A comparison of resonance tuning with positive versus negative sensory feedback

We used a computational model of rhythmic movement to analyze how the connectivity of sensory feedback affects the tuning of a closed-loop neuromechanical system to the mechanical resonant frequency (ω_r). Our model includes a Matsuoka half-center oscillator for a central pattern generator (CPG) and a linear, one-degree-of-freedom system for a mechanical component. Using both an open-loop frequency response analysis and closed-loop simulations, we compared resonance tuning with four different feedback configurations as the mechanical resonant frequency, feedback gain, and mechanical damping varied. The feedback configurations consisted of two negative and two positive feedback connectivity schemes. We found that with negative feedback, resonance tuning predominantly occurred when ω_r was higher than the CPG’s endogenous frequency (ω_CPG). In contrast, with the two positive feedback configurations, resonance tuning only occurred if ω_r was lower than ω_CPG. Moreover, the differences in resonance tuning between the two positive (negative) feedback configurations increased with increasing feedback gain and with decreasing mechanical damping. Our results indicate that resonance tuning can be achieved with positive feedback. Furthermore, we have shown that the feedback configuration affects the parameter space over which the endogenous frequency of the CPG or resonant frequency the mechanical dynamics dominates the frequency of a rhythmic movement.     

Entrainment, Instability, Quasi-periodicity, and Chaos in a Compound Neural Oscillator

We studied the dynamical behavior of a class of compound central pattern generator (CPG) models consisting of a simple neural network oscillator driven by both constant and periodic inputs of varying amplitudes, frequencies, and phases. We focused on a specific oscillator composed of two mutually inhibiting types of neuron (inspiratory and expiratory neurons) that may be considered as a minimal model of the mammalian respiratory rhythm generator. The simulation results demonstrated how a simple CPG model— with a minimum number of neurons and mild nonlinearities— may reproduce a host of complex dynamical behaviors under various periodic inputs. In particular, the network oscillated spontaneously only when both neurons received adequate and proportionate constant excitations. In the presence of a periodic source, the spontaneous rhythm was overriden by an entrained oscillation of varying forms depending on the nature of the source. Stable entrained oscillations were inducible by two types of inputs: (1) anti-phase periodic inputs with alternating agonist-antagonist drives to both neurons and (2) a single periodic drive to only one of the neurons. In-phase inputs, which exert periodic drives of similar magnitude and phase relationships to both neurons, resulted in varying disruptions of the entrained oscillations including magnitude attenuation, harmonic and phase distortions, and quasi-periodic interference. In the absence of significant phasic feedback, chaotic motion developed only when the CPG was driven by multiple periodic inputs. Apneic episodes with repetitive alternation of active (intrinsic oscillation) and inactive (cessation of oscillation) states developed when the network was driven by a moderate periodic input of low frequency. %and amplitudes of intermediate strength, Similar results were demonstrated in other, more complex oscillator models (that is, half-center oscillator and three-phase respiratory network model). These theoretical results may have important implications in elucidating the mechanisms of rhythmogenesis in the mature and developing respiratory CPG as well as other compound CPGs in mammalian and invertebrate nervous systems.     

Recovery of Dynamics and Function in Spiking Neural Networks with Closed-Loop Control

There is a growing interest in developing novel brain stimulation methods to control disease-related aberrant neural activity and to address basic neuroscience questions. Conventional methods for manipulating brain activity rely on open-loop approaches that usually lead to excessive stimulation and, crucially, do not restore the original computations performed by the network. Thus, they are often accompanied by undesired side-effects. Here, we introduce delayed feedback control (DFC), a conceptually simple but effective method, to control pathological oscillations in spiking neural networks (SNNs). Using mathematical analysis and numerical simulations we show that DFC can restore a wide range of aberrant network dynamics either by suppressing or enhancing synchronous irregular activity. Importantly, DFC, besides steering the system back to a healthy state, also recovers the computations performed by the underlying network. Finally, using our theory we identify the role of single neuron and synapse properties in determining the stability of the closed-loop system.     

