Correction to: Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding

Correction to: Biological Cybernetics https://doi.org/10.1007/s00422-021-00877-7

Due to a miscommunication, uncorrected page proofs were published that contained several minor errors. A corrected version of the article is available online. We provide the following corrigendum to clarify the correct forms of the mathematical expressions that were incorrectly typed. These corrections do not alter the argument, conclusions, or correctness of the theorems and other results given in the original paper.

• 1. Figure 1 compares voltage traces recorded from Purkinje cells from a wild-type and a leaner mutant mouse. The caption for panel 1(b) incorrectly states that the variability of the mutant (CV = 30%) is “twice” the variability of the wild-type (CV = 3.9%). The correct statement is simply that the mutant shows greater variability than the wild-type.

• 2. In §2.5 (Small-noise expansions), equation (27) is written with unbalanced parentheses. The correct expression should read

  $$\begin{aligned} \mathbb {E}^\mathbf {x}\left[ \phi (\mathbf {X}(\tau ))\right] =\phi (\mathbf {x})+\mathbb {E}^\mathbf {x}\left[ \int _0^\tau \mathcal {L}^\dagger \left[ \phi (\mathbf {X}(s))\right] \,ds \right] . \end{aligned}$$

• 3. In §3.1 (Assumptions for decomposition of the full noise model), in stating assumption A2, the text refers to channel state occupancy $q_{i(t)}$. It should instead refer to $q_{i(k)}$, namely, the fractional occupancy of the channel state i(k) which is the source node for the kth transition.

• 4. In §E.1 (Comparison with Giacomin et al. (2018)), we state an approximation of the form “$d\tilde{\theta }\approx 1+\varepsilon ^2 b+\varepsilon \sigma \, dW_t$, for some constants a and b...”. The correct expression should be “$d\tilde{\theta }\approx \left( 1+\varepsilon ^2 b\right) \,dt+\varepsilon \sigma \, dW_t$, for some constants $\sigma$ and b...”. There is no constant a associated with the relevant equation.

• 5. In §B (Diffusion matrix of the 14D model), the matrices $S_\text {K}$ and $S_\text {Na}$ were typed incorrectly. The correct expressions are as follows:

  $$\begin{aligned} S_\text {K}=&\frac{1}{\sqrt{N_\text {ref}}}\left[ \begin{array}{cccc} -\sqrt{4\alpha _n n_0}&{} \sqrt{\beta _n n_1}&{}0&{}0\\ \sqrt{4\alpha _n n_0}&{} -\sqrt{\beta _n n_1}&{}-\sqrt{3\alpha _n n_1}&{}\sqrt{2\beta _n n_2} \\ 0 &{}0 &{}\sqrt{3\alpha _n n_1}&{}-\sqrt{2\beta _n n_2}\\ 0 &{}0&{}0&{}0 \\ 0&{}0&{}0&{}0 \\ \end{array} \right. \cdots \\&\quad \cdots \left. \begin{array}{cccc} 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0 \\ -\sqrt{2\alpha _n n_2}&{}\sqrt{3\beta _n n_3}&{}0&{}0\\ \sqrt{2\alpha _n n_2}&{}-\sqrt{3\beta _n n_3}&{}-\sqrt{\alpha _n n_3}&{}\sqrt{4\beta _n n_4} \\ 0&{}0&{}\sqrt{\alpha _n n_3}&{}-\sqrt{4\beta _n n_4} \\ \end{array} \right] , \end{aligned}$$
  $$\begin{aligned}&S^{(1:5)}_\text {Na}=\frac{1}{\sqrt{M_\text {ref}}}\left[ \begin{array}{ccccc} -\sqrt{\alpha _h m_{00}}&{} \sqrt{\beta _h m_{01}}&{}-\sqrt{3\alpha _m m_{00}}&{}\sqrt{\beta _m m_{10}}&{} 0\\ 0&{} 0&{}\sqrt{3\alpha _m m_{00}}&{}-\sqrt{\beta _m m_{10}}&{} -\sqrt{\alpha _h m_{10}} \\ 0 &{}0 &{}0&{}0&{}0\\ 0 &{}0&{}0&{}0&{}0\\ \sqrt{\alpha _h m_{00}}&{}-\sqrt{\beta _h m_{01}}&{}0&{}0&{}0 \\ 0&{}0&{}0&{}0&{} \sqrt{\alpha _h m_{10}}\\ 0&{}0&{}0&{}0&{}0 \\ 0&{}0&{}0&{}0&{}0 \\ \end{array} \right] \\&S_\text {Na}^{(6:10)}=\frac{1}{\sqrt{M_\text {ref}}}\left[ \begin{array}{ccccc} 0&{}0&{}0&{}0&{}0\\ \sqrt{\beta _h m_{11}} &{}-\sqrt{2\alpha _m m_{10}}&{}\sqrt{2\beta _m m_{20}}&{}0&{}0 \\ 0 &{}\sqrt{2\alpha _m m_{10}}&{}-\sqrt{2\beta _m m_{20}}&{}-\sqrt{\alpha _h m_{20}}&{}\sqrt{\beta _h m_{21}}\\ 0&{}0&{}0 &{}0&{}0\\ 0&{}0&{}0&{}0&{}0 \\ -\sqrt{\beta _h m_{11}}&{}0&{}0&{}0&{}0 \\ 0&{}0&{}0&{}\sqrt{\alpha _h m_{20}}&{}-\sqrt{\beta _h m_{21}} \\ 0&{}0&{}0&{}0&{}0 \\ \end{array} \right] \\&S^{(11:15)}_\text {Na}=\frac{1}{\sqrt{M_\text {ref}}}\left[ \begin{array}{ccccc} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ -\sqrt{\alpha _m m_{20}}&{} \sqrt{3\beta _m m_{30}}&{}0&{}0 &{}0\\ \sqrt{\alpha _m m_{20}}&{} -\sqrt{3\beta _m m_{30}} &{}-\sqrt{\alpha _h m_{30}} &{}\sqrt{\beta _h m_{31}}&{}0\\ 0&{}0&{}0&{}0&{}-\sqrt{3\alpha _m m_{01}}\\ 0&{}0&{}0&{}0&{}\sqrt{3\alpha _m m_{01}}\\ 0&{}0&{}0&{}0 &{}0\\ 0&{}0&{}\sqrt{\alpha _h m_{30}}&{}-\sqrt{\beta _h m_{31}} &{}0\\ \end{array} \right] \\&S^{(16:20)}_\text {Na}=\frac{1}{\sqrt{M_\text {ref}}}\left[ \begin{array}{ccccc} 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0\\ \sqrt{\beta _m m_{11}}&{}0&{}0&{}0&{}0\\ -\sqrt{\beta _m m_{11}}&{}-\sqrt{2\alpha _m m_{11}}&{}\sqrt{2\beta _m m_{21}}&{}0&{}0\\ 0&{}\sqrt{2\alpha _m m_{11}}&{}-\sqrt{2\beta _m m_{21}}&{}-\sqrt{\alpha _m m_{21}}&{}\sqrt{3\beta _m m_{31}}\\ 0&{}0&{}0&{}\sqrt{\alpha _m m_{21}}&{}-\sqrt{3\beta _m m_{31}}\\ \end{array} \right] , \end{aligned}$$

  where $S^{(i:j)}_\text {Na}$ is the i$^{th}$-j$^{th}$ column of $S_\text {Na}$.

The original article has been updated.