This paper presents an innovative technique to study postural control. Our translating platform, the Sliding Linear Investigative Platform For Analyzing Lower Limb Stability and Simultaneous Tracking, EMG and Pressure mapping (SLIP-FALLS-STEPm), makes precise, vibration movements under controlled conditions. We look at the psychophysical thresholds to the perception of a sinusoidally induced sway. In the Sine Lock experiments described, an induced sinusoidal perturbation locks the subject's natural sway pattern at the frequency of the perturbation. The input / output system is treated as an Amplitude Shift Key (ASK) modulated signal modulating a carrier frequency (at or about a subject's natural sway frequency). The Position signal (input) and the Anterior-Posterior Center of Pressure (APCOP) signal (output) or the ankle angle are demodulated by mixing them with the pure sine wave carrier at the frequency of underlying oscillation and then low-pass filtering it to detect the amplitude envelope. These detected envelopes elucidate that the square pulse increase in the position sine wave amplitude yields a triangular increase in APCOP demodulated signal.
The Phase-Locked Loop (PLL) is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.
non-linear systems; phase-locked loop; optimization
Postural sway is considered to have two fundamental stochastic components, a slow non-oscillatory component and a faster damped-oscillatory component. The slow component has been shown to account for the majority of sway variance during quiet stance. Postural control is generally viewed as a feedback loop in which sway is detected by sensory systems and appropriate motor commands are generated to stabilize the body’s orientation. Whereas the mechanistic source for the damped-oscillatory sway component is most likely feedback control of an inverted pendulum, the underlying basis for the slow component is less clear. We investigated whether the slow process was inside or outside the feedback loop by providing standing subjects with sum-of-sines visual motion. Linear stochastic models were fit to the experimental sway trajectories to determine the stochastic structure of sway as well as the transfer function from visual motion to sway. The results supported a fifth-order stochastic model, consisting of a slow process and two damped-oscillatory components. Importantly, the slow process was determined to be inside the feedback loop. This supports the hypothesis that the slow component is due to errors in state estimation, since state estimation is inside the feedback loop, rather than a moving reference point or an exploratory process outside the feedback loop.
Quartz tuning forks are extremely good resonators and their use is growing in scanning probe microscopy. Nevertheless, only a few studies on soft biological samples have been reported using these probes. In this work, we present the methodology to develop and use these nanosensors to properly work with biological samples. The working principles, fabrication and experimental setup are presented. The results in the nanocharacterization of different samples in different ambients are presented by using different working modes: amplitude modulation with and without the use of a Phase-Locked Loop (PLL) and frequency modulation. Pseudomonas aeruginosa bacteria are imaged in nitrogen using amplitude modulation. Microcontact printed antibodies are imaged in buffer using amplitude modulation with a PLL. Finally, metastatic cells are imaged in air using frequency modulation.
tuning fork; atomic force microscopy; nanocharacterization
For reasons of stealth and other operational advantages,
distributed aerospace wireless systems have received much attention in recent years. In a distributed aerospace wireless system, since the transmitter and receiver placed on separated platforms which use independent master oscillators, there is no cancellation of low-frequency phase noise as in the monostatic cases. Thus, high accurate time and frequency synchronization techniques are required for distributed wireless systems. The use of a dedicated synchronization link to quantify and compensate oscillator frequency instability is investigated in this paper. With the mathematical statistical models of phase noise, closed-form analytic expressions for the synchronization link performance are derived. The possible error contributions including oscillator, phase-locked loop, and receiver noise are quantified. The link synchronization performance is predicted by utilizing the knowledge of the statistical models, system error contributions, and sampling considerations. Simulation results show that effective synchronization error compensation can be achieved by using this dedicated synchronization link.
Postural sensitivity to moving visual environments in patients with anxiety disorders was studied. We hypothesized that patients with anxiety disorders would have greater sway in response to a moving visual environment compared to healthy adults, especially if they have space and motion discomfort (SMD). Twenty one patients with generalized anxiety without panic (NPA), and 38 patients with panic and agoraphobia (PAG) were compared to 22 healthy controls. SMD was evaluated in all subjects via questionnaire. Subjects stood on a force platform that was either fixed or rotating with the subject (i.e. sway referenced) during exposure to a sinusoidally moving visual surround. Center of Pressure (COP) data were computed from force transducers in the platform as a measure of sway. Results showed that patients swayed significantly more in response to the moving visual scene compared to control subjects, with no differences between the NPA and PAG groups. SMD was a predictor of sway response in the patients: patients with high SMD swayed significantly more than both Controls and anxiety patients with low SMD. These results indicate that patients with anxiety disorders, particularly those with SMD, are more visually dependent for balance. This subgroup of patients may be amenable to treatment used for patients with balance disorders (i.e. vestibular rehabilitation) that focuses on sensory re-integration processes that address visual sensitivity.
