441 lines
68 KiB
TeX
441 lines
68 KiB
TeX
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\documentclass[utf8]{FrontiersinHarvard} % for articles in journals using the Harvard Referencing Style (Author-Date), for Frontiers Reference Styles by Journal: https://zendesk.frontiersin.org/hc/en-us/articles/360017860337-Frontiers-Reference-Styles-by-Journal
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%\documentclass[utf8]{frontiersinFPHY_FAMS} % Vancouver Reference Style (Numbered) for articles in the journals "Frontiers in Physics" and "Frontiers in Applied Mathematics and Statistics"
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\externaldocument{Koch_SupplementaryMaterial}
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%\newcommand{\Kv}{\(\textrm{K}_{\textrm{V}}\textrm{1.1}\)\xspace}
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%\newcommand{\IKv}{\(\textrm{I}_{\textrm{K}_{\textrm{V}}\textrm{1.1}}\)\xspace}
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\newcommand{\Kv}{\(\text{K}_{\text{V}}\text{1.1}\)\xspace}
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\newcommand{\IKv}{\(\text{I}_{\text{K}_{\text{V}}\text{1.1}}\)\xspace}
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\newcommand{\drheo}{\(\Delta\)rheobase\xspace}
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\newcommand{\ndAUC}{normalized \(\Delta\)AUC\xspace}
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\def\keyFont{\fontsize{8}{11}\helveticabold }
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\def\firstAuthorLast{Koch {et~al.}} %use et al only if is more than 1 author
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\def\Authors{Nils A. Koch\,$^{1,2}$, Lukas Sonnenberg\,$^{1,2}$, Ulrike B.S. Hedrich\,$^{3}$, Stephan Lauxmann\,$^{1,3}$ and Jan Benda\,$^{1,2,*}$}
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% Affiliations should be keyed to the author's name with superscript numbers and be listed as follows: Laboratory, Institute, Department, Organization, City, State abbreviation (USA, Canada, Australia), and Country (without detailed address information such as city zip codes or street names).
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% If one of the authors has a change of address, list the new address below the correspondence details using a superscript symbol and use the same symbol to indicate the author in the author list.
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\def\Address{$^{1}$Laboratory X, Institute X, Department X, Organization X, City X , State XX (only USA, Canada and Australia), Country X \\
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$^{2}$Laboratory X, Institute X, Department X, Organization X, City X , State XX (only USA, Canada and Australia), Country X }
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\def\Address{$^{1}$Institute for Neurobiology, University of Tuebingen, 72072 Tuebingen, Germany \\
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$^{2}$Bernstein Center for Computational Neuroscience Tuebingen, 72076 Tuebingen, Germany \\
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$^{3}$Department of Neurology and Epileptology, Hertie Institute for Clinical Brain Research, University of Tuebingen, 72076 Tuebingen, Germany}
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% The Corresponding Author should be marked with an asterisk
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\def\corrAuthor{Jan Benda}
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\def\corrEmail{ jan.benda@uni-tuebingen.de}
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\begin{document}
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\onecolumn
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\firstpage{1}
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\title {Loss or Gain of Function? Effects of Ion Channel Mutations on Neuronal Firing Depend on the Cell Type}
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%\extraAuth{corresponding Author2 \\ Laboratory X2, Institute X2, Department X2, Organization X2, Street X2, City X2 , State XX2 (only USA, Canada and Australia), Zip Code2, X2 Country X2, email2@uni2.edu}
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\maketitle
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\begin{abstract}
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%%% Leave the Abstract empty if your article does not require one, please see the Summary Table for full details.
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\section{}
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%For full guidelines regarding your manuscript please refer to \href{https://www.frontiersin.org/guidelines/author-guidelines}{Author Guidelines}.
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%As a primary goal, the abstract should render the general significance and conceptual advance of the work clearly accessible to a broad readership. References should not be cited in the abstract. Leave the Abstract empty if your article does not require one, please see the Article Types on every Frontiers journal page for full details
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Clinically relevant mutations to voltage-gated ion channels, called channelopathies, alter ion channel function, properties of ionic current and neuronal firing. The effects of ion channel mutations are routinely assessed and characterized as loss of function (LOF) or gain of function (GOF) at the level of ionic currents. Emerging personalized medicine approaches based on LOF/GOF characterization have limited therapeutic success. Potential reasons are that the translation from this binary characterization to neuronal firing especially when considering different neuronal cell types is currently not well understood. Here we investigate the impact of neuronal cell type on the firing outcome of ion channel mutations with simulations of a diverse collection of neuron models. We systematically analyzed the effects of changes in ion current properties on firing in different neuronal types. Additionally, we simulated the effects of mutations in the \textit{KCNA1} gene encoding the \Kv potassium channel subtype associated with episodic ataxia type~1 (EA1). These simulations revealed that the outcome of a given change in ion channel properties on neuronal excitability is cell-type dependent. As a result, cell-type specific effects are vital to a full understanding of the effects of channelopathies on neuronal excitability and present an opportunity to further the efficacy and precision of personalized medicine approaches.
