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ISRN Computational Biology
Volume 2013 (2013), Article ID 230571, 11 pages
All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others
Department of Physics and Astronomy, College of Charleston, 66 George Street, Charleston, SC 29424, USA
Received 22 August 2013; Accepted 19 November 2013
Academic Editors: P. Durrens and A. Fedorov
Copyright © 2013 Sorinel A. Oprisan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Phase resetting curves (PRCs) are phenomenological and quantitative tools that tabulate the transient changes in the firing period of endogenous neural oscillators as a result of external stimuli, for example, presynaptic inputs. A brief current perturbation can produce either a delay (positive phase resetting) or an advance (negative phase resetting) of the subsequent spike, depending on the timing of the stimulus. We showed that any planar neural oscillator has two remarkable points, which we called neutral points, where brief current perturbations produce no phase resetting and where the PRC flips its sign. Since there are only two neutral points, all PRCs of planar neural oscillators are bimodal. The degree of bimodality of a PRC, that is, the ratio between the amplitudes of the delay and advance lobes of a PRC, can be smoothly adjusted when the bifurcation scenario leading to stable oscillatory behavior combines a saddle node of invariant circle (SNIC) and an Andronov-Hopf bifurcation (HB).
1. Introduction
Neural oscillators are excitable cells which means that as soon as a parameter, such as an external bias current or an ionic conductance, crosses a threshold, then the system switches from a stable rest state characterized by small dampened excursions of the membrane potential to a high amplitude excursion of the membrane potential, called action potential (AP). Repetitive firing excitable cells were categorized by Hodgkin [1] as Class 1, that is, the firing frequency is continuously tunable by adjusting a bias current and such neurons can fire with arbitrarily low frequencies, or Class 2, that is, discontinuous curve with a nonzero firing frequency threshold.
Excitable cells generate APs when a certain control parameter crosses a threshold; that is, a bifurcation occurs in their state space [2]. It became clear that threshold phenomena, such as the AP, are related to fundamental properties of the mathematical model describing the neuron [3] and its corresponding bifurcation diagram [4]. Class 1 excitability was ascribed to a saddle node of invariant circle (SNIC) bifurcation and Class 2 excitability to an Andronov-Hopf bifurcation (HB) (see [5, 6] and references therein). Since neurons are excitable cells operating near a bifurcation point, it is mathematically possible to significantly reduce the number of variables describing the state of a neuron to a few essential ones by using the so-called canonical forms (see [2] for an extensive review of existing canonical models in neuroscience). For example, the four-dimensional conductance based Hodgkin-Huxley (HH) model [1], which was originally introduced as a model of the giant squid axon, includes mathematical descriptions of voltage-gated inward (depolarizing) sodium and an outward (hyperpolarizing) potassium currents. However, using additional assumptions, for example, "instantaneous" activation/inactivation of some ionic channels, HH model can be reduced to two-dimensional models, such as FitzHugh [7] or Morris and Lecar [8]. In particular, Morris-Lecar model used in this paper relies on an instantaneously activating and voltage-gated inward current and a voltage-gated outward delayed rectifier channel (see the appendix for detailed equations). While dimension reduction results in a low-dimensional and sometimes even analytically tractable model, the qualitative aspects of neural dynamics are preserved in canonical models since near the bifurcation points some dimensions become trivial (see [6, 9] for a review of methods and applications of dimension reduction). The most significant benefit of dimension reduction is a better understanding of the general principles that govern AP generation through phase space analysis (see [4, 9] and references therein).
The characteristics of repetitive firing of neural oscillators, such as the intrinsic period , the firing threshold, or the duration of the AP, are determined by their biophysical details, such as the morphology and the ionic channels expressed. However, there are circumstances, for example, in the study of phase locked modes and synchrony of neural populations, when a phenomenological understanding of neural activity suffices. A phase resetting curve (PRC) is not concerned with the biophysical details of neurons but rather tabulates the transient changes in the firing period due to an external perturbation (usually a brief current or synaptic conductance pulse). The most significant effect of a perturbation occurs during the cycle that contains the perturbation and is quantified by the first order PRC; that is, , where is the intrinsic period of oscillation of isolated neuron and is the transiently modified duration of the current cycle due to a perturbation applied at the stimulus time or phase (see Figure 1(a)). Higher order PRCs are defined in a similar way in order to account for long-lasting effects of presynaptic inputs on the second and subsequent cycles (see [10] and the references therein).
Despite its limitations, the PRC is an invaluable tool in predicting the phase locked and synchronous modes of neural networks. The PRC-based approaches to treat Parkinson's disease and epilepsy are at the forefront of practical applications of the extensive theory of phase resetting. It is believe that resting tremor is caused by a cluster of neurons located in the thalamus and the basal ganglia which fire synchronously [11–13]. By using the PRC method, it is possible to predict the shape and timing of the necessary stimuli that destabilize a synchronous state of a large population of neurons with a single pulse [14], double pulses [15], or bipolar pulses [16]. Deep brain stimulation uses permanently implanted electrodes into a target area of the brain with the purpose of delivering precise current pulses aiming at correcting abnormal synchronization among neurons through careful phase resetting [15–18]. Epileptic seizures can also be suppressed by applying brief current stimuli in order to reset the phases of some neurons and break the synchrony of the entire network [19–21].
The shape of the PRC plays a crucial role in the way neurons synchronize or desynchronize when they are part of a neural network. It has been shown that neurons characterized by a Type II PRC, which has an almost sinusoidal shape and comparable sizes of the positive and negative lobes, synchronize more readily due to their ability to both slow down (if the oscillator is ahead) and speed up (if the oscillator is behind) [22–24]. In contrast, the neurons endowed with a Type I PRC, which have a disproportionate ratio of the two PRC lobes, can predominantly speed up (or only slow down) their rhythm when coupled with other neurons. Figure 1(b) shows an example of a Type I PRC with a disproportionate ratio between the amplitudes of the positive and negative lobes. Traditionally, if the ratio of the amplitudes of the two lobes was less than [25] or other arbitrary number, then the PRC was classified as "unimodal" (see Figure 1(b)). The distinction between Type I and Type II PRCs is more profound than the simple visual appearance and was linked to the bifurcation mechanisms leading to stable oscillations.
It was long advocated that the PRCs are either of Type I (see Figure 1(b)) when the bifurcation mechanism leading to oscillatory behavior is a saddle node of invariant circle (SNIC) or of Type II when the bifurcation is of Andronov-Hopf (HB) type [22–24, 26, 27]. Recent theoretical studies suggested that a SNIC bifurcation can also lead to bimodal PRCs [28–30].
In this paper we shown that all PRCs are bimodal and the degree of bimodality of a PRC can be smoothly adjusted if the bifurcation scenario leading to stable limit cycle oscillations combines a SNIC and a HB. Furthermore, our current study extends previous results [28, 30] by introducing the concept of neutral points, that is, phases at which external perturbations produce no phase resetting, and showing that a SNIC that ends with a HB leads to a PRC that is a linear combination of both Type I and Type II PRCs.
2. Theory
2.1. Natural Coordinates for Limit Cycle Oscillators
Biological oscillators with a relatively stable periodic activity are mathematically described as limit cycle (LC) oscillators (see [28] and references therein). In order to gain insight into the oscillatory behavior of nonlinear oscillators and their corresponding PRCs, we used a natural reference frame, that is, the tangent and normal to the LC frame, instead of the usual fixed reference frame [29]. In such a natural reference frame, any external perturbation of the LC oscillator determines either a tangent displacement, that is, a sudden phase jump, or a normal (to the LC) perturbation. While the contribution of the tangent perturbations to the PRC is straightforward to account for, the phase resetting induced by normal perturbations is notoriously harder to account for [30, 31] and, according to [32], "the (A/N: canonical) model above may be of limited value since it fails to reflect the relaxation nature of the dynamics."
Without reducing the generality of our results, here we only focused on planar neural oscillators whose vector flows are , where are the fixed reference frame variables, which in the case of Morris-Lecar model [27, 33] are the membrane potential, , and the potassium conductance, , and the superscript means transposed (see the appendix for model's equations).
The analysis of the response of a dynamical system to external perturbations is significantly clearer in the moving reference frame of the figurative point traveling along the LC, , than in the fixed frame of the original system of differential equations (see Figure 2(a)). The choice of the unitary matrix that switches between the reference frames, that is, , is not unique [34–36]. However, the unit vector of the tangent direction is uniquely defined by and then we chose one of the two possible orientations of the normal vector to be .
In the fixed reference frame, , the temporal evolution of small perturbations, , from is described by the linearized equations of the dynamical system; that is, , where is the Jacobian matrix evaluated along . Using the unitary transform U, the same perturbation with respect to the moving reference frame, , is given by [29, 34] where , + , , and (we also corrected a typo in of [34]). The coefficient measures the effect along the direction for a perturbation applied along the direction with respect to . For example, the coefficient quantifies the effect of a pure normal perturbation measured along the tangent direction; that is, the amount of phase resetting induced along tangent direction to by relaxation processes. In all planar systems , which means that a perturbation tangent to never induces a normal displacement of the figurative point.
While the Jacobian matrix describes the stability of a solution to small perturbations with respect to the fixed reference frame , the matrix (see (1)) determines the stability with respect to normal and tangent perturbations at different phases along . A negative value of any coefficient means that the perturbation decays exponentially along the respective direction, that is, the LC is stable to perturbations applied at that phase.