Anxiety; Panic; Optic flow; Posture; Sway; Sensory Integration; Balance
Active hair bundle motility has been proposed to underlie the amplification mechanism in the auditory endorgans of non-mammals and in the vestibular systems of all vertebrates, and to constitute a crucial component of cochlear amplification in mammals. We used semi-intact in vitro preparations of the bullfrog sacculus to study the effects of elastic mechanical loading on both natively coupled and freely oscillating hair bundles. For the latter, we attached glass fibers of different stiffness to the stereocilia and observed the induced changes in the spontaneous bundle movement. When driven with sinusoidal deflections, hair bundles displayed phase-locked response indicative of an Arnold Tongue, with the frequency selectivity highest at low amplitudes and decreasing under stronger stimulation. A striking broadening of the mode-locked response was seen with increasing stiffness of the load, until approximate impedance matching, where the phase-locked response remained flat over the physiological range of frequencies. When the otolithic membrane was left intact atop the preparation, the natural loading of the bundles likewise decreased their frequency selectivity with respect to that observed in freely oscillating bundles. To probe for signatures of the active process under natural loading and coupling conditions, we applied transient mechanical stimuli to the otolithic membrane. Following the pulses, the underlying bundles displayed active movement in the opposite direction, analogous to the twitches observed in individual cells. Tracking features in the otolithic membrane indicated that it moved in phase with the bundles. Hence, synchronous active motility evoked in the system of coupled hair bundles by external input is sufficient to displace large overlying structures.
A novel design of force to rebalance control for a hemispherical resonator gyro (HRG) based on FPGA is demonstrated in this paper. The proposed design takes advantage of the automatic gain control loop and phase lock loop configuration in the drive mode while making full use of the quadrature control loop and rebalance control loop in controlling the oscillating dynamics in the sense mode. First, the math model of HRG with inhomogeneous damping and frequency split is theoretically analyzed. In addition, the major drift mechanisms in the HRG are described and the methods that can suppress the gyro drift are mentioned. Based on the math model and drift mechanisms suppression method, four control loops are employed to realize the manipulation of the HRG by using a FPGA circuit. The reference-phase loop and amplitude control loop are used to maintain the vibration of primary mode at its natural frequency with constant amplitude. The frequency split is readily eliminated by the quadrature loop with a DC voltage feedback from the quadrature component of the node. The secondary mode response to the angle rate input is nullified by the rebalance control loop. In order to validate the effect of the digital control of HRG, experiments are carried out with a turntable. The experimental results show that the design is suitable for the control of HRG which has good linearity scale factor and bias stability.
Hemispherical Resonator Gyro (HRG); force-to-rebalance control; FPGA; quadrature error; rate sensor
Spontaneous respiration influences the body’s center-of-mass when standing. We contend that the healthy postural control system actively adapts to respiration, thereby minimizing its effect on postural sway. We therefore examined the interaction between respiration and postural sway, as measured by center-of-pressure (COP) oscillations, and quantified the extent to which this interaction resulted in “posturo-respiratory synchronization.” We hypothesized that synchronization would be stronger in elderly subjects and those with stroke, and when standing with eyes closed as compared to open, due to alterations in the physiologic mechanisms that normally regulate postural sway. Twenty-five subjects with chronic hemispheric infarction and 38 controls (50–80yrs) stood on a force platform for 3min with eyes-open and 3min with eyes-closed. Respiratory flow and COP dynamics were simultaneously recorded. The dominant oscillatory mode of respiration and the corresponding oscillatory modes of anterioposterior and mediolateral COP dynamics were extracted using ensemble empirical mode decomposition. The strength of posturo-respiratory synchronization was quantified from the regularity of instantaneous phase shifts between extracted respiratory and COP oscillations. Significant posturo-respiratory synchronization was only present in the anterioposterior direction. The strength of synchronization increased with age (p<0.01). Closing the eyes increased synchronization strength in both groups (p=0.01), but more so in stroke patients (p=0.01). These observations suggest that a control system actively regulates the effects of respiration on sagittal-plane postural sway, particularly during eyes-open standing. As evidenced by increased posturo-respiratory synchronization with advanced age and central lesion, this novel metric may be used as a clinical marker of altered postural control.
