Caenorhabditis elegans and the network control framework-FAQs

A network control theory pipeline for studying the dynamics of the structural connectome

Network control theory (NCT) is a simple and powerful tool for studying how network topology informs and constrains the dynamics of a system. Compared to other structure-function coupling approaches, the strength of NCT lies in its capacity to predict the patterns of external control signals that may alter the dynamics of a system in a desired way. An interesting development for NCT in the neuroscience field is its application to study behavior and mental health symptoms. To date, NCT has been validated to study different aspects of the human structural connectome. NCT outputs can be monitored throughout developmental stages to study the effects of connectome topology on neural dynamics and, separately, to test the coherence of empirical datasets with brain function and stimulation. Here, we provide a comprehensive pipeline for applying NCT to structural connectomes by following two procedures. The main procedure focuses on computing the control energy associated with the transitions between specific neural activity states. The second procedure focuses on computing average controllability, which indexes nodes' general capacity to control the dynamics of the system. We provide recommendations for comparing NCT outputs against null network models, and we further support this approach with a Python-based software package called 'network control theory for python'. The procedures in this protocol are appropriate for users with a background in network neuroscience and experience in dynamical systems theory.

Neural network architecture of a mammalian brain

Connectomics research is making rapid advances, although models revealing general principles of connectional architecture are far from complete. Our analysis of 106 published connection reports indicates that the adult rat brain interregional connectome has about 76,940 of a possible 623,310 axonal connections between its 790 gray matter regions mapped in a reference atlas, equating to a network density of 12.3%. We examined the sexually dimorphic network using multiresolution consensus clustering that generated a nested hierarchy of interconnected modules/subsystems with three first-order modules and 157 terminal modules in females. Top-down hierarchy analysis suggests a mirror-image primary module pair in the central nervous system's rostral sector (forebrain-midbrain) associated with behavior control, and a single primary module in the intermediate sector (rhombicbrain) associated with behavior execution; the implications of these results are considered in relation to brain development and evolution. Bottom-up hierarchy analysis reveals known and unfamiliar modules suggesting strong experimentally testable hypotheses. Global network analyses indicate that all hubs are in the rostral module pair, a rich club extends through all three primary modules, and the network exhibits small-world attributes. Simulated lesions of all regions individually enabled ranking their impact on global network organization, and the visual path from the retina was used as a specific example, including the effects of cyclic connection weight changes from the endogenous circadian rhythm generator, suprachiasmatic nucleus. This study elucidates principles of interregional neuronal network architecture for a mammalian brain and suggests a strategy for modeling dynamic structural connectivity.

Innate Orientating Behavior of a Multi-Legged Robot Driven by the Neural Circuits of C. elegans

The objective of this research is to achieve biologically autonomous control by utilizing a whole-brain network model, drawing inspiration from biological neural networks to enhance the development of bionic intelligence. Here, we constructed a whole-brain neural network model of Caenorhabditis elegans (C. elegans), which characterizes the electrochemical processes at the level of the cellular synapses. The neural network simulation integrates computational programming and the visualization of the neurons and synapse connections of C. elegans, containing the specific controllable circuits and their dynamic characteristics. To illustrate the biological neural network (BNN)'s particular intelligent control capability, we introduced an innovative methodology for applying the BNN model to a 12-legged robot's movement control. Two methods were designed, one involving orientation control and the other involving locomotion generation, to demonstrate the intelligent control performance of the BNN. Both the simulation and experimental results indicate that the robot exhibits more autonomy and a more intelligent movement performance under BNN control. The systematic approach of employing the whole-brain BNN for robot control provides biomimetic research with a framework that has been substantiated by innovative methodologies and validated through the observed positive outcomes. This method is established as follows: (1) two integrated dynamic models of the C. elegans' whole-brain network and the robot moving dynamics are built, and all of the controllable circuits are discovered and verified; (2) real-time communication is achieved between the BNN model and the robot's dynamical model, both in the simulation and the experiments, including applicable encoding and decoding algorithms, facilitating their collaborative operation; (3) the designed mechanisms using the BNN model to control the robot are shown to be effective through numerical and experimental tests, focusing on 'foraging' behavior control and locomotion control.

