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@ -146,7 +146,7 @@ All modelling and simulation was done in parallel with custom written Python 3.8
% Linux 3.10.0-123.e17.x86_64.
\subsection*{Different Cell Models}
A group of neuronal models representing the major classes of cortical and thalamic neurons including regular spiking pyramidal (RS pyramidal), regular spiking inhibitory (RS inhibitory), and fast spiking (FS) cells were used \citep{pospischil_minimal_2008}. To each of these models, a \Kv current (\IKv); \cite{ranjan_kinetic_2019}) was added. A cerebellar stellate cell model from \cite{alexander_cerebellar_2019} is used (Cb stellate). This model was also used with a \Kv current (\IKv; \cite{ranjan_kinetic_2019}) in addition to the A-type potassium current (Cb stellate +\Kv) or replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv). A subthalamic nucleus neuron model as described by \cite{otsuka_conductance-based_2004} are used (STN) and with a \Kv current (\IKv; \cite{ranjan_kinetic_2019}) in addition to the A-type potassium current (STN +\Kv) or replacing the A-type potassium current (STN \(\Delta\)\Kv). The properties and conductances of each model are detailed in \Cref{tab:g} and the gating properties are unaltered from the original Cb stellate and STN models. For comparability to typical electrophysiological data fitting reported and for ease of further gating curve manipulations, a Boltzmann function
A group of neuronal models representing the major classes of cortical and thalamic neurons including regular spiking pyramidal (RS pyramidal), regular spiking inhibitory (RS inhibitory), and fast spiking (FS) cells were used \citep{pospischil_minimal_2008}. To each of these models, a \Kv current (\IKv); \cite{ranjan_kinetic_2019}) was added. A cerebellar stellate cell model from \citet{alexander_cerebellar_2019} is used (Cb stellate). This model was also used with a \Kv current (\IKv; \cite{ranjan_kinetic_2019}) in addition to the A-type potassium current (Cb stellate +\Kv) or replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv). A subthalamic nucleus neuron model as described by \citet{otsuka_conductance-based_2004} are used (STN) and with a \Kv current (\IKv; \cite{ranjan_kinetic_2019}) in addition to the A-type potassium current (STN +\Kv) or replacing the A-type potassium current (STN \(\Delta\)\Kv). The properties and conductances of each model are detailed in \Cref{tab:g} and the gating properties are unaltered from the original Cb stellate and STN models. For comparability to typical electrophysiological data fitting reported and for ease of further gating curve manipulations, a Boltzmann function
\begin{equation}\label{eqn:Boltz}
x_\infty = {\left(\frac{1-a}{1+{exp[{\frac{V-V_{1/2}}{k}}]}} +a\right)^j}
@ -160,7 +160,7 @@ with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}
\input{gating_table}
\subsection*{Firing Frequency Analysis}
The membrane responses to 200 equidistant two second long current steps were simulated using the forward-Euler method with a \(\Delta \textrm{t} = 0.01\) ms from steady state. Current steps ranged from 0 to 1 nA for all models except for the RS inhibitory neuron models where a range of 0 to 0.35 nA was used to ensure repetitive firing across the range of input currents. For each current step, action potentials were detected as peaks with at least 50 mV prominence and a minimum interspike interval of 1 ms. The interspike interval was computed and used to determine the instantaneous firing frequencies elicited by the current step. The steady-state firing frequency were defined as the mean firing frequency in 0.5 seconds after the first action potential in the last second of the current step respectively and was used to construct frequency-current (fI) curves.
The membrane responses to 200 equidistant two second long current steps were simulated using the forward-Euler method with a \(\Delta \textrm{t} = 0.01\)\,ms from steady state. Current steps ranged from 0 to 1\,nA for all models except for the RS inhibitory neuron models where a range of 0 to 0.35 nA was used to ensure repetitive firing across the range of input currents. For each current step, action potentials were detected as peaks with at least 50\,mV prominence and a minimum interspike interval of 1\,ms. The interspike interval was computed and used to determine the instantaneous firing frequencies elicited by the current step. The steady-state firing frequency were defined as the mean firing frequency in 0.5\,s after the first action potential in the last second of the current step respectively and was used to construct frequency-current (fI) curves.
The smallest current at which steady state firing occurs was identified and the current step interval preceding the occurrence of steady state firing was simulated at higher resolution (100 current steps) to determine the current at which steady state firing began. Firing was simulated with 100 current steps from this current upwards for 1/5 of the overall current range.
Over this range a fI curve was constructed and the integral, or area under the curve (AUC), of the fI curve over this interval was computed with the composite trapezoidal rule and used as a measure of firing rate independent from rheobase.
