simple ‘on’ signal.

3.2. A global, linear system with control reconstructs entire time series

The majority of the variance of C. elegans neural dynamics can be described via a few PCA modes, and the first two for a single individual are shown in figure 4a. These dynamics cannot be described by a purely linear model due to the presence of multiple stable global behaviours, as shown in figure 4b. Specifically, linear models can only admit a single fixed state. However, the majority of neurons can be reconstructed using our controlled, global, linear dynamical system due to the sparse transition signals as shown in figure 4c for expert hand-labelled signals and figure 4d for signals learned from data. Each time snapshot of these data is reconstructed analogously to equation (2.3), and then projected onto the two dominant PCA modes of the original data so that each panel in figure 4a–d is in the same coordinate space. This comparison is only possible due to identification of neurons with stereotyped identities, as described in [4]. Because this is a global linear model that uses a single framework for the entire state space, the need for additional nonlinear modelling can be constrained to particular groups of neurons and well-defined time windows.

Figure 4.

A global model reconstructs the dominant structure of the data using only the initial condition and control signals. Two-dimensional PCA projections of (a) data, (b) an uncontrolled ‘null’ model, (c) a ‘supervised’ model using expert-determined control signals, and (d) an ‘unsupervised’ model that uses control signals learned via algorithm 1. The governing equations matrices are all learned from data, either uncontrolled (b, equation (2.1)) or controlled (c,d, equation (2.2)). These data are colour-coded by state: black for unknown, yellow for reversal neurons, green for forward and light (dark) blue for dorsal (ventral) turns. (e,f) Example neuron datasets (blue) with reconstructions (orange, via equation (2.3)) from the supervised model. A reversal-active (AVAL) and a dorsal-turn-active neuron (SMDDL) are shown. (g,h) The same neurons shown with reconstructions from the unsupervised model. (i–k) Correlations across datasets between data and reconstructions, split up into four different neuron groupings for interpretability. (i) The left is a straight line fit; the right-hand side is the uncontrolled model in panel (b). (j) Full models with either expert/supervised control signals, (c) above, or learned/unsupervised control signals, (d) above. For each neuron grouping the expert signals produce significantly better fits than the null models in (i). (k) Partial supervised models, as more signals are added. For the reversal (left-hand side) set, a ‘baseline’ of a straight-line fit is subtracted. Shown are additive improvements, i.e. how much better each partial model is than the one immediately to the left.

In particular, across individuals the reversal class of neurons is captured very well by the supervised control signal as shown in figure 4j and thus, up to encoding the transition signal itself, the relevant subnetwork does not appear to require nonlinearities. This means that future efforts related to nonlinear modelling may be most productive if they concentrate on the small window of time during the onset of the behaviour, instead of the entire neural trace where linear models are sufficient. In addition, the type of nonlinearity required to more fully model this class of neurons is characterized: fast and short-lived spike-like activations.

Turns are also largely captured, as shown by the high correlation for the light and dark blue boxplots in figure 4j. The neurons involved in turning have a large number of smaller events, as shown in the reconstruction of the neuron SMDDL in figure 4f; these do not lead to one of the four state transitions identified by experimentalists in this dataset, but may correspond to recently described fast time-scale states [47]. However, the unsupervised method does pick up on these smaller events and reconstructs them well as in figure 4h, but over all datasets there is much more variability as shown in figure 4j.

The last group of neurons, those related to forward motion, has a very large variability of correlation between the data and reconstructions. This implies that these neurons require non-trivial nonlinearities throughout the time series, not just at the onset, for full reconstruction. Although it is well known that different, dedicated subnetworks of neurons are active in forward and backwards motion [48], the functional implications have not been fully modelled. Some recent experimental work [10] characterizes an asymmetry between forward and reversal states as due to intrinsic bias towards the forward state. In addition and unlike the reversal-active neurons that require only a simple ‘on’ transition signal, the forward-active neurons may be continuously parametrized by speed, or contain additional behaviours like steering [49], tracking [50] or head casting [47]. Moreover, Kaplan et al. show that many neurons exhibit diverse faster time-scale fluctuations particularly during forward states [47]. Our work is consistent with these experimental results, and adds that this complexity is functionally different from that of the reversal or turn states.