Respiration; Postural Control; Balance; Coupling
Poor postural balance is one of the major risk factors for falling in normal pressure hydrocephalus (NPH). Postural instability in the clinic is commonly assessed based upon force platform posturography. In this study we focused on the identification of changes in sway characteristics while standing quiet in patients with NPH before and after shunt implantation. Postural sway area and sway radius were analyzed in a group of 9 patients and 46 controls of both genders. Subject's spontaneous sway was recorded while standing quiet on a force platform for 30-60 s, with eyes open and then closed. Both analyzed sway descriptors identified between-group differences and also an effect of shunt implantation in the NPH group. Sway radius and sway area in patients exhibited very high values compared with those in the control group. Importantly, the effect of eyesight in patients was not observed before shunt implantation and reappeared after the surgical treatment. The study documents that static force platform posturography may be a reliable measure of postural control improvement due to shunt surgery.
posture; sway; normal pressure hydrocephalus
We examined changes in the motor organization of postural control in response to continuous, variable amplitude oscillations evoked by a translating platform and explored whether these changes reflected implicit sequence learning. The platform underwent random amplitude (maximum ± 15 cm) and constant frequency (0.5 Hz) oscillations. Each trial was composed of three 15-second segments containing seemingly random oscillations. Unbeknownst to participants, the middle segment was repeated in each of 42 trials on the first day of testing and in an additional seven trials completed approximately 24 hours later. Kinematic data were used to determine spatial and temporal components of total body centre of mass (COM) and joint segment coordination. Results showed that with repeated trials, participants reduced the magnitude of horizontal body COM displacement, shifted from a COM phase lag to a phase lead relative to platform motion and increased correlations between ankle/platform motion and hip/platform motion as they evolved from an ankle strategy to a multi-segment control strategy involving the ankle and hip. Maintenance of these changes across days provided evidence for learning. Similar improvements for the random and repeated segments, however, indicate that participants did not exploit the sequence of perturbations to improve balance control. Rather, the central nervous system (CNS) may have been tuning into more general features of platform motion. These findings provide important insight into the generalizabilty of improved compensatory balance control with training.
platform translation; balance; learning; continuous perturbation; postural coordination; implicit sequence learning
We study spontaneous dynamics and signal transduction in a model of active hair bundle mechanics of sensory hair cells. The hair bundle motion is subjected to internal noise resulted from thermal fluctuations and stochastic dynamics of mechano-electrical transduction ion channels. Similar to other studies we found that in the presence of noise the coherence of stochastic oscillations is maximal at a point on the bifurcation diagram away from the Andronov-Hopf bifurcation and is close to the point of maximum sensitivity of the system to weak periodic mechanical perturbations. Despite decoherent effect of noise the stochastic hair bundle oscillations can be synchronized by external periodic force of few pN amplitude in a finite range of control parameters. We then study effects of receptor potential oscillations on mechanics of the hair bundle and show that the hair bundle oscillations can be synchronized by oscillating receptor voltage. Moreover, using a linear model for the receptor potential we show that bi-directional coupling of the hair bundle and the receptor potential results in significant enhancement of the coherence of spontaneous oscillations and of the sensitivity to the external mechanical perturbations.
Cortical neurons are often classified by current–frequency relationship. Such a static description is inadequate to interpret neuronal responses to time-varying stimuli. Theoretical studies suggested that single-cell dynamical response properties are necessary to interpret ensemble responses to fast input transients. Further, it was shown that input-noise linearizes and boosts the response bandwidth, and that the interplay between the barrage of noisy synaptic currents and the spike-initiation mechanisms determine the dynamical properties of the firing rate. To test these model predictions, we estimated the linear response properties of layer 5 pyramidal cells by injecting a superposition of a small-amplitude sinusoidal wave and a background noise. We characterized the evoked firing probability across many stimulation trials and a range of oscillation frequencies (1–1000 Hz), quantifying response amplitude and phase-shift while changing noise statistics. We found that neurons track unexpectedly fast transients, as their response amplitude has no attenuation up to 200 Hz. This cut-off frequency is higher than the limits set by passive membrane properties (∼50 Hz) and average firing rate (∼20 Hz) and is not affected by the rate of change of the input. Finally, above 200 Hz, the response amplitude decays as a power-law with an exponent that is independent of voltage fluctuations induced by the background noise.
dynamics; frequency response; noise; oscillations; pyramidal cell; somatosensory cortex
Cardiac tissue can be entrained when subjected to sinusoidal stimuli, often responding with action potentials sustained for the duration of the stimulus. To investigate mechanisms responsible for both entrainment and extended action potential duration, computer simulations of a two-dimensional grid of cardiac cells subjected to sinusoidal extracellular stimulation were performed.