In the PRC theory, the actual perturbation only affects the membrane potential equation; that is, with respect to the fixed reference frame, (see thick horizontal arrow in Figure 2(a)). The corresponding tangent and normal projections of the perturbation can be found using the unitary matrix : Equation (2) gives the initial conditions with which we integrated the perturbation equations (1). Since for any planar dynamical systems, , the differential equation for normal perturbations, that is, , has the analytic solution: where the initial condition is determined by the injected current of amplitude ; that is, . The differential equation for tangent perturbation, which actually determines the total amount of phase resetting due to the current perturbation , contains both pure tangent perturbations (through coefficient) and normal perturbations (through ); that is, which has the analytic solution: where .
Those fundamental theoretical results highlighted by (3) and (5) relate to the local stability coefficients of the normal and tangent geometrical displacements from the unperturbed limit cycle . In the subsequent section, we will confer these geometric (phase space) displacements into temporal advances or delays of the perturbed figurative point, that is, phase resetting.
3. Results
3.1. Why Are All PRCs Bimodal?
The fact that all PRCs are bimodal results from the existence of two remarkable points along —the neutral points or zero phase resetting points. A perturbation applied at a neutral point induces no phase resetting. For planar systems, the existence of the two neutral points can be explain in terms of the stability coefficients of the LC. Indeed, let us assume for a moment that the crossover coefficient vanishes. This means that the only contribution to the PRC is due to tangent perturbations through . When the applied perturbation is perpendicular to , its corresponding tangent projection vanishes; therefore, its contribution to the PRC also vanishes because the initial tangent displacement in (5) is zero. Since the normal perturbation does not contribute to the tangent displacement (because ), the points where the brief (infinitesimal) voltage perturbation is normal to are the neutral points, that is, the points "" and "" in Figure 2(b). In other words, if , then the maximum and the minimum of the membrane potential are the neutral points of the limit cycle.
If , the exact positions (or phases) of the two neutral points and in Figure 2(b) can be determined from the requirement that the contribution of the pure tangential perturbations to (5), that is, , is exactly balanced by the contribution of the normal relaxation processes through . Furthermore, for planar systems, there are two and only two neutral points along ; that is, the PRC is only bimodal. Indeed, assuming again for a moment that there is no normal-to-tangent coupling between perturbations (), then the only contribution to the PRC is due to . On the other hand, contribution vanishes at local minima or maxima of the membrane potential because the tangent projection of the perturbation, , vanishes. For a planar system, there is only one maximum and only one minimum of the membrane potential; otherwise at least one more independent variable (in addition to the membrane potential and one slow variable ) is required in order to further modulate the membrane potential and produce more local minima and maxima, such as bursting activity. Furthermore, at neutral points the sign of the PRC flips. Indeed, assuming for a moment no normal-to-tangent perturbation coupling (), then the tangent projection of a membrane potential perturbation (see Figure 2(b)) always delays the next spike if applied during the repolarization of the AP around the two neutral points of the limit cycle (see point in Figure 2(b)). On the other hand, outside the repolarization region of the AP, a brief depolarization always advances the next spike. Therefore, the PRC must flip its sign at neutral points.
3.2. Are PRCs Unimodal Near a SNIC Bifurcation?
The theoretical arguments of our conjecture regarding the bimodality of all PRCs are supported by numerical simulations. We found that a brief depolarization applied during the repolarization phase of the AP always delays the next spike of ML model neuron (lengthening the duration of the current cycle), whereas a brief hyperpolarization during the repolarization phase always advances the next spike (see Figure 3). All PRCs in Figure 3 were numerically obtained by integrating the differential equations of the ML model (see the appendix) with a current perturbation in the shape of a rectangular pulse of amplitude (in absolute units, which is ≈2.45% of the bias current at the SNIC bifurcation point) and a duration of .
Close to the SNIC bifurcation point (see Figure 3(A1)) the positive portion of the PRC is hardly visible in print unless it is magnified 1000 times (see the solid dots in Figure 3(A2)). When the magnified PRC is plotted together with the AP (Figure 3(A2)), we notice that the delay is only induced during a small portion of the repolarization phase that starts shortly after the membrane potential maximum (the first neutral point in the absence of normal-to-tangent coupling) and ends before the membrane potential minimum (the second neutral point in the absence of normal-to-tangent coupling). We also notice from Figure 3 that, very close to the SNIC bifurcation point, small perturbations induce significant phase resettings. Indeed, stimulus amplitude has a very large effect (over maximum phase resetting) close to the SNIC bifurcation point (Figure 3(A1)), whereas for a stimulus amplitude that is only about away from the SNIC bifurcation the amount of phase resetting is reduced 100 times (Figure 3(A2)). The first column of Figure 3 clearly supports the conjecture that Type I PRC, which is characteristic to stable oscillations emerging through a SNIC bifurcation, is always bimodal and its bimodality becomes more apparent while the system moves away from the SNIC bifurcation point. At the same time, the positive portion of the PRC, that is, the delay of the subsequent spike in response to a brief depolarization, always occurs during the repolarization phase (second column in Figure 3), as we conjectured.
The curve shows the characteristic Class I excitability behavior (Figure 4(a)); that is, the firing frequency can be continuously adjusted arbitrarily close to zero [1], when the bias current approaches the SNIC bifurcation, up to about 120 Hz (Figure 4(a)). We used the freely available software package AUTO [37] to obtain the bifurcation diagram (Figure 4(b)) for a fixed value of calcium conductance () and variable bias current. The bifurcation diagram has a branch of hyperpolarized stable steady states for low values of bias current (continuous line), which becomes unstable at the SNIC bifurcation point (see point in Figure 4(b)). The emerging LC at SNIC bifurcation is stable with the peak-to-peak amplitude represented by the pairs of solid circles for each value of the bias current up to the maximum that corresponds to the limit point in Figure 4(b). To further support our conjecture that all PRCs are bimodal, we numerically computed the ratio of the PRC lobes (see Figure 1(b)). Very close to the SNIC bifurcation point the amplitude of the smaller lobe is a few orders of magnitude smaller than the magnitude of the other lobe (Figure 4(c)), which justifies the name "unimodal" for Type I PRC, associated with SNIC bifurcations.
3.3. Why the Bimodality of the PRCs Can Be Smoothly Tuned?
While it is clear that every PRC is bimodal, one question is still open: why does the bimodality of the PRCs change with the control parameters? For example, in Figure 4(c) we notice that the ratio of the amplitudes of the two lobes smoothly increases with the bias current until they are comparable.
How is it possible for the PRC of a system for which the oscillatory behavior emerges through a SNIC bifurcation (and whose PRC was supposed to be "unimodal") to be so bimodal that we cannot distinguish it from a Type II PRC (which is characteristic for HB)? We believe that the answer must take into account the entire structure of the bifurcation diagram and not only the fact that at some point the stable oscillatory behavior was generated by a SNIC. For example, the complete bifurcation diagram for ML model shown in Figure 4(b) has two bifurcation points, both of which generate stable limit cycle oscillations: on the hyperpolarized stable steady state branch there is indeed a SNIC bifurcation point for low values of the bias current, and on the depolarized branch there is an Andronov-Hopf (HB) point (Figure 4(b)). The oscillatory behavior emerges through a SNIC bifurcation by slowly increasing the bias current along the hyperpolarized branch ( point in Figure 4(b)), which produces a PRC that can be approximated with the "unimodal" Type I expression in the close proximity of the SNIC bifurcation [2, 22, 26]. On the other hand, stable oscillations emerge as soon as the HB point is reached (HB point in Figure 4(b)) by slowly decreasing the bias current along the depolarized branch. As such, the close proximity of the HB produces a PRC that can be approximated with the "bimodal" Type II expression with being a constant phase [2, 22, 26]. We conjectured that no PRC is purely of Type I or Type II but a (linear) combination of the two types. To check this conjecture, we fitted the numerically generated PRCs with a linear combination of the and ; that is, , where and are fitting coefficients. Ideally, we expect that the PRC is entirely captured by a Type I iPRC very close to a SNIC bifurcation; that is, .
By the same token, we expect that the PRC is entirely captured by a Type II iPRC very close to an Andronov-Hopf bifurcation point; that is, . Since the amplitudes and depend on the characteristics of the injected current perturbation (amplitude and duration), we computed their ratio , which is insensitive to such variations (see Figure 4(d)). Our numerical simulations show that indeed the contribution of is one order of magnitude larger than that of near a SNIC bifurcation point whereas the situation is reversed near a HB point (see Figure 4(d)). Furthermore, by setting an arbitrary ratio for classifying the PRCs into "unimodal" or "bimodal", for example, 0.175 [25] or , we can actually use Figure 4(d) to estimate the range of the bias currents for which such an arbitrary classification is valid.
3.4. How Robust Is the SNIC-HB Scenario That Produces Tunable PRC Bimodality?