Cytosolic calcium oscillators

Many cells display oscillations in intracellular calcium resulting from the periodic release of calcium from intracellular reservoirs. Frequencies are varied, but most oscillations have periods ranging from 5 to 60 s. For any given cell, frequency can vary depending on external conditions, particularly the concentration of natural stimuli or calcium. This cytosolic calcium oscillator is particularly sensitive to those stimuli (neurotransmitters, hormones, growth factors) that hydrolyze phosphoinositides to give diacylglycerol and inositol 1,4,5-trisphosphate (Ins1,4,5P3). The ability of Ins1,4,5P3 to mobilize intracellular calcium is a significant feature of many of the proposed models that are used to explain oscillatory activity. Receptor-controlled oscillator models propose that there are complex feedback mechanisms that generate oscillations in the level of Ins1,4,5P3. Second messenger-controlled oscillator models demonstrate that the oscillator is a component of the calcium reservoir, which is induced to release calcium by a constant input of either Ins1,4,5P3 or calcium itself. In the latter case, the process of calcium-induced calcium release might be the basis of oscillatory activity in many cell types. The function of calcium oscillations is still unknown. Because oscillator frequency can vary with agonist concentration, calcium transients might be part of a frequency-encoded signaling system. When an external stimulus arrives at the cell surface the information is translated into a train of calcium spikes, i.e., the signal is digitized. Certain cells may then convey information by varying the frequency of this digital signal.     

Formal analysis of resonance entrainment by central pattern generator

The neuronal circuit controlling the rhythmic movements in animal locomotion is called the central pattern generator (CPG). The biological control mechanism appears to exploit mechanical resonance to achieve efficient locomotion. The objective of this paper is to reveal the fundamental mechanism underlying entrainment of CPGs to resonance through sensory feedback. To uncover the essential principle, we consider the simplest setting where a pendulum is driven by the reciprocal inhibition oscillator. Existence and properties of stable oscillations are examined by the harmonic balance method, which enables approximate but insightful analysis. In particular, analytical conditions are obtained under which harmonic balance predicts existence of an oscillation at a frequency near the resonance frequency. Our result reveals that the resonance entrainment can be maintained robustly against parameter perturbations through two distinct mechanisms: negative integral feedback and positive rate feedback.     

Emergence of oscillations in a model of weakly coupled two Bonhoeffer-van der Pol equations

Bifurcations of periodic solutions in a model of weakly coupled two Bonhoeffer-van der Pol equations are studied. The model realizes a half-center model with reciprocal inhibition, a typical model used in the field of neural motor control to account for the generation of alternating rhythmic bursts observed in motoneurons and spinal neural networks. Several oscillatory solutions such as in-phase, anti-phase as well as out-of-phase solutions emerge from the model's equilibrium as one of the parameters of the model changes. Among the variety of bifurcations exhibited by the model, we analyze Hopf bifurcations, by which several periodic solutions emerge, and illustrate generation mechanisms of alternating oscillations in the model.     

Autoreceptors do not regulate routinely neurotransmitter release: focus on adrenergic systems

The theory that neurotransmitter release is regulated locally at the individual terminals of neurons has achieved a rapid and seemingly secure status in our understanding of neuronal function both in the periphery and in the central nervous system. This concept of negative feedback control through the monitoring of the perineuronal concentration of previously released transmitter has been extended to a multiplicity of transmitters and utilized to explain the mechanisms of action of diverse classes of drugs, ranging from antihypertensives to antidepressants. It is my view that negative feedback by terminal and by somadendritic receptors cannot account for the existing body of experimental work. Analyses of the profiles of action of agonists and antagonists, and of the per pulse release of transmitter in the absence of drugs in a variety if peripheral organ systems, as well as in superfused brain slices, demonstrates the need for alternate interpretations of the available data. Evidence is provided that the actions of agonists to inhibit transmitter release and that of antagonists to enhance release occur at different cellular loci and that the purported unitary action of these two classes that is so central to the validity of presynaptic theory is unsupportable.     

