Half-center Oscillator Research Articles

A modeling exploration of how synaptic feedback to descending projection neurons shapes the activity of an oscillatory network.

Rhythmic activity which underlies motor output is often initiated and controlled by descending modulatory projection pathways onto central pattern generator (CPG) networks. In turn, these descending pathways receive synaptic feedback from their target CPG network, which can influence the CPG output. However, the mechanisms underlying such bi-directional synaptic interactions are mostly unexplored. We develop a reduced mathematical model, including both feed-forward and feedback circuitry, to examine how the synaptic interactions involving two projection neurons, MCN1 and CPN2, can produce and shape the activity of the gastric mill CPG in the crab stomatogastric nervous system. We use simplifying assumptions that are based on the behavior of the biological system to reduce this model down to 2 dimensions, which allows for phase plane analysis of the model output. The model shows a distinct activity for the gastric mill rhythm that is elicited when MCN1 and CPN2 are co-active compared to the rhythm elicited by MCN1 activity alone. Furthermore, the presence of feedback to the projection neuron CPN2 provides a distinct locus of pattern generation in the model which does not require reciprocally inhibitory interactions between the gastric mill CPG neurons, but is instead based on a half-center oscillator that occurs through a tri-synaptic pathway that includes CPN2. Our modeling results show that feedback to projection pathways may provide additional mechanisms for the generation of motor activity. These mechanisms can have distinct dependence on network parameters and may therefore provide additional flexibility for the rhythmic motor output.

Protective role of the half-center oscillator connectivity against external perturbations

Central pattern generators (CPGs) are neuronal circuits that control rhythmic movements in animals. The heartbeat of the medicinal leech is controlled by a CPG that is distributed over several ganglia. In ganglia 3 and 4, there are half-center oscillators (HCOs): pairs of reciprocally inhibitory heart interneurons (HNs). These HCOs form the kernel of the heartbeat CPG, and each HCO is composed of a pair of endogenously bursting heart interneurons [1]. A canonical model of the HN accurately reproduces experimental results [1,2]. It also exhibits bistability of bursting and silence in single HNs and the HCO [1,3]. The single HN and HCO model have been expanded into a family of models that satisfy biological constraints [2]. Multistability is highly prevalent in this database [4]. The coexistence of bursting and silent regimes could present a life threatening condition for the animal. If a perturbation were able to trigger a switch in the activity of the CPG to a silent state, circulation of blood in the leech would be crippled. Our hypothesis is that the HCO connectivity protects functional pattern of activity against an external perturbation. Here, we investigated the perturbations that switch the activity of the HN and the HCO from bursting to silence. Over a series of simulations, we applied a 30 ms pulse of current with various amplitudes and phases of application in the period of bursting activity. This protocol provided a set of pulse parameters in the (phase, amplitude) parameter space for which a switch was triggered. In the canonical model of a single HN for which the leak conductance was 9.59 nS, both depolarizing and hyperpolarizing pulses could trigger switches (Figure ​(Figure1).1). We observed contiguous sets of pulse parameters where triggered switches occurred, but there were also fine structures in the parameter space (Figure ​(Figure1).1). We estimated the susceptibility of bursting activity in the HN to perturbations by the proportion of pulse parameter sets that would trigger a switch to the total number of trials. We found that 20.84% of pulses triggered a switch (Figure ​(Figure1).1). We repeated this pulse protocol in the HCO for the same parameter values of the ionic conductances, and found that no pulses elicited a switch from bursting to silence. Our results suggest that in a network consisting of bistable elements, the half-center oscillator connectivity protects the functional regime against external perturbations. Figure 1 Properties of pulses triggering a switch from bursting to silence. Red or blue points indicate a switch or failure to switch. The yellow trajectory is the bursting waveform that defines the phase. The green dashed line indicates 0 nA.

