Synaptic patterning of left-right alternation in a computational model of the rodent hindlimb central pattern generator

The spinal premotor network driving scratching flexor and extensor alternation

Rhythmic motor behaviors are generated by neural networks termed central pattern generators (CPGs). Although locomotor CPGs have been extensively characterized, it remains unknown how the neuronal populations composing them interact to generate adaptive rhythms. We explored the non-linear cooperation dynamics among the three main populations of ipsilaterally projecting spinal CPG neurons - V1, V2a, V2b neurons - in scratch reflex rhythmogenesis. Ablation of all three neuronal subtypes reduced the oscillation frequency. Activation of excitatory V2a neurons enhanced the oscillation frequency, while activating inhibitory V1 neurons caused atonia. These findings required the development of a novel neuromechanical model that consists of flexor and extensor modules coupled via inhibition, in which rhythm in each module is generated by self-bursting excitatory populations and accelerated by intra-module inhibition. Inter-module inhibition coordinates the phases of flexor and extensor activity and slows the oscillations, while facilitation mechanisms in excitatory neurons explain the V2a activation-driven increase in frequency.

Error Function Optimization to Compare Neural Activity and Train Blended Rhythmic Networks

We present a novel set of quantitative measures for "likeness" (error function) designed to alleviate the time-consuming and subjective nature of manually comparing biological recordings from electrophysiological experiments with the outcomes of their mathematical models. Our innovative "blended" system approach offers an objective, high-throughput, and computationally efficient method for comparing biological and mathematical models. This approach involves using voltage recordings of biological neurons to drive and train mathematical models, facilitating the derivation of the error function for further parameter optimization. Our calibration process incorporates measurements such as action potential (AP) frequency, voltage moving average, voltage envelopes, and the probability of post-synaptic channels. To assess the effectiveness of our method, we utilized the sea slug Melibe leonina swim central pattern generator (CPG) as our model circuit and conducted electrophysiological experiments with TTX to isolate CPG interneurons. During the comparison of biological recordings and mathematically simulated neurons, we performed a grid search of inhibitory and excitatory synapse conductance. Our findings indicate that a weighted sum of simple functions is essential for comprehensively capturing a neuron's rhythmic activity. Overall, our study suggests that our blended system approach holds promise for enabling objective and high-throughput comparisons between biological and mathematical models, offering significant potential for advancing research in neural circuitry and related fields.

Computational Modeling of Spinal Locomotor Circuitry in the Age of Molecular Genetics

Neuronal circuits in the spinal cord are essential for the control of locomotion. They integrate supraspinal commands and afferent feedback signals to produce coordinated rhythmic muscle activations necessary for stable locomotion. For several decades, computational modeling has complemented experimental studies by providing a mechanistic rationale for experimental observations and by deriving experimentally testable predictions. This symbiotic relationship between experimental and computational approaches has resulted in numerous fundamental insights. With recent advances in molecular and genetic methods, it has become possible to manipulate specific constituent elements of the spinal circuitry and relate them to locomotor behavior. This has led to computational modeling studies investigating mechanisms at the level of genetically defined neuronal populations and their interactions. We review literature on the spinal locomotor circuitry from a computational perspective. By reviewing examples leading up to and in the age of molecular genetics, we demonstrate the importance of computational modeling and its interactions with experiments. Moving forward, neuromechanical models with neuronal circuitry modeled at the level of genetically defined neuronal populations will be required to further unravel the mechanisms by which neuronal interactions lead to locomotor behavior.

Recent Insights into the Rhythmogenic Core of the Locomotor CPG

In order for locomotion to occur, a complex pattern of muscle activation is required. For more than a century, it has been known that the timing and pattern of stepping movements in mammals are generated by neural networks known as central pattern generators (CPGs), which comprise multiple interneuron cell types located entirely within the spinal cord. A genetic approach has recently been successful in identifying several populations of spinal neurons that make up this neural network, as well as the specific role they play during stepping. In spite of this progress, the identity of the neurons responsible for generating the locomotor rhythm and the manner in which they are interconnected have yet to be deciphered. In this review, we summarize key features considered to be expressed by locomotor rhythm-generating neurons and describe the different genetically defined classes of interneurons which have been proposed to be involved.