The complete bifurcation diagram shown in Figure 4(b) was obtained for fixed calcium conductance (). Since the pair SNIC-HB of bifurcation points is crucial to a tunable bimodality of the PRC, we also explored the reproducibility of such a bifurcation diagram while changing . For example, for large values of calcium conductance, , the bifurcation scenario is always the same: while increasing the bias current from very low values (below 0.08), stable LC oscillations emerge through a SNIC bifurcation (point in Figure 5(a)), but while decreasing the bias current from large values (above 0.3), stable limit cycle oscillations emerge through an Andronov-Hopf bifurcation (HB point in Figure 5(a)). It is this particular bifurcation scenario, with a SNIC bifurcation at one end and an Andronov-Hopf bifurcation at the other end of the bifurcation diagram, that leads to a PRC which is a linear combination of a Type I iPRC (typically near a SNIC bifurcation) and a Type II iPRC (typically near an Andronov-Hopf bifurcation) as shown above. For a critical value of calcium conductance (close to 0.6), both ends of the stable LC oscillations are marked by Andronov-Hopf bifurcation points (Figure 5(b)). The SNIC-HB bifurcation scenario changes to a HB-HB when the two limit points and of the SNIC fold collide and give way to a HB point (see Figure 5(b)). The bifurcation diagrams (Figure 5) also show that large calcium conductances lead to larger peak-to-peak membrane potential oscillations, especially through a significant increase of the depolarization overshoot. This effect is due to the fact the a larger calcium conductance drives the depolarization of the cell more effectively as soon as the membrane potential exceeds the half activation threshold of calcium channels (the parameter of the model described in the appendix) and leads to a rapid and strong overshoot (Figure 5(a)). Large calcium currents, due to a large value of , require a relatively modest depolarizing bias current compensation to sustain oscillations. This is the reason why the bias current range is narrower for larger in the SNIC-HB bifurcation diagrams. On the other hand, for small calcium conductances (Figure 5(b)) the oscillations have small amplitude since calcium channels do not reach their half activation voltage and the dynamics of the system are dominated by potassium channels. Indeed, small hyperpolarizations induced by potassium immediately and dramatically decrease the number of open voltage-gated calcium channels (since the calcium channels already work below their half activation threshold). At the same time, the bias current required to compensate the dynamics described by a HB-HB scenario, which is dominated by the outward potassium current, is significantly larger than the bias required for a SNIC-HB scenario (compare bias current values in Figure 5(a) with those in Figure 5(b)).
4. Discussions
Network models showed that selectively blocking some channels in critical inhibitory neurons can lead to switches from unsynchronized high frequency to synchronized low frequency network oscillations [40]. General anesthetics are selective blockers that could also induce dramatic change in the PRC shape and switch the brain from firing patterns associated with consciousness to states associated with general anesthesia [41–43]. Therefore, the conductances of voltage-gated channels, such as calcium in our ML example, could be pharmacologically manipulated to produce a dramatic change in the bimodality of the PRC. Although this paper focuses on understanding the structure of PRCs and the effect of biophysically relevant control parameters, such as the bias current and calcium conductance, at the single cell level, the results are relevant actually at network level. In the end, the PRC is a tool used for predicting neural population behavior [14, 44].
We analyzed the mechanisms leading to a smooth and continuous change of PRC's bimodality in a planar model. We used a ML model neuron not only because it represents a biophysically realistic model of an actual neuron [8, 27, 33], but also because, by changing only two parameters, that is, the calcium conductance and the bias current, it displays both SNIC and Andronov-Hopf bifurcations. Some authors used the ratio of the positive and the negative amplitudes of the PRCs to classify them as unimodal or bimodal (see [26, 28] and references therein). In this paper, we showed that all PRCs are bimodal regardless of the bifurcation mechanism that led to oscillatory behavior. Moreover, we showed that there are two remarkable points on any planar limit cycle, which we called the neutral or zero resetting points, at which the PRC flips its sign (see Figure 2). Although the phases of the two neutral points slightly change with the parameters of the external perturbation (amplitude and duration of stimulus), their location can be inferred from the model's equations using the stability coefficients of the LC. The phase at which a brief depolarization can induce a delay (positive resetting) is situated along the repolarization region of the AP (see Figure 3). Since for fast spiking neurons the repolarization covers only a minute fraction of the [30], it may raise concerns regarding real-world applicability of such weak bimodality. Experimental studies showed that a brief pulse can desynchronize a fully synchronized cluster of neural oscillators if the pulse hits the cluster in a very narrow phase range (≈5% ) [14, 16, 45]. It is also possible for the two neutral points to actually collide, in which case the PRC becomes truly unimodal. We showed that a ML model can mimic Class I excitability (with a continuous frequency versus current curve—see Figure 4(a)) with a SNIC-HB bifurcation diagram (Figure 4(a)). The tunable bimodality of the PRC is a result of the presence of both SNIC and HB, which allows fine mixing of Type I character (specific to a SNIC bifurcation) with Type II character (specific to a HB) into the final PRC (Figure 4(c)).
Oscillatory activity is a hallmark of neuronal network function in various brain regions, including the olfactory bulb [46–48], thalamus [49, 50], hippocampus [51, 52], and neocortex [53]. The existence and stability of synchronous and phase locked modes can be predicted using PRCs (see [10, 31, 54–56] and references therein). Neural networks oscillate over more than three orders of magnitude and include delta (0.5–3 Hz), theta (3–8 Hz), gamma (30–90 Hz), and ultrafast (90–200 Hz) oscillations [57, 58]. Stable oscillatory activity at single cell or population levels was associated with higher brain functions [59, 60], such as gamma rhythm that is believed to be the basis of temporal encoding [61, 62], sensory binding of features into a coherent perception [59, 63], and storage and recall of information [64–66]. Disruption of synchronous rhythms underlies some psychiatric disorders, such as schizophrenia [67, 68]. The ability to smoothly change the PRC from a Type I to a Type II through experimentally available control parameters, such as bias currents or pharmacologically controlled ionic channels, has potential application in shaping the activity of neural cells and networks.
We used a dimensionless Morris-Lecar model [27, 33] described by the following equations: where is the membrane potential, is the slow potassium activation, and all ionic currents are described by , where is the conductance of the voltage-gated channel and is the corresponding reversal potential. In particular, the calcium current is , the potassium current is , and the leak current is . The reversal potentials for calcium, potassium, and leak currents are , , and , respectively. The steady state activation function for calcium channels is , where , , the steady state activation function for potassium channels is , where , , the inverse time constant of potassium channels is , the potassium and leak conductances are , , respectively, and the .
The two control parameters that can switch the bifurcation diagram from a SNIC-HB type to a HB-HB type (see Figure 5) are the calcium conductance and the bias current . For example, if and , (A.1) describe what was classified by Hodgkin as Class I excitable cells. For example, if and , (A.1) describe what was classified by Hodgkin as Class II excitable cells.
The two nullclines of (A.1) are two-dimensional curves and , respectively. A fixed point of (A.1) is any crossing of the two nullclines, that is, phase space points where both the rate of change of membrane potential and the potassium activation vanish. The linear stability of the fixed points is determined by the two eigenvalues of the Jacobian matrix of (A.1). Stable steady states have eigenvalues with negative real part and are represented by a solid line in any bifurcation diagram (see Figures 4(b) and 5). Saddle or hyperbolic points have real eigenvalues with opposed signs, whereas Andronov-Hopf points have pairs of purely imaginary eigenvalues.
This research was supported by the National Science Foundation Grant CAREER Award IOS 1054914.
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text:
TAMPA — No one enjoys paying library fines, yet some Hillsborough school librarians are drawing smiles this season with a catchy phrase: "Can Your Fines!"
Read that literally. Canned food has become a popular replacement for late fees at some middle and high schools. Owe a dollar on a late-returned library book? All can be forgiven for a can of beans.
The district does not set rules on the matter, leaving it to librarians to work out the details of can-for-money substitutions. The idea appears to have taken off by word-of-mouth endorsements.
"For some kids, I know paying the fines is probably not the easiest thing to do," said Darlene Meginnis, media specialist at New Tampa's Liberty Middle School, which started collecting cans last winter after hearing that another school did it. "I still want to hold them responsible for the late fee."
She makes a point of telling students that donating cans to charity is just as good — maybe even better — than paying into a media center fund. Some students donate out of need, others because they want to.
Thirteen-year-old Jasmine Morro didn't even have overdue books. Still, the seventh-grader recently brought canned turkey, peas, potatoes and bread to the Monroe Middle School media center.
"Not that many people around Tampa get to have a wonderful Thanksgiving, and I already know that I'm going to," she said. "I would like other people to have a wonderful Thanksgiving, too."
Like other schools, South Tampa's Monroe finds the pitch has special appeal during the holiday season.
Ferrell Middle School's Mike Saltzgaver began a fine-forgiveness can drive before Thanksgiving. He didn't set a firm stop date. He knows that cash can be hard for students to come by at an east Tampa school, where four out of five students qualify for federally subsidized lunches.
Instead, he tells students who racked up late fees to go into Mom's pantry and "take out stuff you don't like — lima beans, spinach, whatever."
"You guys are healing the hungry people out there in the world," Saltzgaver tells students when they bring in the cans, trying to promote community activism.
Other school libraries still stick with the traditional late book fine — generally 5 cents per day in middle and high schools.
But the canned food fine concept is catching on.
At Walker Middle School in Odessa, the media center has collected cans instead of fines during February for the past two years. The school promotes it as "Have a Heart Month."
High school media specialist Janet Drake has asked students to channel late fees into a can drive for as many as 10 years. She can't recall exactly where she got the idea to start program when she worked at Hillsborough High.
After that, she brought it to Chamberlain High, where a morning show announcement kicked off the "Can Your Fine" program. She asks students to surrender one can for every 50 cents in late fees. Yet the thought counts more than hard and fast rules on pennies owed and donation deadlines.
"I have such a big heart," Drake said. "When they're still bringing cans in, I don't say, 'No, we're back to fines.' So I end up collecting them through Christmas."
Letitia Stein can be reached at [email protected] or (813) 226-3400. For more education news, visit the Gradebook at blogs.tampabay.com/schools.