Neuromechanical model of reflexes and locomotor rhythms in the crayfish leg

The role of sensory feedback in the production of locomotor rhythms has been discussed for a hundred years [1], but a detailed description has emerged only recently with the coupling of neurophysiological and behavioral experiments and neuromechanical models [2]. In both cats and stick insects, this combination has helped clarify the role of proprioceptive feedback in determining the stance-swing transition during walking. The role of sensory feedback in posture and locomotor control of the crayfish leg has been intensively investigated over the last 25 years, and the neural circuitry governing leg elevation and depression movements during static posture and walking has been described [3]. We have developed a hybrid neuromechanical experiment consisting of an AnimatLab neuromechanical model (http://www.animatlab.com) of the crayfish thorax and 5th leg linked to the corresponding sensory and motor nerves of the living crayfish thoracic nervous system. We found that lifting the model leg would evoke resistance reflexes in tonically active preparations similar to those seen in intact resting animals, whereas the reflex reversed sign to become assistive during a locomotor rhythm. Moreover, the rhythm had a much higher frequency with the sensory feedback closed than when the loop was opened. To help determine whether the known circuitry can account for postural reflexes and reflex-assisted walking, we linked the same neuromechanical model of the crayfish leg to a model of the neural circuit that controls leg levation and depression (Figure 1). We found that an appropriately 'tuned' model could replicate the hybrid experimental results, and that this depended on the proper balance between the strength (gain) of the assistance reflex and the strength and stability of the half-center oscillator. To increase the intrinsic CPG frequency, the leg levation assistance reflex had to be sufficiently sensitive and strong to reset the CPG earlier in the interval between depression and levation motor bursts. The CPG had to be sufficiently robust to produce a stable low-frequency burst pattern without reflex input, but be able to reset in response to appropriately timed input. Figure 1 Neuromechanical model. Body model is at left, showing thorax and left 5th leg with hinges, CBCO stretch receptor, and Dep (red) and Lev (blue) muscles identified. Circuit is at right, with Dep MNs, INs, and muscles in red, and Lev in blue. Muscles and ... Figure 2 Model responses under open loop (left) and closed loop (right) conditions. Closing the feedback loop increased the burst frequency of the Lev/Dep CPG.

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.

High Prevalence of Multistability of Rest States and Bursting in a Database of a Model Neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.

Phase response properties of half-center oscillators

We examine the phase response properties of half-center oscillators (HCOs) that are modeled by a pair of Morris-Lecar-type neurons connected by strong fast inhibitory synapses. We find that the two basic mechanisms for half-center oscillations, "release" and "escape", give rise to strikingly different phase response curves (PRCs). Release-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the active to the suppressed state, and PRCs are dominated by a large negative peak (phase delays) at corresponding phases. On the other hand, escape-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the suppressed to the active state, and PRCs are dominated by a large positive peak (phase advances) at corresponding phases. By analyzing the phase space structure of Morris-Lecar-type HCO models with fast synaptic dynamics, we identify the dynamical mechanisms underlying the shapes of the PRCs. To demonstrate the significance of the different shapes of the PRCs for the release-type and escape-type HCOs, we link the shapes of the PRCs to the different frequency modulation properties of release-type and escape-type HCOs, and we show that the different shapes of the PRCs for the release-type and escape-type HCOs can lead to fundamentally different phase-locking dynamics.

Dynamical Systems Guided Design and Analysis of Silicon Oscillators for Central Pattern Generators

In this paper, a dynamical systems (DS) approach is proposed for the analysis and design of bio-inspired silicon central pattern generator (CPG) systems. Based on this approach, a new leaky-integrate-and-leaky-discharge oscillator circuit is proposed that has dynamical properties closer to biological half-center oscillators while being power and area efficient. The membrane potential charges and discharges through a single resistor eliminating mismatch in charging and discharging phases. Switched-capacitor (SC) and floating-gate wide input linear range operational transconductance amplifier (FGOTA) based approaches have been proposed to implement the resistor. Both approaches enable controllable and large resistances in a small area. Oscillation frequency can be easily controlled by the frequency of switching in SC based and bias current in FGOTA based implementations, which are very useful for global change of oscillation frequency in an array of oscillators. Dynamical systems analysis has shown that when it is used as a single oscillator, the proposed circuit is able to produce a phase response curve (PRC) close to that of a lamprey CPG system. By applying averaging theory to a system of coupled oscillators, the averaged <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> functions for unidirectional and bidirectional coupling cases are obtained. Analysis of these functions shows our circuit's superior capability to achieve entrainment when driven by a periodic input (e.g., from sensory feedback) and reach equilibrium even with high frequency mismatch.