Methods and Results
The tissue is represented as a bidomain with unequal anisotropy ratios. Cardiac membrane dynamics are governed by a modified Beeler-Reuter model. The stimulus, delivered by a bipolar electrode, has a duration of 750 to 1,000 msec, an amplitude range of 800 to 3,200 μA/cm, and a frequency range of 10 to 60 Hz. The applied stimuli create virtual electrode polarization (VEP) throughout the sheet. The simulations demonstrate that periodic extracellular stimulation results in entrainment of the tissue. This phase-locking of the membrane potential to the stimulus is dependent on the location in the sheet and the magnitude of the stimulus. Near the electrodes, the oscillations are 1:1 or 1:2 phase-locked; at the middle of the sheet, the oscillations are 1:2 or 1:4 phase-locked and occur on the extended plateau of an action potential. The 1:2 behavior near the electrodes is due to periodic change in the voltage gradient between VEP of opposite polarity; at the middle of the sheet, it is due to spread of electrotonic current following the collision of a propagating wave with refractory tissue.
The simulations suggest that formation of VEP in cardiac tissue subjected to periodic extracellular stimulation is of paramount importance to tissue entrainment and formation of an extended oscillatory action potential plateau.
defibrillation; AC stimulation; entrainment; action potential duration; computer simulation; virtual electrode polarization
Excessive sway during quiet standing is a common sequela of chronic alcoholism even with prolonged sobriety. Whether alcoholic men and women who have remained abstinent from alcohol for weeks to months differ from each other in the degree of residual postural instability and biomechanical control mechanisms has not been directly tested.
We used a force platform to characterize center-of-pressure biomechanical features of postural sway, with and without stabilizing conditions from touch, vision, and stance, in 34 alcoholic men, 15 alcoholic women, 22 control men, and 29 control women. Groups were matched in age (49.4 years), general intelligence, socioeconomic status, and handedness. Each alcoholic group was sober for an average of 75 days.
Analysis of postural sway when using all 3 stabilizing conditions vs. none revealed diagnosis and sex differences in ability to balance. Alcoholics had significantly longer sway paths, especially in the anterior-posterior direction, than controls when maintaining erect posture without balance aids. With stabilizing conditions the sway paths of all groups shortened significantly, especially those of alcoholic men, who demonstrated a 3.1-fold improvement in sway path difference between the easiest and most challenging conditions; the remaining 3 groups, each showed a ~2.4-fold improvement. Application of a mechanical model to partition sway paths into open-loop and closed-loop postural control systems revealed that the sway paths of the alcoholic men but not alcoholic women were characterized by greater short-term (open-loop) diffusion coefficients without aids, often associated with muscle stiffening response. With stabilizing factors, all four groups showed similar long-term (closed loop) postural control. Correlations between cognitive abilities and closed-loop sway indices were more robust in alcoholic men than alcoholic women.
Reduction in sway and closed-loop activity during quiet standing with stabilizing factors shows some differential expression in men and women with histories of alcohol dependence. Nonetheless, enduring deficits in postural instability of both alcoholic men and alcoholic women suggest persisting liability for falling.
balance; alcohol; alcoholism; gender; posturography; postural stability; musculoskeletal response mechanism
To acquire images of dynamic scenes from multiple points of view simultaneously, the acquisition time of vision sensors should be synchronized. In this paper, an illumination-based synchronization derived from the phase-locked loop (PLL) mechanism based on the signal normalization method is proposed and evaluated. To eliminate the system dependency due to the amplitude fluctuation of the reference illumination, which may be caused by the moving objects or relative positional distance change between the light source and the observed objects, the fluctuant amplitude of the reference signal is normalized framely by the estimated maximum amplitude between the reference signal and its quadrature counterpart to generate a stable synchronization in highly dynamic scenes. Both simulated results and real world experimental results demonstrated successful synchronization result that 1,000-Hz frame rate vision sensors can be successfully synchronized to a LED illumination or its reflected light with satisfactory stability and only 28-μs jitters.