subset: commoncrawl


text:
i already pre-ordered my license for us$39.99, and now parallels is throwing in its compressor product for free to those of us who already ordered and those who end up purchasing parallels workstation for the mac within 30 days of its full version release. compressor is priced at $49.99 as a stand-alone product, so this is an awesome deal.
compressor is a virtual hard disk management tool that can reduce the size of this disk by 50% or more. i was also very happy to see the rc2 release, because the last release kept stealing my ip address from my mac, so i couldn't browse the internet with both the mac os and windows xp os running at the same time. this is now fixed, and other new features and improvements include improved usb performance and device support, customizable full screen mode, improved folder share performance, and more.
user comments 6 comment(s)
|time to talk? (1:21pm est fri jun 02 2006)
this is ben rudolph, marketing manger at parallels…i'd love to talk with you a bit more about your experience with paralells desktop. email me at firstname.lastname@example.com if you have a free minute! – by ben
|can one uncompress? (12:45am est sun jun 04 2006)
after compressing the windows xp vm i get constantly the message not enough disk space and installing additional programms is rejected. is there a way to uncompress a vm or is a new installation required? – by walter
|i�ll wait on this (5:24pm est mon jun 05 2006)
i installed boot camp on an imac and it runs without a hitch � stable, compatible and fast. the reboot requirement is only a minor issue because it only takes a few seconds. maybe products like parallel will one day be the future direction. but the state of development today favors the use of boot camp.
|fine for you, not for me (6:14pm est wed jun 07 2006)
for me, it is simply not acceptable to reboot to switch between os's. with parallels i can be working on my windows machine and switch over instantly to the mac when an email comes in or i want to look something up using safari. i'm not seeing any performance hit for my applications, which admittedly are not graphics intensive, but they are cpu intensive. so while i can understand why bootcamp is sufficient or even superior for some workflows, it sure isn't for mine. – by michael grant
|vm/bootcamp and stuff. (6:59am est thu jun 08 2006)
i'm an old user of vmware on windows, i've always done all my work in vmware, leaving my host machine free of clutter (with the exceptions of one or two games.)
i've bought vmware 3 and 5 so far.
ive now switched to mac, and apart from not being able to move over my vmware vms to parallels, i'm amazed at the performance!
it's rock-solid (vmware style) and well worth the money. (i use fixed-sized images, so compressor is wasted on me)
on the matter of bootcamp… i have it installed as well (but only for games). i cannot afford to reboot in the work environment… i cannot close my browser/e-mail etc… so as far as bootcamp is concerned, it is only for games. parallels is much faster than i am used to with vmware, it's almost native cpu speed, although slower graphics.
i'm trying to move over to mac native anyway, so i'm sorta phasing out parallels anyway, but for now it is indispensable!
|looking forward for 2 missing great features: (9:15am est thu jun 08 2006)
this is what would make parallels really excellent:
1. ability to load / convert vmware machines.
2. ability to load as a machine the bootcamp partition of windows (or any other partition – based installations). this way you get the best of both worlds, i.e. reboot the machine and run the partition -based os (e.g. windows xp) at full speed and with 3d graphics acceleration etc. or use that same disk-space of the partition within parallels while you are in macos x, with the only limitation of graphics & speed but with huge saving in time for reinstalls and disk space..
it may be tricky to have to re-emulate all the hardware in windows, but should be possible. – by yannis