@ -173,7 +173,7 @@ with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}
% Sensitivity analyses enable investigation into how different sources of uncertainty in a model result in uncertainty in model outputs \citep{saltelli_sensitivity_2002} and provide information on the relative impact of model inputs \citep{saltelli_why_2019}. We recently used a one-factor-at-a-time (OFAT) sensitivity analysis approach to evaluate the relative impacts of currents on neuronal firing and developed a scoring system for SCN8A mutations that correlated (p = 0.0077, r = 0.64) with the clinical severity of epilepsy in patients with these mutations \citep{johannesen_genotype-phenotype_2021}. This was done in an isolated neuronal model and suggests that even with disregard of network level effects of mutations, the single cell level outcomes of mutations are relevant to disease phenotypes. OFAT sensitivity analyses indicate which factors have or do not have influence, with uninfluential factors never detected as relevant \citep{saltelli_how_2010}. OFAT sensitivity analyses can be used to screen factors that are influential on model outcomes and provide a mechanism by which factors and their relative influence can be easily identified and used in predictive applications.
Current properties of currents common to all models (\(\textrm{I}_{\textrm{Na}}\), \(\textrm{I}_{\textrm{K}}\), \(\textrm{I}_{\textrm{A}}\)/\IKv, and \(\textrm{I}_{Leak}\)) were systematically altered in a one-factor-at-a-time sensitivity analysis for all models. The gating curves for each current were shifted (\(\Delta V_{1/2}\)) from -10 to 10 mV in increments of 1 mV. The slope (\(k\)) of the gating curves were altered from half to twice the initial slope. Similarly, the maximal current conductance (\(g\)) was also scaled from half to twice the initial value. For both slope and conductance alterations, alterations consisted of 21 steps spaced equally on a \(\textrm{log}_2\) scale.
Current properties of currents common to all models (\(\textrm{I}_{\textrm{Na}}\), \(\textrm{I}_{\textrm{K}}\), \(\textrm{I}_{\textrm{A}}\)/\IKv, and \(\textrm{I}_{Leak}\)) were systematically altered in a one-factor-at-a-time sensitivity analysis for all models. The gating curves for each current were shifted (\(\Delta V_{1/2}\)) from -10 to 10\,mV in increments of 1\,mV. The slope (\(k\)) of the gating curves were altered from half to twice the initial slope. Similarly, the maximal current conductance (\(g\)) was also scaled from half to twice the initial value. For both slope and conductance alterations, alterations consisted of 21 steps spaced equally on a \(\textrm{log}_2\) scale.
\subsection*{Model Comparison}
Changes in rheobase (\(\Delta rheobase\)) are calculated in relation to the original model rheobase. The contrast of each AUC value (\(AUC_i\)) was computed in comparison to the AUC of the unaltered wild type model (\(AUC_{wt}\))
@ -186,7 +186,7 @@ with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}
The Kendall's \(\tau\) coefficient, a non-parametric rank correlation, is used to describe the relationship between the magnitude of the alteration and AUC or rheobase values. A Kendall \(\tau\) value of -1 or 1 is indicative of monotonically decreasing and increasing relationships respectively.
\subsection*{KCNA1/\Kv Mutations}\label{subsec:mut}
Known episodic ataxia type 1 associated KCNA1 mutations and their electrophysiological characterization reviewed in \cite{lauxmann_therapeutic_2021}. The mutation-induced changes in \IKv amplitude and activation slope (\(k\)) were normalized to wild type measurements and changes in activation \(V_{1/2}\) were used relative to wild type measurements. The effects of a mutation were also applied to \(\textrm{I}_{\textrm{A}}\) when present as both potassium currents display prominent inactivation. In all cases, the mutation effects were applied to half of the \Kv or \(\textrm{I}_{\textrm{A}}\) under the assumption that the heterozygous mutation results in 50\% of channels carrying the mutation. Frequency-current curves for each mutation in each model were obtained through simulation and used to characterize firing behaviour as described above. For each model the differences in mutation AUC to wild type AUC were normalized by wild type AUC (\(AUC_{contrast}\)) and mutation rheobases are compared to wild type rheobase values (\(\Delta rheobase\)). Pairwise Kendall rank correlations (Kendall \(\tau\)) are used to compare the the correlation in the effects of \Kv mutations on AUC and rheobase between models.
Known episodic ataxia type 1 associated KCNA1 mutations and their electrophysiological characterization reviewed in \citet{lauxmann_therapeutic_2021}. The mutation-induced changes in \IKv amplitude and activation slope (\(k\)) were normalized to wild type measurements and changes in activation \(V_{1/2}\) were used relative to wild type measurements. The effects of a mutation were also applied to \(\textrm{I}_{\textrm{A}}\) when present as both potassium currents display prominent inactivation. In all cases, the mutation effects were applied to half of the \Kv or \(\textrm{I}_{\textrm{A}}\) under the assumption that the heterozygous mutation results in 50\% of channels carrying the mutation. Frequency-current curves for each mutation in each model were obtained through simulation and used to characterize firing behaviour as described above. For each model the differences in mutation AUC to wild type AUC were normalized by wild type AUC (\(AUC_{contrast}\)) and mutation rheobases are compared to wild type rheobase values (\(\Delta rheobase\)). Pairwise Kendall rank correlations (Kendall \(\tau\)) are used to compare the the correlation in the effects of \Kv mutations on AUC and rheobase between models.