A physical model of sensorimotor interactions during locomotion

In this paper, we describe the development of a bipedal robot that models the neuromuscular architecture of human walking. The body is based on principles derived from human muscular architecture, using muscles on straps to mimic agonist/antagonist muscle action as well as bifunctional muscles. Load sensors in the straps model Golgi tendon organs. The neural architecture is a central pattern generator (CPG) composed of a half-center oscillator combined with phase-modulated reflexes that is simulated using a spiking neural network. We show that the interaction between the reflex system, body dynamics and CPG results in a walking cycle that is entrained to the dynamics of the system. We also show that the CPG helped stabilize the gait against perturbations relative to a purely reflexive system, and compared the joint trajectories to human walking data. This robot represents a complete physical, or ‘neurorobotic’, model of the system, demonstrating the usefulness of this type of robotics research for investigating the neurophysiological processes underlying walking in humans and animals.

Exploring the relationships between neuronal parameters and network activity

This study constitutes our next step toward a full investigation of how intrinsic membrane and synaptic parameters affect the electrical activity of a half-center oscillator (HCO) model and how different parameter regimes influence stability and modulatability of the HCO model’s output. In our previous study [3], we constructed an efficient relational database table with the resulting characteristics of Hill et al.’s [2] HCO single-compartment conductance-based model. This model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat CPG under a variety of experimental conditions. We systematically [4] explored the parameter space of this HCO model by varying a set of eight selected parameters (maximal conductances of intrinsic and synaptic currents) in all combinations possible, resulting in 10,485,760 simulated model instances (10,321,920 HCO instances and 163,840 corresponding isolated neuron instances). We classified within a group the simulated instances showing similar electrical activity. The HCOs were split into ten groups labeled: spiking, silent, asymmetric activity, plateau, irregular spikes, asymmetric bursting, one burst, irregular period, unbalanced, and functional (1,202,139 instances); the isolated neuron instances were split into seven groups: spiking, silent, bistable, plateau, irregular spikes, irregular period, and regular bursting [3]. By querying the database we started analyzing the relationships between the neuronal parameters of the model instances within the same group and across groups. Model instances whose neurons are synaptically isolated are called isolated neuron instances. We found that regularly bursting (isolated) neuron instances compose only a small minority of functional HCOs (424 instances); the vast majority was composed of spiking neurons (32,568 instances). Also, our data suggest that regular bursting isolated neurons form functional HCOs that are more robust to variations in synaptic parameters than those formed by spiking isolated neurons. Approximately 8.2% (99,066 instances) of the functional HCOs have bursting characteristics that are similar to those of the living animal [1]. We used dimensional stacking (the NDVis tool) to visualize the distribution of each group of model instances versus the neuronal parameters. While each group has its own distribution within the parameter space, the groups of bistable and regular bursting isolated neuron instances separate large zones of spiking and silent model instances. To explore in depth the interaction of parameters that lead to the different activity groups we have identified, we used the Principal Component Analysis (PCA) method. When we applied the PCA to the group of regular bursting isolated neuron instances, we found that only 4 (out of 6) components are important for this group (sum of variances > 95%) and that the 1st component has the greatest impact (variance > 60%). This component is mostly influenced by the maximal conductances of NaP, K2 and Leak, and the ELeak (linear combination of the parameters). Then, we applied PCA to our group of functional HCOs showing realistic leech bursting characteristics and we found that the first 6 (out of 8) are the most important principal components for this group (sum of variances > 96%). However, unlike in the regular bursting group, none of these principal components is dominant. This suggests that this group has complex non-linear relationships between its parameters and to explore them we need to use more sophisticate computational methods that the (linear) PCA.