text:
Why Us Rates and Agreements Get Here
Due to Hurricane Harvey, Country Villa Mobile Home & RV Park sustained serious damage to the property and its units. We regret the loss of serving our tenants over the years and hope to see you as future tenants to the rebuilt Country Villa RV Park. As we pick up the pieces and make improvements, we are temporarily closed until tentatively October 1, 2017. We look forward serving your residential needs in the near future.
Located on FM 1069 between Aransas Pass and Ingleside, Texas is Country Villa Mobile Home and RV Park. Whether you need a mobile home to stay or to pull in your RV for seasonal work or a permanent residency, we are within easy driving distance of fabulous restaurants and great recreational activities in Corpus Christi, Portland, Gregory, Port Aransas, Refugio, and Rockport. We offer on-site laundry facility, showers, and a maintenance shop for the hobbyist. Feel free to call us to schedule a tour or just stop on by. We believe you will enjoy this serene and centrally located mobile home and RV Park.
All residents have access to the following amenities:
Monthly - $280 to $310 plus electricWeekly - $150 plus electricNightly - $30
Includes all the amenities above; $125 for a 30 amp deposit and $175 for 50 amp
Country Villa Mobile Home & RV Park have put in place park rules and agreements to protect and enhance the value of community resident properties. We here at Country Villa strive to ensure that everyone has a peaceful and well-maintained neighborhood. We have provided the following rules and agreements online. Please read and understand all documents.
NOTE: No refunds for partial rental usage. All rates are subject to change.
Contact Information
Phone(361) 776-7534
AddressCountry Villa Mobile Home & R.V. Park3780 FM 1069Aransas Pass, Texas 78336
Leasing Office Houses
Monday thru Friday - 8:00am-6:00pmSaturday - 8:00am-12:00pmSunday - Closed
Country Villa Mobile Home and RV Park is a member of the Aransas Pass Chamber of Commerce and the Ingleside Chamber of Commerce


text:
Background speech recognition increases MT productivity by partially automating the traditional, labor-intensive transcription process. Increases in lines per hour can be significant.
To benefit fully from this increase and help motivate MTs to learn and master the powerful editing shortcuts that come with speech solutions, many healthcare organizations are re-evaluating existing transcription compensation plans and performance standards.
Nuance representatives understand the challenges of change, and can work with you and your organization to develop a plan that helps you achieve your goals.
Higher productivity, lower costs
Nuance's background speech recognition platforms, eScription and Dictaphone® Enterprise Speech System, help healthcare organizations reduce costs in medical transcription by increasing MT productivity.
Compensation that works
Organizations that adjust their compensation plans to reward MTs for both high production and quality output as they transition to editors achieve higher productivity gains and resulting cost savings. Our experience working with these healthcare leaders has helped us hone our expertise to share with you.
Raising the standards
Nuance recommends increasing MT performance standards to reflect increased MT productivity. Nuance representatives can work with you and your organization to develop a plan that helps you achieve your goals.
© 2002-2012 Nuance Communications, Inc. Alle Rechte vorbehalten.


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Calculus for Business and Social Science
Office Hours: MW 9:00am -9:30am, 12:15pm - 1:00pm
TR 8:30am - 9:30am, 1:00pm - 2:15pm
PLACEMENT REQUIREMENTS: Math 134 College Algebra or permission of the instructor
COURSE DESCRIPTION: This course is designed primarily for students majoring in business or social and behavioral sciences. Topics of study will include: limits, continuity, definition of a derivative, rule of differentiation, maxima and minima, and indefinite and definite integrals.
TEXTBOOK: Applied Calculus, by Waner and Costenoble (3rd edition, 2004)
EVALUATION: Six online exams will be given during the semester. 100 pts. Each.
Test 1 Chapter 1 and 2 Functions and Linear/Nonlinear Models
Test 2 Chapter 3.1-3.4 Introduction to the Derivative
Test 3 Chapter 3.5-3.8 Introduction to the Limit
Test 4 Chapter 4 Techniques of Differentiation
Test 5 Chapter 5 Application of the Derivative
Test 6 Chapter 6 The Integral
Test 7 Chapter 7.1-7.4 Further Techniques and Applications of the Integral
Bonus Points: 50 bonus points maximum will be possible for work completed in development of chapter summaries and explanations with examples.
Thus, TOTAL POINTS FOR THE CLASS would be 700 pts.
Grades will be assigned as follows:
630 700 A, 560 629 B, 490 559 C, 420 489 D, below 420 F
CHEATING POLICY: If caught cheating in any way, the student will receive an F for the final grade.
ATTENDANCE POLICY: To be successful in a math course, attendance would be very important, almost critical. If more than two weeks of classes are missed without a valid excuse ( death in family, hospitalization, nuclear blast, etc.) I reserve the right to withdraw you from class with an F. If you know in advance that you cannot attend class on a certain day, you may possibly get my prior approval. There are no make-up exams or quizzes. If you come to class late, you will not receive extra time for exams or quizzes.
Learning Outcomes
At the end of this course students will be able to:


text:
Chris Powell, executive vice president for human resources at Scripps Networks Interactive (NYSE: SNI), joins some of the nation's most accomplished business leaders on Savoy Magazine's 2012 list of the "Top 100 Most Influential Blacks in Corporate America." Powell is responsible for the strategy, development and delivery of initiatives, programs and policies to recruit, develop, retain and reward employees for Scripps Networks Interactive. He also leads the company's efforts related to diversity and other key strategic initiatives.
Chris Powell, executive vice president, human resources at Scripps Networks Interactive, earned a spot on Savoy Magazine's "Top 100 Blacks in Corporate America." (Photo: Business Wire)The Savoy Top 100 is the definitive listing of African-American influencers and achievers impacting corporate America. Selection of the "Top 100 Blacks in Corporate America" begins by examining the landscape of spheres of influence impacting Savoy's readership including: corporate sector influence, scholastic achievement, career growth, community outreach and recognition. "Chris' accomplishments are vital in developing the diverse, inclusive environment that makes our company so successful," said Ken Lowe, chairman, CEO and president of SNI. "This recognition is well-deserved and rightly places Chris among the elite in American business." In addition to his duties as the head of human resources, Powell chairs the company's Benefits Committee, serves on its Executive Diversity and Management Committees. He is vice chair of the Knoxville Area Urban League, 2012 Leadership Knoxville Program participant and the University of Tennessee Chancellor's Associate Council. Powell serves on the board of directors for Women in Cable & Telecommunications (WICT) and is board vice president/president-elect for the Cable and Telecommunications Human Resources Association (CTHRA).


