Fig 1.
Model dynamics and thresholds.
A. The two species activator (u)-inhibitor (v) system with external input r. B. Phase-plane representation with the activator and inhibitor nullclines in green and red respectively with the critical concentrations marked. A typical suprathreshold response trajectory is shown by the grey arrows. C. Computing the response (final value) with increasing rstep step input magnitudes reveal a threshold value for an output to be elicited in the case of the bistable system (solid curve). The response curve for the excitable system is shown in the dotted curve. The threshold D. Time responses (blue line) of three cases a, b and c with Δt as the duration of the input pulse (red curve). E. Computing the response (final value) with increasing Δt pulses of the unit magnitude reveal a minimum pulse duration for an output to be elicited in the case of the bistable system (solid curve). The response curve for the excitable system is shown in the dotted curve. F. Plot of the minimum pulse duration for different input pulse strengths.
Fig 2.
Modulating the minimum pulse duration.
A. The bistable nullcline (f(u, v0)) plotted as a function of the state u (green curve), with increasing initial inhibition (v0) levels as indicated by the arrow. A zoomed in version of the critical region is shown in the box in order to highlight the raising of the nullclines. B. (blue) Graph plotting the minimum pulse duration needed to elicit a full response with increasing levels of inhibition. (purple) Location of eigenvalue corresponding to the new equilibrium state with each level of inhibition. Only one pertinent eigenvalue is shown. C. Graph plotting how the step (blue) and pulse (purple) input thresholds changed with altering level of inhibition. D. The bistable nullcline (f(ku, kv0)) plotted against u with increasing values of k causing a contraction of the nullcline as indicated by the red arrow (dark red: k = 0.5, lighter red: k = 1, light red: k = 1.5). E. (blue) Plot showing the minimum pulse duration needed to elicit a full response with increasing scaling coefficient k. (purple) Location of the eigenvalue corresponding to the new equilibrium state with each k. F. Graph plotting how the step (blue) and pulse (purple) input thresholds changed with altering scaling coefficient.
Fig 3.
Benefits of an excitable system in directed migration.
A. Examples of the responses of the excitable system (left) and switch (right). The responses have been normalized. B. The number of suprathreshold firings (peak ≥ 0.1, as indicated by the dashed black line in the example responses as a function of the noise variance for both systems that occurred in 200 simulation time units. Error-bars were calculated using 10 simulations. C. Schematic of simulations of cell migration, in which a uniform force (ffront) is applied to the front of the cell while the back fires stochastically, with σnoise serving as the input for both systems. D. The input-output pathway where the decision making occurs through either a switch (purple) or an excitable system (EN, orange). The output of both is fed to a common cytoskeletal excitable network (CON) from which the output activity is considered. E. One dimensional simulations shown via kymographs where activity is indicated through patches that are graded in color from blue to yellow corresponding to low to high activity respectively. The front of the cell (indicated in brown) oscillates identically for both cases (top—switch, bottom—excitable network) while the rest of the perimeter fires stochastically depending on the output from D. F. The kymographs in E were used to obtain the net displacement of the center of mass of the cell for both cases. Error-bars were calculated using 10 simulations. G. Level set simulations for the kymographs indicated in E, shown through overlayed frames. The blue dot indicates the chemoattractant source while the position of the cell boundary with time is shown through lighter shades of red. The initial center of the cell is denoted by the black plus sign which acts as a fixed spatial fiduciary.
Fig 4.
The control mechanism using the decoupled thresholds.
A. The excitable system (f(ku, kv)) represented in phase space. The activator nullcline plotted for k = 0.5 (right) and k = 2 (left). The inhibitor nullcline is denoted in red. The constant step input threshold is indicated between the two dashed lines. The separatrix for one system is shown in orange for reference. B. Quantification of the magnitude of the step needed to elicit a response (blue curve) and the minimum pulse duration needed to elicit a response (purple curve) for the different scaling factors in an excitable system. C. Bee-swarm plot of the number of suprathreshold firings for the cases in A shows a significant change in the firing rate (p-value obtained from student’s t-test). Number of firings were calculated in 200 simulation time units for a noise variance of 0.5. D. Response to a step input. (left) Output from the switch (purple, from Fig 3D, threshold value indicated as hth) to a step input (top) and a stochastic input (bottom). (right) Output from the excitable system (orange, from Fig 3D, step and pulse threshold values indicated as hth and sth respectively) to a step input (top) and a stochastic input (bottom). E. Response to a stochastic input. Same output scheme as in D just with an increased step input threshold for the switch and an increased pulse input threshold for the excitable system.
Fig 5.
Control of chemotaxis signaling.
A. Lowering the number of firings of the excitable system (left) using the scaling factor k, and of the ultra-sensitive switch (right) using the hard threshold value. The scaling factor and the threshold value were increased till zero firings were obtained for a noise variance of 0.2. Number of firings were calculated in 200 simulation time units. B. Illustration of the effect of the control mechanism on cell migration response. Quantification of the mean activity at the front (top) and back (bottom) of a migrating cell using the control mechanism (middle). The step inputs (middle) are aligned in time with the front and back response plots. The back activity is presented as a percentage of the total front response, which is normalized. The step response magnitude chosen for this illustration was 0.3. After a point in time (vertical dashed line) the firings were reduced by altering the scaling factor for the excitable system and the threshold value for the switch as dictated by A. C. The number of firings during the two step responses—before and after firings were reduced, were quantified. Errorbars were from 10 simulations. The ratio between the two firing rates are presented for both the front and back of the cell. The step magnitude chosen here was 0.4 for a q1 = 30. D. The excitable system nullclines (green—activator, brown—inhibitor) plotted in phase space, showing the ideal scenario where just the pulse input threshold (dashed black lines) is altered while the step input threshold (dashed red lines) and the rest of the nullcline remains conserved.