Modeling study of a Central Pattern Generator in the Melibe seaslug

Behaving animals show correlated neuronal activities in their nervous system. These observations lead to the question, how are the specific patterns of activities generated and used to control behaviors? Central Pattern Generators (CPGs) are small networks of neurons that are experimentally identified as necessary for expressions of behaviors, and the core group capable of controlling various aspects of the behavior. Mechanistic understanding of CPG functions is under intense investigation [1]. In this study, we investigate the role of network configurations of an experimentally based CPG for swimming behavior in the marine invertebrate Melibe leonine [2]. Repetitive nature of activity patterns of CPGs can be examined through the relative phase relations between bursting interneurons. Experimentally and computationally, such phase-lags are measured with respect to a reference neuron, thus allowing for an analysis of various activity patterns in a systematic way. For the Melibe CPG model under consideration, we consider four HodgkinHuxley type square-wave bursters, connected via fast nondelayed inhibitory synapses represented by fast threshold modulating function. Due to intrinsic symmetry, the network can be treated as two pairs of half center oscillators (HCOs). In a HCO both neurons reciprocally inhibit each other, leading to alternate, anti-phase bursting patterns, which can coexist with in-phase bursting when the HCO is inhibited externally [3,4]. In the given model, one HCO inhibits the other one and gets ap ositive feedback in the form of excitatory drive. We will further enhance the CPG model by including additional interneurons to reflect physiological fine details observed in-vitro. The exploration of stable phase-locked states, perturbed externally by applied current pulses, allows us to suggest plausible mechanisms of the CPG and come up with predictions that may be verified by electrophysiological experiments. The phase-locked pattern, observed in the experiment, is identified as an attractor in numerical simulations. We find that specific distributions of synaptic conductance shift the attractor so that it merges with a saddle corresponding to unstable phase-locked state. Investigation indicates that weak excitatory connections play no significant role in the generation of the pattern but may have stabilizing effect. Electrical coupling, which appears to be very weak and negligible in the physiological context, may play a critical role in altering the behavior of the network. Conclusions We conclude that the idiosyncratic phase-locking seen in the experimental setup is due to a specific network configuration of the CPG model where a driving HCO uni-directionally inhibits a nearly identical one, provided that coupling strengths stay rather weak. Future investigations will be focused on enhanced realistic CPG models with additional connections and participating interneurons.

Analyzing how neuronal parameters influence network activity

A half-center oscillator (HCO) is a common circuit building block of central pattern generator (CPG) networks that produce rhythmic motor patterns in animals. Rhythmic activity in the heartbeat CPG of the leech is based on alternating bursting in two pairs of inhibitory interneurons that make reciprocal spike-mediated and graded synapses across the ganglionic midline. Our study here constitutes the next step toward a full investigation of how intrinsic membrane and synaptic parameters affect the electrical activity of a half-center oscillator (HCO) model and how different parameter regimes influence stability and modulatability of the HCO model’s output. In our previous study [2], we constructed an efficient relational database table with the resulting characteristics of Hill et al.’s [1] HCO single-compartment conductance-based model. The model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat CPG under a variety of experimental conditions. To systematically explore the parameter space of this model, we used a brute-force approach. We varied a set of eight selected parameters (maximal conductances of intrinsic and synaptic currents) in all combinations resulting in a parameter space of 10,485,760 simulated models (10,321,920 HCO models and 163,840 corresponding isolated neuron models). After performing all the simulations, we built a SQL database table for their firing characteristics [3,4], which we can use to ask fundamental questions about the activity of HCOs. We began our analysis by classifying these HCO and isolated neuron model simulations by their activity characteristics: models showing the same electrical activity are segregated to the same group. The HCOs were split into ten groups labeled: spiking, silent, asymmetric activity, plateau, irregular spikes, asymmetric bursting, one burst, irregular period, unbalanced, and functional; the isolated neuron models were split into seven groups: spiking, silent, bistable, plateau, irregular spikes, irregular period, and regular bursting[2]. By querying the database, we compared the activity characteristics of the groups of our simulated HCO models with those of our simulated isolated neuron models and found that regularly bursting neurons compose only a small minority of functional HCO models (419 models); the vast majority was composed of spiking neurons (32,568 models). This finding indicates that, within the parameter space considered, spiking neurons are highly prevalent, suggesting that this mode of activity is robust, and that regular bursting neurons are rare, suggesting that this state requires a delicate balance of intrinsic membrane conductances (parameters) and that they may not be robust to parameter changes. Also, our data suggest that regular bursting isolated neurons form functional HCOs that are more robust to variations in synaptic parameters than those formed by spiking isolated neurons. We will now use the entire database to explore in depth the interaction of parameters that lead to the different activity groups we have identified. We will be particularly interested in parameter changes which correspond to known neuromodulations such as the modulation of h current by myomodulin [5].

