A spiral attractor network drives rhythmic locomotion

Analysis of High-Dimensional Coordination in Human Movement Using Variance Spectrum Scaling and Intrinsic Dimensionality

Full-body movement involving multi-segmental coordination has been essential to our evolution as a species, but its study has been focused mostly on the analysis of one-dimensional data. The field is poised for a change by the availability of high-density recording and data sharing. New ideas are needed to revive classical theoretical questions such as the organization of the highly redundant biomechanical degrees of freedom and the optimal distribution of variability for efficiency and adaptiveness. In movement science, there are popular methods that up-dimensionalize: they start with one or a few recorded dimensions and make inferences about the properties of a higher-dimensional system. The opposite problem, dimensionality reduction, arises when making inferences about the properties of a low-dimensional manifold embedded inside a large number of kinematic degrees of freedom. We present an approach to quantify the smoothness and degree to which the kinematic manifold of full-body movement is distributed among embedding dimensions. The principal components of embedding dimensions are rank-ordered by variance. The power law scaling exponent of this variance spectrum is a function of the smoothness and dimensionality of the embedded manifold. It defines a threshold value below which the manifold becomes non-differentiable. We verified this approach by showing that the Kuramoto model obeys the threshold when approaching global synchronization. Next, we tested whether the scaling exponent was sensitive to participants' gait impairment in a full-body motion capture dataset containing short gait trials. Variance scaling was highest in healthy individuals, followed by osteoarthritis patients after hip replacement, and lastly, the same patients before surgery. Interestingly, in the same order of groups, the intrinsic dimensionality increased but the fractal dimension decreased, suggesting a more compact but complex manifold in the healthy group. Thinking about manifold dimensionality and smoothness could inform classic problems in movement science and the exploration of the biomechanics of full-body action.

Motor cortex latent dynamics encode spatial and temporal arm movement parameters independently

The fluid movement of an arm requires multiple spatiotemporal parameters to be set independently. Recent studies have argued that arm movements are generated by the collective dynamics of neurons in motor cortex. An untested prediction of this hypothesis is that independent parameters of movement must map to independent components of the neural dynamics. Using a task where monkeys made a sequence of reaching movements to randomly placed targets, we show that the spatial and temporal parameters of arm movements are independently encoded in the low-dimensional trajectories of population activity in motor cortex: Each movement's direction corresponds to a fixed neural trajectory through neural state space and its speed to how quickly that trajectory is traversed. Recurrent neural network models show this coding allows independent control over the spatial and temporal parameters of movement by separate network parameters. Our results support a key prediction of the dynamical systems view of motor cortex, but also argue that not all parameters of movement are defined by different trajectories of population activity.

Dynamics of brain-muscle networks reveal effects of age and somatosensory function on gait

Walking is a complex motor activity that requires coordinated interactions between the sensory and motor systems. We used mobile EEG and EMG to investigate the brain-muscle networks involved in gait control during overground walking in young people, older people, and individuals with Parkinson's disease. Dynamic interactions between the sensorimotor cortices and eight leg muscles within a gait cycle were assessed using multivariate analysis. We identified three distinct brain-muscle networks during a gait cycle. These networks include a bilateral network, a left-lateralized network activated during the left swing phase, and a right-lateralized network active during the right swing. The trajectories of these networks are contracted in older adults, indicating a reduction in neuromuscular connectivity with age. Individuals with the impaired tactile sensitivity of the foot showed a selective enhancement of the bilateral network, possibly reflecting a compensation strategy to maintain gait stability. These findings provide a parsimonious description of interindividual differences in neuromuscular connectivity during gait.