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\tiny
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\keyFont{ \section{Keywords:} Channelopathy, Epilepsy, Ataxia, Potassium Current, Neuronal Simulations, Conductance-based Models, Neuronal heterogeneity } %All article types: you may provide up to 8 keywords; at least 5 are mandatory.
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\end{abstract}
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\section{Introduction}
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%
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%For Original Research Articles \citep{conference}, Clinical Trial Articles \citep{article}, and Technology Reports \citep{patent}, the introduction should be succinct, with no subheadings \citep{book}. For Case Reports the Introduction should include symptoms at presentation \citep{chapter}, physical exams and lab results \citep{dataset}.
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The properties and combinations of voltage-gated ion channels are vital in determining neuronal excitability \citep{bernard_channelopathies_2008, carbone_ion_2020, rutecki_neuronal_1992, pospischil_minimal_2008}. However, ion channel function can be disturbed, for instance through genetic alterations, resulting in altered neuronal firing behavior \citep{carbone_ion_2020}. In recent years, next generation sequencing has led to an increase in the discovery of clinically relevant ion channel mutations and has provided the basis for pathophysiological studies of genetic epilepsies, pain disorders, dyskinesias, intellectual disabilities, myotonias, and periodic paralyses \citep{bernard_channelopathies_2008, carbone_ion_2020}.
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Ongoing efforts of many research groups have contributed to the current understanding of underlying disease mechanism in channelopathies, however a complex pathophysiological landscape has emerged for many channelopathies and is likely a reason for limited therapeutic success with standard care.
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Ion channel variants are frequently classified in heterologous expression systems as either a loss of function (LOF) or a gain of function (GOF) in the respective ionic current \citep{Musto2020, Kullmann2002, Waxman2011, Kim2021}. This LOF/GOF classification is often directly used to predict the effects on neuronal firing \citep{Niday2018, Wei2017, Wolff2017,Masnada2017}, which in turn is important for understanding the pathophysiology of these disorders and for identification of potential therapeutic targets \citep{Orsini2018, Yang2018, Colasante2020, Yu2006}. Experimentally, the effects of channelopathies on neuronal firing are assessed using primary neuronal cultures \citep{Scalmani2006, Smith2018, Liu2019} or \textit{in vitro} recordings from slices of transgenic mouse lines \citep{Mantegazza2019, Xie2010,Lory2020, Habib2015, Hedrich14874} but are restricted to limited number of neuronal types. Different neuron types differ in their composition of ionic currents \citep{yao2021taxonomy, Cadwell2016, BICCN2021, Scala2021} and therefore likely respond differently to changes in the properties of a single ionic current. Expression level of an affected gene \citep{Layer2021} and relative amplitudes of ionic currents \citep{rutecki_neuronal_1992, pospischil_minimal_2008,Kispersky2012, golowasch_failure_2002, barreiro_-current_2012} indeed dramatically influence the firing behavior and dynamics of neurons. Mutations in different sodium channel genes have been experimentally shown to affect firing in a cell-type specific manner based on differences in expression levels of the affected gene \citep{Layer2021}, but also on other cell-type specific mechanisms \citep{Hedrich14874, makinson_scn1a_2016}.
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Cell-type specificity is likely vital for successful precision medicine treatment approaches. For example, Dravet syndrome was identified as the consquence of LOF mutations in \textit{SCN1A} \citep{Claes2001,Fujiwara2003,Ohmori2002}, however limited success in the treatment of Dravet syndrome persisted \citep{Claes2001,Oguni2001} in part due to lack of understanding that inhibitory interneurons and not pyramidal neurons had altered excitability as a result of LOF \textit{SCN1A} mutations \citep{Yu2006, Colasante2020}.
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Taken together, these examples demonstrate the need to study the effects of ion channel mutations in many different cell types --- a daunting if not impossible experimental challenge. In the context of this diversity, simulations of conductance-based neuronal models are a powerful tool bridging the gap between altered ionic currents and firing in a systematic and efficient way. Furthermore, simlutions allow to predict the potential effects of drugs needed to alleviate the pathophysiology of the respective mutation \citep{johannesen_genotype-phenotype_2021, lauxmann_therapeutic_2021, Bayraktar}.