To further characterize the effects of the control signals on the ability of this framework to capture the neural dynamics, partial models were created with a subset of control signals. Partial models using cumulative subsets of the expert-labelled control signals are shown in figure 4k. Adding reversal-onset signals alone does not produce a model that captures the data better than a straight-line fit to the data, but the combination of reversal and turning signals is significantly better. However, subsequent addition of forward control signals is, remarkably, useless. In summary, there are several related functional observations that further work may connect to physical differences in the reversal and forward neuronal subnetworks: the lack of discovery of sparse forward onset signals, which is corroborated by the ineffectiveness of the experimentally known onset times, and the poor reconstruction of forward-related neurons using linear dynamics.

3.3. Transitions are encoded in previously unidentified neurons

Having shown the control signals to contribute significantly to the reconstruction of the data, we reconstruct the control signals themselves as shown in step 3 of figure 1. As described in [4,13], each of the four interpretable transition signals shown in figure 3 are hand-labelled using the activity of certain well-known neurons. Thus, it is not surprising that these signals can be reconstructed from data when those well-known neurons are included. In particular, as they were used to define the dorsal turn behavioural states, like SMDVL/R which define ventral turns [47], an excellent validation of this sparse selection method is that the SMDDL/R and SMDVL/R pairs of left/right neurons consistently encodes this control signal, as figure 5a shows.

Mathematically, this encoding uses time-delay data matrices and sparse linear models according to equation (2.4). In this case, U is the expert-labelled behavioural transitions because they most accurately reconstruct the data, as described in figure 4h–j. However, as the elimination path (equation (2.5)) is explored further, it is revealed that these well-known neurons can be eliminated from the sparse models and the transition signals can still be reconstructed as shown in figure 5b. Indeed, figures 5a and 5b look nearly identical, and figure 5c quantifies this using the percentage of false positives and negatives. Figure 5c also shows more of the elimination path and when the reconstructions eventually break down. Figure 5d,e shows the how K matrices in equation (2.4) change as neurons are removed. Taken together, these results reveal previously unidentified neurons that can successfully predict control signals, which are themselves shown to be important to reconstructing the full neural manifold. However, only rows with names are neurons that have been connected to the stereotyped C. elegans connectome and can thus be identified across individuals; rows with numbers cannot be so compared. In summary, this work identifies sets of neurons previously implicated in transitions and also reveals new potential candidates for actuators of behavioural transitions. In addition, the time of encoding is revealed, which can inform further study.

4. Discussion

We have presented the first data-driven model that uses a single set of linear intrinsic dynamics that can reconstruct the multiple behavioural regimes present in a real animal and transitions between them. In this study, we have analysed neuronal recordings from C. elegans that lacked any acutely delivered and time-varying sensory inputs, therefore behavioural transitions are likely internally driven [4] and governed by stochastic processes [5]. Thus, control signals cannot be inferred simply from sensory neuron activity. To overcome this challenge, we provide an unsupervised approach for identifying such internally driven control signals and their underlying neuronal identities. The fact that this deterministic, controlled, linear model accurately reproduces both short- and long-time-scale dynamics has two implications: first, it places clear restrictions on the need, specifically the lack thereof, for nonlinear or stochastic elements, in this system and provides hypotheses about neurons that may contain nonlinearities (i.e. poorly reconstructed sets). However, it is important to note that in general, the presence of poorly reconstructed subsets might destabilize an entire model. We hypothesize that this did not happen in this case because the activity of the poorly reconstructed neurons is largely mutually exclusive with the well-reconstructed subsets. Second, the control signals can be studied as objects in their own right, producing candidates for their encoding and a new simplified representation of this high-dimensional dataset. In future work, this framework can be generalized to other complex datasets or to include hypothesized nonlinearities.

Much excitement has been generated by the availability of the C. elegans anatomical connectome, and one aim of data-driven modelling efforts is to produce a functional connectome that can complement the anatomical data. The DMDc algorithm in this paper is similar to several algorithms in the engineering literature that attempt similar network reconstruction tasks, namely system identification [51]. One strategy to fully disambiguate the effects of the intrinsic dynamics and the external control signals uses known external perturbations should be applied and the system response measured. Such perturbations are not generally available in biological systems and thus the data collected are ‘uninformative’ [52] in the sense that the underlying structure cannot be determined. In the same way as the dynamics matrix A, the sparse variable selection methods used to determine candidate neurons that encode these transitions are associative and do not reveal a causal connection. However, follow-up experimental work can take these identified sets of candidates and, through perturbative studies, determine such causal connections.

