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The second slow variable allows for the trajectory to move between the SNIC (green arrow in Fig. 5c) and the HB (magenta arrow) in a direction parallel to the lower fold, but in a region of state space beyond the extent of the lower surface, removing the limitation on the spike after-hyperpolarization.
The relative speed between this trajectory and N v, t is large enough to allow the limit trajectory to move in an almost horizontal direction, but, due to the limit cycle trajectory's lower speed, N v, t "returns" before this trajectory covers a large enough distance.
The resonance frequency f res is the input frequency for which their relative speed is optimal in the sense that it allows the limit cycle trajectory to move in a direction that maximizes (over the range of input frequencies f) the distance (in the v direction) the limit cycle trajectory can cover before intersecting the moving v-nullcline, at which time they are forced to reverse direction.
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The nonlinear amplification of the voltage response results from the ability of the limit cycle trajectories to move beyond the dashed-red line with slope η = − 1.
For ϵ = 0.01 (panel a, left) the system is fast-slow causing trajectories to move fast along horizontal fibers as compared to larger values of ϵ (middle and right panels), and thus the limit cycle trajectory corresponding to f res reaches a higher value of v max (the voltage response is more amplified).
These oscillations correspond to the projected trajectory trying to move back and forth between the left branches of two extreme nullsurfaces as v oscillates between its minimum and maximum during the spiking phase of each burst (Fig. 4B and C); the trajectory cannot make it all the way to the (v_{max}) surface because the dynamcs of (ca_{i}) is slower than that of v.
We observed that negative and positive symptom trajectories tended to move in parallel over the 1-year study for most of the 11 patterns of combined latent trajectory classes, and correlation between the changes of PANSS negative and positive symptoms was significant as demonstrated by Pearson correlation coefficients.
For Slack to continue its growth trajectory, it had to move on from being a tool for discrete teams to one that could deal with substantially more users, and provide some security and governance as it connected to other departments, repositories and programs across the enterprise.
For Slack to continue its growth trajectory, it had to move on from being a tool for discrete teams to one that could deal with substantially more users, and provide some security and governance as it connected to other departments, repositories and programs across the enterprise.
For small and large enough values of f the corresponding limit cycle trajectories are constrained to move along quasi-one-dimensional directions (w-nullcline and v-axis, respectively).
Traversing on a trajectory enables the navigator to move reliably into featureless space, based on local headings formed by visible landmarks that are robust to positional and orientational errors.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com