TEMPORAL PATTERN OF LH SECRETION: REGULATION BY MULTIPLE ULTRADIAN OSCILLATORS VERSUS A SINGLE CIRCADIAN OSCILLATOR

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399–412, 2001)     

Temporal pattern of LH secretion: regulation by multiple ultradian oscillators versus a single circadian oscillator

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399-412, 2001)     

A platform for dynamic simulation and control of movement based on OpenSim and MATLAB

Numerical simulations play an important role in solving complex engineering problems and have the potential to revolutionize medical decision making and treatment strategies. In this paper, we combine the rapid model-based design, control systems and powerful numerical method strengths of MATLAB/Simulink with the simulation and human movement dynamics strengths of OpenSim by developing a new interface between the two software tools. OpenSim is integrated with Simulink using the MATLAB S-function mechanism, and the interface is demonstrated using both open-loop and closed-loop control systems. While the open-loop system uses MATLAB/Simulink to separately reproduce the OpenSim Forward Dynamics Tool, the closed-loop system adds the unique feature of feedback control to OpenSim, which is necessary for most human movement simulations. An arm model example was successfully used in both open-loop and closed-loop cases. For the open-loop case, the simulation reproduced results from the OpenSim Forward Dynamics Tool with root mean square (RMS) differences of 0.03° for the shoulder elevation angle and 0.06° for the elbow flexion angle. MATLAB's variable step-size integrator reduced the time required to generate the forward dynamic simulation from 7.1s (OpenSim) to 2.9s (MATLAB). For the closed-loop case, a proportional-integral-derivative controller was used to successfully balance a pole on model's hand despite random force disturbances on the pole. The new interface presented here not only integrates the OpenSim and MATLAB/Simulink software tools, but also will allow neuroscientists, physiologists, biomechanists, and physical therapists to adapt and generate new solutions as treatments for musculoskeletal conditions.     

The role of proprioception for motor learning in locust flight

In a muscle-specific flight simulator (simulator driven by muscle action potentials) locusts (Locusta migratoria) show motor learning by which steering performance of the closed-loop muscles is improved. The role of proprioceptive feedback for this motor learning has been studied. Closed-loop muscles were cut in order to disable proprioceptive feedback of their contractions. Since there are no proprioceptors within the muscles, this is a muscle-specific deafferentation. Cut muscles are still activated during flight and their action potentials can be used for controlling the flight simulator. With cut muscles in closed-loop, steering is less reliable as can be seen from the frequent oscillations of the yaw angle. However, periods of stable flight indicate that deafferented muscles are still, in principle, functional for steering. Open-loop yaw stimuli reveal that steering reactions in cut muscles are weaker and have a longer delay than intact muscles. This is responsible for the oscillations observed in closed-loop flight. Intact muscles can take over from cut muscles in order to re-establish stable closed-loop flight. This shows that proprioceptive mechanisms for learning are muscle specific. A hypothetical scheme is presented to explain the role of proprioception for motor learning.     

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.     

Mechanical and Electrical Oscillations in Cardiac Muscle of the Turtle

During contractures of the turtle ventricle rapid changes in length induce sinusoidal oscillations under isotonic conditions. They are due to delayed responses to stretching and release, which can be demonstrated also under isometric conditions. Oscillations of two distinct frequencies are produced under different conditions and are distinguished as high- and low-frequency oscillations. In depolarized muscles the frequency is such that the duration of one cycle is about the same as that of a normal twitch, while in high-Ca solutions the duration can be the same as in high-K solutions or about six times lower. As reported previously, twitches are followed by weak mechanical and electrical oscillations. Their frequency agrees with the high-frequency oscillations. The same effects can also be induced by stretching and release. It is suggested that the phenomena observed are due to feedback mechanisms which originate in the contractile mechanism. The high-frequency oscillations are similar to those observed previously in other muscles, particularly insect fibrillar muscle, and are not due to changes in Ca concentration. The other mechanisms involve the membrane and possibly the intracellular Ca stores.     