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In this study, we therefore investigated how the outcome of ionic current kinetic changes on firing depend on neuronal cell type by (1) characterizing firing responses with two measures, (2) simulating the response of a repertoire of different neuronal models to changes in single current parameters as well as (3) to more complex changes in this case as they were observed for specific \textit{KCNA1} mutations that are associated with episodic ataxia type~1 \citep{Browne1994, Browne1995, lauxmann_therapeutic_2021}.
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\section{Material and Methods}
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All modelling and simulation was done in parallel with custom written Python 3.8 (Python Programming Language; RRID:SCR\_008394) software, run on a Cent-OS 7 server with an Intel(R) Xeon (R) E5-2630 v2 CPU.
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% @ 2.60 GHz Linux 3.10.0-123.e17.x86_64.
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\subsection{Different Cell Models}
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A group of neuronal models representing the major classes of cortical and thalamic neurons including regular spiking pyramidal (RS pyramidal; model D), regular spiking inhibitory (RS inhibitory; model B), and fast spiking (FS; model C) cells were used \citep{pospischil_minimal_2008}. Additionally, a \Kv current (\IKv; \citealt{ranjan_kinetic_2019}) was added to each of these models (RS pyramidal +\Kv; model H, RS inhibitory +\Kv; model E, and FS +\Kv; model G respectively). A cerebellar stellate cell model from \citet{alexander_cerebellar_2019} is used (Cb stellate; model A) in this study. This cell model was also extended by a \Kv current \citep{ranjan_kinetic_2019}, either in addition to the A-type potassium current (Cb stellate +\Kv; model F) or by replacing the A-type potassium current (Cb stellate \(\Delta\)\Kv; model J). A subthalamic nucleus (STN; model L) neuron model as described by \citet{otsuka_conductance-based_2004} was also used. The STN cell model (model L) was additionally extended by a \Kv current \citep{ranjan_kinetic_2019}, either in addition to the A-type potassium current (STN +\Kv; model I) or by replacing the A-type potassium current (STN \(\Delta\)\Kv; model K). Model letter naming corresponds to panel lettering in Figure \ref{fig:diversity_in_firing}. The properties and maximal conductances of each model are detailed in Table \ref{tab:g} and the gating properties are unaltered from the original Cb stellate (model A) and STN (model L) models \citep{alexander_cerebellar_2019, otsuka_conductance-based_2004}. For enabling the comparison of models with the typically reported electrophysiological data fitting reported and for ease of further gating curve manipulations, a modified Boltzmann function
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\begin{equation}\label{eqn:Boltz}
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x_\infty = {\left(\frac{1-a}{1+{\exp\left[{\frac{V-V_{1/2}}{k}}\right]}} +a\right)^j}
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\end{equation}
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with slope \(k\), voltage for half-maximal activation or inactivation (\(V_{1/2}\)), exponent \(j\), and persistent current \(0 \leq a \leq 1\) were fitted to the original formulism for RS pyramidal (model D), RS inhibitory (model B) and FS (model C) models from \citet{pospischil_minimal_2008}. The properties of \IKv were fitted to the mean wild type biophysical parameters of \Kv described in \citet{lauxmann_therapeutic_2021}. \textcolor{red}{The fitted gating parameters are detailed in Table \ref{tab:gating}}. Each of the original single-compartment models used here can reproduce physiological firing behavior of the neurons they represent (Figure \ref{fig:diversity_in_firing}; \citealt{pospischil_minimal_2008, alexander_cerebellar_2019, otsuka_conductance-based_2004}) and capture key aspects of the dynamics of these cell types.
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\subsection{Firing Frequency Analysis}
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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 (step size 5\,pA) for all models except for the RS inhibitory neuron models where a range of 0 to 0.35\,nA (step size 1.75\,pA) 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, or the relative height above the lowest contour line encircling it, 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.
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To ensure accurate firing frequencies at low firing rates and reduced spike sampling bias, steady-state firing was defined as the mean firing frequency in a 500\,ms window in the last second of the current steps starting at the inital action potential in this last second.
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Firing characterization was performed in the last second of current steps to ensure steady-state firing is captured and adaptation processes are neglected in our analysis. Alteration in current magnitudes can have different effects on rheobase and the initial slope of the fI curve \citep{Kispersky2012}.