Fibration symmetries and cluster synchronization in the Caenorhabditis elegans connectome

Capturing how the Caenorhabditis elegans connectome structure gives rise to its neuron functionality remains unclear. It is through fiber symmetries found in its neuronal connectivity that synchronization of a group of neurons can be determined. To understand these we investigate graph symmetries and search for such in the symmetrized versions of the forward and backward locomotive sub-networks of the Caenorhabditi elegans worm neuron network. The use of ordinarily differential equations simulations admissible to these graphs are used to validate the predictions of these fiber symmetries and are compared to the more restrictive orbit symmetries. Additionally fibration symmetries are used to decompose these graphs into their fundamental building blocks which reveal units formed by nested loops or multilayered fibers. It is found that fiber symmetries of the connectome can accurately predict neuronal synchronization even under not idealized connectivity as long as the dynamics are within stable regimes of simulations.

Fibration symmetries and cluster synchronization in the Caenorhabditis elegans connectome

Capturing how the Caenorhabditis elegans connectome structure gives rise to its neuron functionality remains unclear. It is through fiber symmetries found in its neuronal connectivity that synchronization of a group of neurons can be determined. To understand these we investigate graph symmetries and search for such in the symmetrized versions of the forward and backward locomotive sub-networks of the Caenorhabditi elegans worm neuron network. The use of ordinarily differential equations simulations admissible to these graphs are used to validate the predictions of these fiber symmetries and are compared to the more restrictive orbit symmetries. Additionally fibration symmetries are used to decompose these graphs into their fundamental building blocks which reveal units formed by nested loops or multilayered fibers. It is found that fiber symmetries of the connectome can accurately predict neuronal synchronization even under not idealized connectivity as long as the dynamics are within stable regimes of simulations.

Macroscopic resting-state brain dynamics are best described by linear models

It is typically assumed that large networks of neurons exhibit a large repertoire of nonlinear behaviours. Here we challenge this assumption by leveraging mathematical models derived from measurements of local field potentials via intracranial electroencephalography and of whole-brain blood-oxygen-level-dependent brain activity via functional magnetic resonance imaging. We used state-of-the-art linear and nonlinear families of models to describe spontaneous resting-state activity of 700 participants in the Human Connectome Project and 122 participants in the Restoring Active Memory project. We found that linear autoregressive models provide the best fit across both data types and three performance metrics: predictive power, computational complexity and the extent of the residual dynamics unexplained by the model. To explain this observation, we show that microscopic nonlinear dynamics can be counteracted or masked by four factors associated with macroscopic dynamics: averaging over space and over time, which are inherent to aggregated macroscopic brain activity, and observation noise and limited data samples, which stem from technological limitations. We therefore argue that easier-to-interpret linear models can faithfully describe macroscopic brain dynamics during resting-state conditions.

Using network control theory to study the dynamics of the structural connectome

Network control theory (NCT) is a simple and powerful tool for studying how network topology informs and constrains dynamics. Compared to other structure-function coupling approaches, the strength of NCT lies in its capacity to predict the patterns of external control signals that may alter dynamics in a desired way. We have extensively developed and validated the application of NCT to the human structural connectome. Through these efforts, we have studied (i) how different aspects of connectome topology affect neural dynamics, (ii) whether NCT outputs cohere with empirical data on brain function and stimulation, and (iii) how NCT outputs vary across development and correlate with behavior and mental health symptoms. In this protocol, we introduce a framework for applying NCT to structural connectomes following two main pathways. Our primary pathway focuses on computing the control energy associated with transitioning between specific neural activity states. Our second pathway focuses on computing average controllability, which indexes nodes' general capacity to control dynamics. We also provide recommendations for comparing NCT outputs against null network models. Finally, we support this protocol with a Python-based software package called network control theory for python (nctpy).

Change of motifs in C. elegans reveals developmental principle of neural network

Revealing the organizing principles of developing neural networks is a difficult but significant task in neuroscience. As a creature with a rather compact and well-studied neural network, C. elegans is an ideal subject for neuroscience study. However, the researches on its developing neural network remain challenging. The changes in specific properties of neural network across development may uncover part of its principles. Motif is a typical structure property that can be well applied to various complex networks. Here, we study the motif changes in C. elegans neural network across development. By counting the occurrence number of all three-node subgraph motif structures in its neural network at different stages of C. elegans development, along with those in corresponding random networks, we determine which of these structures are motifs for C. elegans, finding out the regular changes of motifs during its development. Combined with the potential function of these subgraph motifs and synaptic information, we gain insight into the organizing principle of neural network during development, which may increase our understanding of neuroscience and inspire the construction of artificial neural network.