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.     

Studying metabolic regulation in human muscle

Phosphorus magnetic resonance spectroscopy, despite some limitations, is a valuable non-invasive window on muscle metabolism in vivo, particularly oxidative ATP synthesis. A number of experiments have shown this to be dominated by closed-loop feedback mechanisms: a well-known model posits regulation by ADP, but there are others, difficult to distinguish experimentally. Moreover the contribution of open-loop control mechanisms ('feed forward' or 'parallel activation') in vivo remains controversial. Progress will require more precise data, better integrated with other measurements (e.g. muscle oxygenation), and improvement of the conceptual tools appropriate to such studies, where data are limited and steady-state assumptions frequently inapplicable.     

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.     

Spatial and temporal aspects of cell signalling

As new techniques are developed to measure intracellular messengers it becomes increasingly apparent that there is a remarkable spatial and temporal organization of cell signalling. Cells possess a small discrete hormone-sensitive pool of inositol lipid. In some cells such as Xenopus oocytes and Limulus photoreceptors this phosphoinositide signalling system is highly concentrated in one region of the cell, so establishing localized calcium gradients. Another example is the hydrolysis of inositol lipids in eggs at the point of sperm entry resulting in a localized increase in Ins(1,4,5)P3 and calcium which spreads like a wave throughout the egg. In hamster eggs this burst of calcium at fertilization recurs at 1-3 min intervals for over 100 min, a particularly dramatic example of spontaneous activity. Spontaneous oscillations in intracellular calcium exist in many different cell types and are often induced by agonists that hydrolyse inositol lipids. We have made a distinction between oscillations that are approximately sinusoidal and occur at a higher frequency where free calcium is probably continuously involved in the oscillatory cycle and those where calcium falls to resting levels for many seconds between transients. In the former case, the oscillations are thought to be induced through a cytoplasmic oscillator based on the phenomenon of calcium-induced calcium release. Such oscillations can be induced in Xenopus oocytes after injection with Ins(1,4,5)P3. A receptor-controlled oscillator based on the periodic formation of Ins(1,4,5)P3 is probably responsible for the generation of the widely spaced calcium transients. The function of such calcium oscillations is currently unknown. They may be a reflection of the feedback interactions that operate to control intracellular calcium. Another possibility emerged from observations that in some cells the frequency of calcium oscillations varied with agonist concentration, suggesting that cells might employ these oscillations as a way of encoding information. One advantage of using such a frequency-dependent mechanism may lie in an increase in fidelity, especially at low agonist concentrations. Whatever these functions might be, it is clear that uncovering the mechanisms responsible for such oscillatory activity will greatly enhance our understanding of the relation between the phosphoinositides and calcium signalling.     

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.     

Reafferent control of optomotor yaw torque inDrosophila melanogaster

Summary In the flight simulator the optomotor response ofDrosophila melanogaster does not operate as a simple feedback loop. Reafferent and exafferent motion stimuli are processed differently. Under open-loop conditions responses to motion are weaker than under closed-loop conditions. It takes the fly less than 100 ms to distinguish reafferent from exafferent motion. In closed-loop conditions, flies constantly generate torque fluctuations leading to small-angle oscillations of the panorama. This reafferent motion stimulus facilitates the response to exafferent motion but does not itself elicit optomotor responses. Reafference control appears to be directionally selective: while a displacement of the patternm by as little as 0.1° against the expected direction leads to a fast syndirectional torque response, displacements in the expected direction have no comparable effect. Based on the behavior of the mutantrol sol, which under open-loop conditions is directionally motion-blind but in closed-loop conditions still performs optomotor balance, a model is proposed in which the fly's endogenous torque fluctuations are an essential part of the course control process. It is argued that the model may also account for wild type optomotor balance in the flight simulator.