Neural Interactions in Developing Rhythmogenic Spinal Networks: Insights From Computational Modeling

The mechanisms involved in generation of rhythmic locomotor activity in the mammalian spinal cord remain poorly understood. These mechanisms supposedly rely on both intrinsic properties of constituting neurons and interactions between them. A subset of Shox2 neurons was suggested to contribute to generation of spinal locomotor activity, but the possible cellular basis for rhythmic bursting in these neurons remains unknown. Ha and Dougherty (2018) recently revealed the presence of bidirectional electrical coupling between Shox2 neurons in neonatal spinal cords, which can be critically involved in neuronal synchronization and generation of populational bursting. Gap junctional connections found between functionally-related Shox2 interneurons decrease with age, possibly being replaced by increasing interactions through chemical synapses. Here, we developed a computational model of a heterogeneous population of neurons sparsely connected by electrical or/and chemical synapses and investigated the dependence of frequency of populational bursting on the type and strength of neuronal interconnections. The model proposes a mechanistic explanation that can account for the emergence of a synchronized rhythmic activity in the neuronal population and provides insights into the possible role of gap junctional coupling between Shox2 neurons in the spinal mechanisms for locomotor rhythm generation.

Dynamics and bifurcations in multistable 3-cell neural networks

We disclose the generality of the intrinsic mechanisms underlying multistability in reciprocally inhibitory 3-cell circuits composed of simplified, low-dimensional models of oscillatory neurons, as opposed to those of a detailed Hodgkin-Huxley type [Wojcik et al., PLoS One 9, e92918 (2014)]. The computational reduction to return maps for the phase-lags between neurons reveals a rich multiplicity of rhythmic patterns in such circuits. We perform a detailed bifurcation analysis to show how such rhythms can emerge, disappear, and gain or lose stability, as the parameters of the individual cells and the synapses are varied.

State-dependent rhythmogenesis and frequency control in a half-center locomotor CPG

The spinal locomotor central pattern generator (CPG) generates rhythmic activity with alternating flexion and extension phases. This rhythmic pattern is likely to result from inhibitory interactions between neural populations representing flexor and extensor half-centers. However, it is unclear whether the flexor-extensor CPG has a quasi-symmetric organization with both half-centers critically involved in rhythm generation, features an asymmetric organization with flexor-driven rhythmogenesis, or comprises a pair of intrinsically rhythmic half-centers. There are experimental data that support each of the above concepts but appear to be inconsistent with the others. In this theoretical/modeling study, we present and analyze a CPG model architecture that can operate in different regimes consistent with the above three concepts depending on conditions, which are defined by external excitatory drives to CPG half-centers. We show that control of frequency and phase durations within each regime depends on network dynamics, defined by the regime-dependent expression of the half-centers' intrinsic rhythmic capabilities and the operating phase transition mechanisms (escape vs. release). Our study suggests state dependency in locomotor CPG operation and proposes explanations for seemingly contradictory experimental data. NEW & NOTEWORTHY Our theoretical/modeling study focuses on the analysis of locomotor central pattern generators (CPGs) composed of conditionally bursting half-centers coupled with reciprocal inhibition and receiving independent external drives. We show that this CPG framework can operate in several regimes consistent with seemingly contradictory experimental data. In each regime, we study how intrinsic dynamics and phase-switching mechanisms control oscillation frequency and phase durations. Our results provide insights into the organization of spinal circuits controlling locomotion.

