Structural and functional properties of a probabilistic model of neuronal connectivity in a simple locomotor network

Mechanisms Underlying the Recruitment of Inhibitory Interneurons in Fictive Swimming in Developing Xenopus laevis Tadpoles

Developing spinal circuits generate patterned motor outputs while many neurons with high membrane resistances are still maturing. In the spinal cord of hatchling frog tadpoles of unknown sex, we found that the firing reliability in swimming of inhibitory interneurons with commissural and ipsilateral ascending axons was negatively correlated with their cellular membrane resistance. Further analyses showed that neurons with higher resistances had outward rectifying properties, low firing thresholds, and little delay in firing evoked by current injections. Input synaptic currents these neurons received during swimming, either compound, unitary current amplitudes, or unitary synaptic current numbers, were scaled with their membrane resistances, but their own synaptic outputs were correlated with membrane resistances of their postsynaptic partners. Analyses of neuronal dendritic and axonal lengths and their activities in swimming and cellular input resistances did not reveal a clear correlation pattern. Incorporating these electrical and synaptic properties into a computer swimming model produced robust swimming rhythms, whereas randomizing input synaptic strengths led to the breakdown of swimming rhythms, coupled with less synchronized spiking in the inhibitory interneurons. We conclude that the recruitment of these developing interneurons in swimming can be predicted by cellular input resistances, but the order is opposite to the motor-strength-based recruitment scheme depicted by Henneman's size principle. This form of recruitment/integration order in development before the emergence of refined motor control is progressive potentially with neuronal acquisition of mature electrical and synaptic properties, among which the scaling of input synaptic strengths with cellular input resistance plays a critical role.SIGNIFICANCE STATEMENT The mechanisms on how interneurons are recruited to participate in circuit function in developing neuronal systems are rarely investigated. In 2-d-old frog tadpole spinal cord, we found the recruitment of inhibitory interneurons in swimming is inversely correlated with cellular input resistances, opposite to the motor-strength-based recruitment order depicted by Henneman's size principle. Further analyses showed the amplitude of synaptic inputs that neurons received during swimming was inversely correlated with cellular input resistances. Randomizing/reversing the relation between input synaptic strengths and membrane resistances in modeling broke down swimming rhythms. Therefore, the recruitment or integration of these interneurons is conditional on the acquisition of several electrical and synaptic properties including the scaling of input synaptic strengths with cellular input resistances.

Locomotion-induced ocular motor behavior in larval Xenopus is developmentally tuned by visuo-vestibular reflexes

Locomotion in vertebrates is accompanied by retinal image-stabilizing eye movements that derive from sensory-motor transformations and predictive locomotor efference copies. During development, concurrent maturation of locomotor and ocular motor proficiency depends on the structural and neuronal capacity of the motion detection systems, the propulsive elements and the computational capability for signal integration. In developing Xenopus larvae, we demonstrate an interactive plasticity of predictive locomotor efference copies and multi-sensory motion signals to constantly elicit dynamically adequate eye movements during swimming. During ontogeny, the neuronal integration of vestibulo- and spino-ocular reflex components progressively alters as locomotion parameters change. In young larvae, spino-ocular motor coupling attenuates concurrent angular vestibulo-ocular reflexes, while older larvae express eye movements that derive from a combination of the two components. This integrative switch depends on the locomotor pattern generator frequency, represents a stage-independent gating mechanism, and appears during ontogeny when the swim frequency naturally declines with larval age.

From decision to action: Detailed modelling of frog tadpoles reveals neuronal mechanisms of decision-making and reproduces unpredictable swimming movements in response to sensory signals

How does the brain process sensory stimuli, and decide whether to initiate locomotor behaviour? To investigate this question we develop two whole body computer models of a tadpole. The "Central Nervous System" (CNS) model uses evidence from whole-cell recording to define 2300 neurons in 12 classes to study how sensory signals from the skin initiate and stop swimming. In response to skin stimulation, it generates realistic sensory pathway spiking and shows how hindbrain sensory memory populations on each side can compete to initiate reticulospinal neuron firing and start swimming. The 3-D "Virtual Tadpole" (VT) biomechanical model with realistic muscle innervation, body flexion, body-water interaction, and movement is then used to evaluate if motor nerve outputs from the CNS model can produce swimming-like movements in a volume of "water". We find that the whole tadpole VT model generates reliable and realistic swimming. Combining these two models opens new perspectives for experiments.

