Estimation of Coupling Strength in Regenerated Lamprey Spinal Cords Based on a Stochastic Phase Model

We present a simple stochastic model of two coupled phase oscillators and a method of fitting the model to experimental spike-train data or to sequences of burst times. We apply the method to data from lesioned isolated lamprey spinal cords. The remaining tracts at the lesion site are either regenerated medial tracts, regenerated lateral tracts, control medial tracts, or control lateral tracts. We show that regenerated tracts on average provide significantly weaker coupling than control tracts. We compare our model-dependent estimate of coupling strength to a measure of coordination based on the size of deflections in the spike-train cross-correlation histogram (CCH). Using simulated data, we show that our estimates are able to detect changes in coupling strength that do not change the size of deflections in the CCH. Our estimates are also more resistant to changes in the level of dynamic noise and to changes in relative oscillator frequency than is the CCH. In simulations with high levels of dynamic noise and in one experimental preparation, we are able detect significant coupling strength although there are no significant deflections in the CCH.     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

Hard-wired central pattern generators for quadrupedal locomotion

Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.     

Intersegmental coordinating system of the lamprey central pattern generator for locomotion

Summary In the lamprey,Ichthyomyzon unicuspis, the wave of activity required for normal swimming movements can be generated by a central pattern generator (CPG) residing in the spinal cord. A constant phase coupling between spinal segments can be organized by intersegmental coordinating neurons intrinsic to the cord. The rostral and caudal segmental oscillators of the CPG have different preferred frequencies when separated from each other. Therefore the system must maintain the segmental oscillators of the locomotor CPG at a single common frequency and with the proper relative timing. Using selective lesions and a split-bath, it is demonstrated that the coordinating system is comprised of at least 3 subsystems, short-axon systems in the lateral and medial tracts and a long axon system in the lateral tracts. Each alone can sustain relatively stable coordinated activity.Abbreviations CPG central pattern generator - NMDA N-methyl-D-aspartate - VR ventral root     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

The Calculation of Frequency-Shift Functions for Chains of Coupled Oscillators, with Application to a Network Model of the Lamprey Locomotor Pattern Generator

Chains of coupled limit-cycle oscillators are considered, in which the coupling is assumed to be weak and only between adjacent oscillators. For such a system the change in frequency of an oscillator due to the coupling can be expressed, up to first order in thecoupling strength, by functions that depend only on the phase difference between the coupled oscillators. In this article a numerical algorithm is developed for the evaluation of these functions (the H-functions) in terms of a single oscillator and the interactions between coupled oscillators. The technique is applied to a connectionist model for the locomotor pattern generator in the lamprey spinal cord.An H-function so derived is compared to a function derived empirically(the C-function) from simulations of the same system. The phase lagsthat develop between adjacent oscillators in a simulated chain are compared with those predicted theoretically, and it is shown that coupling thatis functionally strong is nonetheless weak enough to behave as predicted.     

Neural network simulations of coupled locomotor oscillators in the lamprey spinal cord

The segmental locomotor network in the lamprey spinal cord was simulated on a computer using a connectionist-type neural network. The cells of the network were identical except for their excitatory levels and their synaptic connections. The synaptic connections used were based on previous experimental work. It was demonstrated that the connectivity of the circuit is capable of generating oscillatory activity with the appropriate phase relations among the cells. Intersegmental coordination was explored by coupling two identical segmental networks using only the cells of the network. Each of the possible couplings of a bilateral pair of cells in one oscillator with a bilateral pair of cells in the other oscillator produced stable phase locking of the two oscillators. The degree of phase difference was dependent upon synaptic weight, and the operating range of synaptic weights varied among the pairs of connections. The coupling was tested using several criteria from experimental work on the lamprey spinal cord. Coupling schemes involving several pairs of connecting cells were found which 1) achieved steadystate phase locking within a single cycle, 2) exhibited constant phase differences over a wide range of cycle periods, and 3) maintained stable phase locking in spite of large differences in the intrinsic frequencies of the two oscillators. It is concluded that the synaptic connectivity plays a large role in producing oscillations in this network and that it is not necessary to postulate a separate set of coordinating neurons between oscillators in order to achieve appropriate phase coupling.     

Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control

Like human walking, passive dynamic walking—i.e. walking down a slope with no actuation except gravity—is energy efficient by exploiting the natural dynamics. In the animal world, neural oscillators termed central pattern generators (CPGs) provide the basic rhythm for muscular activity in locomotion. We present a CPG model, which automatically tunes into the resonance frequency of the passive dynamics of a bipedal walker, i.e. the CPG model exhibits resonance tuning behavior. Each leg is coupled to its own CPG, controlling the hip moment of force. Resonance tuning above the endogenous frequency of the CPG—i.e. the CPG’s eigenfrequency—is achieved by feedback of both limb angles to their corresponding CPG, while integration of the limb angles provides resonance tuning at and below the endogenous frequency of the CPG. Feedback of the angular velocity of both limbs to their corresponding CPG compensates for the time delay in the loop coupling each limb to its CPG. The resonance tuning behavior of the CPG model allows the gait velocity to be controlled by a single parameter, while retaining the energy efficiency of passive dynamic walking.     

Resonance tuning in a neuro-musculo-skeletal model of the forearm

In rhythmic movements, humans activate their muscles in a robust and energy efficient way. These activation patterns are oscillatory and seem to originate from neural networks in the spinal cord, called central pattern generators (CPGs). Evidence for the existence of CPGs was found for instance in lampreys, cats and rats. There are indications that CPGs exist in humans as well, but this is not proven yet. Energy efficiency is achieved by resonance tuning: the central nervous system is able to tune into the resonance frequency of the limb, which is determined by the local reflex gains. The goal of this study is to investigate if the existence of a CPG in the human spine can explain the resonance tuning behavior, observed in human rhythmic limb movement. A neuro-musculo-skeletal model of the forearm is proposed, in which a CPG is organized in parallel to the local reflexloop. The afferent and efferent connections to the CPG are based on clues about the organization of the CPG, found in literature. The model is kept as simple as possible (i.e., lumped muscle models, groups of neurons are lumped into half-centers, simple reflex model), but incorporates enough of the essential dynamics to explain behavior—such as resonance tuning—in a qualitative way. Resonance tuning is achieved above, at and below the endogenous frequency of the CPG in a highly non-linear neuro- musculo-skeletal model. Afferent feedback of muscle lengthening to the CPG is necessary to accomplish resonance tuning above the endogenous frequency of the CPG, while feedback of muscle velocity is necessary to compensate for the phase lag, caused by the time delay in the loop coupling the limb to the CPG. This afferent feedback of muscle lengthening and velocity represents the Ia and II fibers, which—according to literature—is the input to the CPG. An internal process of the CPG, which integrates the delayed muscle lengthening and feeds it to the half-center model, provides resonance tuning below the endogenous frequency. Increased co-contraction makes higher movement frequencies possible. This agrees with studies of rhythmic forearm movements, which have shown that co-contraction increases with movement frequency. Robustness against force perturbations originates mainly from the CPG and the local reflex loop. The CPG delivers an increasing part of the necessary muscle activation for increasing perturbation size. As far as we know, the proposed neuro-musculo-skeletal model is the first that explains the observed resonance tuning in human rhythmic limb movement.     

Fast subsystem bifurcations in strongly coupled heterogeneous collections of excitable cells

A continuum model for a heterogeneous collection of excitable cells electrically coupled through gap junctions is introduced and analysed using spatial averaging, asymptotic and numerical techniques. Heterogeneity is modelled by imposing a spatial dependence on parameters which define the single cell model and a diffusion term is used to model the gap junction coupling. For different parameter values, single cell models can exhibit bursting, beating and a myriad of other complex oscillations. A procedure for finding asymptotic estimates of the thresholds between these (synchronous) behaviors in the cellular aggregates is described for the heterogeneous case where the coupling strength is strong. This procedure is tested on a model of a strongly coupled heterogeneous collection of bursting and beating cells. Since isolated pancreatic β-cells have been observed to both burst and beat, this test of the spatial averaging techniques provides a possible explanation to measured discrepancies between the electrical activities of isolated β-cells and coupled collections (islets) of β-cells. This work was supported by the National Science Foundation Grant DMS-97-04-966.     