Latent neural population dynamics underlying breathing, opioid-induced respiratory depression and gasping

Breathing is vital and must be concurrently robust and flexible. This rhythmic behavior is generated and maintained within a rostrocaudally aligned set of medullary nuclei called the ventral respiratory column (VRC). The rhythmic properties of individual VRC nuclei are well known, yet technical challenges have limited the interrogation of the entire VRC population simultaneously. Here we characterize over 15,000 medullary units using high-density electrophysiology, opto-tagging and histological reconstruction. Population dynamics analysis reveals consistent rotational trajectories through a low-dimensional neural manifold. These rotations are robust and maintained even during opioid-induced respiratory depression. During severe hypoxia-induced gasping, the low-dimensional dynamics of the VRC reconfigure from rotational to all-or-none, ballistic efforts. Thus, latent dynamics provide a unifying lens onto the activities of large, heterogeneous populations of neurons involved in the simple, yet vital, behavior of breathing, and well describe how these populations respond to a variety of perturbations.

A Single Central Pattern Generator for the Control of a Locomotor Rolling Wave in Mollusc Aplysia

Locomotion in mollusc Aplysia is implemented by a pedal rolling wave, a type of axial locomotion. Well-studied examples of axial locomotion (pedal waves in Drosophila larvae and body waves in leech, lamprey, and fish) are generated in a segmented nervous system via activation of multiple coupled central pattern generators (CPGs). Pedal waves in molluscs, however, are generated by a single pedal ganglion, and it is unknown whether there are single or multiple CPGs that generate rhythmic activity and phase shifts between different body parts. During locomotion in intact Aplysia, bursting activity in the parapedal commissural nerve (PPCN) was found to occur during tail contraction. A cluster of 20 to 30 P1 root neurons (P1Ns) on the ventral surface of the pedal ganglion, active during the pedal wave, were identified. Computational cluster analysis revealed that there are 2 phases to the motor program: phase I (centered around 168°) and phase II (centered around 357°). PPCN activity occurs during phase II. The majority of P1Ns are motoneurons. Coactive P1Ns tend to be electrically coupled. Two classes of pedal interneurons (PIs) were characterized. Class 1 (PI1 and PI2) is active during phase I. Their axons make a loop within the pedal ganglion and contribute to locomotor pattern generation. They are electrically coupled to P1Ns that fire during phase I. Class 2 (PI3) is active during phase II and innervates the contralateral pedal ganglion. PI3 may contribute to bilateral coordination. Overall, our findings support the idea that Aplysia pedal waves are generated by a single CPG.

The centrality of population-level factors to network computation is demonstrated by a versatile approach for training spiking networks

Neural activity is often described in terms of population-level factors extracted from the responses of many neurons. Factors provide a lower-dimensional description with the aim of shedding light on network computations. Yet, mechanistically, computations are performed not by continuously valued factors but by interactions among neurons that spike discretely and variably. Models provide a means of bridging these levels of description. We developed a general method for training model networks of spiking neurons by leveraging factors extracted from either data or firing-rate-based networks. In addition to providing a useful model-building framework, this formalism illustrates how reliable and continuously valued factors can arise from seemingly stochastic spiking. Our framework establishes procedures for embedding this property in network models with different levels of realism. The relationship between spikes and factors in such networks provides a foundation for interpreting (and subtly redefining) commonly used quantities such as firing rates.

Local networks from different parts of the human cerebral cortex generate and share the same population dynamic

A major goal of neuroscience is to reveal mechanisms supporting collaborative actions of neurons in local and larger-scale networks. However, no clear overall principle of operation has emerged despite decades-long experimental efforts. Here, we used an unbiased method to extract and identify the dynamics of local postsynaptic network states contained in the cortical field potential. Field potentials were recorded by depth electrodes targeting a wide selection of cortical regions during spontaneous activities, and sensory, motor, and cognitive experimental tasks. Despite different architectures and different activities, all local cortical networks generated the same type of dynamic confined to one region only of state space. Surprisingly, within this region, state trajectories expanded and contracted continuously during all brain activities and generated a single expansion followed by a contraction in a single trial. This behavior deviates from known attractors and attractor networks. The state-space contractions of particular subsets of brain regions cross-correlated during perceptive, motor, and cognitive tasks. Our results imply that the cortex does not need to change its dynamic to shift between different activities, making task-switching inherent in the dynamic of collective cortical operations. Our results provide a mathematically described general explanation of local and larger scale cortical dynamic.

