The rectified distribution of open channels results in differences between the two

The Hodgkin-Huxley model describes the collective response of a channel population. At every time point the current passing through the channels is proportional to the fraction of open channels in the overall population, given by the activation and inactivation variables and their respective exponents. This fraction is voltage dependent, because the transition rates between the open and closed channel states are voltage dependent, and it remains fixed at an equilibrium value for every steady-state voltage level. As a result, the response of the system (macroscopic membrane voltage and current) is steady, until a new event takes place. On the other hand, in the Markov model the transitions between the states are stochastic. Therefore, at any time point the transition between states is still voltage dependent, as in the Hodgkin-Huxley model, but whether it actually occurs is a stochastic event. Hence, for the same steady-state the voltage and channel current responses are not fixed, but fluctuate around a mean point, the mean response of the population. Furthermore, the behaviour of the stochastic model is affected by changes in the number of channels. The mean current measured in the stochastic model is the average of all single (unitary) channel currents. For the same channel population current per unit membrane area, a larger single channel conductance implies a smaller total number of channels, and a larger unitary current. Thus, channel populations with a larger single channel conductance will exhibit greater current flow fluctuations (Donnell and Nolan, 2014; Kole et al., 2006; White et al., 2000).

The distribution of the open channels in the Markov model can be described by a binomial distribution B(N, p), where the number of trials N is defined by the number of channels and the success probability p in each trial is the probability of a channel to be open. The mean current I through a population of channels is the product of the unitary channel current i, the total number of channels N and their open probability, namely, I = iNp, where the product Np corresponds to the mean number of open channels. Thus, I can also be described by a binomial distribution with the standard deviation of the unitary current fluctuations 𝜎 = 𝑖√𝑁𝑝(1 − 𝑝) (Donnell and Nolan, 2014; White et al., 2000). The standard deviation σ describes the absolute amplitude of channel current noise for a given channel population. Therefore, for larger channel populations, and smaller single channel conductances, the current fluctuations will decrease, and the stochastic channel model behaviour will converge with the deterministic channel behaviour. This is due to the relationships of the σ with i and N. Smaller single channel Figure 5.11Tuning NaF channel models at gNaF = 11 x 10-3 S/cm2 and γNaF = 10-11 S. Right panels:

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

conductance, γ, results in smaller i, since i = γ(V-Echannel), which in turn will decrease the σ. On

the other hand, the concurrent increase in Ν¸ could increase the σ. However, σ is not directly proportional with the Ν, as it is with the i, but grows proportional with the √𝑁. Thus, the σ decreases faster with i than N (Figure 5.12).

The NaF channel models exhibit this anticipated behaviour (Figure 5.7). However, the temporal means of the voltage and current responses do not exactly match the deterministic responses under stimulus-free conditions. The stochastic model has a higher membrane voltage and current response compared to the deterministic one. The cause of this discrepancy might lie in a rectification effect emerging from a combined effect of a little overlap between the Hodgkin–Huxley steady-state activation and inactivation variables and a biologically realistic feature of the stochastic NaF channel model. There is a voltage range where the steady-state values of both sodium channel gating variables are not zero (Figure 5.12). Consequently, in this restricted voltage range, a sodium window current flows into the cell, slightly depolarizing the membrane and affecting the equilibrium value of the membrane voltage. In the case of the stochastic NaF model, the voltage (and current response) will fluctuate around the equilibrium value because the number of open channels also fluctuates, due to the probabilistic transition rates.

The stochastic current equilibrium point is the average of all fluctuating unitary channel currents. In order to be at the same level as the deterministic current equilibrium, symmetrical fluctuations around this equilibrium value would need to occur. However, this is not possible as it would require the existence of a negative number of channels, which is a biologically unrealistic condition. Consequently, the stochastic equilibrium point is estimated by averaging only the biologically allowed fluctuations, namely, a rectification effect is applied, which shifts the stochastic equilibrium up. Hence, for the observed rectification effect to take place, three conditions should apply: (a) the channel kinetics and state variable characteristics allow the generation of a window current, (b) the current equilibrium is close to zero and (c) the number of open channels exhibit fluctuations with a magnitude greater than their mean (which would be clipped at zero) (Figure 5.12).

An alternative way to depict this phenomenon is to study the distribution of open stochastic channels. Since a negative number of channels is biologically unrealistic, the distribution of the open stochastic channels is ‘forced’ to move to more positive values (for instance, 0, 1, 2), causing a rectification effect. As a result, the average fraction of open

stochastic channels is higher compared to the average deterministic fraction without any parameter correction. Without such an adjustment of the total conductance the behaviour of the two models cannot match exactly (Figure 5.8, 5.9, 5.10 and 5.12).

Figure 5.12 Rectification effect due to clipped distribution of open NaF channels. Upper panels:

Distribution of the open channels when no current is injected (gNaF = 11 x 10-3 S/cm2 and γNaF = 10-11 S.

Left panel: Channel population behaviour for all experimental trials with the stochastic NaF channel model. Right panel: Distribution for the first experimental trial and its probability density function (PDF). The distribution is clipped because there cannot be negative number of channels. Lower panels: Voltage dependence of the steady-state activation and inactivation variables of the NaF channel (left panel) and fraction of open NaF channels for both the stochastic and deterministic models (right panel). The average fraction of open stochastic channels is larger compared to the average deterministic fraction, and without any parameter correction (adjustment of the gNaF), the behaviour of the two