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Number of Active Network ClientsWe define an active network client as an organization that has implemented one of more of solutions but has not implemented either the Conversations or PowerReviews Enterprise platforms. Such solutions may include our Connections solutions, Media solutions or Express platform. We count organizations that are closely related as one client, even if they have signed separate contractual agreements. We believe that our network client base in combination with our enterprise client base is an indicator of the reach of our network. Non-GAAP Financial Measures Adjusted EBITDA discussed in this press release is defined as net loss adjusted for stock-based expense, contingent consideration related to acquisition, adjusted depreciation and amortization (which excludes amortization of capitalized internal-use software development costs), integration and other costs related to acquisitions, other non-business costs and benefits, income tax expense and other (income) expense, net. Non-GAAP net loss, which is used to calculate non-GAAP net loss per share, is defined as our GAAP net loss adjusted to exclude stock-based expense, contingent consideration related to acquisition, amortization of acquired intangible assets, integration and other costs related to acquisitions, and other non-business costs and benefits along with the associated income tax effect of these adjustments. Non-GAAP basic and diluted loss per share for the fourth fiscal quarter and fiscal year ended April 30, 2012 has been calculated assuming the conversion of all outstanding shares of our preferred stock into 27,897,031 shares of our common stock as of the first day of the beginning of the period. Management presents these non-GAAP financial measures because it considers them to be important supplemental measures of core operating performance. Management uses the non-GAAP financial measures for planning purposes, including analysis of the company's operating performance against prior periods and the effectiveness of our business strategies, the preparation of operating budgets and to determine appropriate levels of operating and capital investments, as well as and in communications with our board of directors concerning our financial performance. Management also believes that the non-GAAP financial measures provide additional insight for securities analysts and investors in evaluating the company's financial and operational performance without regard to items that can vary substantially from company to company depending upon their financing, capital structures and the method by which assets were acquired. However, these non-GAAP financial measures have limitations as an analytical tool, and you should not consider them in isolation or as a substitute for analysis of our results of operations as reported under GAAP. Furthermore, these non-GAAP financial measures may not be comparable to similarly titled measures of other organizations because other organizations may not calculate these non-GAAP financial measures in the same manner. We intend to provide these non-GAAP financial measures as part of our future financial results discussions and, therefore, the inclusion of these non-GAAP financial measures will provide consistency in our financial reporting. A reconciliation of these non-GAAP measures to GAAP is provided in the accompanying tables.
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text: Monticello School Board eliminates half-day kindergarten program Published February 10, 2014 at 6:27 pm Education » The Monticello School Board Monday night unanimously decided Monday night there will be no half-day kindergarten starting next year. Parents will retain the option to take children home after the halfway point is reached during the school day. See this week's Monticello Times for an update.


text:
Building the geometric models in CAD, Describe the approaches.This question is related with CAD/CAM Engineering.
In computer aided design (CAD) systems the geometry of components and products is described in terms of mathematical data and equations called models. And the system of representing these models are modellers. These models are stored in computer memory and used for designing as well representing the products and components in engineering drawing and in producing their isometric views. These are also used for other design related calculations such as calculating volumes, and surface area.
In CAD systems, different types of modellers are used that use different system of geometric representing to provide different levels of details and sophistication. There are four main type of Modellers.
- Wire-frame modellers
- Surface modellers
- Solid modellers
- Feature based modellers
Wire frame modellers represent just the edges of parts and their corners (or vertices), where the lines meet. The vertices are represented as coordinates on a three dimensional axis. And lines are represented as equation of straight lines, circles and arcs.
Surface modellers are able to represent continuously curved surfaces such as, say, body of a car. Cross- and longitudinal-sections of surfaces are taken and lines of these sections are represented by polynomial equations. Alternately the surfaces may be divides in small patches and each patch represented by corners of the patches and slope of the surface at the corners.
Solid modellers provide a true three dimensional representation of the part. These type of models are created bu taking a two dimensional shapes and projecting them in the third dimension.
Feature based modellers are basically solid modellers with added facility for representing specific feature common in parts - for example holes, pockets, flanges ribs.


text:
A friend recently told me about Eight O'Clock Ranch and I'm interested in joining their CSA, but I wanted to find out if anyone else has had any experience with their CSA or their meats in general.
I'm trying to determine if I should get a full share or half share of the mixed meats and what cuts of meat I might get in a share.
Also, if anyone knows where I could buy their meats in NYC I would love to give their stuff a try before committing to the CSA.
Any other feedback anyone has would be greatly appreciated!


text: I got mine today....and I can't stop looking at him. I think he's my new favorite beast too, and I'm not a big 200x style fan.....although in this line it's slowly starting to grow on me (Zodak, BGEL, 200x Stinkor head). I can't imagine how the 4H could out do this guy. He almost sets the bar too high for future beasts.


text:
Report: Texas Rangers open to trade offers for Ian Kinsler, Elvis Andrus or Jurickson Profar
MLB.com's T.R. Sullivan reports that the Rangers have reached out to other teams to let them know that they are open to discussing trading one of their middle infielders: Ian Kinsler, Elvis Andrus, or Jurickson Profar.
According to the report, the Rangers are not urgently trying to trade anyone but merely open to conversations. However, the Rangers are adamant that should Kinsler—who SportsDay's Evan Grant named last month as the most likely chip to move—be dealt, they will not pick up any of Kinsler's $62 million in guaranteed money left on the remaining five years of his deal. Such a deal would free up more money to spend on needs at other positions, with the primary target being Atlanta's Brian McCann, who could serve time at catcher, DH, and first base.
On October 7, Evan Grant wrote that the Rangers are unlikely to send Jurickson Profar back down to the minors next season, which would almost certainly require Kinsler to move positions or be dealt to another team.
Kinsler had an OPS of .757 last season, good enough for eighth in the Majors among second basemen. However, his below-average baserunning and defense only rank him as the 14th-best second basemen in WAR.
There are more than a few trade partners for a potential Kinsler deal. Should the Yankees or Tigers not re-sign Robinson Cano or Omar Infante respectively, they will have an open slot at second base. Sullivan reports that the Blue Jays, Royals, Orioles, and Cubs are also interested in acquiring help at second base.
Kinsler has a no-trade clause, though, and he has expressed his desire to stay in Texas.
Baseball's annual winter meetings are scheduled for next week in Orlando.