A set of hub neurons and non-local connectivity features support global brain dynamics in C. elegans

The wiring architecture of neuronal networks is assumed to be a strong determinant of their dynamical computations. An ongoing effort in neuroscience is therefore to generate comprehensive synapse-resolution connectomes alongside brain-wide activity maps. However, the structure-function relationship, i.e., how the anatomical connectome and neuronal dynamics relate to each other on a global scale, remains unsolved. Systematically, comparing graph features in the C. elegans connectome with correlations in nervous system-wide neuronal dynamics, we found that few local connectivity motifs and mostly other non-local features such as triplet motifs and input similarities can predict functional relationships between neurons. Surprisingly, quantities such as connection strength and amount of common inputs do not improve these predictions, suggesting that the network's topology is sufficient. We demonstrate that hub neurons in the connectome are key to these relevant graph features. Consistently, inhibition of multiple hub neurons specifically disrupts brain-wide correlations. Thus, we propose that a set of hub neurons and non-local connectivity features provide an anatomical substrate for global brain dynamics.

Age-associated changes to neuronal dynamics involve a disruption of excitatory/inhibitory balance in C. elegans

In the aging brain, many of the alterations underlying cognitive and behavioral decline remain opaque. C. elegans offers a powerful model for aging research, with a simple, well-studied nervous system to further our understanding of the cellular modifications and functional alterations accompanying senescence. We perform multi-neuronal functional imaging across the aged C. elegans nervous system, measuring an age-associated breakdown in system-wide functional organization. At single-cell resolution, we detect shifts in activity dynamics toward higher frequencies. In addition, we measure a specific loss of inhibitory signaling that occurs early in the aging process and alters the systems critical excitatory/inhibitory balance. These effects are recapitulated with mutation of the calcium channel subunit UNC-2/CaV2a. We find that manipulation of inhibitory GABA signaling can partially ameliorate or accelerate the effects of aging. The effects of aging are also partially mitigated by disruption of the insulin signaling pathway, known to increase longevity, or by a reduction of caspase activation. Data from mammals are consistent with our findings, suggesting a conserved shift in the balance of excitatory/inhibitory signaling with age that leads to breakdown in global neuronal dynamics and functional decline.

Motor cortex activity across movement speeds is predicted by network-level strategies for generating muscle activity

Learned movements can be skillfully performed at different paces. What neural strategies produce this flexibility? Can they be predicted and understood by network modeling? We trained monkeys to perform a cycling task at different speeds, and trained artificial recurrent networks to generate the empirical muscle-activity patterns. Network solutions reflected the principle that smooth well-behaved dynamics require low trajectory tangling. Network solutions had a consistent form, which yielded quantitative and qualitative predictions. To evaluate predictions, we analyzed motor cortex activity recorded during the same task. Responses supported the hypothesis that the dominant neural signals reflect not muscle activity, but network-level strategies for generating muscle activity. Single-neuron responses were better accounted for by network activity than by muscle activity. Similarly, neural population trajectories shared their organization not with muscle trajectories, but with network solutions. Thus, cortical activity could be understood based on the need to generate muscle activity via dynamics that allow smooth, robust control over movement speed.

Fixed Point Attractor Theory Bridges Structure and Function in C. elegans Neuronal Network

Understanding the structure–function relationship in a neuronal network is one of the major challenges in neuroscience research. Despite increasing researches at circuit connectivity and neural network structure, their structure-based biological interpretability remains unclear. Based on the attractor theory, here we develop an analytical framework that links neural circuit structures and their functions together through fixed point attractor in Caenorhabditis elegans. In this framework, we successfully established the structural condition for the emergence of multiple fixed points in C. elegans connectome. Then we construct a finite state machine to explain how functions related to bistable phenomena at the neural activity and behavioral levels are encoded. By applying the proposed framework to the command circuit in C. elegans, we provide a circuit level interpretation for the forward-reverse switching behaviors. Interestingly, network properties of the command circuit and first layer amphid interneuron circuit can also be inferred from their functions in this framework. Our research indicates the reliability of the fixed point attractor bridging circuit structure and functions, suggesting its potential applicability to more complex neuronal circuits in other species.

