Potential role of a ventral nerve cord central pattern generator in forward and backward locomotion in Caenorhabditis elegans

Milking a spherical cow: Toy models in neuroscience

There are many different kinds of models, and they play many different roles in the scientific endeavour. Neuroscience, and biology more generally, has understandably tended to emphasise empirical models that are grounded in data and make specific, experimentally testable predictions. Meanwhile, strongly idealised or 'toy' models have played a central role in the theoretical development of other sciences such as physics. In this paper, we examine the nature of toy models and their prospects in neuroscience.

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.

The Conscious Nematode: Exploring Hallmarks of Minimal Phenomenal Consciousness in Caenorhabditis Elegans

While subcellular components of cognition and affectivity that involve the interaction between experience, environment, and physiology -such as learning, trauma, or emotion- are being identified, the physical mechanisms of phenomenal consciousness remain more elusive. We are interested in exploring whether ancient, simpler organisms such as nematodes have minimal consciousness. Is there something that feels like to be a worm? Or are worms blind machines? 'Simpler' models allow us to simultaneously extract data from multiple levels such as slow and fast neural dynamics, structural connectivity, molecular dynamics, behavior, decision making, etc., and thus, to test predictions of the current frameworks in dispute. In the present critical review, we summarize the current models of consciousness in order to reassess in light of the new evidence whether Caenorhabditis elegans, a nematode with a nervous system composed of 302 neurons, has minimal consciousness. We also suggest empirical paths to further advance consciousness research using C. elegans.

Emergence of In Vitro Preparations and Their Contribution to Understanding the Neural Control of Behavior in Vertebrates

One of the longstanding goals of the field of neuroscience is to understand the neural control of behavior in both invertebrate and vertebrate species. A series of early discoveries showed that certain motor patterns like locomotion could be generated by neuronal circuits without sensory feedback or descending control systems. These were called fictitious, or "fictive", motor programs because they could be expressed by neurons in the absence of movement. This finding lead investigators to isolate central nervous system tissue and maintain it in a dish in vitro to better study mechanisms of motor pattern generation. A period of rapid development of in vitro preparations from invertebrate species that could generate fictive motor programs from the activity of central pattern generating circuits (CPGs) emerged that was gradually followed by the introduction of such preparations from vertebrates. Here, I will review some of the notable in vitropreparations from both mammalian and non-mammalian vertebrate species developed to study the neural circuits underlying a variety of complex behaviors. This approach has been instrumental in delineating not only the cellular substrates underlying locomotion, respiration, scratching, and other behaviors, but also mechanisms underlying the modifiability of motor pathways through synaptic plasticity. In vitro preparations have had a significant impact on the field of motor systems neuroscience and the expansion of our understanding of how nervous systems control behavior. The field is ready for further advancement of this approach to explore neural substrates for variations in behavior generated by social and seasonal context, and the environment.

Codimension-2 parameter space structure of continuous-time recurrent neural networks

If we are ever to move beyond the study of isolated special cases in theoretical neuroscience, we need to develop more general theories of neural circuits over a given neural model. The present paper considers this challenge in the context of continuous-time recurrent neural networks (CTRNNs), a simple but dynamically universal model that has been widely utilized in both computational neuroscience and neural networks. Here, we extend previous work on the parameter space structure of codimension-1 local bifurcations in CTRNNs to include codimension-2 local bifurcation manifolds. Specifically, we derive the necessary conditions for all generic local codimension-2 bifurcations for general CTRNNs, specialize these conditions to circuits containing from one to four neurons, illustrate in full detail the application of these conditions to example circuits, derive closed-form expressions for these bifurcation manifolds where possible, and demonstrate how this analysis allows us to find and trace several global codimension-1 bifurcation manifolds that originate from the codimension-2 bifurcations.

