Biological clockwork underlying adaptive rhythmic movements.

Owing to the complexity of neuronal circuits, precise mathematical descriptions of brain functions remain an elusive ambition. A more modest focus of many neuroscientists, central pattern generators, are more tractable neuronal circuits specialized to generate rhythmic movements, including locomotion. The relative simplicity and well-defined motor functions of these circuits provide an opportunity for uncovering fundamental principles of neuronal information processing. Here we present the culmination of mathematical analysis that captures the adaptive behaviors emerging from interactions between a central pattern generator, the body, and the physical environment during locomotion. The biologically realistic model describes the undulatory motions of swimming leeches with quantitative accuracy and, without further parameter tuning, predicts the sweeping changes in oscillation patterns of leeches undulating in air or swimming in high-viscosity fluid. The study demonstrates that central pattern generators are capable of adapting oscillations to the environment through sensory feedback, but without guidance from the brain.

The brain matters: effects of descending signals on motor control.

The ability of nerve cords and spinal cords to exhibit fictive rhythmic locomotion in the absence of the brain is well-documented in numerous species. Although the brain is important for modulating the fictive motor output, it is broadly assumed that the functional properties of neuronal circuits identified in simplified preparations are conserved with the brain attached. We tested this assumption by examining the properties of a novel interneuron recently identified in the leech (Hirudo verbana) nerve cord. This neuron, cell E21, initiates and drives stereotyped fictive swimming activity in preparations of the isolated leech nerve cord deprived of the head brain. We report that, contrary to expectation, the motor output generated when cell E21 is stimulated in preparations with the brain attached is highly variable. Swim frequency and episode duration are increased in some of these preparations and decreased in others. Cell E21 controls swimming, in part, via excitatory synaptic interactions with cells 204, previously identified gating neurons that reliably initiate and strongly enhance leech swimming activity when the brain is absent. We found that in preparations with the brain present, the magnitude of the synaptic interaction from cell E21 to cell 204 is reduced by 50% and that cell 204-evoked responses also were highly variable. Intriguingly, most of this variability disappeared in semi-intact preparations. We conclude that neuronal circuit properties identified in reduced preparations might be fundamentally altered from those that occur in more physiological conditions.

Forced desynchronization of activity rhythms in a model of chronic jet lag in mice.

We studied locomotor activity rhythms of C57/Bl6 mice under a chronic jet lag (CJL) protocol (ChrA(6/2) ), which consisted of 6-hour phase advances of the light-dark schedule (LD) every 2 days. Through periodogram analysis, we found 2 components of the activity rhythm: a short-period component (21.01 ± 0.04 h) that was entrained by the LD schedule and a long-period component (24.68 ± 0.26 h). We developed a mathematical model comprising 2 coupled circadian oscillators that was tested experimentally with different CJL schedules. Our simulations suggested that under CJL, the system behaves as if it were under a zeitgeber with a period determined by (24 - [phase shift size/days between shifts]). Desynchronization within the system arises according to whether this effective zeitgeber is inside or outside the range of entrainment of the oscillators. In this sense, ChrA(6/2) is interpreted as a (24 - 6/2 = 21 h) zeitgeber, and simulations predicted the behavior of mice under other CJL schedules with an effective 21-hour zeitgeber. Animals studied under an asymmetric T = 21 h zeitgeber (carried out by a 3-hour shortening of every dark phase) showed 2 activity components as observed under ChrA(6/2): an entrained short-period (21.01 ± 0.03 h) and a long-period component (23.93 ± 0.31 h). Internal desynchronization was lost when mice were subjected to 9-hour advances every 3 days, a possibility also contemplated by the simulations. Simulations also predicted that desynchronization should be less prevalent under delaying than under advancing CJL. Indeed, most mice subjected to 6-hour delay shifts every 2 days (an effective 27-hour zeitgeber) displayed a single entrained activity component (26.92 ± 0.11 h). Our results demonstrate that the disruption provoked by CJL schedules is not dependent on the phase-shift magnitude or the frequency of the shifts separately but on the combination of both, through its ratio and additionally on their absolute values. In this study, we present a novel model of forced desynchronization in mice under a specific CJL schedule; in addition, our model provides theoretical tools for the evaluation of circadian disruption under CJL conditions that are currently used in circadian research.

Mechanisms underlying rhythmic locomotion: interactions between activation, tension and body curvature waves.