Diversity of molecularly defined spinal interneurons engaged in mammalian locomotor pattern generation

Mapping the expression of transcription factors in the mouse spinal cord has identified ten progenitor domains, four of which are cardinal classes of molecularly defined, ventrally located interneurons that are integrated in the locomotor circuitry. This review focuses on the properties of these interneuronal populations and their contribution to hindlimb locomotor central pattern generation. Interneuronal populations are categorized based on their excitatory or inhibitory functions and their axonal projections as predictors of their role in locomotor rhythm generation and coordination. The synaptic connectivity and functions of these interneurons in the locomotor central pattern generators (CPGs) have been assessed by correlating their activity patterns with motor output responses to rhythmogenic neurochemicals and sensory and descending fibers stimulations as well as analyzing kinematic gait patterns in adult mice. The observed complex organization of interneurons in the locomotor CPG circuitry, some with seemingly similar physiological functions, reflects the intricate repertoire associated with mammalian motor control and is consistent with high transcriptional heterogeneity arising from cardinal interneuronal classes. This review discusses insights derived from recent studies to describe innovative approaches and limitations in experimental model systems and to identify missing links in current investigational enterprise.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

Novel Methods to Enhance Precision and Reliability in Muscle Synergy Identification during Walking

Muscle synergies are hypothesized to reflect modular control of muscle groups via descending commands sent through multiple neural pathways. Recently, the number of synergies has been reported as a functionally relevant indicator of motor control complexity in individuals with neurological movement disorders. Yet the number of synergies extracted during a given activity, e.g., gait, varies within and across studies, even for unimpaired individuals. With no standardized methods for precise determination, this variability remains unexplained making comparisons across studies and cohorts difficult. Here, we utilize k-means clustering and intra-class and between-level correlation coefficients to precisely discriminate reliable from unreliable synergies. Electromyography (EMG) was recorded bilaterally from eight leg muscles during treadmill walking at self-selected speed. Muscle synergies were extracted from 20 consecutive gait cycles using non-negative matrix factorization. We demonstrate that the number of synergies is highly dependent on the threshold when using the variance accounted for by reconstructed EMG. Beyond use of threshold, our method utilized a quantitative metric to reliably identify four or five synergies underpinning walking in unimpaired adults and revealed synergies having poor reproducibility that should not be considered as true synergies. We show that robust and unreliable synergies emerge similarly, emphasizing the need for careful analysis in those with pathology.

Organization of the Mammalian Locomotor CPG: Review of Computational Model and Circuit Architectures Based on Genetically Identified Spinal Interneurons(1,2,3)

The organization of neural circuits that form the locomotor central pattern generator (CPG) and provide flexor–extensor and left–right coordination of neuronal activity remains largely unknown. However, significant progress has been made in the molecular/genetic identification of several types of spinal interneurons, including V0 (V0_D and V0_V subtypes), V1, V2a, V2b, V3, and Shox2, among others. The possible functional roles of these interneurons can be suggested from changes in the locomotor pattern generated in mutant mice lacking particular neuron types. Computational modeling of spinal circuits may complement these studies by bringing together data from different experimental studies and proposing the possible connectivity of these interneurons that may define rhythm generation, flexor–extensor interactions on each side of the cord, and commissural interactions between left and right circuits. This review focuses on the analysis of potential architectures of spinal circuits that can reproduce recent results and suggest common explanations for a series of experimental data on genetically identified spinal interneurons, including the consequences of their genetic ablation, and provides important insights into the organization of the spinal CPG and neural control of locomotion.

Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: a mathematical modeling view

The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized “hopping” pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0_D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0_V subtype) maintained the left–right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0_D, and V0_V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0_D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0_V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

Movements of left and right limbs in mammals during locomotion are controlled by distinct rhythm-generating neuronal circuits in the spinal cord. Complex interactions between these circuits provide flexible coordination of limb movements in different gaits. It was shown that interactions between left and right spinal circuits are mediated by commissural interneurons. Genetic ablation of a particular type of these interneurons, called V0, leads to switching from a regular, left-right alternating “walking” activity to a left-right synchronous “hopping” pattern. Moreover, the V0 commissural interneurons have excitatory and inhibitory subtypes that appear to play different roles in the left-right coordination depending on locomotor speed. In this theoretical study, we build a simplified mathematical model of spinal circuits that describes left and right rhythm generators interacting bilaterally via several types of commissural connections. Using this model, we simulate different experimental manipulations, analyze the resultant alternating and synchronous regimes of activity, and propose explanations for the results of experimental studies. We show that although both excitatory and inhibitory V0 commissural pathways support left-right alternation, the resultant locomotor pattern and gait depend on the balance between different commissural interactions, which in turn may depend on the level of neuronal excitation and locomotor speed.

Key bifurcations of bursting polyrhythms in 3-cell central pattern generators

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.

Toward robust phase-locking in Melibe swim central pattern generator models

Small groups of interneurons, abbreviated by CPG for central pattern generators, are arranged into neural networks to generate a variety of core bursting rhythms with specific phase-locked states, on distinct time scales, which govern vital motor behaviors in invertebrates such as chewing and swimming. These movements in lower level animals mimic motions of organs in higher animals due to evolutionarily conserved mechanisms. Hence, various neurological diseases can be linked to abnormal movement of body parts that are regulated by a malfunctioning CPG. In this paper, we, being inspired by recent experimental studies of neuronal activity patterns recorded from a swimming motion CPG of the sea slug Melibe leonina, examine a mathematical model of a 4-cell network that can plausibly and stably underlie the observed bursting rhythm. We develop a dynamical systems framework for explaining the existence and robustness of phase-locked states in activity patterns produced by the modeled CPGs. The proposed tools can be used for identifying core components for other CPG networks with reliable bursting outcomes and specific phase relationships between the interneurons. Our findings can be employed for identifying or implementing the conditions for normal and pathological functioning of basic CPGs of animals and artificially intelligent prosthetics that can regulate various movements.

Phase locking asymmetries at flexor-extensor transitions during fictive locomotion

The motor output for walking is produced by a network of neurons termed the spinal central pattern generator (CPG) for locomotion. The basic building block of this CPG is a half-center oscillator composed of two mutually inhibitory sets of interneurons, each controlling one of the two dominant phases of locomotion: flexion and extension. To investigate symmetry between the two components of this oscillator, we analyzed the statistics of natural variation in timing during fictive locomotion induced by stimulation of the midbrain locomotor region in the cat. As a complement to previously published analysis of these data focused on burst and cycle durations, we present a new analysis examining the strength of phase locking at the transitions between flexion and extension. Across our sample of nerve pairs, phase locking at the transition from extension to flexion (E to F) is stronger than at the transition from flexion to extension (F to E). This pattern did not reverse when considering bouts of fictive locomotion that were flexor vs. extensor dominated, demonstrating that asymmetric locking at the transitions between phases is dissociable from which phase dominates cycle duration. We also find that the strength of phase locking is correlated with the mean latency between burst offset and burst onset. These results are interpreted in the context of a hypothesis where network inhibition and intrinsic oscillatory mechanisms make distinct contributions to flexor-extensor alternation in half-center networks.

Sensitivity of spinal neurons to GABA and glycine during voluntary movement in behaving monkeys

GABAergic and glycinergic inhibition play key roles in the function of spinal motor pathways. However, there is little direct information on the extent to which inhibition controls the activity of spinal neurons during behavior or the relative effectiveness of GABA and glycine on cell activity under normal conditions. These issues were investigated in three macaque monkeys trained to perform voluntary ramp-and-hold wrist movements and grip. Pipettes with an extracellular recording electrode and iontophoresis barrels were used to eject GABA, glycine, and/or their respective antagonists, bicuculline and strychnine, as the activity of single neurons was recorded in the C6-T1 spinal segments during hand movements. The firing rate of the vast majority of neurons decreased when an inhibitory neurotransmitter was ejected from the electrode, suggesting that most movement-related spinal neurons are sensitive to both GABA and glycine. Most movement-related neurons exhibited increased activity during iontophoresis of an antagonist, suggesting that both GABAergic and glycinergic inhibition actively regulate the majority of spinal neurons during movement. These conclusions were supported by the responses of neurons tested with both agonists or both antagonists. Bicuculline and strychnine produced the largest increases in firing rate during dynamic movements (ramp phase), smaller increases during maintained torque/force (hold phase), and the smallest increase during the rest period. Since excitatory inputs also tend to increase progressively from rest to static to dynamic muscle contractions, this result is consistent with coupled excitatory and inhibitory inputs to spinal neurons during movement.