high-speed vision; robot vision; vision chip; camera synchronization; phase-locked loop; signal processing; visible light communication; signal normalization; quadrature detection
Back muscle fatigue decreases the postural stability during quiet standing, but it is not known whether this fatigue-induced postural instability is due to an altered proprioceptive postural control strategy. Therefore, the aim of the study was to evaluate if acute back muscle fatigue may be a mechanism to induce or sustain a suboptimal proprioceptive postural control strategy in people with and without recurrent low back pain (LBP). Postural sway was evaluated on a force platform in 16 healthy subjects and 16 individuals with recurrent LBP during a control (Condition 1) and a back muscle fatigue condition (Condition 2). Back muscle fatigue was induced by performing a modified Biering-Sørensen test. Ankle and back muscle vibration, a potent stimulus for muscle spindles, was used to differentiate proprioceptive postural control strategies during standing on a stable and unstable support surface, where the latter was achieved by placing a foam pad under the feet. Ankle signals were predominantly used for postural control in all subjects although, in each condition, their influence was greater in people with LBP compared to healthy subjects (p < 0.001). The latter group adapted their postural control strategy when standing on an unstable surface so that input from back muscles increased (p < 0.001). However, such adaptation was not observed when the back muscles were fatigued. Furthermore, people with LBP continued to rely strongly on ankle proprioception regardless of the testing conditions. In conclusion, these findings suggest that impaired back muscle function, as a result of acute muscle fatigue or pain, may lead to an inability to adapt postural control strategies to the prevailing conditions.
Postural strategy; Proprioception; Muscle vibration; Sensory reweighting; Postural balance
Speech elicits a phase-locked response in the auditory cortex that is dominated by theta (3–7 Hz) frequencies when observed via magnetoencephalography (MEG). This phase-locked response is potentially explained as new phase-locked activity superimposed on the ongoing theta oscillation or, alternatively, as phase-resetting of the ongoing oscillation. The conventional method used to distinguish between the two hypotheses is the comparison of post- to prestimulus amplitude for the phase-locked frequency across a set of trials. In theory, increased amplitude indicates the presence of additive activity, while unchanged amplitude points to phase-resetting. However, this interpretation may not be valid if the amplitude of ongoing background activity also changes following the stimulus. In this study, we employ a new approach that circumvents this problem. Specifically, we utilize a fine-grained time–frequency analysis of MEG channel data to examine the co-modulation of amplitude change and phase coherence in the post-stimulus theta-band response. If the phase-locked response is attributable solely to phase-resetting of the ongoing theta oscillation, then amplitude and phase coherence should be uncorrelated. In contrast, additive activity should produce a positive correlation. We find significant positive correlation not only during the onset response but also throughout the response period. In fact, transient increases in phase coherence are accompanied by transient increases in amplitude in accordance with a “signal plus background” model of the evoked response. The results support the hypothesis that the theta-band phase-locked response to attended speech observed using MEG is dominated by additive phase-locked activity.
Auditory; Speech; MEG; Evoked; Phase; Amplitude
In nonlinear dynamic systems, synchrony through oscillation and frequency modulation is a general control strategy to coordinate multiple modules in response to external signals. Conversely, the synchrony information can be utilized to infer interaction. Increasing evidence suggests that frequency modulation is also common in transcription regulation.
In this study, we investigate the potential of phase locking analysis, a technique to study the synchrony patterns, in the transcription network modeling of time course gene expression data. Using the yeast cell cycle data, we show that significant phase locking exists between transcription factors and their targets, between gene pairs with prior evidence of physical or genetic interactions, and among cell cycle genes. When compared with simple correlation we found that the phase locking metric can identify gene pairs that interact with each other more efficiently. In addition, it can automatically address issues of arbitrary time lags or different dynamic time scales in different genes, without the need for alignment. Interestingly, many of the phase locked gene pairs exhibit higher order than 1:1 locking, and significant phase lags with respect to each other. Based on these findings we propose a new phase locking metric for network reconstruction using time course gene expression data. We show that it is efficient at identifying network modules of focused biological themes that are important to cell cycle regulation.
Our result demonstrates the potential of phase locking analysis in transcription network modeling. It also suggests the importance of understanding the dynamics underlying the gene expression patterns.