Photodiode-Based Optical Imaging for Recording Network Dynamics with Single-Neuron Resolution in Non-Transgenic Invertebrates

The development of transgenic invertebrate preparations in which the activity of specifiable sets of neurons can be recorded and manipulated with light represents a revolutionary advance for studies of the neural basis of behavior. However, a downside of this development is its tendency to focus investigators on a very small number of "designer" organisms (e.g., C. elegans and Drosophila), potentially negatively impacting the pursuit of comparative studies across many species, which is needed for identifying general principles of network function. The present article illustrates how optical recording with voltage-sensitive dyes in the brains of non-transgenic gastropod species can be used to rapidly (i.e., within the time course of single experiments) reveal features of the functional organization of their neural networks with single-cell resolution. We outline in detail the dissection, staining, and recording methods used by our laboratory to obtain action potential traces from dozens to ~150 neurons during behaviorally relevant motor programs in the CNS of multiple gastropod species, including one new to neuroscience - the nudibranch Berghia stephanieae. Imaging is performed with absorbance voltage-sensitive dyes and a 464-element photodiode array that samples at 1,600 frames/second, fast enough to capture all action potentials generated by the recorded neurons. Multiple several-minute recordings can be obtained per preparation with little to no signal bleaching or phototoxicity. The raw optical data collected through the methods described can subsequently be analyzed through a variety of illustrated methods. Our optical recording approach can be readily used to probe network activity in a variety of non-transgenic species, making it well suited for comparative studies of how brains generate behavior.

Cerebellar Neurodynamics Predict Decision Timing and Outcome on the Single-Trial Level

Goal-directed behavior requires the interaction of multiple brain regions. How these regions and their interactions with brain-wide activity drive action selection is less understood. We have investigated this question by combining whole-brain volumetric calcium imaging using light-field microscopy and an operant-conditioning task in larval zebrafish. We find global, recurring dynamics of brain states to exhibit pre-motor bifurcations toward mutually exclusive decision outcomes. These dynamics arise from a distributed network displaying trial-by-trial functional connectivity changes, especially between cerebellum and habenula, which correlate with decision outcome. Within this network the cerebellum shows particularly strong and predictive pre-motor activity (>10 s before movement initiation), mainly within the granule cells. Turn directions are determined by the difference neuroactivity between the ipsilateral and contralateral hemispheres, while the rate of bi-hemispheric population ramping quantitatively predicts decision time on the trial-by-trial level. Our results highlight a cognitive role of the cerebellum and its importance in motor planning.

Low-Dimensional Motor Cortex Dynamics Preserve Kinematics Information During Unconstrained Locomotion in Nonhuman Primates

The dynamical systems view of movement generation in motor cortical areas has emerged as an effective way to explain the firing properties of populations of neurons recorded from these regions. Recently, many studies have focused on finding low-dimensional representations of these dynamical systems during voluntary reaching and grasping behaviors carried out by the forelimbs. One such model, the Poisson linear-dynamical-system (PLDS) model, has been shown to extract dynamics which can be used to decode reaching kinematics. However, few have investigated these dynamics, especially in non-human primates, during behaviors such as locomotion, which may involve motor cortex to a lesser degree. Here, we focused on unconstrained quadrupedal locomotion, and investigated whether unsupervised latent state-space models can extract low-dimensional dynamics while preserving information about hind-limb kinematics. Spiking activity from the leg area of primary motor cortex of rhesus macaques was recorded simultaneously with hind-limb joint positions during ambulation across a corridor, ladder, and on a treadmill at various speeds. We found that PLDS models can extract stereotyped low-dimensional neural trajectories from these neurons phase-locked to the gait cycle, and that distinct trajectories emerge depending on the speed and class of behavior. Additionally, it was possible to decode both the hind-limb kinematics and the gait phase from these inferred trajectories just as well or better than from the full neural population (18-80 neurons) with only 12 dimensions. Our results demonstrate that kinematics and gait phase during various locomotion tasks are well represented in low-dimensional latent dynamics inferred from motor cortex population activity.