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For this reason, we quantified neuronal firing using the rheobase as well as the area under the curve (AUC) of the initial portion of the fI curve as a measure of the initial slope of the fI curve Figure \ref{fig:firing_characterization}~A.
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The smallest current at which steady state firing occured 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.
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To obtain the rheobase at a higher current resolution than the fI curve, the current step interval preceding the occurrence of action potentials was explored at higher resolution with 100 current steps spanning the interval (step sizes of 0.05 pA and 0.0175 pA, respectively). Membrane responses to these current steps were then analyzed for action potentials and the rheobase was considered the lowest current step for which an action potential was elicited.
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All models exhibited tonic steady-state firing with default parameters. In limited instances, variations of parameters elicited periodic bursting, however these instances were excluded from further analysis.
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\subsection{Sensitivity Analysis and Comparison of Models}
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Properties of ionic currents common to all models (\(\text{I}_{\text{Na}}\), \(\text{I}_{\text{K}}\), \(\text{I}_{\text{A}}\)/\IKv, and \(\text{I}_{\text{Leak}}\)) 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 voltage dependence of the time constant associated with the shifted gating curve was correspondingly shifted. 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. We neglected the variation of time constants for the practical reason that estimation and assessment of time constants and changes to them is not straightforward \citep{Clerx2019, Whittaker2020}.
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\subsection{Model Comparison}
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Changes in rheobase (\drheo) were 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 (\(AUC_{wt}\))
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\begin{equation}\label{eqn:AUC_contrast}
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\textrm{normalized } \Delta \text{AUC} = \frac{AUC_i - AUC_{wt}}{AUC_{wt}}
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\end{equation}
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To assess whether the effects of a given alteration on \ndAUC or \drheo were robust across models, the correlation between \ndAUC or \drheo and the magnitude of the alteration of a current property was computed for each alteration in each model and compared across alteration types.
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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.
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\subsection{\textit{KCNA1} Mutations}\label{subsec:mut}
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Known episodic ataxia type~1 associated \textit{KCNA1} mutations and their electrophysiological characterization have been 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. Although initially described to lack fast activation, \Kv displays prominent inactivation at physiologically relevant temperatures \citep{ranjan_kinetic_2019}. The effects of a mutation were also applied to \(\text{I}_{\text{A}}\) when present as both potassium currents display inactivation. In all cases, the mutation effects were applied to half of the \Kv or \(\text{I}_{\text{A}}\) 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 behavior as described above. For each model the differences in mutation AUC to wild type AUC were normalized by wild type AUC (\ndAUC) and mutation rheobases were compared to wild type rheobase values (\drheo). Pairwise Kendall rank correlations (Kendall \(\tau\)) were used to compare the correlation in the effects of \Kv mutations on AUC and rheobase between models.
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\subsection{Code Accessibility}
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%The code/software described in the paper is freely available online at [URL redacted for double-blind review]. The code is available as \Cref{code_zip}.
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The code/software described in the paper is freely available online at \newline \href{https://github.com/nkoch1/LOFGOF2023}{https://github.com/nkoch1/LOFGOF2023}. %The code is available as \ref{code_zip}.
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\section{Results}
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To examine the role of cell-type specific ionic current environments on the impact of altered ion currents properties on firing behavior:
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(1) firing responses were characterized with rheobase and \(\Delta\)AUC, (2) a set of neuronal models was used and properties of channels common across models were altered systematically one at a time, and (3) the effects of a set of episodic ataxia type~1 associated \textit{KCNA1} mutations on firing was then examined across different neuronal models with different ionic current environments.
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\subsection{Variety of model neurons}
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Neuronal firing is heterogenous across the CNS and a set of neuronal models with heterogenous firing due to different ionic currents is desirable to reflect this heterogeneity. The set of single-compartment, conductance-based neuronal models used here has considerable diversity as evident in the variability seen across neuronal models both in spike trains and their fI curves (Figure \ref{fig:diversity_in_firing}). The models chosen for this study all fire tonically and do not exhibit bursting (see methods for details and naming of the models). Models are qualitatively sorted based on their firing curves and labeled model A through L accordingly. Some models, such as models A and B, display type I firing, whereas others such as models J and L exhibit type II firing. Type I firing is characterized by continuous fI curves (i.e. firing rate increases from 0 in a continuous fashion) whereas type II firing is characterized by a discontinuity in the fI curve (i.e. a jump occurs from no firing to firing at a certain frequency; \citealt{ermentrout_type_1996, Rinzel_1998}). The other models used here lie on a continuum between these prototypical firing classifications. Most neuronal models exhibit hysteresis with ascending and descending ramps eliciting spikes at different current thresholds. However, the models I, J, and K have large hysteresis \textcolor{red}{ (Figure \ref{fig:diversity_in_firing, ramp_firing})}. Different types of underlying current dynamics are known to generate these different firing types and hysteresis \cite{ERMENTROUT2002, ermentrout_type_1996, Izhikevich2006}.