Bringing rest into consideration: analyzing a database of computational models for multistability of oscillatory and stationary regimes

Biophysically realistic neuronal models are defined in high-dimensional parameter spaces. Brute-force database approaches have proven themselves fruitful in characterizing neuronal models [1,3,5]. In a brute-force database approach, the corresponding activities and parameters of the simulated neuronal models are stored in a database, which is further efficiently searched for insightful information about the models. We present a novel technique which combines a brute-force approach with bifurcation continuation methods from the theory of dynamical systems. We considered a database of single compartment neuron model that can display, silence, tonic spiking and bursting activity [3]. A one-parameter bifurcation analysis allowed us to determine stable and unstable stationary states for each case from the database. Using this novel database technique, we systematically tested models for the coexistence of stable stationary states and bursting activity. Specifically we used this technique to extend the database describing the model of the isolated leech heart interneuron to include stationary states [3]. We took into consideration only those cases that were shown to produce functional half-center oscillators (HCOs) [2-4]. By using the leak conductance (gleak) as the controlling parameter for bifurcation analysis, we showed that the model could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. For different initial conditions as many as three different attracting regimes could be exhibited by a case of the model. We found cases demonstrating either bursting activity, hyperpolarized stationary state or depolarized stationary state. We investigated the 2,387 cases that produced regular bursting activity, and found that 421 (18%) were multistable. We found that among the latter there were 361 cases which exhibited bistability of bursting and a hyperpolarized stationary state and 47 cases which showed bistability of bursting and a depolarized stationary state. Out of those 361 cases, one case showed bistability of bursting and a second, slightly less hyperpolarized stationary state. Twelve cases demonstrated tristability of coexisting bursting and two stable stationary states, hyperpolarized and depolarized. Our method also allowed us to determine that 2,128 out of 2,387 (89%) cases are multistable for some range of gleak values. Among these cases, the majority, 2,078 (98%), showed at least one range of gleak values with two attracting regimes, bursting and a hyperpolarized stationary state. We also explored the activity of HCOs built of multistable regular bursting model neurons (421 described above) in response to perturbations of the state variables. For a given set of initial conditions, 353 (84%) cases showed functional activity. We further explored the activity of these 353 cases. After 100 seconds of functional activity the state of one of the neurons was reset to be close to a stable stationary state, and the synaptic conductances determining the mutual inhibition of the HCO were set to zero for 5 seconds. This interruption of the normal HCO dynamics produced a perturbation of the state variables. In 96% of these cases, the HCOs exhibited functional bursting, proving itself robust against this perturbation too. Multistability of oscillatory regimes and stationary states poses a threat to functionality of central pattern generators which are required to produce a persistent pattern. The half-center oscillator configuration could serve as network-based protective mechanism against malfunctional multistability. On the other hand multistability of the dynamics of neurons could be functional providing a substrate for memory units, toggle switches, and elements of multifunctional central pattern generators. BM is supported by the Brazilian agency FAPESP. GC and WB are supported by NSF grant PHY-0750456. AD-M and RLC are supported by NIH grant NS024072

Understanding the mechanisms underlying phase-locking behavior in the crayfish swimmeret system

One of the fundamental challenges in neuroscience is to understand how the intrinsic properties of neurons and the properties of neural networks combine to produce behavior. Networks that produce rhythmic motor behaviors, such as locomotion, provide important model systems to address this problem. A particularly good model for this purpose is the neural circuit underlying the coordinated rhythmic limb movements in the crayfish swimmeret system. During forward swimming, rhythmic movements of swimmerets on different segments of the crayfish abdomen progress from back to front with the same period, but neighboring swimmerets are phase-lagged by 25% of the period. This coordination of limb movements is maintained over a wide range of frequency. The exact mechanisms underlying this robustly stable phase-locking are not known. Here, we use mathematical modeling and analysis in conjunction with recent experimental results to obtained insight into these mechanisms. The rhythmic behavior of each swimmeret is driven by a local pattern generating circuit consisting of a half-center oscillator (HCO). These local pattern generating circuits are connected through well-described intersegmental connections. We model the neural circuitry of the swimmeret system as a chain of HCOs. First, we examine the phase response properties of HCOs for two fundamentally different mechanisms that produce anti-phase activity in HCOs: the “escape” and “release” mechanisms. We demonstrate that the “escape” and “release” mechanisms give rise to very different phase response properties, and we use phase plane arguments to explain the different shapes of the phase response curves. We then examine a chain of four nearest-neighbor coupled HCOs. We use the coupled oscillator theory and symmetry arguments to show that the phase-locking 25% phase-locking observed in the crayfish swimmeret system arises naturally from the network connectivity, but this phase-locking is robustly stable for only some combinations of connectivity and the escape/release mechanisms.