The importance of identified neurons in gastropod molluscs to neuroscience

Gastropod molluscs have large neurons that are uniquely identifiable across individuals and across species based on neuroanatomical and neurochemical criteria, facilitating research into neural signaling and neural circuits. Novel neuropeptides have been identified through RNA sequencing and mass spectroscopic analysis of single neurons. The roles of peptides and other signaling molecules including second messengers have been placed in the context of small circuits that control simple behaviors. Despite the stereotypy, neurons vary over time in their activity in large ensembles. Furthermore, there is both intra-species and inter-species variation in synaptic properties and gene expression. Research on gastropod identified neurons highlights the features that might be expected to be stable in more complex systems when trying to identify cell types.

Adaptive, locally linear models of complex dynamics

The dynamics of complex systems generally include high-dimensional, nonstationary, and nonlinear behavior, all of which pose fundamental challenges to quantitative understanding. To address these difficulties, we detail an approach based on local linear models within windows determined adaptively from data. While the dynamics within each window are simple, consisting of exponential decay, growth, and oscillations, the collection of local parameters across all windows provides a principled characterization of the full time series. To explore the resulting model space, we develop a likelihood-based hierarchical clustering, and we examine the eigenvalues of the linear dynamics. We demonstrate our analysis with the Lorenz system undergoing stable spiral dynamics and in the standard chaotic regime. Applied to the posture dynamics of the nematode Caenorhabditis elegans, our approach identifies fine-grained behavioral states and model dynamics which fluctuate about an instability boundary, and we detail a bifurcation in a transition from forward to backward crawling. We analyze whole-brain imaging in C. elegans and show that global brain dynamics is damped away from the instability boundary by a decrease in oxygen concentration. We provide additional evidence for such near-critical dynamics from the analysis of electrocorticography in monkey and the imaging of a neural population from mouse visual cortex at single-cell resolution.

Sensorimotor integration in Caenorhabditis elegans: a reappraisal towards dynamic and distributed computations

The nematode Caenorhabditis elegans is a tractable model system to study locomotion, sensory navigation and decision-making. In its natural habitat, it is thought to navigate complex multisensory environments in order to find food and mating partners, while avoiding threats like predators or toxic environments. While research in past decades has shed much light on the functions and mechanisms of selected sensory neurons, we are just at the brink of understanding how sensory information is integrated by interneuron circuits for action selection in the worm. Recent technological advances have enabled whole-brain Ca2+ imaging and Ca2+ imaging of neuronal activity in freely moving worms. A common principle emerging across multiple studies is that most interneuron activities are tightly coupled to the worm's instantaneous behaviour; notably, these observations encompass neurons receiving direct sensory neuron inputs. The new findings suggest that in the C. elegans brain, sensory and motor representations are integrated already at the uppermost sensory processing layers. Moreover, these results challenge a perhaps more intuitive view of sequential feed-forward sensory pathways that converge onto premotor interneurons and motor neurons. We propose that sensorimotor integration occurs rather in a distributed dynamical fashion. In this perspective article, we will explore this view, discuss the challenges and implications of these discoveries on the interpretation and design of neural activity experiments, and discuss possible functions. Furthermore, we will discuss the broader context of similar findings in fruit flies and rodents, which suggest generalizable principles that can be learnt from this amenable nematode model organism.This article is part of a discussion meeting issue 'Connectome to behaviour: modelling C. elegans at cellular resolution'.