Undulatory animal locomotion arises from three closely related propagating waves that sweep rostrocaudally along the body: activation of segmental muscles by motoneurons (MNs), strain of the body wall, and muscle tension induced by activation and strain. Neuromechanical models that predict the relative propagation speeds of neural/muscle activation, muscle tension and body curvature can reveal crucial underlying control features of the central nervous system and the power-generating mechanisms of the muscle. We provide an analytical explanation of the relative speeds of these three waves based on a model of neuromuscular activation and a model of the body-fluid interactions for leech anguilliform-like swimming. First, we deduced the motoneuron spike frequencies that activate the muscle and the resulting muscle tension during swimming in intact leeches from muscle bending moments. Muscle bending moments were derived from our video-recorded kinematic motion data by our body-fluid interaction model. The phase relationships of neural activation and muscle tension in the strain cycle were then calculated. Our study predicts that the MN activation and body curvature waves have roughly the same speed (the ratio of curvature to MN activation speed ≈0.84), whereas the tension wave travels about twice as fast. The high speed of the tension wave resulting from slow MN activation is explained by the multiplicative effects of MN activation and muscle strain on tension development. That is, the product of two slower waves (activation and strain) with appropriate amplitude, bias and phase can generate a tension wave with twice the propagation speed of the factors. Our study predicts that (1) the bending moment required for swimming is achieved by minimal MN spike frequency, rather than by minimal muscle tension; (2) MN activity is greater in the mid-body than in the head and tail regions; (3) inhibitory MNs not only accelerate the muscle relaxation but also reduce the intrinsic tonus tension during one sector of the swim cycle; and (4) movements of the caudal end are passive during swimming. These predictions await verification or rejection through further experiments on swimming animals.

Specialized brain regions and sensory inputs that control locomotion in leeches.

Locomotor systems are often controlled by specialized cephalic neurons and undergo modulation by sensory inputs. In many species, dedicated brain regions initiate and maintain behavior and set the duration and frequency of the locomotor episode. In the leech, removing the entire head brain enhances swimming, but the individual roles of its components, the supra- and subesophageal ganglia, in the control of locomotion are unknown. Here we describe the influence of these two structures and that of the tail brain on rhythmic swimming in isolated nerve cord preparations and in nearly intact leeches suspended in an aqueous, "swim-enhancing" environment. We found that, in isolated preparations, swim episode duration and swim burst frequency are greatly increased when the supraesophageal ganglion is removed, but the subesophageal ganglion is intact. The prolonged swim durations observed with the anterior-most ganglion removed were abolished by removal of the tail ganglion. Experiments on the nearly intact leeches show that, in these preparations, the subesophageal ganglion acts to decrease cycle period but, unexpectedly, also decreases swim duration. These results suggest that the supraesophageal ganglion is the primary structure that constrains leech swimming; however, the control of swim duration in the leech is complex, especially in the intact animal.

Modeling two-oscillator circadian systems entrained by two environmental cycles.

Several experimental studies have altered the phase relationship between photic and non-photic environmental, 24 h cycles (zeitgebers) in order to assess their role in the synchronization of circadian rhythms. To assist in the interpretation of the complex activity patterns that emerge from these "conflicting zeitgeber" protocols, we present computer simulations of coupled circadian oscillators forced by two independent zeitgebers. This circadian system configuration was first employed by Pittendrigh and Bruce (1959), to model their studies of the light and temperature entrainment of the eclosion oscillator in Drosophila. Whereas most of the recent experiments have restricted conflicting zeitgeber experiments to two experimental conditions, by comparing circadian oscillator phases under two distinct phase relationships between zeitgebers (usually 0 and 12 h), Pittendrigh and Bruce compared eclosion phase under 12 distinct phase relationships, spanning the 24 h interval. Our simulations using non-linear differential equations replicated complex non-linear phenomena, such as "phase jumps" and sudden switches in zeitgeber preferences, which had previously been difficult to interpret. Our simulations reveal that these phenomena generally arise when inter-oscillator coupling is high in relation to the zeitgeber strength. Manipulations in the structural symmetry of the model indicated that these results can be expected to apply to a wide range of system configurations. Finally, our studies recommend the use of the complete protocol employed by Pittendrigh and Bruce, because different system configurations can generate similar results when a "conflicting zeitgeber experiment" incorporates only two phase relationships between zeitgebers.

Mechanisms underlying rhythmic locomotion: dynamics of muscle activation.

