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The discrepancy between two deterministic and stochastic channel models brings into question why these differences arise and whether there are conditions under which the two models could exhibit the same behaviour. The NaF channel is chosen for further investigation. Figure 5.6 An example of deterministic and stochastic models with corresponding behaviour. The

fKdr channel models showed similar behaviour under different simulation conditions. Left panels: Voltage (upper panel) and current (lower panel) response of the fKdr deterministic and stochastic models, when no current is injected. Right panels: Voltage (upper panel) and current (lower panel) response of the fKdr models when a 200 pA current pulse was injected for 1 s.

When no current is injected into the single compartmental model, both the deterministic and stochastic channel models eventually reach a steady-state or equilibrium voltage (and corresponding current). Due to the nature of the models, the equilibrium state of each model behaves differently. In the deterministic channel model, the equilibrium is achieved when the membrane potential stays at the exact same level of depolarization until a new event (for example, an external stimulus) is presented to the system. In the stochastic channel model, the depolarization level of the membrane alters through time as a result of the probabilistic channel gating and consequently electrical noise, expressed as fluctuations around the equilibrium point, is introduced. However, the stochastic temporal mean of the voltage (or current) gives an estimation of the average equilibrium point over a time window. Alternatively, an equilibrium value, equivalent to the stochastic temporal mean, could be reached with a very large number of channels, because the resulting fluctuations would even out.

Figure 5.7 An example of deterministic and stochastic models with different behaviours. The NaF

channel models under different simulation conditions. Left panels: Voltage (upper panel) and current (lower panel) response of the NaF deterministic and stochastic models, when no current is injected. Right panels: Voltage (upper panel) and current (lower panel) response of the NaF models when a 100 pA current pulse was injected for 1 s.

The deterministic models of the channels used in this study are taken from the CN neuron model, which is turn is tuned to in vitro electrophysiological experimental data(Steuber et al., 2011). Thus, the approach applied here is to tune the stochastic channel model so that it matches the deterministic one (and therefore experimental data). The current through the open channels is proportional to the total channel conductance in both formalisms (see Equation 5.1 and 5.4). Therefore, different channel current responses indicate that the two models have different total channel conductances and consequently the specific conductance of the stochastic model is adjusted to replicate the behaviour of the deterministic model.

The ratio of the deterministic current mean (Ideterministic) to the stochastic temporal

current mean (Istochastic) is estimated for different values of the specific channel conductance in

the stochastic model, when both models are equilibrated, and no current injection is applied. A current mean ratio close to one indicates that the channel models exhibit similar macroscopic current flow for this particular value of the total channel conductance (Figure 5.8).

Using the same experimental conditions (only the NaF channel type inserted into a single compartmental neuronal model), fixed current injection simulations are performed for a specific NaF conductance range of 10 - 16 x 10-3 S/cm2. The behaviour of the two models is studied for the pre- and during stimulus period. For the time window when no current stimulus Figure 5.8 Effect of the specific or single NaF channel conductance Left panel: Adjusting the

specific NaF channel conductance. The difference between the stochastic NaF channel model and the deterministic one is estimated in the absence of current injection, for the same single channel conductance (γNaF = 10-11 S). The numbers under the bars indicate the specific channel conductance (in

S/cm2_{). Right panel: Testing different single NaF channel conductances. The effect of different single}

NaF channel conductances is estimated in the absence of current injection, and for the same specific NaF channel conductance (gNaF = 11 x 10-3 S/cm2). The number of channels is inversely proportional

to the single channel conductance (10-9 _{S corresponds to 169 total channels and accordingly 10}-10 _{S to}

is delivered and both channel models are equilibrated, a total specific NaF conductance of 11 x 10-3 _S/cm2 _{is enough to match the behaviour of the stochastic model to the deterministic one}

(Figure 5.8 and 5.9, left panels). However, during stimulus application the responses of the two models deviate, with the stochastic model showing a smaller depolarization of the membrane voltage for the same current injection (Figure 5.9, right panels). Increasing the specific channel conductance leads to a larger behavioural deviation in the pre-stimulus window, with the stochastic model having a lower temporal current mean and a higher temporal voltage response, and to a better match during the stimulus, with the stochastic model gradually getting faster. At a specific channel sodium conductance of 15 x 10-3 S/cm2, the response to the stimulus of the two NaF channel models is similarly fast (Figure 5.10). Therefore, the two models could exhibit similar behaviours under different stimulation conditions, but not for the same specific channel conductance value at all conditions.

Figure 5.9 Tuning NaF channel models at gNaF = 11 x 10-3 S/cm2. Right panels: Voltage (upper

panel) and current (lower panel) response of the deterministic and stochastic NaF models, when no current is injected (left panes) and during stimulus (100 pA current, for 1 s) (right panel).

5.2.8 Effect of the single channel conductance on the behaviour of the