Serial-section atlas of the Tritonia pedal ganglion

The pedal ganglion of the nudibranch gastropod Tritonia diomedea has been the focus of neurophysiological studies for more than 50 yr. These investigations have examined the neural basis of behaviors as diverse as swimming, crawling, reflex withdrawals, orientation to water flow, orientation to the earth's magnetic field, and learning. Despite this sustained research focus, most studies have confined themselves to the layer of neurons that are visible on the ganglion surface, leaving many neurons, which reside in deeper layers, largely unknown and thus unstudied. To facilitate work on such neurons, the present study used serial-section light microscopy to generate a detailed pictorial atlas of the pedal ganglion. One pedal ganglion was sectioned horizontally at 2-µm intervals and another vertically at 5-µm intervals. The resulting images were examined separately or combined into stacks to generate movie tours through the ganglion. These were also used to generate 3D reconstructions of individual neurons and rotating movies of digitally desheathed whole ganglia to reveal all surface neurons. A complete neuron count of the horizontally sectioned ganglion yielded 1,885 neurons. Real and virtual sections from the image stacks were used to reveal the morphology of individual neurons, as well as the major axon bundles traveling within the ganglion to and between its several nerves and connectives. Extensive supplemental data are provided, as well as a link to the Dryad Data Repository site, where the complete sets of high-resolution serial-section images can be downloaded. NEW & NOTEWORTHY Because of the large size and relatively low numbers of their neurons, gastropod mollusks are widely used for investigations of the neural basis of behavior. Most studies, however, focus on the neurons visible on the ganglion surface, leaving the majority, located out of sight below the surface, unexamined. The present light microscopy study generates the first detailed visual atlas of all neurons of the highly studied Tritonia pedal ganglion.

A comparison of hatchery-rearing in exercise to wild animal physiology and reflex behavior in Aplysia californica

Aplysia californica was hatchery-reared in two turbulence protocols intended to imitate the intermittent turbulence of the native habitat and to promote development of the foot muscle from the exercise of adhering to the substrate. Hatchery-reared animals in turbulence regimes were compared to siblings reared in quiet water, and to wild animals, using noninvasive assessments of the development of the foot muscle. The objective was to learn if the turbulence-reared phenotype mimicked laboratory-targeted aspects of the wild phenotype, that is, reflex behavior, swim tunnel performance, and resting oxygen consumption (MO₂). No group exhibited different MO₂. MO₂ values for all of the compared groups of animals followed the power law, with an exponent of 0.69, consistent with this relationship throughout the animal kingdom. Turbulence-induced exercise did not affect the righting reflex or the tail withdrawal reflex, standard behavioral tests that involve the foot muscle, compared to quiet water-reared siblings. Wild individuals had significantly shorter time-to-right than all hatchery reared animals, however, wild animals did not perform better in flume tests. That turbulence-reared hatchery- or wild animals lacked superior flume performance suggests that this species may shelter from intertidal wave energy to remain near its optimal feeding areas.

Neuronal Population Activity in Spinal Motor Circuits: Greater Than the Sum of Its Parts

The core elements of stereotypical movements such as locomotion, scratching and breathing are generated by networks in the lower brainstem and the spinal cord. Ensemble activities in spinal motor networks had until recently been merely a black box, but with the emergence of ultra-thin Silicon multi-electrode technology it was possible to reveal the spiking activity of larger parts of the network. A series of experiments revealed unexpected features of spinal networks, such as multiple spiking regimes and lognormal firing rate distributions. The lognormality renders the widespread idea of a typical firing rate ± standard deviation an ill-suited description, and therefore these findings define a new arithmetic of motor networks. Focusing on the population activity behind motor pattern generation this review summarizes this advance and discusses its implications.

Dynamical networks: Finding, measuring, and tracking neural population activity using network science

Systems neuroscience is in a headlong rush to record from as many neurons at the same time as possible. As the brain computes and codes using neuron populations, it is hoped these data will uncover the fundamentals of neural computation. But with hundreds, thousands, or more simultaneously recorded neurons come the inescapable problems of visualizing, describing, and quantifying their interactions. Here I argue that network science provides a set of scalable, analytical tools that already solve these problems. By treating neurons as nodes and their interactions as links, a single network can visualize and describe an arbitrarily large recording. I show that with this description we can quantify the effects of manipulating a neural circuit, track changes in population dynamics over time, and quantitatively define theoretical concepts of neural populations such as cell assemblies. Using network science as a core part of analyzing population recordings will thus provide both qualitative and quantitative advances to our understanding of neural computation.