We have studied the dynamical properties of tension development in leech longitudinal muscle during swimming. A new method is proposed for modeling muscle properties under functionally relevant conditions where the muscle is subjected to both periodic activation and rhythmic length changes. The 'dual-sinusoid' experiments were conducted on preparations of leech nerve cord and body wall. The longitudinal muscle was activated periodically by injection of sinusoidal currents into an identified motoneuron. Simultaneously, sinusoidal length changes were imposed on the body wall with prescribed phase differences (12 values equally spaced over 2π radians) with respect to the current injection. Through the singular value decomposition of appropriately constructed tension data matrices, the leech muscle was found to have a multiplicative structure in which the tension was expressed as the product of activation and length factors. The time courses of activation and length factors were determined from the tension data and were used to develop component models. The proposed modeling method is a general one and is applicable to contractile elements for which the effects of series elasticity are negligible.

Local-distributed integration by a novel neuron ensures rapid initiation of animal locomotion.

Animals are adapted to respond quickly to threats in their environment. In many invertebrate and some vertebrate species, the evolutionary pressures have resulted in rapidly conducting giant axons, which allow short response times. Although neural circuits mediating escape behavior are identified in several species, little attention has been paid to this behavior in the medicinal leech, a model organism whose neuronal circuits are well known. We present data that suggest an alternative to giant axons for the rapid initiation of locomotion. A novel individual neuron, cell E21, appears to be one mediator of this short-latency action in the leech. In isolated nerve cord and semi-intact preparations, cell E21 excitation initiates and extends swimming and reduces the cycle period. The soma of this cell is located caudally, but its axon extends nearly the entire length of the nerve cord. We found that cell E21 fires impulses following local sensory inputs anywhere along the body and makes excitatory synapses onto the gating cells that drive swimming behavior. These distributed input-output sites minimize the distance information travels to initiate swimming behavior, thus minimizing the latency between sensory input and motor output. We propose that this single cell E21 functions to rapidly initiate or modulate locomotion through its distributed synaptic connections.

Neuronal control of swimming behavior: comparison of vertebrate and invertebrate model systems.

Swimming movements in the leech and lamprey are highly analogous, and lack homology. Thus, similarities in mechanisms must arise from convergent evolution rather than from common ancestry. Despite over 40 years of parallel investigations into this annelid and primitive vertebrate, a close comparison of the approaches and results of this research is lacking. The present review evaluates the neural mechanisms underlying swimming in these two animals and describes the many similarities that provide intriguing examples of convergent evolution. Specifically, we discuss swim initiation, maintenance and termination, isolated nervous system preparations, neural-circuitry, central oscillators, intersegmental coupling, phase lags, cycle periods and sensory feedback. Comparative studies between species highlight mechanisms that optimize behavior and allow us a broader understanding of nervous system function.

Corrections to the theory and the optimal line in the swimming diagram of Taylor (1952).

The analysis of undulatory swimming gaits requires knowledge of the fluid forces acting on the animal body during swimming. In his classical 1952 paper, Taylor analysed this problem using a 'resistive-force' theory. The theory was used to characterize the undulatory gaits that result in the smallest energy dissipation to the fluid for a given swim velocity. The optimal gaits thus found were compared with data recorded from movies of a snake and a leech swimming. This report identifies and corrects a mathematical error in Taylor's paper, showing that his theory applies even better to animals of circular cross section.

Dissociation of circadian and light inhibition of melatonin release through forced desynchronization in the rat.

Pineal melatonin release exhibits a circadian rhythm with a tight nocturnal pattern. Melatonin synthesis is regulated by the master circadian clock within the hypothalamic suprachiasmatic nucleus (SCN) and is also directly inhibited by light. The SCN is necessary for both circadian regulation and light inhibition of melatonin synthesis and thus it has been difficult to isolate these two regulatory limbs to define the output pathways by which the SCN conveys circadian and light phase information to the pineal. A 22-h light-dark (LD) cycle forced desynchrony protocol leads to the stable dissociation of rhythmic clock gene expression within the ventrolateral SCN (vlSCN) and the dorsomedial SCN (dmSCN). In the present study, we have used this protocol to assess the pattern of melatonin release under forced desynchronization of these SCN subregions. In light of our reported patterns of clock gene expression in the forced desynchronized rat, we propose that the vlSCN oscillator entrains to the 22-h LD cycle whereas the dmSCN shows relative coordination to the light-entrained vlSCN, and that this dual-oscillator configuration accounts for the pattern of melatonin release. We present a simple mathematical model in which the relative coordination of a single oscillator within the dmSCN to a single light-entrained oscillator within the vlSCN faithfully portrays the circadian phase, duration and amplitude of melatonin release under forced desynchronization. Our results underscore the importance of the SCN's subregional organization to both photic input processing and rhythmic output control.