Forward and backward locomotion patterns in C. elegans generated by a connectome-based model simulation

Caenorhabditis elegans (C. elegans) can produce various motion patterns despite having only 69 motor neurons and 95 muscle cells. Previous studies successfully elucidate the connectome and role of the respective motor neuron classes related to movement. However, these models have not analyzed the distribution of the synaptic and gap connection weights. In this study, we examined whether a motor neuron and muscle network can generate oscillations for both forward and backward movement and analyzed the distribution of the trained synaptic and gap connection weights through a machine learning approach. This paper presents a connectome-based neural network model consisting of motor neurons of classes A, B, D, AS, and muscle, considering both synaptic and gap connections. A supervised learning method called backpropagation through time was adapted to train the connection parameters by feeding teacher data composed of the command neuron input and muscle cell activation. Simulation results confirmed that the motor neuron circuit could generate oscillations with different phase patterns corresponding to forward and backward movement, and could be switched at arbitrary times according to the binary inputs simulating the output of command neurons. Subsequently, we confirmed that the trained synaptic and gap connection weights followed a Boltzmann-type distribution. It should be noted that the proposed model can be trained to reproduce the activity patterns measured for an animal (HRB4 strain). Therefore, the supervised learning approach adopted in this study may allow further analysis of complex activity patterns associated with movements.

Inhibition Underlies Fast Undulatory Locomotion in Caenorhabditis elegans

Inhibition plays important roles in modulating the neural activities of sensory and motor systems at different levels from synapses to brain regions. To achieve coordinated movement, motor systems produce alternating contractions of antagonist muscles, whether along the body axis or within and among limbs, which often involves direct or indirect cross-inhibitory pathways. In the nematode Caenorhabditis elegans, a small network involving excitatory cholinergic and inhibitory GABAergic motoneurons generates the dorsoventral alternation of body-wall muscles that supports undulatory locomotion. Inhibition has been suggested to be necessary for backward undulation because mutants that are defective in GABA transmission exhibit a shrinking phenotype in response to a harsh touch to the head, whereas wild-type animals produce a backward escape response. Here, we demonstrate that the shrinking phenotype is exhibited by wild-type as well as mutant animals in response to harsh touch to the head or tail, but only GABA transmission mutants show slow locomotion after stimulation. Impairment of GABA transmission, either genetically or optogenetically, induces lower undulation frequency and lower translocation speed during crawling and swimming in both directions. The activity patterns of GABAergic motoneurons are different during low-frequency and high-frequency undulation. During low-frequency undulation, GABAergic VD and DD motoneurons show correlated activity patterns, while during high-frequency undulation, their activity alternates. The experimental results suggest at least three non-mutually exclusive roles for inhibition that could underlie fast undulatory locomotion in C. elegans, which we tested with computational models: cross-inhibition or disinhibition of body-wall muscles, or neuronal reset.

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.

A Neuromechanical Model of Multiple Network Rhythmic Pattern Generators for Forward Locomotion in C. elegans

Multiple mechanisms contribute to the generation, propagation, and coordination of the rhythmic patterns necessary for locomotion in Caenorhabditis elegans. Current experiments have focused on two possibilities: pacemaker neurons and stretch-receptor feedback. Here, we focus on whether it is possible that a chain of multiple network rhythmic pattern generators in the ventral nerve cord also contribute to locomotion. We use a simulation model to search for parameters of the anatomically constrained ventral nerve cord circuit that, when embodied and situated, can drive forward locomotion on agar, in the absence of pacemaker neurons or stretch-receptor feedback. Systematic exploration of the space of possible solutions reveals that there are multiple configurations that result in locomotion that is consistent with certain aspects of the kinematics of worm locomotion on agar. Analysis of the best solutions reveals that gap junctions between different classes of motorneurons in the ventral nerve cord can play key roles in coordinating the multiple rhythmic pattern generators.

Molecular mechanisms of homeostatic plasticity in central pattern generator networks

Central pattern generator (CPG) networks rely on a balance of intrinsic and network properties to produce reliable, repeatable activity patterns. This balance is maintained by homeostatic plasticity where alterations in neuronal properties dynamically maintain appropriate neural output in the face of changing environmental conditions and perturbations. However, it remains unclear just how these neurons and networks can both monitor their ongoing activity and use this information to elicit homeostatic physiological responses to ensure robustness of output over time. Evidence exists that CPG networks use a mixed strategy of activity-dependent, activity-independent, modulator-dependent, and synaptically regulated homeostatic plasticity to achieve this critical stability. In this review, we focus on some of the current understanding of the molecular pathways and mechanisms responsible for this homeostatic plasticity in the context of central pattern generator function, with a special emphasis on some of the smaller invertebrate networks that have allowed for extensive cellular-level analyses that have brought recent insights to these questions.