Modeling spinal locomotor circuits for movements in developing zebrafish

Many spinal circuits dedicated to locomotor control have been identified in the developing zebrafish. How these circuits operate together to generate the various swimming movements during development remains to be clarified. In this study, we iteratively built models of developing zebrafish spinal circuits coupled to simplified musculoskeletal models that reproduce coiling and swimming movements. The neurons of the models were based upon morphologically or genetically identified populations in the developing zebrafish spinal cord. We simulated intact spinal circuits as well as circuits with silenced neurons or altered synaptic transmission to better understand the role of specific spinal neurons. Analysis of firing patterns and phase relationships helped to identify possible mechanisms underlying the locomotor movements of developing zebrafish. Notably, our simulations demonstrated how the site and the operation of rhythm generation could transition between coiling and swimming. The simulations also underlined the importance of contralateral excitation to multiple tail beats. They allowed us to estimate the sensitivity of spinal locomotor networks to motor command amplitude, synaptic weights, length of ascending and descending axons, and firing behavior. These models will serve as valuable tools to test and further understand the operation of spinal circuits for locomotion.

A simple method defines 3D morphology and axon projections of filled neurons in a small CNS volume: Steps toward understanding functional network circuitry

BACKGROUND

Fundamental to understanding neuronal network function is defining neuron morphology, location, properties, and synaptic connectivity in the nervous system. A significant challenge is to reconstruct individual neuron morphology and connections at a whole CNS scale and bring together functional and anatomical data to understand the whole network.

NEW METHOD

We used a PC controlled micropositioner to hold a fixed whole mount of Xenopus tadpole CNS and replace the stage on a standard microscope. This allowed direct recording in 3D coordinates of features and axon projections of one or two neurons dye-filled during whole-cell recording to study synaptic connections. Neuron reconstructions were normalised relative to the ventral longitudinal axis of the nervous system. Coordinate data were stored as simple text files.

RESULTS

Reconstructions were at 1 μm resolution, capturing axon lengths in mm. The output files were converted to SWC format and visualised in 3D reconstruction software NeuRomantic. Coordinate data are tractable, allowing correction for histological artefacts. Through normalisation across multiple specimens we could infer features of network connectivity of mapped neurons of different types.

COMPARISON WITH EXISTING METHODS

Unlike other methods using fluorescent markers and utilising large-scale imaging, our method allows direct acquisition of 3D data on neurons whose properties and synaptic connections have been studied using whole-cell recording.

CONCLUSIONS

This method can be used to reconstruct neuron 3D morphology and follow axon projections in the CNS. After normalisation to a common CNS framework, inferences on network connectivity at a whole nervous system scale contribute to network modelling to understand CNS function.

The decision to move: response times, neuronal circuits and sensory memory in a simple vertebrate

All animals use sensory systems to monitor external events and have to decide whether to move. Response times are long and variable compared to reflexes, and fast escape movements. The complexity of adult vertebrate brains makes it difficult to trace the neuronal circuits underlying basic decisions to move. To simplify the problem, we investigate the nervous system and responses of hatchling frog tadpoles which swim when their skin is stimulated. Studying the neuron-by-neuron pathway from sensory to hindbrain neurons, where the decision to swim is made, has revealed two simple pathways generating excitation which sums to threshold in these neurons to initiate swimming. The direct pathway leads to short, and reliable delays like an escape response. The other includes a population of sensory processing neurons which extend firing to introduce noise and delay into responses. These neurons provide a brief, sensory memory of the stimulus, that allows tadpoles to integrate stimuli occurring within a second or so of each other. We relate these findings to other studies and conclude that sensory memory makes a fundamental contribution to simple decisions and is present in the brainstem of a basic vertebrate at a surprisingly early stage in development.

Cross-Frequency Coupling in Descending Motor Pathways: Theory and Simulation

Coupling of neural oscillations is essential for the transmission of cortical motor commands to motoneuron pools through direct and indirect descending motor pathways. Most studies focus on iso-frequency coupling between brain and muscle activities, i.e., cortico-muscular coherence, which is thought to reflect motor command transmission in the mono-synaptic corticospinal pathway. Compared to this direct pathway, indirect corticobulbospinal motor pathways involve multiple intermediate synaptic connections via spinal interneurons. Neuronal processing of synaptic inputs can lead to modulation of inter-spike intervals which produces cross-frequency coupling. This theoretical study aims to evaluate the effect of the number of synaptic layers in descending pathways on the expression of cross-frequency coupling between supraspinal input and the cumulative output of the motoneuron pool using a computer simulation. We simulated descending pathways as various layers of interneurons with a terminal motoneuron pool using Hogdkin–Huxley styled neuron models. Both cross- and iso-frequency coupling between the supraspinal input and the motorneuron pool output were computed using a novel generalized coherence measure, i.e., n:m coherence. We found that the iso-frequency coupling is only dominant in the mono-synaptic corticospinal tract, while the cross-frequency coupling is dominant in multi-synaptic indirect motor pathways. Furthermore, simulations incorporating both mono-synaptic direct and multi-synaptic indirect descending pathways showed that increased reliance on a multi-synaptic indirect pathway over a mono-synaptic direct pathway enhances the dominance of cross-frequency coupling between the supraspinal input and the motorneuron pool output. These results provide the theoretical basis for future human subject study quantitatively assessing motor command transmission in indirect vs. direct pathways and its changes after neurological disorders such as unilateral brain injury.

