Fig. 5.

Phase plane analysis of model performance during “hypercapnia”. (a) Projection of model trajectory to the (V₅,h₅) phase plane together with V₅ nullclines corresponding to intermediate and high values of d₃ (solid and dashed blue curves, respectively) and the nullcline for the slow variable, h₅ (red curve). The upper V₅ nullclines correspond to higher inhibition than do the lower ones. The trajectory shown undergoes three excursions, namely a failed burst (lowest h₅), a successful burst that is prematurely terminated by the return of inhibition (middle h₅), and a successful burst that temporarily survives the return of inhibition (highest h₅). Vertical dashed black line marked “synapses on” shows the lower threshold of the voltage-to-activity function f(V) defined by Eq. (12). (b) Trajectories during 1:1 and 2:1 activity regimes (black and grey curves, respectively) projected to the (h₅,inh₅) plane, together with curves for the right (dashed) and left (solid) knees of the V₅ nullcline. Different curves of knees correspond to d₃ values that produce 2:1 (red) and 1:1 (green) regimes. As shown in a, increasing inhibition moves the nullcline to lower voltages and makes the cubic-like shape more pronounced, thus raising the value of h₅ at each knee. A trajectory must cross the curve of left knees for the late-E neuron to become active. Curves of critical points of the (V₅,h₅) equations on the left branch of the V₅ nullcline are shown in blue (1:1 regime) and light blue (2:1 regime). (c) and (d) Trajectories projected to show early-I (c) and post-I (d) dynamics, along with curves of critical points of the (V₂,m₂) and (V₃,m₃) equations (blue) and synaptic thresholds (red). The synaptic thresholds represent the values of input₂ and m₂ or inh₃ and m₃ at which each neuron will reach voltages of −50 (“on”, upper dashed), −35 (“half max”, solid red), and −20 (“max”, lower dashed, not shown in (d) if it evolves on its voltage nullcline. Each m variable increases while the corresponding neuron is active (rightward arrows) and decreases while the neuron is silent (leftward arrows)