text:
By SELWYN RAAB
Published: March 25, 1988
Twenty-eight current and former New York City health inspectors and supervisors were charged yesterday with extorting hundreds of thousands of dollars from restaurants by threatening to close them or cite them for sanitation violations.
More than 300 restaurants were forced to make payoffs ranging from $50 to $1,400 in a systematic corruption scheme by Health Department employees that authorities said dated back at least to the early 1980's.
The head of the Federal Bureau of Investigation in the city, James M. Fox, said the payoffs had doubled the incomes of most of the inspectors and allowed several of them to lead ''lavish life styles.'' Cadillacs and Rolexes
''Some restaurant owners dreaded to see them coming, driving up in Cadillacs and wearing Rolex watches,'' Mr. Fox said in announcing the filing of the Federal charges. In a single day, a team of two inspectors obtained $3,000 of payoffs from six restaurants, Mr. Fox asserted.
Andrew J. Maloney, the United States Attorney in Brooklyn, said the evidence suggested that usually the inspectors, rather than overlook unsanitary conditions, had threatened to cite restaurants falsely for nonexistent violations unless payments were made. Mr. Maloney said restaurateurs ''dreaded'' financial losses if cited for health code irregularities in a weekly list that was issued by the Health Department and published in newspapers.
The department said routine inspections of restaurants would be suspended, except for emergencies, until at least next week.
The New York Times, The Daily News and New York Newsday said yesterday that they would not print the current list of health code violations. [ Page B6. ] Law-enforcement officials declined to identify any restaurant that was said to have made payoffs. Most of the 300, Mr. Fox said, were on the East Side of Manhattan. He noted, however, that inspectors ''shook down'' restaurants in all boroughs.
Dr. Stephen C. Joseph, the City Health Commissioner, said there was no indication that the improper inspections and payoffs had caused any public health problem. But he said routine restaurant inspections in the city would be suspended until a new inspection system is put into operation. He said he expected normal inspections to be resumed ''sometime next week.''
A department spokesman, Marvin Bogner, said that while inspections are suspended ''inspectors would be sent out immediately'' in cases where there are possible emergency sanitation problems at a restaurant.
A court document specifying the charges said that one inspector was known to his colleagues as Hungry Joe because ''of his greed in extorting money.'' Joint Inquiry
The undercover inquiry of restaurant inspectors was jointly conducted by the F.B.I. and the city's Department of Investigation.
Mr. Maloney, whose office is prosecuting the case, said that the restaurant owners were apparently victims and that some of them were cooperating in the inquiry.
The arrests were the latest accusation of municipal corruption to trouble the administration of Mayor Koch in the last three years. Other agencies in which officials or large numbers of employees have been accused of extortion include the Parking Violations Bureau, the Housing Authority, the Department of Environmental Protection, the Department of General Services, the Board of Education and the Taxi and Limousine Commission.
The disclosures about widespread corruption in the restaurant inspections were made in complaints filed in Federal District Court in Brooklyn against 4 current supervisors, 18 inspectors and 6 former department employees. All were accused of violating the Federal Hobbs Act, which forbids public employees from obtaining payments while performing their duties.
Those employees who could be reached would not comment.
The 22 department employees were suspended without pay. Their annual salaries range from $23,213 for inspectors, to $32,735 for employees who supervise inspectors. Koch Urges Stiff Penalties
The 22 employees accounted for 20 percent of 110 Health Department employees involved in restaurant inspections. Each of the city's 15,000 restaurants is normally inspected once a year or whenever a complaint is received, Mr. Bogner said.
In a news conference at F.B.I. headquarters in lower Manhattan, Mayor Koch urged that judges impose prison terms, ''not weekend sentences,'' for employees convicted of corruption.
''No section of our society'' is without corruption, Mr. Koch said, adding: ''You can't prevent the fact there is corruption. Our job is to apprehend those engaged in it.''
Kevin Frawley, the city's Investigation Commissioner, said in an interview that the restaurant inquiry began in 1985 after a new group of Health Department inspectors had reported that they were frequently offered bribes by restaurant owners. To determine if veteran inspectors were implicated, Mr. Frawley said, an undercover agent with the cooperation of a restaurant owner posed as one of the restaurant's employees.
In May 1986, the F.B.I. began a similar undercover inquiry in Brooklyn after a former agent, who was working as a security official for a group of restaurants, reported systematic extortion problems. False Citations Possible


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Millions of documents from the Guatemalan national police archive, shedding light on torture, forced disappearances and murders committed during the1960-1996 counterinsurgency war in this country, are now available on-line thanks to a collaboration with the University of Texas at Austin.
The Politics of Memory conference held Friday Dec. 2 at the University of Texas unveiled a digital archive hosted by the university, holding 12 million of the roughly 80 million pages of national police documents discovered by chance in 2005.
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My friend's daughter went for an underarm bleach in one derma clinic but was prevailed upon by the nurse attendant to undergo a glycolic peel instead. She is only 15 years old. A few minutes after, she felt pain and itchiness in her unerarms. She was told it was only normal. When they got home, the burned sensation worsened in that her daughter can hardly close her arms and from the looks of it, it was more of a first degree burn. The percentage of glycolic acid used was 25%.
What is the Percentage Glycolic Acid That Should Be Used on Minors?
Doctor Answers 1
Peeling teenagers?
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Superficial peels like glycolic and salicylic acid are used routinely for acne, darker color and other skin disorders. The percentage is chosen by the dermatologist to get the best result with the least possibility of side effect and complications. There is no age determinant per se.
These answers are for educational purposes and should not be relied upon as a substitute for medical advice you may receive from your physician. If you have a medical emergency, please call 911. These answers do not constitute or initiate a patient/doctor relationship.


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For a neutral -based optical frequency standard we report a fractional frequency instability of in 1 s, which represents a fivefold improvement over existing atomic frequency standards. Using the technique of optical Bordé–Ramsey spectroscopy with a sample of trapped atoms, we have resolved linewidths as narrow as 200 Hz (FWHM). With colder atoms this system could potentially achieve an instability as low as in 1 s. Such low instabilities are important for frequency standards and precision tests of fundamental physics.
© 2000 Optical Society of AmericaFull Article | PDF Article


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Will my Implants Shrink?
I had transumbical procedure today. I really thought from what we discussed i would be bigger. i already was a c, now i feel like a slightly bigger c.... READ MORE
Saline Implant Deflation? (photo)
I am concerned with my right breast. It seems to have gotten smaller over the last 2 years...especially this last year. I have had my saline implants... READ MORE
10 Year Old Implants: Too Far Apart and Rippling - Can They be Fixed?
I have 10 yr old saline, under muscle implants. I've nursed three children. If I get them redone, can they make them closer together? They are so... READ MORE
Can You Break Your Silicone Implant While Massaging It Too Forcefully? (photo)
I was given instructions to massage and I really spent much effort on one boob that wasn't moving freely and looking odd/small. Now when i feel that... READ MORE
9 weeks post op, is this the final result? Seems like they've shrunk. I'm even more unhappy with them. Planning revision (Photo)
I had 500cc silicone mod+ implants unders 9 weeks (almost 10) , wanting a revision to HP or UHP 750cc because my results were incredibly small and I'm... READ MORE
Do Memory Gel Implants Shrink?
I was wondering if Memory Gel Implants will shrink over time? I've heard that Saline implants does shrink as time goes on. I am not talking about... READ MORE
Are my Silicon Implants Leaking?
I'm four weeks post op and my c cup breasts that I came out of surgery with are now a small B cup. It looks like they are shrinking in size every... READ MORE
9 Years Post-BA, One Breast Has Shrunk. What Is Going On?
I had implants about 9 years ago. I have now noticed one breast has become slightly smaller, is there a leak or something to worry about. I have no... READ MORE
I Think I Have a Leak As Well As Bottoming Out!? (photo)
I am 1 yr 9 months PO and my left breast seems to be shrinking! I know that I have bottomed out but my left keeps getting smaller but very slowly.... READ MORE
How much will the breast shrink after getting implants?
I am due to have my BA on March 19 and I am still unsure about size. I am a petite woman, 5'0, about 105lbs. I am a small A looking to go to a small C... READ MORE
I Got Breast Silicone Augmentation 5 Months Ago and Now I Lost Weight and my Breasts Seem to Get Smaller?
3 months post op - one breast is smaller. Does this look normal or should I be concerned? (Photo)
Hi there, I'm currently 3 months post op with 345cc silicone under the muscle and this morning I woke up to find my right breast implant has shrank in... READ MORE
Boob Size Shrunk and Implants Rippled After Pregnancy?
Before I had my breasts enlarged I told the surgeon I was trying to get pregnant & asked if I should wait & he said no. I went from an A to a... READ MORE
2 Months Post BA. Why Are They Getting Smaller?
Its been 2 months since my BA. I was cup A before PS due to breast-feeding 2 kids for 12 months each. Before having kids I was cup B. My PS... READ MORE
Can I Go As Big As I Did While I Was Nursing? (photo)
While nursing, I was a large D. I loved the way they made me feel. Now that I am no longer breast feeding (stoped a year ago) my breasts have shrunk... READ MORE
Will 385cc get me to a D cup? (Photos)
I was a 34b I am 26 5"4 weigh 125 pounds and never had kids. I got a BA 3 weeks ago they shrunk a little because the swelling subsided but I recieved... READ MORE
How to Lose Extra Breast Fat when You Have Breast Implants?
Im 28, I have 550cc saline mentor implants, under the muscle. I want to lose some of my natural breast fat. TRUST ME IM ADDICTED TO CARDIO, and ALSO... READ MORE
It has been about 1 1/2 years since my augmentation and I'm beginning to feel like I'm shrinking. Is this possible?
I got 350cc under the muscle and for a while it appeared that I was a full C and was pleased, but now it seems as though I have gotten smaller,... READ MORE
Will my breasts shrink after swelling goes down?
I just had my BA 4 days ago. I LOVE my size and that worries me because I am afraid I may have went too small. I was around a 32B and I got saline... READ MORE