Decay of connection probabilities with distance in 2D and 3D neuronal networks

We study connectivity in a model of a growing neuronal network in dimensions 2 and 3. Although the axon-to-dendrite proximity is an insufficient condition for establishing a functional synapse, it is still a necessary one. Therefore we study connection probabilities at short distances between the randomly grown axon trees and somas as probabilities of potential connections between the corresponding neurons. Our results show that, contrary to a common belief, these probabilities do not necessarily decay polynomially or exponentially in distance, but there are regimes of parameter values when the probability of proximity is not sensitive to the distance. In particular, in 3 dimensions the Euclidean distance between the neuronal cell body centers of neurons seems to play a very subtle role, as the probabilities of connections are practically constant within a certain finite range of distance. The model has a sufficient number of parameters to assess networks of neurons of different types. Our results give a firm basis for further modelling of the neuronal connectivity taking into account some realistic bouton distributions for establishing synaptic connections.

Networks of random trees as a model of neuronal connectivity

We provide an analysis of a randomly grown 2-d network which models the morphological growth of dendritic and axonal arbors. From the stochastic geometry of this model we derive a dynamic graph of potential synaptic connections. We estimate standard network parameters such as degree distribution, average shortest path length and clustering coefficient, considering all these parameters as functions of time. Our results show that even a simple model with just a few parameters is capable of representing a wide spectra of architecture, capturing properties of well-known models, such as random graphs or small world networks, depending on the time of the network development. The introduced model allows not only rather straightforward simulations but it is also amenable to a rigorous analysis. This provides a base for further study of formation of synaptic connections on such networks and their dynamics due to plasticity.

Dynamic Computational Model of the Human Spinal Cord Connectome

Connectomes abound, but few for the human spinal cord. Using anatomical data in the literature, we constructed a draft connectivity map of the human spinal cord connectome, providing a template for the many calibrations of specialized behavior to be overlaid on it and the basis for an initial computational model. A thorough literature review gleaned cell types, connectivity, and connection strength indications. Where human data were not available, we selected species that have been studied. Cadaveric spinal cord measurements, cross-sectional histology images, and cytoarchitectural data regarding cell size and density served as the starting point for estimating numbers of neurons. Simulations were run using neural circuitry simulation software. The model contains the neural circuitry in all ten Rexed laminae with intralaminar, interlaminar, and intersegmental connections, as well as ascending and descending brain connections and estimated neuron counts for various cell types in every lamina of all 31 segments. We noted the presence of highly interconnected complex networks exhibiting several orders of recurrence. The model was used to perform a detailed study of spinal cord stimulation for analgesia. This model is a starting point for workers to develop and test hypotheses across an array of biomedical applications focused on the spinal cord. Each such model requires additional calibrations to constrain its output to verifiable predictions. Future work will include simulating additional segments and expanding the research uses of the model.

Bifurcations of Limit Cycles in a Reduced Model of the Xenopus Tadpole Central Pattern Generator

We present the study of a minimal microcircuit controlling locomotion in two-day-old Xenopus tadpoles. During swimming, neurons in the spinal central pattern generator (CPG) generate anti-phase oscillations between left and right half-centres. Experimental recordings show that the same CPG neurons can also generate transient bouts of long-lasting in-phase oscillations between left-right centres. These synchronous episodes are rarely recorded and have no identified behavioural purpose. However, metamorphosing tadpoles require both anti-phase and in-phase oscillations for swimming locomotion. Previous models have shown the ability to generate biologically realistic patterns of synchrony and swimming oscillations in tadpoles, but a mathematical description of how these oscillations appear is still missing. We define a simplified model that incorporates the key operating principles of tadpole locomotion. The model generates the various outputs seen in experimental recordings, including swimming and synchrony. To study the model, we perform detailed one- and two-parameter bifurcation analysis. This reveals the critical boundaries that separate different dynamical regimes and demonstrates the existence of parameter regions of bi-stable swimming and synchrony. We show that swimming is stable in a significantly larger range of parameters, and can be initiated more robustly, than synchrony. Our results can explain the appearance of long-lasting synchrony bouts seen in experiments at the start of a swimming episode.