The expanding horizons of network neuroscience: From description to prediction and control

The field of network neuroscience has emerged as a natural framework for the study of the brain and has been increasingly applied across divergent problems in neuroscience. From a disciplinary perspective, network neuroscience originally emerged as a formal integration of graph theory (from mathematics) and neuroscience (from biology). This early integration afforded marked utility in describing the interconnected nature of neural units, both structurally and functionally, and underscored the relevance of that interconnection for cognition and behavior. But since its inception, the field has not remained static in its methodological composition. Instead, it has grown to use increasingly advanced graph-theoretic tools and to bring in several other disciplinary perspectives-including machine learning and systems engineering-that have proven complementary. In doing so, the problem space amenable to the discipline has expanded markedly. In this review, we discuss three distinct flavors of investigation in state-of-the-art network neuroscience: (i) descriptive network neuroscience, (ii) predictive network neuroscience, and (iii) a perturbative network neuroscience that draws on recent advances in network control theory. In considering each area, we provide a brief summary of the approaches, discuss the nature of the insights obtained, and highlight future directions.

Network controllability mediates the relationship between rigid structure and flexible dynamics

Precisely how the anatomical structure of the brain supports a wide range of complex functions remains a question of marked importance in both basic and clinical neuroscience. Progress has been hampered by the lack of theoretical frameworks explaining how a structural network of relatively rigid interareal connections can produce a diverse repertoire of functional neural dynamics. Here, we address this gap by positing that the brain's structural network architecture determines the set of accessible functional connectivity patterns according to predictions of network control theory. In a large developmental cohort of 823 youths aged 8 to 23 years, we found that the flexibility of a brain region's functional connectivity was positively correlated with the proportion of its structural links extending to different cognitive systems. Notably, this relationship was mediated by nodes' boundary controllability, suggesting that a region's strategic location on the boundaries of modules may underpin the capacity to integrate information across different cognitive processes. Broadly, our study provides a mechanistic framework that illustrates how temporal flexibility observed in functional networks may be mediated by the controllability of the underlying structural connectivity.

Computing Temporal Sequences Associated With Dynamic Patterns on the C. elegans Connectome

Understanding how the structural connectivity and spatial geometry of a network constrains the dynamics it is able to support is an active and open area of research. We simulated the plausible dynamics resulting from the known C. elegans connectome using a recent model and theoretical analysis that computes the dynamics of neurobiological networks by focusing on how local interactions among connected neurons give rise to the global dynamics in an emergent way. We studied the dynamics which resulted from stimulating a chemosensory neuron (ASEL) in a known feeding circuit, both in isolation and embedded in the full connectome. We show that contralateral motorneuron activations in ventral (VB) and dorsal (DB) classes of motorneurons emerged from the simulations, which are qualitatively similar to rhythmic motorneuron firing pattern associated with locomotion of the worm. One interpretation of these results is that there is an inherent-and we propose-purposeful structural wiring to the C. elegans connectome that has evolved to serve specific behavioral functions. To study network signaling pathways responsible for the dynamics we developed an analytic framework that constructs Temporal Sequences (TSeq), time-ordered walks of signals on graphs. We found that only 5% of TSeq are preserved between the isolated feeding network relative to its embedded counterpart. The remaining 95% of signaling pathways computed in the isolated network are not present in the embedded network. This suggests a cautionary note for computational studies of isolated neurobiological circuits and networks.