Cortical neurons are capable of generating trains of action potentials in response to current injections. These discharges can take different forms, e.g. repetive firing that adapts during the period of current injection or bursting behaviors. We have used a combined experimental and computational approach to characterize the dynamics leading to action potential responses in single neurons. Specifically we investigated the origin of complex firing patterns in response to sinusoidal current injections. Using a reduced model, the theta neuron, alongside recordings from cortical pyramidal cells we show that both real and simulated neurons show phase locking to sine wave stimuli up to a critical frequency, above which period skipping and 1-to-x phase locking occurs. The locking behavior follows a complex “devil’s staircase” phenomena, where locked modes are interleaved with irregular firing. We further show that the critical frequency depends on the time scale of spike generation and on the level of spike frequency adaptation. These results suggest that phase locking of neuronal responses to complex input patterns can be explained by basic properties of the spike generating machinery.
bifurcation theory; devil’s staircase; endogenous oscillators
Oscillatory activity in neuronal networks correlates with different behavioral states throughout the nervous system, and the frequency-response characteristics of individual neurons are believed to be critical for network oscillations. Recent in vivo studies suggest that neurons experience periods of high membrane conductance, and that action potentials are often driven by membrane-potential fluctuations in the living animal. To investigate the frequency-response characteristics of CA1 pyramidal neurons in the presence of high conductance and voltage fluctuations, we performed dynamic-clamp experiments in rat hippocampal brain slices. We drove neurons with noisy stimuli that included a sinusoidal component ranging, in different trials, from 0.1 to 500 Hz. In subsequent data analysis, we determined action potential phase-locking profiles with respect to background conductance, average firing rate, and frequency of the sinusoidal component. We found that background conductance and firing rate qualitatively change the phase-locking profiles of CA1 pyramidal neurons vs. frequency. In particular, higher average spiking rates promoted band-pass profiles, and the high-conductance state promoted phase-locking at frequencies well above what would be predicted from changes in the membrane time constant. Mechanistically, spike-rate adaptation and frequency resonance in the spike-generating mechanism are implicated in shaping the different phase-locking profiles. Our results demonstrate that CA1 pyramidal cells can actively change their synchronization properties in response to global changes in activity associated with different behavioral states.
We explore mode-locking of spontaneous oscillations of saccular hair cell bundles to periodic mechanical deflections. A simple dynamic systems framework is presented that captures the main features of the experimentally observed behavior in the form of an Arnold Tongue. We propose that the phase-locking transition can proceed via different bifurcations. At low stimulus amplitudes F, the transition to mode-locking as a function of the stimulus frequency ω has the character of a saddle-node bifurcation on an invariant circle. At higher stimulus amplitudes, the mode-locking transition has the character of a supercritical Andronov-Hopf bifurcation.
The cell cycle oscillator, based on a core negative feedback loop and modified extensively by positive feedback, cycles with a frequency that is regulated by environmental and developmental programs to encompass a wide range of cell cycle times. We discuss how positive feedback allows frequency tuning, how size and morphogenetic checkpoints regulate oscillator frequency, and how extrinsic oscillators such as the circadian clock gate cell cycle frequency. The master cell cycle regulatory oscillator in turn controls the frequency of peripheral oscillators controlling essential events. A recently proposed phase-locking model accounts for this coupling.
Gamma oscillations in the dentate gyrus and hippocampal CA3 show variable coherence in vivo, but the mechanisms and relevance for information flow are unknown. We found that carbachol-induced oscillations in rat CA3 have biphasic phase-response curves, consistent with the ability to couple with oscillations in afferent projections. Differences in response to stimulation of either the intrinsic feedback circuit or the dentate gyrus were well described by varying an impulse vector in a two-dimensional dynamical system, representing the relative input to excitatory and inhibitory neurons. Responses to sinusoidally modulated optogenetic stimulation confirmed that the CA3 network oscillation can entrain to periodic inputs, with a steep dependence of entrainment phase on input frequency. CA3 oscillations are therefore suited to coupling with oscillations in the dentate gyrus over a broad range of frequencies.
Boundary value formulations are presented for exact and efficient sensitivity analysis, with respect to model parameters and initial conditions, of different classes of oscillating systems. Methods for the computation of sensitivities of derived quantities of oscillations such as period, amplitude and different types of phases are first developed for limit-cycle oscillators. In particular, a novel decomposition of the state sensitivities into three parts is proposed to provide an intuitive classification of the influence of parameter changes on period, amplitude and relative phase. The importance of the choice of time reference, i.e., the phase locking condition, is demonstrated and discussed, and its influence on the sensitivity solution is quantified. The methods are then extended to other classes of oscillatory systems in a general formulation. Numerical techniques are presented to facilitate the solution of the boundary value problem, and the computation of different types of sensitivities. Numerical results are verified by demonstrating consistency with finite difference approximations and are superior both in computational efficiency and in numerical precision to existing partial methods.
periodic system; limit cycle; nonlinear ODEs; boundary value problem; amplitude sensitivity; period sensitivity; phase sensitivity; phase locking condition