Repetitive Action Potential Firing

Abstract

The biophysical basis of repetitive action potential firing can be understood in terms of the opposing positive and negative feedback processes that generate the repetitive activity. The relative timescales of these processes, which can include ion channel activation and inactivation as well as ion accumulation within a cell, are key factors in the generation of repetitive single‐spike firing, intrinsic burst firing and repetitive activity resulting from network interactions. Bifurcation theory, in concert with the steady state current–voltage relationship, explains why the onset of repetitive firing can be gradual or abrupt and accounts for different excitability types and distinct types of bistability. Moreover, bifurcation theory allows for the classification of different mechanisms underlying rhythmic bursting.

Key Concepts

  • A branch of mathematics called bifurcation theory can explain initiation of repetitive firing without reference to the specific ionic currents involved.
  • Two major bifurcations can underlie the initiation of repetitive firing.
  • Initiation of repetitive firing is gradual if the underlying bifurcation is a saddle‐node on an invariant circle (SNIC).
  • A SNIC requires that the steady state IV curve intersect the zero current axis in three places.
  • Initiation of repetitive firing is abrupt at a threshold frequency if due to a Hopf bifurcation.
  • Most rhythmic bursting can be classified as square wave, parabolic, or elliptical, depending upon the bifurcations that initiate and terminate spiking.
  • Rhythmic bursting requires at least one slow process, and parabolic bursting requires two.
  • Repetitive firing can also arise via network mechanisms, in addition to the intrinsic ones described above.

Keywords: bursting; computational modelling; dynamical system; electrophysiology; excitability; pacemaker; rhythm generation; neural oscillations

Figure 1. Monotonic and cubic steady state I–V curves and their associated f–I curves. (a1) Monotonic steady state (black solid curve) and instantaneous I–V curve (dashed red curve). The single intersection of the zero current axis is a stable resting potential (black filled circle) that can be destabilised by making the applied current more depolarising and activating the negative slope resistance region in the instantaneous I–V curve. (a2) The f–I curve gives frequency as a function of the applied stimulus current. For the model with a monotonic I–V, this curve has two overlapping but discontinuous branches that produce the abrupt onset of firing at a finite frequency characteristic of type 2 excitability. (b) Cubic steady state I–V curve. (b1) Low values of the applied stimulus current produce a cubic curve with three intersections with the zero current axis, each corresponding to a fixed point of the system. The point at −64 mV is stable, while the points at −58 and −37 mV are unstable and correspond to a saddle and a focus respectively. Increasing the bias current shifts the axis (inset) such that the two lower intersections merge in a saddle‐node, then disappear. Models with a cubic IV curve can produce two types of f–I curves. (b2a) One is an f–I curve with bistability. The abrupt onset of firing at a finite frequency as the applied current is increased characteristic of type 2 excitability. (b2b) The same cubic steady state IV can also produce a continuously varying f–I curve with arbitrarily low frequencies is characteristic of type 1 excitability.
Figure 2. Phase portraits for monotonic steady state I–V curve in Figure a. (a) Quiescent but excitable (a1) 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 (red dashed curve). A sufficiently large depolarisation will cause a jump to the rightmost stable branch resulting in an action potential before returning to rest. (a2) The time course of the depolarising perturbations in a1, the blue arrow indicating the time of the perturbation. Voltage traces for subthreshold (dashed) and superthreshold (solid) perturbations. (b) Bistable (b1) Increasing the bias current shifts the fixed point toward 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) Membrane potential traces for bistable dynamics. The arrows indicate the transient current pulses used to switch the cell between quiescence and repetitive firing by crossing the unstable limit cycle. (c) Pacemaker (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.
Figure 3. Phase plane portraits for cubic steady state I–V curve. (a) Bistable (at decreased τw) (a1) Three intersections of the voltage (green) and w (blue) nullclines define three fixed points. The rightmost fixed point is an unstable focus (red filled circle) inside a stable limit cycle (red curve). The leftmost fixed point is a stable node (black filled circle) and the middle one is a saddle (open bisected circle). The separatrix (dashed black curve) passes through the saddle along its attractive axis and separates repetitive firing along the limit cycle from quiescence at the stable fixed point corresponding to the rest potential (see inset). (a2) Bistability associated with a cubic I–V. Perturbations (arrows that indicate depolarising and hyperpolarising respectively) switch between quiescence and pacemaking. (b) Increasing the bias current raises the hump until the two intersections collide as a saddle‐node (inset in Figure b). The saddle‐node collides with a limit cycle (red curve), but the velocity of the dynamics is zero at the saddle‐node, giving an infinite period (0 frequency). The slower time course of w allows for a deeper after‐hyperpolarisation that does not allow the limit cycle to coexist with the fixed point as it did in (a).
Figure 4. Three types of intrinsic burst firing. (a) Square wave bursting. (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 (upper knee) and the saddle‐node (lower knee), at each value of calcium concentration, 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 concentration decreases. At low values of calcium concentration, the left branch terminates at a knee (SN), so the trajectory jumps toward the right branch, which is associated with repetitive firing and along which calcium concentration is increasing. At high enough values of calcium concentration, 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 be revealed in the limit where the fast subsystem is set to its steady state. (b) Parabolic bursting. (b1) The voltage is plotted against the two slow variables calcium concentration (c) and calcium channel activation (s). Starting at the homoclinic bifurcation at left, s increases due to depolarisation associated with spiking. This, in turn, causes calcium to accumulate, curving the trajectory back towards the unstable manifold (black outlined surface), where it again intersects at the right homoclinic bifurcation. The calcium concentration decays as the calcium activation decreases sufficiently following repolarisation. (b2) Parabolic bursting (red) along with a subthreshold oscillation revealed by setting INa to zero (dashed trace) (c) 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 INa 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 concentration increases, causing the trajectory (red) to enter the region between the homoclinic (HB) bifurcation and the saddle‐node of periodics (SNP) in which there is both a stable repetitive firing solution and a stable quiescent solution. When calcium concentration increases above the SNP, the repetitive firing solution ceases to exist. The trajectory then tracks the quiescent solution along the potential nullcline, with calcium concentration 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 repetitive firing. (a) Model circuit. Two identical mutually inhibitory neurons. (b) Single spike firing via PIR. (b1) The h nullcline (blue) intersects at a stable fixed point (black filled dot) with the potential nullcline for an uninhibited neuron (solid green). The activation of the synapse by a presynaptic spike perturbs the cell (black arrow) across the quasi‐threshold (black dashed line) producing a spike. (b2) An initial pulse (blue arrow) initiates a spike in cell 1, which initiates a spike in cell 2 via postinhibitory rebound from the transient hyperpolarisation. This continues until another pulse of sustained inhibition to both cells (blue bar) terminates the oscillation. (b3) Level of activation of synaptic conductances during spiking in b2. (c) Network burst firing via release. (c1) 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. (c2) The release mechanism is shown for two identical mutually inhibitory oscillators (red and black solid lines) with the ‘synaptic threshold’ indicated by black dashed line. (d) Network burst firing via escape. (d1) 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. (d2) The release is mechanism shown for two identical mutually inhibitory oscillators (red and black solid lines) with the synaptic threshold indicated by black dashed line.
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Knowlton, Christopher J, Baxter, Douglas A, Byrne, John H, and Canavier, Carmen C(Jun 2020) Repetitive Action Potential Firing. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0000084.pub3]