Quantifying the strength of the linear causal coupling in closed loop interacting cardiovascular variability signals

 The coherence function measures the amount of correlation between two signals x and y as a function of the frequency, independently of their causal relationships. Therefore, the coherence function is not useful in deciding whether an open-loop relationship between x and y is set (x acts on y, but the reverse relationship is prevented) or x and y interact in a closed loop (x affects y, and vice versa). This study proposes a method based on a bivariate autoregressive model to derive the strength of the causal coupling on both arms of a closed loop. The method exploits the definition of causal coherence. After the closed-loop identification of the model coefficients, the causal coherence is calculated by switching off separately the feedback or the feedforward path, thus opening the closed loop and fixing causality. The method was tested in simulations and applied to evaluate the degree of the causal coupling between two variables known to interact in a closed loop mainly at a low frequency (LF, around 0.1 Hz) and at a high frequency (HF, at the respiratory rate): the heart period (RR interval) and systolic arterial pressure (SAP). In dogs at control, the RR interval and the SAP are highly correlated at HF. This coupling occurs in the causal direction from the RR interval to the SAP (the mechanical path), while the coupling on the reverse causal direction (the baroreflex path) is not significant, thus pointing out the importance of the direct effects of respiration on the RR interval. Total baroreceptive denervation, by opening the closed loop at the level of the influences of SAP on RR interval, does not change these results. In elderly healthy men at rest, the RR interval and SAP are highly correlated at the LF and the HF. At the HF, a significant coupling in both causal directions is found, even though closed-loop interactions are detected in few cases. At the LF, the link on the baroreflex pathway is negligible with respect to that on the reverse mechanical one. In heart transplant recipients, in which SAP variations do not cause RR interval changes as a result of the cardiac denervation, the method correctly detects a significant coupling only on the pathway from the RR interval to the SAP. Received: 28 June 2001 / Accepted in revised form: 23 October 2001     

Diffusively coupled bursters: Effects of cell heterogeneity

The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity. An erratum to this article is available at .     

Multivariable harmonic balance analysis of the neuronal oscillator for leech swimming

Biological systems, and particularly neuronal circuits, embody a very high level of complexity. Mathematical modeling is therefore essential for understanding how large sets of neurons with complex multiple interconnections work as a functional system. With the increase in computing power, it is now possible to numerically integrate a model with many variables to simulate behavior. However, such analysis can be time-consuming and may not reveal the mechanisms underlying the observed phenomena. An alternative, complementary approach is mathematical analysis, which can demonstrate direct and explicit relationships between a property of interest and system parameters. This paper introduces a mathematical tool for analyzing neuronal oscillator circuits based on multivariable harmonic balance (MHB). The tool is applied to a model of the central pattern generator (CPG) for leech swimming, which comprises a chain of weakly coupled segmental oscillators. The results demonstrate the effectiveness of the MHB method and provide analytical explanations for some CPG properties. In particular, the intersegmental phase lag is estimated to be the sum of a nominal value and a perturbation, where the former depends on the structure and span of the neuronal connections and the latter is roughly proportional to the period gradient, communication delay, and the reciprocal of the intersegmental coupling strength.

Phase Resetting Curves Allow for Simple and Accurate Prediction of Robust N:1 Phase Locking for Strongly Coupled Neural Oscillators

Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves. Our theoretical methods are general in the sense that no assumptions about the strength of coupling, type of synaptic coupling, and model are made. These methods were then tested using two reciprocally inhibitory Wang and Buzsáki model neurons. The existence of bands of 2:1, 3:1, 4:1, and 5:1 phase locking in the relative frequency parameter space was predicted correctly, as was the phase of the slow neuron's spike within the cycle of the fast neuron in which it occurred. For weak coupling the bands are very narrow, but strong coupling broadens the bands. The predictions of the pulse coupled method agreed with weak coupling methods in the weak coupling regime, but extended predictability into the strong coupling regime. We show that our prediction method generalizes to pairs of neural oscillators coupled through excitatory synapses, and to networks of multiple oscillatory neurons. The main limitation of the method is the central assumption that the effect of each input dies out before the next input is received.     