Smooth changes in the EMG patterns during gait transitions under body weight unloading

During gradual speed changes, humans exhibit a sudden discontinuous switch from walking to running at a specific speed, and it has been suggested that different gaits may be associated with different functioning of neuronal networks. In this study we recorded the EMG activity of leg muscles at slow increments and decrements in treadmill belt speed and at different levels of body weight unloading. In contrast to normal walking at 1 g, at lower levels of simulated gravity (<0.4 g) the transition between walking and running was generally gradual, without systematic abrupt changes in either intensity or timing of EMG patterns. This phenomenon depended to a limited extent on the gravity simulation technique, although the exact level of the appearance of smooth transitions (0.4-0.6 g) tended to be lower for the vertical than for the tilted body weight support system. Furthermore, simulations performed with a half-center oscillator neuromechanical model showed that the abruptness of motor patterns at gait transitions at 1 g could be predicted from the distinct parameters anchored already in the normal range of walking and running speeds, whereas at low gravity levels the parameters of the model were similar for the two human gaits. A lack of discontinuous changes in the pattern of speed-dependent locomotor characteristics in a hypogravity environment is consistent with the idea of a continuous shift in the state of a given set of central pattern generators, rather than the activation of a separate set of central pattern generators for each distinct gait.

A database of computational models of a half-center oscillator for analyzing how neuronal parameters influence network activity.

A half-center oscillator (HCO) is a common circuit building block of central pattern generator networks that produce rhythmic motor patterns in animals. Here we constructed an efficient relational database table with the resulting characteristics of the Hill et al.'s (J Comput Neurosci 10:281-302, 2001) HCO simple conductance-based model. The model consists of two reciprocally inhibitory neurons and replicates the electrical activity of the oscillator interneurons of the leech heartbeat central pattern generator under a variety of experimental conditions. Our long-range goal is to understand how this basic circuit building block produces functional activity under a variety of parameter regimes and how different parameter regimes influence stability and modulatability. By using the latest developments in computer technology, we simulated and stored large amounts of data (on the order of terabytes). We systematically explored the parameter space of the HCO and corresponding isolated neuron models using a brute-force approach. We varied a set of selected parameters (maximal conductance of intrinsic and synaptic currents) in all combinations, resulting in about 10 million simulations. We classified these HCO and isolated neuron model simulations by their activity characteristics into identifiable groups and quantified their prevalence. By querying the database, we compared the activity characteristics of the identified groups of our simulated HCO models with those of our simulated isolated neuron models and found that regularly bursting neurons compose only a small minority of functional HCO models; the vast majority was composed of spiking neurons.

Phase response properties of an idealized half-center oscillator

Half-center oscillators are widely used as models for central pattern generators. They consist of two cells (or groups of cells) connected by relatively strong reciprocal inhibition. Typically, the individual cells are phase-locked in an anti-phase state, where one cell is active while the other is suppressed, and vice versa. In central pattern generators, this allows for the alternating order of two usually opposing behaviors. Often, central pattern generators are coupled, forming networks that coordinate rhythmic activity, such as in the coordinated limb movement during locomotion. Understanding the mechanisms underlying this coordination is an important issue in neuroscience. The neural circuitry of the crayfish swimmeret system is ideal for exploring coordination, and phase-locking in general, in coupled central pattern generators. It is composed of four bilateral segments each containing two modules equipped with a half-center oscillator that controls a corresponding limb. The half-center oscillator of each module is capable of oscillations independent of other modules, however all modules are connected to each other through well-described inter-segmental coupling. The coupling between cells in the half-center oscillators is relatively strong, whereas the inter-segmental coupling between the half-center oscillators is relatively weak. Therefore, it should be possible to use the theory of weakly coupled oscillators to deduce how phase-locking is achieved in the swimmeret system. However, before this theory can be applied, we need to know the phase response properties of the half-center oscillators that comprise the individual modules. Furthermore, to fully understand the mechanisms underlying phase-locking in these systems, we need to understand how the intrinsic dynamics of the cells and the coupling properties interact to produce the phase response properties of the half-center oscillator. To gain insight into these issues, we examine the phase response properties of idealized models of half-center oscillators. The dynamics of the cells in the half-center oscillator are described by either the leaky integrate-and-fire model or a general one-variable phase model; the coupling between the cells is described by either delta-function pulse coupling or current-based exponential synapses. We obtain analytical or semi-analytical solutions for the phase response curves (PRCs) of these idealized half-center oscillators, which show how the cells’ individual PRC and the coupling dynamics produce the PRC of the half-center oscillator. We compare our analytical results to numerical simulations of more realistic models and experimental data from the crayfish swimmeret system. We also discuss the implications for phase-locking in networks of half-center oscillators.