Patterns of inspiratory phase-dependent activity in the in vitro respiratory network

Mechanistic descriptions of rhythmogenic neural networks have often relied on ball-and-stick diagrams, which define interactions between functional classes of cells assumed to be reasonably homogenous. Application of this formalism to networks underlying respiratory rhythm generation in mammals has produced increasingly intricate models that have generated significant insight, but the underlying assumption that individual cells within these network fall into distinct functional classes has not been rigorously tested. In the present study we used multiunit extracellular recording in the in vitro pre-Bötzinger complex to identify and characterize the rhythmic activity of 951 cells. Inspiratory phase-dependent activity was estimated for all cells, and the data set as a whole was analyzed with principal component analysis, nonlinear dimensionality reduction, and hierarchical clustering techniques. None of these techniques revealed categorically distinct functional cell classes, indicating instead that the behavior of these cells within the network falls along several continua of spiking behavior.

Multistable phase regulation for robust steady and transitional legged gaits

We develop robust methods that allow the specification, control, and transition of a multi-legged robot’s stepping pattern, its ‘gait’, during active locomotion over natural terrain. Resulting gaits emerge through the introduction of controllers that impose appropriately placed repellors within the space of gaits, the torus of relative leg phases, thereby mitigating against dangerous patterns of leg timing. Moreover, these repellors are organized with respect to a natural cellular decomposition of gait space and result in limit cycles with associated basins that are well characterized by these cells, thus conferring a symbolic character upon the overall behavioral repertoire. These ideas are particularly applicable to four- and six-legged robots, for which a large variety of interesting and useful (and, in many cases, familiar) gaits exist, and whose tradeoffs between speed and reliability motivate the desire for transitioning between them during active locomotion. We provide an empirical instance of this gait regulation scheme by application to a climbing hexapod, whose ‘physical layer’ sensor-feedback control requires adequate grasp of a climbing surface but whose closed-loop control perturbs the robot from its desired gait. We document how the regulation scheme secures the desired gait and permits operator selection of different gaits as required during active climbing on challenging surfaces.

Functional characterization of dI6 interneurons in the neonatal mouse spinal cord

Our understanding of the neural control of locomotion has been greatly enhanced by the ability to identify and manipulate genetically defined populations of interneurons that comprise the locomotor central pattern generator (CPG). To date, the dI6 interneurons are one of the few populations that settle in the ventral region of the postnatal spinal cord that have not been investigated. In the present study, we utilized a novel transgenic mouse line to electrophysiologically characterize dI6 interneurons located close to the central canal and study their function during fictive locomotion. The majority of dI6 cells investigated were found to be rhythmically active during fictive locomotion and could be divided into two electrophysiologically distinct populations of interneurons. The first population fired rhythmic trains of action potentials that were loosely coupled to ventral root output and contained several intrinsic membrane properties of rhythm-generating neurons, raising the possibility that these cells may be involved in the generation of rhythmic activity in the locomotor CPG. The second population fired rhythmic trains of action potentials that were tightly coupled to ventral root output and lacked intrinsic oscillatory mechanisms, indicating that these neurons may be driven by a rhythm-generating network. Together these results indicate that dI6 neurons comprise an important component of the locomotor CPG that participate in multiple facets of motor behavior.