A limitation of this model is that it is not generative; it cannot be used to predict a system response that includes transitions to novel stimuli. To accomplish this, the transition signals must be written as a function of the data. Step three of our method does this with a linear encoding and demonstrates that the signals can be successfully reconstructed with all neurons to a certain level of accuracy. If this level of accuracy were sufficient, then the system would be fully linear and an uncontrolled model would produce a good reconstruction, as is clearly not the case. Recent methods for incorporating nonlinearities into controlled systems (e.g. [53,54]) have the potential to create a fully closed-loop feedback system and this is an active area of further research [55].

This methodology uses two key hypotheses about this system. First, although the true biological system includes many thousands of nonlinear interactions, we assume that the leading order or dominant balance dynamics of the dynamical system within certain regimes is simple. Recent work in the fluids community [56] has shown that even when the full global model is perfectly known, almost every region of phase space is well described by a simpler model with fewer or no nonlinear terms. In the same way, this paper posits that a single set of linear dynamics capture the dominant activity of large regions of phase space. The second observation is of a time-scale separation between activity within a state, and transitions between states. This leads to a hypothesis about when complex nonlinearities may be active: sparsely, during transitions between behaviours. Thus, this methodology is directly applicable when trajectories follow simplified, linear dynamics produced by intrinsic neural dynamics in large regions of phase space, with short periods of complex dynamics. An area of future work is to explore how these control signals could be produced biologically, and a strength of this method is that the control signals may be well modelled by an intermediate spike-type thresholding of a more complex signal.

A potential criticism of this method is that we have used discrete labelled states in our model, despite ongoing debate regarding how uniform ‘states’ in C. elegans are across instances, and if they should be subdivided or are simply continuous [45,46,57]. We have contributed to this debate by providing evidence that the reversal and turn states in fact appear to be simple and have well-defined initiation signals. However, the forward ‘state’ is much more complex, and breaks the assumptions of our model. Specifically, the intrinsic dynamics may be different in the forward state as compared to the rest of the phase space, and may be a different linear system as posited in [15] or nonlinear as in [55]. Related, the ‘transition’ into this state may not follow the sparsity assumptions of algorithm 1, perhaps due to a continuum of states as opposed to a discrete transition. Moreover, Kaplan et al. [47] show that during the forward state many neurons fluctuate at faster time scales, contributing to this complexity. We argue that this is a strength of this methodology: because this is not a method that can universally approximate arbitrary dynamics, the fact that a state and its transition cannot be reconstructed gives additional information about that state, and about its complexity in relation to other states. However, it is conceivable that failure to identify control signals during the forward state, and subsequent low reconstruction quality, is simply due to lack of sufficient data. As more neurons are imaged or longer recording times become experimentally feasible, so far undiscovered control signals and neuronal candidates during the forward state may be revealed.

An alternative approach to modelling complex systems in order to understand structure is to use locally linear models [15,25–27]. In this methodology, the initial network as described by the matrix ${\mathbf{A}}_{\mathbf{k}}$ is replaced by a new matrix, ${\mathbf{A}}_{\mathbf{k}\mathbf{+}\mathbf{1}}$, at certain change points. These have achieved great success in reconstructing nonlinear datasets, and in fact can reconstruct arbitrary dynamics given enough change points, and is an active field in machine learning research. However, it is difficult to interpret what such a replacement of the underlying dynamics would mean biologically, particularly if many separate matrices ${\mathbf{A}}_{\mathbf{k}}$ are required. On the other hand, the language of control theory from engineering meshes directly with the biological intuition that certain states are initiated by relatively unique signals produced by a small number of neurons. We propose that our framework for constructing a single, global model of the dynamics of this neural system is promising not only in its ready generalizability to include nonlinearities, but also in its biological interpretability.

We have produced the first, to our knowledge, global data-driven model of both the intrinsic and control dynamics of C. elegans.

Here, we provide a framework for the identification of neurons critical in actuating network transitions, which can be tested in future experiments.

Data accessibility

Data used in this paper are available at https://osf.io/a64uz/ and are associated with two previous experimental papers [4,13]. All code used in this paper is written in MATLAB, and is available in a user-friendly package at https://github.com/Charles-Fieseler/Learning_Control_Signals_MATLAB [58].

Authors' contributions

C.F. conceived of the study, drafted the manuscript and wrote the data analysis pipeline for the project. J.N.K. helped conceive of the study, provided funding, and gave guidance for the theoretical and mathematical issues and context. M.Z. provided the experimental data and gave feedback on biological significance. All authors contributed critical feedback and edits.

Competing interests

We declare we have no competing interests.

Funding

We acknowledge support from the Air Force Office of Scientific Research MURI 1FA9550-19-1-0386.