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.

Muscle function in animal movement: passive mechanical properties of leech muscle.

We investigated passive properties of leech body wall as part of a larger project to understand better mechanisms that control locomotion and to establish mathematical models that predict such dynamical behavior. In tests of length-tension relationships in 2-segment-long preparations of body wall through step-stretch manipulations (step size = 1 mm), we discovered that these relationships are nonlinear, with significant hysteresis, even for the relatively small changes in length that occur during swimming. We developed a mathematical model comprising three nonlinear springs, two in series with nonlinear dashpots that describe well the tension statics and dynamics for step-stretch experiments. This model suggested that body wall dynamics are slow enough to be neglected when predicting the tension generated by imposed sinusoidal length changes (about +/-10% of nominal) at 1-3 Hz, mimicking swimming. We derived a static model, comprising one nonlinear spring, which predicts sinusoidal data accurately, even when preparations were exposed to serotonin (0.1-10 microM). Preparations bathed in saline-serotonin had significantly reduced steady-state and peak tensions, without alterations in tension dynamics. Anesthetizing preparations (8% ethanol) reduced body wall tension by 77%, indicating that passive tension in the obliquely striated longitudinal muscles of leeches results primarily from a resting tonus.

Leech locomotion: swimming, crawling, and decisions.

Research on the neuronal control of locomotion in leeches spans almost four decades. Recent advances reviewed here include discoveries that: (1) interactions between multiple hormones modulate initiation of swimming; (2) stretch receptors associated with longitudinal muscles interact with the central oscillator circuit via electrical junctions; (3) intersegmental interactions, according to theoretical analyses, must be relatively weak compared to oscillator interactions within ganglia; and (4) multiple interacting neurons control the expression of alternative modes of locomotion. The innovative techniques that facilitated these advances include optical recording of membrane potential changes, simultaneous intracellular injection of high and low molecular weight fluorescent dyes, and detailed modeling via an input-output systems engineering approach.

Characterization of central axon terminals of putative stretch receptors in leeches.

Sensory feedback from stretch receptors, neurons that detect position or tension, is crucial for generating normal, robust locomotion. Among the eight pairs of putative stretch receptors associated with longitudinal muscles in midbody segments of medicinal leeches, only the ventral stretch receptor has been characterized in detail. To achieve the identification of all such receptors, we penetrated large axons in the nerve roots of nerve cords from adult leeches with dye-filled (Alexa Fluor hydrazide) electrodes. We identified the terminal arborizations of two additional putative stretch receptors with axons in anterior nerve roots and four more such receptors with axons in posterior roots of midbody ganglia. The axons are nonspiking and are individually identifiable by their entry point into the CNS; their projections within the neuropile; the pattern, extent, and orientation of their terminal branches; and the characteristics of small "spike-like" events. At least two of these axons undergo membrane potential oscillations that are phase locked to the swimming rhythm expressed in nerve cord-body wall preparations and, at a different phase angle, also in isolated nerve cords. Thus the membrane potentials of at least two axons are phasically modulated by the periphery and hence could provide cycle-by-cycle sensory input to coordinate swimming activity. One of these neurons has a soma associated with the dorsal body wall and hence is a putative stretch receptor in dorsal longitudinal muscle. Thus the traveling body wave expressed by swimming leeches may be regulated by sensory feedback from both ventral and dorsal longitudinal muscles.

Neuronal control of leech behavior.

The medicinal leech has served as an important experimental preparation for neuroscience research since the late 19th century. Initial anatomical and developmental studies dating back more than 100 years ago were followed by behavioral and electrophysiological investigations in the first half of the 20th century. More recently, intense studies of the neuronal mechanisms underlying leech movements have resulted in detailed descriptions of six behaviors described in this review; namely, heartbeat, local bending, shortening, swimming, crawling, and feeding. Neuroethological studies in leeches are particularly tractable because the CNS is distributed and metameric, with only 400 identifiable, mostly paired neurons in segmental ganglia. An interesting, yet limited, set of discrete movements allows students of leech behavior not only to describe the underlying neuronal circuits, but also interactions among circuits and behaviors. This review provides descriptions of six behaviors including their origins within neuronal circuits, their modification by feedback loops and neuromodulators, and interactions between circuits underlying with these behaviors.

Entrainment of leech swimming activity by the ventral stretch receptor.