Modeling brain, symptom, and behavior in the winds of change

Neuropsychopharmacology addresses pressing questions in the study of three intertwined complex systems: the brain, human behavior, and symptoms of illness. The field seeks to understand the perturbations that impinge upon those systems, either driving greater health or illness. In the pursuit of this aim, investigators often perform analyses that make certain assumptions about the nature of the systems that are being perturbed. Those assumptions can be encoded in powerful computational models that serve to bridge the wide gulf between a descriptive analysis and a formal theory of a system's response. Here we review a set of three such models along a continuum of complexity, moving from a local treatment to a network treatment: one commonly applied form of the general linear model, impulse response models, and network control models. For each, we describe the model's basic form, review its use in the field, and provide a frank assessment of its relative strengths and weaknesses. The discussion naturally motivates future efforts to interlink data analysis, computational modeling, and formal theory. Our goal is to inspire practitioners to consider the assumptions implicit in their analytical approach, align those assumptions to the complexity of the systems under study, and take advantage of exciting recent advances in modeling the relations between perturbations and system function.

Network structure of cascading neural systems predicts stimulus propagation and recovery

OBJECTIVE

Many neural systems display spontaneous, spatiotemporal patterns of neural activity that are crucial for information processing. While these cascading patterns presumably arise from the underlying network of synaptic connections between neurons, the precise contribution of the network's local and global connectivity to these patterns and information processing remains largely unknown.

APPROACH

Here, we demonstrate how network structure supports information processing through network dynamics in empirical and simulated spiking neurons using mathematical tools from linear systems theory, network control theory, and information theory.

MAIN RESULTS

In particular, we show that activity, and the information that it contains, travels through cycles in real and simulated networks.

SIGNIFICANCE

Broadly, our results demonstrate how cascading neural networks could contribute to cognitive faculties that require lasting activation of neuronal patterns, such as working memory or attention.

Path-dependent connectivity, not modularity, consistently predicts controllability of structural brain networks

The human brain displays rich communication dynamics that are thought to be particularly well-reflected in its marked community structure. Yet, the precise relationship between community structure in structural brain networks and the communication dynamics that can emerge therefrom is not well understood. In addition to offering insight into the structure-function relationship of networked systems, such an understanding is a critical step toward the ability to manipulate the brain's large-scale dynamical activity in a targeted manner. We investigate the role of community structure in the controllability of structural brain networks. At the region level, we find that certain network measures of community structure are sometimes statistically correlated with measures of linear controllability. However, we then demonstrate that this relationship depends on the distribution of network edge weights. We highlight the complexity of the relationship between community structure and controllability by performing numerical simulations using canonical graph models with varying mesoscale architectures and edge weight distributions. Finally, we demonstrate that weighted subgraph centrality, a measure rooted in the graph spectrum, and which captures higher order graph architecture, is a stronger and more consistent predictor of controllability. Our study contributes to an understanding of how the brain's diverse mesoscale structure supports transient communication dynamics.

Models of communication and control for brain networks: distinctions, convergence, and future outlook

Recent advances in computational models of signal propagation and routing in the human brain have underscored the critical role of white-matter structure. A complementary approach has utilized the framework of network control theory to better understand how white matter constrains the manner in which a region or set of regions can direct or control the activity of other regions. Despite the potential for both of these approaches to enhance our understanding of the role of network structure in brain function, little work has sought to understand the relations between them. Here, we seek to explicitly bridge computational models of communication and principles of network control in a conceptual review of the current literature. By drawing comparisons between communication and control models in terms of the level of abstraction, the dynamical complexity, the dependence on network attributes, and the interplay of multiple spatiotemporal scales, we highlight the convergence of and distinctions between the two frameworks. Based on the understanding of the intertwined nature of communication and control in human brain networks, this work provides an integrative perspective for the field and outlines exciting directions for future work.

A practical guide to methodological considerations in the controllability of structural brain networks

OBJECTIVE

Predicting how the brain can be driven to specific states by means of internal or external control requires a fundamental understanding of the relationship between neural connectivity and activity. Network control theory is a powerful tool from the physical and engineering sciences that can provide insights regarding that relationship; it formalizes the study of how the dynamics of a complex system can arise from its underlying structure of interconnected units.

APPROACH

Given the recent use of network control theory in neuroscience, it is now timely to offer a practical guide to methodological considerations in the controllability of structural brain networks. Here we provide a systematic overview of the framework, examine the impact of modeling choices on frequently studied control metrics, and suggest potentially useful theoretical extensions. We ground our discussions, numerical demonstrations, and theoretical advances in a dataset of high-resolution diffusion imaging with 730 diffusion directions acquired over approximately 1 h of scanning from ten healthy young adults.

