Fig 1.
Spikes, bursts and the effect of noise.
(A) The duration of an event is determined by the time that V spends above some threshold (chosen throughout as −45 mV). Events are classified as spikes if the event duration is less than 100 ms and there are no oscillations in V (e.g. first, yellow event). Events of longer duration or events with oscillations are identified as bursts (e.g. second, orange event). (B) The deterministic model with gBK = 0.5 nS leads to pure periodic spiking. (C) The stochastic model with the same parameters shows that noise can convert some spikes into bursts. (D) Increasing gBK to 1 nS leads to pure bursting in the deterministic model. (E) The equivalent stochastic model shows a mixture of spikes and bursts, with noise responsible for converting some bursts into spikes. (F) The deterministic model with gCa doubled to 4 nS simply sits at a depolarised steady state (gBK = 0.5 nS). (G) However, the stochastic model with the same parameters shows bursts as noise continually drives the system away from the steady state.
Fig 2.
The bursting fraction against total BK conductance.
In the deterministic model (dashed curve), the bursting fraction (BF) shows a sharp jump from pure spiking (BF = 0) to pure bursting (BF = 1) at a critical value of gBK. However, with channel noise (solid curve), this step function is smoothed out and no longer reaches pure bursting even for large gBK. Here, g1,BK has been fixed throughout so that gBK is proportional to NBK. Note that NBK values only have a physical meaning in the stochastic model.
Fig 3.
The effect on bursting of the total number of channels and the BK time constant.
(A) For gBK = 0.5 nS, corresponding to pure spiking in the deterministic model, BF gradually decreases as the total number of channels increases. However, for gBK = 0.6 nS, corresponding to pure bursting deterministically, increasing σ leads to an initially drop in BF before an increase towards BF = 1. (B) Higher values of τBK lead to less bursting as BK channels start to act more like K channels. Conversely smaller τBK, i.e. even faster acting BK channels, lead to more bursting. (C) The overall sigmoidal BF-against-gBK behaviour seen in Fig 2 is the same for other values of τBK. However, sufficiently large values of τBK only rarely lead to bursting even for very large total BK conductances.
Fig 4.
Relative noise contributions for BK versus non-BK channels.
(A) Plot of dV/dt against V with gBK = 0.5 nS and noise only in BK channels (Ca, K and SK behave deterministically) for a short 600 ms period during which two events occur (a spike followed by a burst). The dashed line shows the entirely deterministic solution (with no noise in any channels), which corresponds to continuous spiking. The black circle labels the initial starting configuration. (B) As in (A) but with noise only in the Ca, K and SK channels (BK acts deterministically). (C) The BF-against-gBK behaviour for different sources of noise showing that, whereas non-BK noise slightly smooths out the deterministic step function, the majority of the difference between the deterministic and fully stochastic models arises from noise in the BK channels.
Fig 5.
The effect of cell size in the deterministic model.
(A) Event duration against cell size for the case where gBK = 0.5 nS at λ = 1. The spiking behaviour switches sharply to bursting at around λ = 1.35. Light shading: pure spiking; dark shading: pure bursting. (B) The critical BK conductance for switching between spiking and bursting as a function of the cell size. The blue curve (left axis) shows the total BK conductance within the cell (gBK), whereas the red curve (right axis) shows the BK conductance area density (gBK/4πR2).
Fig 6.
The effect of cell size in the stochastic model.
(A) Examples of the membrane potential from simulations of five different cell sizes, showing mostly spiking for the smallest cells (λ = 0.45) and gradually becoming almost complete bursting for large cells (λ = 3.16). For each trace, the y-axis ranges from −70 to 0 mV. (B) The bursting fraction as a function of the cell size based on a 10 μm cell (λ = 1) having 5 BK channels. The dashed line shows the equivalent plot for the deterministic model. (C) The bursting fraction as a function of the gBK area density for three cell sizes. Panel (B) corresponds to BF measured along the dashed vertical line in Panel (C).
Fig 7.
The effect of opening and closing individual channels.
(A) Ranges over which opening or closing channels in the deterministic model causes spikes to become bursts. The blue curve shows the (non-perturbed) deterministic V profile, which corresponds to a spike. The horizontal axis shows the point during the action potential when the perturbation (here implemented as an extra current) is applied. Red/yellow regions: times from Vmax when adding/removing a current converts spiking to bursting. Top/bottom: perturbation applied in BK/Ca channels. In each case the perturbation is applied for 5 ms. (B) Effect on BF in the stochastic model of opening/closing a single BK channel at different points during the action potential. The horizontal axis shows the point when the perturbation is applied: a time from Vmax corresponds to a particular value of the potential in the deterministic model (blue curves in (A)); the perturbation is applied when the potential first rises above or drops below this value. Blue horizontal dashed line: value of BF with no perturbation.