Half-center oscillator computational models: the influence of neuronal parameters

The rhythmic activity of the heartbeat neuronal network of the leech is based on pairs of inhibitory interneurons that make reciprocal spike-mediated and graded synapses across the ganglionic midline. Here we build upon our previous research, where we modeled such a pair of HN(4) reciprocally inhibitory interneurons, known as a half-center oscillator model [1]. We aim at investigating the changes in this model’s oscillatory activity and bursting characteristics based on cellular and synaptic parameters. To achieve this, we varied eight parameters in all combinations by using a brute-force approach, which resulted in a parameter space of 10,485,760 models. After changing a parameter, a model was run for 100 s to allow the system to establish stable activity, and then, it was run for another 100 s, from which the data were recorded and analyzed. We performed all the simulations and we built a SQL database table for their firing characteristics [2,3]. Now, we use the entire database to ask fundamental questions about the activity of half-center oscillators. First, we subdivide the models into those in which the component cells are intrinsically silent, spiking or bursting, and then, ask whether or not oscillators of these different types respond to parameter changes similarly. The results we have so far show that in approximately 36.46% (3,823,240 simulations) of the models both cells were silent, in 27.72% (2,906,249) both cells were spiking, and in 22.03% (2,310,359) both cells were bursting. The rest of the simulations (13.79% or 1,445,912) did not show symmetric activity in the two model cells. Out of the bursting models, in 18.96% (438,041) both cells showed irregular activity (Figure ​(Figure11 A), in 18.19% (420,307) the component cells produced spikeless plateau potentials (Figure ​(Figure11 B), in 8.18% (189,041) the two cells showed asymmetric bursting activity, and in 54.67% (1,262,970) both cells were bursting with standard bursting activity (Figure ​(Figure11 C). We will now use this last sample of bursting models and then the entire database to ask mechanistic questions about their alternating activity. We will be particularly interested in parameter changes which correspond to known neuromodulations such as the modulation of h current by myomodulin [4]. Figure 1 Half-center computational models.A. Irregular; B. Plateau; C. Regular.

Do conventional measures of intrinsic properties predict a neuron's behavior in a network?

We pharmacologically isolated STG neurons by blocking neuromodulatory input from descending pathways as well as local glutamatergic synaptic contacts. We measured the intrinsic properties of four types of STG neurons: dorsal gastric (DG), gastric mill (GM), lateral pyloric (LP) and pyloric dilator (PD) neurons. Creating half-center oscillators in hybrid circuits Neurons may be classified based on their morphology, gene expression, innervation patterns, receptor expression, electrical excitability and more. Specifically, electrical excitability has been used to classify neurons based on the idea that these properties confer information about each neuron's functional role. It isn't clear, however, whether any or all of these electrical properties predict or correlate to a neuron's behavior in a circuit. We are therefore examining the following questions: -To what extent do traditionally measured intrinsic properties predict how a neuron will behave in a network? -Can we create a more functionally relevant measure of intrinsic excitability?

Synchronization in a bursting half-center oscillator with slow-to-fast reciprocal inhibition

Synchronization in a bursting half-center oscillator with slow-to-fast reciprocal inhibition

Analyzing how varying neuronal parameters influence network activity using a database of computational models of a half-center oscillator

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