Figure 1. Monotonic steady-state I–V curve. The instantaneous I–V curve (dashed red line) illustrates the current required to hold the membrane at a given voltage with the restorative process w set to zero, thus showing only the effect of the autocatalytic dependence of m_¥ on potential. If the restorative process w is also set to its steady-state value, the steady-state current voltage curve results (solid red line). The intersection of the steady-state I–V with the zero-injected current x-axis (dashed blue line) determines the resting potential of the cell (filled circle). The I–V curve is relatively flat in the region of the intersection, so the inset is plotted on a different scale to illustrate that the monotonicity that implies a single fixed point also holds near the resting potential.

Figure 2. Nullcline portraits for monotonic steady-state I–V curve. (a1) This corresponds to the I–V curve in Figure 1. The intersection with the w nullcline (blue) occurs on the left-hand branch of the potential nullcline (green), therefore the fixed point is stable and small perturbations will return to the fixed point without a regenerative increase in membrane potential (solid red trace). A perturbation large enough will cause a jump (red dashed line) to the rightmost stable branch. (a2) The time course of the two perturbations described in a1 are shown: the solid red line illustrates a decay of the small perturbation with time, whereas the dashed red line illustrates an action potential. (b1) The fixed point is shifted towards the boundary between the stable right branch and the unstable middle branch (black filled circle), and retains its stability. However, a stable limit cycle (red curve) also appears, and an unstable limit cycle (dashed black line) forms the boundary between the two stable solutions. (b2) The black line illustrates the quiescent solution associated with the filled black circle in b1. The repetitive action potential firing shown in red corresponds to the red limit cycle shown in b1. The arrows indicate the transient current pulses that were used to switch the cell between quiescence and repetitive firing. (c1) If the fixed point occurs on the unstable middle branch (filled red circle), the system trajectory jumps between stable branches cyclically (red curve) while avoiding the region surrounding the unstable branch. (c2) The tonic firing that results is shown in red. Note difference in time scale for parts a2, b2 and c2.

Figure 3. Cubic steady-state I–V curve and resultant phase portrait. (a) The steady-state current (solid line) has three fixed points. The two leftmost fixed points are situated on either side of the hump shown in the inset. As a constant hyperpolarizing bias current of increasing magnitude is applied, the two points move towards each other, collide and disappear as the hump slips below the zero current axis. The black dashed line illustrates a cubic I–V curve that is sufficiently hyperpolarized so that only a single intersection (red filled circle) remains. (b1) The leftmost intersection (filled black circle) of the potential nullcline (green) with the w nullcline (blue) falls on the left branch of the potential nullcline and is stable (quiescent). The rightmost intersection (red filled circle) falls on the middle branch, is unstable and is associated with a limit cycle (red curve). The middle intersection (red cross) also falls in the middle branch and on the boundary (black dashed line) between the domains of the other two fixed points. (b2) The black line shows the quiescent resting potential associated with the black-filled circle in b1, and the red shows the repetitive firing associated with the red curve in b1. The arrows indicate transient current pulses that were applied to switch between quiescence and repetitive firing.

Figure 4. Three types of intrinsic burst firing. (a) Type-I square wave. (a1) The calcium nullcline (blue) and the potential nullcline (green) intersect (filled red circle) in the unstable middle branch of the potential nullcline. Between the homoclinic bifurcation (dashed line) and the saddle node (SN), at each value of calcium there are two stable solutions of the fast subsystem: quiescence and repetitive spiking, associated with the leftmost and rightmost branches, respectively, of the potential nullcline. While the trajectory (red curve) moves along the left branch, the cell is quiescent and calcium decreases. At low values of calcium, the left branch terminates at a knee (SN), so the trajectory jumps towards the right branch, which is associated with repetitive firing and along which calcium is increasing. At high-enough values of calcium, the trajectory collides with the unstable middle branch and becomes quiescent again. (a2) Burst firing corresponding to the red curve in a1 is shown in red. A subthreshold oscillation (black dotted trace) can sometimes be revealed by blocking I_Na. (b) Type-II parabolic bursting. (b1) Although there are two slow variables, only calcium is plotted versus voltage (see trajectory in red). The effect of the second slow variable s is shown by plotting the potential nullcline with s fixed at its value at burst initiation (dashed green) and burst termination (solid green). The variation in a second slow variable allows burst termination and initiation via a homoclinic mechanism when the trajectory collides with the unstable middle branch. Since the value of s varies along the trajectory, the only true intersections in the full phase space are indicated by the homoclinic bifurcations. (b2) Parabolic bursting (red) along with a subthreshold oscillation (black dashed trace) revealed by blocking I_Na. (c) Type-III elliptic bursting. (c1) The intersection (filled red circle) of the potential nullcline (green) and the calcium nullcline (blue) is unstable due to the positive feedback associated with I_Na and is surrounded by a stable limit cycle (in a fast subsystem plane that is not shown) associated with repetitive firing. During repetitive firing, calcium increases, causing the trajectory (red) to enter the region between the homoclinic bifurcation (HB) and the saddle node of periodics (SNP) in which there is both a stable repetitive firing solution and a stable quiescent solution. When calcium increases above the SNP, the repetitive firing solution ceases to exist. The trajectory then tracks the quiescent solution along the potential nullcline, with calcium decreasing until the quiescent solution ceases to exist at the HB, and spiking is re-established. (c2) Elliptic burst firing (red) corresponding to trajectory in c1.

Figure 5. Network mechanisms for burst firing. (a) Model circuit. Two identical mutually inhibitory neurons. (b) Release. (b1) The h nullcline (blue) intersects (black filled dots) with both the potential nullcline for an uninhibited neuron (solid green) and for an inhibited neuron (dashed green) below the synaptic threshold (black dashed line). An oscillatory trajectory (red) can also occur. (b2) The release mechanism shown for two identical mutually inhibitory oscillators (red and black solid lines) with the ‘synaptic threshold’ indicated by black dashed line. (c) Escape. (c1) The potential nullcline is cubic for the inhibited (green dashed curve) case, but this cubic nature is not expressed in the uninhibited or ‘free’ (green) case. The intersections (black filled circles) with the h nullcline (blue) occur in stable regions, but on opposite sides of the synaptic threshold (black dashed line). A sustained oscillation (red) is also stable. (c2) The release mechanism shown for two identical mutually inhibitory oscillators (red and black solid lines) with the synaptic threshold indicated by black dashed line. See text for additional details.