Analysis of a distributed model of leg coordination

Using tools from discrete dynamical systems theory, we begin a systematic analysis of a distributed model of leg coordination with both biological and robotic applications. In this paper, we clarify the role of individual coordination mechanisms by studying a system of two leg oscillators coupled in one direction by each of the three major mechanisms that have been described for the stick insect Carausius morosus. For each mechanism, we derive analytical return maps, and analyze the behavior of these return maps under iteration in order to determine the asymptotic phase relationship between the two legs. We also derive asymptotic relative phase densities for each mechanism and compare these densities to those obtained from numerical simulations of the model. Our analysis demonstrates that, although each of these mechanisms can individually compress a range of initial conditions into a narrow band of relative phase, this asymptotic relative phase relationship is, in general, only neutrally stable. We also show that the nonlinear dependence of relative phase on walking speed along the body in the full hexapod model can be explained by our analysis. Finally, we provide detailed parameter charts of the range of behavior that each mechanism can produce as coupling strength and walking speed are varied. Received: 22 June 1999 / Accepted in revised form: 7 September 1999     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Automatic classification of interference patterns in driven event series: application to single sympathetic neuron discharge forced by mechanical ventilation

This study proposes a method for the automatic classification of nonlinear interactions between a strictly periodical event series modelling the activity of an exogenous oscillator working at a fixed and well-known rate and an event series modelling the activity of a self-sustained oscillator forced by the exogenous one. The method is based on a combination of several well-known tools (probability density function of the cyclic relative phase, probability density function of the count of forced events per forcing cycle, conditional entropy of the cyclic relative phase sequence and a surrogate data approach). Classification is reached via a sequence of easily applicable decision rules, thus rendering classification virtually user-independent and fully reproducible. The method classifies four types of dynamics: full uncoupling, quasiperiodicity, phase locking and aperiodicity. In the case of phase locking, the coupling ratio (i.e. n:m) and the strength of the coupling are calculated. The method, validated on simulations of simple and complex phase-locking dynamics corrupted by different levels of noise, is applied to data derived from one anesthetized and artificially ventilated rat to classify the nonlinear interactions between mechanical ventilation and: (1) the discharges of two (contemporaneously recorded) single postganglionic sympathetic neurons innervating the caudal ventral artery in the tail and (2) arterial blood pressure. Under central apnea, the activity of the underlying sympathetic oscillators is perturbed by means of five different lung inflation rates (0.58, 0.64, 0.76, 0.95, 1.99 Hz). While ventilation and arterial pressure are fully uncoupled, ventilation is capable of phase locking sympathetic discharges, thus producing 40% of phase-locked patterns (one case of 2:5, 1:1, 3:2 and 2:2) and 40% of aperiodic dynamics. In the case of phase-locked patterns, the coupling strength is low, thus demonstrating that this pattern is sliding. Non-stationary interactions are observed in 20% of cases. The two discharges behave differently, suggesting the presence of a population of sympathetic oscillators working at different frequencies.     

Coupled Van der Pol oscillators — A model of excitatory and inhibitory neural interactions

A system of mutually coupled Van der Pol equations is derived from an extended version of the Wilson and Cowan model for the dynamics of a number of excitatory and inhibitory neural subsets. In the lowest order of approximation, interactions between excitatory and inhibitory subsets appear as linear elastic coupling, while those within and between excitatory and excitatory subsets appear as nonlinear frictional coupling. The case of two coupled oscillators is investigated by the method of averaging and the stability conditions for two mode oscillations are obtained. Internal resonance is also discussed briefly in the case of identical oscillators.     

Neural networks for computing eigenvalues and eigenvectors

We consider two electrically coupled oscillators described by modified Fitzhugh-Nagumo equations. We study the relative influence of the individual cellular characteristics and the electrical coupling on the behavior of the coupled system. We show that, for similar oscillators, the load effect of the slow oscillator increases with the coupling strength. We prove that an asymmetry between the uncoupled bursters can accelerate the system with respect to the free cells, this effect depending on the characteristics of the coupling.On leave from Centre de Physique Théoruique (UPR A0014 CNRS), Palaiseau, France