From head to tail: a neuromechanical model of forward locomotion in Caenorhabditis elegans

With 302 neurons and a near-complete reconstruction of the neural and muscle anatomy at the cellular level, Caenorhabditis elegans is an ideal candidate organism to study the neuromechanical basis of behaviour. Yet despite the breadth of knowledge about the neurobiology, anatomy and physics of C. elegans, there are still a number of unanswered questions about one of its most basic and fundamental behaviours: forward locomotion. How the rhythmic pattern is generated and propagated along the body is not yet well understood. We report on the development and analysis of a model of forward locomotion that integrates the neuroanatomy, neurophysiology and body mechanics of the worm. Our model is motivated by experimental analysis of the structure of the ventral cord circuitry and the effect of local body curvature on nearby motoneurons. We developed a neuroanatomically grounded model of the head motoneuron circuit and the ventral nerve cord circuit. We integrated the neural model with an existing biomechanical model of the worm's body, with updated musculature and stretch receptors. Unknown parameters were evolved using an evolutionary algorithm to match the speed of the worm on agar. We performed 100 evolutionary runs and consistently found electrophysiological configurations that reproduced realistic control of forward movement. The ensemble of successful solutions reproduced key experimental observations that they were not designed to fit, including the wavelength and frequency of the propagating wave. Analysis of the ensemble revealed that head motoneurons SMD and RMD are sufficient to drive dorsoventral undulations in the head and neck and that short-range posteriorly directed proprioceptive feedback is sufficient to propagate the wave along the rest of the body.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Caenorhabditis elegans excitatory ventral cord motor neurons derive rhythm for body undulation

The intrinsic oscillatory activity of central pattern generators underlies motor rhythm. We review and discuss recent findings that address the origin of Caenorhabditis elegans motor rhythm. These studies propose that the A- and mid-body B-class excitatory motor neurons at the ventral cord function as non-bursting intrinsic oscillators to underlie body undulation during reversal and forward movements, respectively. Proprioception entrains their intrinsic activities, allows phase-coupling between members of the same class motor neurons, and thereby facilitates directional propagation of undulations. Distinct pools of premotor interneurons project along the ventral nerve cord to innervate all members of the A- and B-class motor neurons, modulating their oscillations, as well as promoting their bi-directional coupling. The two motor sub-circuits, which consist of oscillators and descending inputs with distinct properties, form the structural base of dynamic rhythmicity and flexible partition of the forward and backward motor states. These results contribute to a continuous effort to establish a mechanistic and dynamic model of the C. elegans sensorimotor system. C. elegans exhibits rich sensorimotor functions despite a small neuron number. These findings implicate a circuit-level functional compression. By integrating the role of rhythm generation and proprioception into motor neurons, and the role of descending regulation of oscillators into premotor interneurons, this numerically simple nervous system can achieve a circuit infrastructure analogous to that of anatomically complex systems. C. elegans has manifested itself as a compact model to search for general principles of sensorimotor behaviours.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Signatures of proprioceptive control in Caenorhabditis elegans locomotion

Animal neuromechanics describes the coordinated self-propelled movement of a body, subject to the combined effects of internal neural control and mechanical forces. Here we use a computational model to identify effects of neural and mechanical modulation on undulatory forward locomotion of Caenorhabditis elegans, with a focus on proprioceptively driven neural control. We reveal a fundamental relationship between body elasticity and environmental drag in determining the dynamics of the body and demonstrate the manifestation of this relationship in the context of proprioceptively driven control. By considering characteristics unique to proprioceptive neurons, we predict the signatures of internal gait modulation that contrast with the known signatures of externally or biomechanically modulated gait. We further show that proprioceptive feedback can suppress neuromechanical phase lags during undulatory locomotion, contrasting with well studied advancing phase lags that have long been a signature of centrally generated, feed-forward control.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Editorial: Bridging Scales and Levels

Network neuroscience strives to understand the networks of the brain on all spatiotemporal scales and levels of observation. Current experimental and theoretical capabilities are beginning to facilitate a more holistic perspective, uniting these networks. This focus feature, "Bridging Scales and Levels," aims to document current research and looks to future progress towards this vision.