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Free memberships at Lifestyle Family Fitness for teens this summer!
We all know that obesity among adolescents and teens is skyrocketing. In fact, just last week I wrote a post about a study that found about 20 percent of 4 year olds in this country are considered obese.
For three years, Lifestyle Family Fitness has offered teens free memberships during the summer months. And they are doing it again this year May 15 through Aug. 15. Kids ages 12-17 can join at any of the 55 Lifestyle locations through the country. There are more than 30 sites in Florida.
"I always dreamed of doing this for teens who may be in the same position I once faced," says Lifestyle's founder Geoff Dyer, who, at age 17, tipped the scales at 248 pounds. "We hope that by inviting the teens to work out for free, we can help them begin a life-long path which includes fitness and healthy living."
Parents can register teens for the free summer program beginning April 15. Teens are allowed to visit any Lifestyle Family Fitness unaccompanied after the registration process is complete. However, they must end their visits by 5 p.m. weekdays.
What a good deal, especially for families that might want to get into the healthy habit together.
-- Sherry Robinson
[Photo: Times files]


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Ohio State Football: 3 Keys to Beating the Hokies in the 'Shoe Saturday NightSeptember 4, 2014
Ohio State's 17-point victory over Navy last Saturday was a little too close for comfort. Buckeye fans are not accustomed to seeing just six points on the board at halftime. Neither is Ohio State head coach Urban Meyer, and he told Bill Landis of the Northeast Ohio Media Group the poor play starts up front.
Very disappointed," Meyer said. "There's a standard set of offensive line play for many, many years, and it's really been enhanced by Coach Warinner the last two years. That didn't resemble an offensive line at Ohio State for the first two quarters.
Meyer, tradition and a defense filled with elite talent led to lofty expectations this season, but it is obvious there is no substitute for experience. With eight new starters on offense, no one should be surprised that it took a couple of quarters to get going against the Midshipmen. The offense won't have the luxury of a slow start this week.
Ohio State takes on Virginia Tech at 8 p.m. ET Saturday in Ohio Stadium. This will be the Buckeyes' first difficult nonconference game under Meyer. The Hokies might be unranked, but they have been one of the most consistently good teams for the past 20 years under head coach Frank Beamer. The Buckeyes are in for a battle, as Virginia Tech wants to end Ohio State's 25-game regular-season winning streak and get back to being among the best in the FBS. Ohio State must play considerably better to hold off the Hokies.
Here are three keys to the winning the game:
J.T. Barrett Playing with Poise
After Braxton Miller was lost for the year, there was a legitimate concern that a special season was in jeopardy. After Barrett's performance against Navy, there should be a collective sigh of relief. He finished the game 12-of-15, passing for 225 yards with two touchdowns and one interception. He also had 50 yards rushing. That is a great start to his career, but Virginia Tech's defense is far superior to Navy's defense. This week's challenge will substantially harder.
Virginia Tech defensive coordinator Bud Foster is a fantastic coach and one of the premier defensive strategists in the game. Expect him to dial up the intensity and disguise his cover packages to confuse Barrett and force him into making costly mistakes.
The Hokies' strength is their secondary, where Kendall Fuller and Brandon Facyson return. Last year, Fuller was the ACC's Rookie of the Year, and both were named first-team freshman All-Americans by 247Sports. They're hard-hitting cornerbacks who cover well. Fuller is ready for the challenge of playing the Buckeyes telling the Associated Press, "It's definitely going to be fun, something we're all looking forward to, playing a team like that that's been successful over the years with a great environment and great college atmosphere," he said.
With Fuller and Facyson flying all over the field, Barrett will have to be quick with his decisions and precise in his throws for the offense to be effective. Avoiding interceptions and getting the ball to his playmakers in space is critical. This is also his first home game in front of 105,000 fans at night. He definitely needs to feed off of that energy. If he plays well, the Buckeyes' offense will be fine.
Attack on Defense
If the coaches and talent are as good as advertised, it is time to unleash the fury. The yearning for the return of the Silver Bullets is palpable, and Buckeye Nation has been anxiously waiting since January to see the impact that co-defensive coordinator Chris Ash and defensive line coach Larry Johnson will have on this team. Their aggressive style will be put to the test against Virginia Tech who runs an uptempo, pro-style offense.
One player who appears ready to show off the new look is sophomore Darron Lee, who will be making just his second start at strong-side linebacker. He finished with seven tackles and had a 61-yard fumble recovery for a touchdown against Navy, earning him the Big Ten Network's Defensive Player of The Week. If his progression continues, it is safe to say the Buckeyes won't miss Ryan Shazier.
Ash's effect on the defense should be most visible in the secondary, which did not get many opportunities to showcase its improvement against Navy's triple option. That won't be the case against Virginia Tech's balanced attack.
Quarterback Michael Brewer had a successful debut against William & Mary last week, finishing the game 23-of-30 for 251 yards and two touchdowns and one interception. The Hokies also rushed for 222 yards against the Tribe.
Tyvis Powell and Vonn Bell have the perfect opportunity to put last year's demons to rest by playing with discipline. Under Ash's base 4-3 over Cover 4 defense, the safeties need to react quickly to the call. With the defensive line and linebackers focusing on their gap assignments and the corners locking down their section of the field, the safeties are the glue that makes the Cover 4 work well. If they play fundamentally sound and are in sync with the corners and linebackers, the defense will earn back its Silver Bullets nickname.
Improve Third-Down Conversion Percentage
Against Navy, the offense was just 2-of-8 on third-down conversion attempts. It needs to be closer to 50 percent to beat the Hokies. The key is getting good production on first and second down to avoid long-yardage situations. Success depends on the offensive line playing much better.
Given his history, it is likely that Ed Warinner will correct the line's mistakes. If he did, they should be able to stop the Hokies' pressure giving the skill players a little time and space to work their magic and put the offense in great field position throughout the game.
This game has been marked on Ohio State's calendar for several years, and it comes at an ideal time. Championship-caliber teams need to face some adversity to test their resolve. Beating the Hokies would give the Buckeyes a tremendous confidence boost. It won't be easy, but they should pull out a 27-17 win.