Rhythmic animal movements originate in CNS oscillator circuits; however, sensory inputs play an important role in shaping motor output. Our recent studies demonstrated that leeches with severed nerve cords swim with excellent coordination between the two ends, indicating that sensory inputs are sufficient for maintaining intersegmental coordination. In this study, we examined the neuronal substrates that underlie intersegmental coordination via sensory mechanisms. Among the identified sensory neurons in the leech, we found the ventral stretch receptor (VSR) to be the best candidate for our study because of its sensitivity to tension in longitudinal muscle. Our experiments demonstrate that (1) the membrane potential of the VSR is depolarized during swimming and oscillates with an amplitude of 1.5-5.0 mV, (2) rhythmic currents injected into the VSR can entrain ongoing swimming over a large frequency range (0.9-1.8 Hz), and (3) large current pulses injected into the VSR shift the phase of the swimming rhythm. These results suggest that VSRs play an important role in generating and modulating the swim rhythm. We propose that coordinated swimming in leech preparations with severed nerve cords results from mutual entrainment between the two ends of the leech mediated by stretch receptors.

Analysis of the optimal channel density of the squid giant axon using a reparameterized Hodgkin-Huxley model.

A reparameterized Hodgkin-Huxley-type model is developed that improves the 1952 model's fit to the biological action potential. In addition to altering Na(+) inactivation and K(+) activation kinetics, a voltage-dependent gating-current mechanism has been added to the model. The resulting improved model fits the experimental trace nearly exactly over the rising phase, and it has a propagation velocity that is within 3% of the experimentally measured value of 21.2 m/s (at 18.5 degrees C). Having eliminated most inaccuracies associated with the velocity-dependent rising phase of the action potential, the model is used to test Hodgkin's maximum velocity hypothesis, which asserts that channel density has evolved to maximize conduction velocity. In fact the predicted optimal channel density is more than twice as high as the actual squid channel density. When the available capacitance is reduced to approximate more modern serial Na(+)-channel models, the optimal channel density is 4 times the actual value. We suggest that, although Hodgkin's maximum velocity hypothesis is acceptable as a first approximation, the microscopic optimization perspective of natural selection will not explain the channel density of the squid unless other constraints are taken into account, for example, the metabolic costs of velocity.

Model for intersegmental coordination of leech swimming: central and sensory mechanisms.

Sensory feedback as well as the coupling signals within the CNS are essential for leeches to produce intersegmental phase relationships in body movements appropriate for swimming behavior. To study the interactions between the central pattern generator (CPG) and peripheral feedback in controlling intersegmental coordination, we have constructed a computational model for the leech swimming system with physiologically realistic parameters. First, the leech swimming CPG is simulated by a chain of phase oscillators coupled by three channels of coordinating signals. The activity phase, the projection direction, and the phase response curve (PRC) of each channel are based on the identified intersegmental interneuron network. Output of this largely constrained model produces stable coordination in the simulated CPG with average phase lags of 8-10 degrees/segment in the period range from 0.5 to 1.5 s, similar to those observed in isolated nerve cords. The model also replicates the experimental finding that shorter chains of leech nerve cords have larger phase lags per segment. Sensory inputs, represented by stretch receptors, were subsequently incorporated into the CPG model. Each stretch receptor with its associated PRC, which was defined to mimic the experimental results of phase-dependent phase shifts of the central oscillator by the ventral stretch receptor, can alter the phase of the local central oscillator. Finally, mechanical interactions between the muscles from neighboring segments were simulated by PRCs linking adjacent stretch receptors. This model shows that interactions between neighboring muscles could globally increase the phase lags to the larger value required for the one-wavelength body form observed in freely swimming leeches. The full model also replicates the experimental observation that leeches with severed nerve cords have larger intersegmental phase lags than intact animals. The similarities between physiological and simulation results demonstrate that we have established a realistic model for the central and peripheral control of intersegmental coordination of leech swimming.

A model for "splitting" of running-wheel activity in hamsters.

Splitting of locomotor activity rhythm in hamsters occurs when the animals are exposed for several weeks to constant light. The authors propose a mathematical model that explains splitting in terms of a switch in the sign of coupling of two oscillators, from positive to negative, due to long-term exposure to constant light. The model assumes that the two oscillators are not identical and that the negative coupling strengths achieved by each individual animal are variable. With these assumptions, the model provides a unified picture of all different splitting patterns presented by the hamsters, provides an explanation for why the two activity components cross each other during many patterns, and explains why the phase difference achieved by the split components is often near 180 degrees.