MAIN RESULTS

Following a didactic introduction of the theory, we probe how a selection of modeling choices affects four common statistics: average controllability, modal controllability, minimum control energy, and optimal control energy. Next, we extend the current state-of-the-art in two ways: first, by developing an alternative measure of structural connectivity that accounts for radial propagation of activity through abutting tissue, and second, by defining a complementary metric quantifying the complexity of the energy landscape of a system. We close with specific modeling recommendations and a discussion of methodological constraints.

SIGNIFICANCE

Our hope is that this accessible account will inspire the neuroimaging community to more fully exploit the potential of network control theory in tackling pressing questions in cognitive, developmental, and clinical neuroscience.

Synthetic ablations in the C. elegans nervous system

Synthetic lethality, the finding that the simultaneous knockout of two or more individually nonessential genes leads to cell or organism death, has offered a systematic framework to explore cellular function, and also offered therapeutic applications. Yet the concept lacks its parallel in neuroscience—a systematic knowledge base on the role of double or higher order ablations in the functioning of a neural system. Here, we use the framework of network control to systematically predict the effects of ablating neuron pairs and triplets on the gentle touch response. We find that surprisingly small sets of 58 pairs and 46 triplets can reduce muscle controllability in this context, and that these sets are localized in the nervous system in distinct groups. Further, they lead to highly specific experimentally testable predictions about mechanisms of loss of control, and which muscle cells are expected to experience this loss.

“Synthetic lethality” in cell biology is an extreme example of the effects of higher order genetic interactions: The simultaneous knockout of two or more individually nonessential genes leads to cell death. We define a neural analog to this concept in relation to the locomotor response to gentle touch in C. elegans. Two or more neurons are synthetic essential if individually they are not required for this behavior, yet their combination is. We employ a network control approach to systematically assess all pairs and triplets of neurons by their effect on body wall muscle controllability, and find that only surprisingly small sets of neurons are synthetic essential. They are highly localized in the nervous system and predicted to affect control over specific sets of muscles.

No Country for Old Worms: A Systematic Review of the Application of C. elegans to Investigate a Bacterial Source of Environmental Neurotoxicity in Parkinson's Disease

While progress has been made in discerning genetic associations with Parkinson's disease (PD), identifying elusive environmental contributors necessitates the application of unconventional hypotheses and experimental strategies. Here, we provide an overview of studies that we conducted on a neurotoxic metabolite produced by a species of common soil bacteria, Streptomyces venezuelae (S. ven), indicating that the toxicity displayed by this bacterium causes stress in diverse cellular mechanisms, such as the ubiquitin proteasome system and mitochondrial homeostasis. This dysfunction eventually leads to age and dose-dependent neurodegeneration in the nematode Caenorhabditis elegans. Notably, dopaminergic neurons have heightened susceptibility, but all of the neuronal classes eventually degenerate following exposure. Toxicity further extends to human SH-SY5Y cells, which also degenerate following exposure. Additionally, the neurons of nematodes expressing heterologous aggregation-prone proteins display enhanced metabolite vulnerability. These mechanistic analyses collectively reveal a unique metabolomic fingerprint for this bacterially-derived neurotoxin. In considering that epidemiological distinctions in locales influence the incidence of PD, we surveyed soils from diverse regions of Alabama, and found that exposure to ~30% of isolated Streptomyces species caused worm dopaminergic neurons to die. In addition to aging, one of the few established contributors to PD appears to be a rural lifestyle, where exposure to soil on a regular basis might increase the risk of interaction with bacteria producing such toxins. Taken together, these data suggest that a novel toxicant within the Streptomyces genus might represent an environmental contributor to the progressive neurodegeneration that is associated with PD.

Connectome to behaviour: modelling Caenorhabditis elegans at cellular resolution

It has been 30 years since the 'mind of the worm' was published in Philosophical Transactions B (White et al 1986 Phil. Trans. R. Soc. Lond. B314, 1-340). Predicting Caenorhabditis elegans' behaviour from its wiring diagram has been an enduring challenge since then. This special theme issue of Philosophical Transactions B combines research from neuroscientists, physicists, mathematicians and engineers to discuss advances in neural activity imaging, behaviour quantification and multiscale simulations, and how they are bringing the goal of whole-animal modelling at cellular resolution within reach.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.