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\subsection{Characterization of Neuronal Firing Properties}
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Neuronal firing is a complex phenomenon, and a quantification of firing properties is required for comparisons across cell types and between different conditions. Here we focus on two aspects of firing: rheobase, the smallest injected current at which the cell fires an action potential, and the shape of the frequency-current (fI) curve as quantified by the area under the curve (AUC) for a fixed range of input currents above rheobase (Figure \ref{fig:firing_characterization}~A). The characterization of the firing properties of a neuron by using rheobase and AUC allows to characterize both a neuron's excitability in the sub-threshold regime (rheobase) and periodic firing in the super-threshold regime (AUC) by two independent measures. Note that AUC is essentially quantifying the slope of a neuron's fI curve.
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Using these two measures we quantified the effects a changed property of an ionic current has on neural firing by the differences in both rheobase, \drheo, and in AUC, \(\Delta\)AUC, relative to the wild type neuron. \(\Delta\)AUC is in addition normalized to the AUC of the wild type neuron, see Eq.~\eqref{eqn:AUC_contrast}. Each fI curve resulting from an altered ionic current is a point in a two-dimensional coordinate system spanned by \drheo and \ndAUC (Figure \ref{fig:firing_characterization}~B). An fI curve similar to the one of the wild type neuron is marked by a point close to the origin. In the upper left quadrant, fI curves become steeper (positive difference of AUC values: \(+\Delta\)AUC) and are shifted to lower rheobases (negative difference of rheobases: \(-\)\drheo), unambigously indicating an increased firing that clearly might be classified as a gain of function (GOF) of neuronal firing. The opposite happens in the bottom right quadrant where the slope of fI curves decreases (\(-\Delta\)AUC) and the rheobase is shifted to higher currents (\(+\)\drheo), indicating a decreased, loss of function (LOF) firing. In the lower left (\(-\Delta\)AUC and \(-\)\drheo) and upper right (\(+\Delta\)AUC and \(+\)\drheo) quadrants, the effects on firing are less clear-cut, because the changes in rheobase and AUC have opposite effects on neuronal firing. Changes in a neuron's fI curves in these two quadrants cannot uniquely be described as a gain or loss of excitability.
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\subsection{Sensitivity Analysis}
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Sensitivity analyses are used to understand how input model parameters contribute to determining the output of a model \citep{Saltelli2002}. In other words, sensitivity analyses are used to understand how sensitive the output of a model is to a change in input or model parameters. One-factor-a-time sensitivity analyses involve altering one parameter at a time and assessing the impact of this parameter on the output. This approach enables the comparison of given alterations in parameters of ionic currents across models.
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For example, when shifting the half activation voltage \(V_{1/2}\) of the delayed rectifier potassium current in the model G to more depolarized values, then the rheobase of the resulting fI curves shifted to lower currents \(-\)\drheo, making the neuron more sensitive to weak inputs, but at the same time the slope of the fI curves was reduced (\(-\)\ndAUC), which resulted in a reduced firing rate (Figure \ref{fig:AUC_correlation}~A). As a result the effect of a depolarizing shift in the delayed rectifier potassium current half activation \(V_{1/2}\) in model G is in the bottom left quadrant of Figure \ref{fig:firing_characterization}~B and characterization as LOF or GOF in excitability is not possible. Plotting the corresponding changes in AUC against the change in half activation potential \(V_{1/2}\) results in a monotonically falling curve (thick orange line in Figure \ref{fig:AUC_correlation}~B). For each of the many models we got a different relation between the changes in AUC and the shifts in half maximal potential \(V_{1/2}\) (thin lines in Figure \ref{fig:AUC_correlation}~B). To further summarize these different dependencies of the various models we characterized each of these curves by a single number, the \( \text{Kendall} \ \tau \) correlation coefficient. A monotonically increasing curve resulted in a \( \text{Kendall} \ \tau \) close to \(+1\) a monotonously decreasing curve in \( \text{Kendall} \ \tau \approx -1 \), and a non-monotonous, non-linear relation in \( \text{Kendall} \ \tau \) close to zero (compare lines in Figure \ref{fig:AUC_correlation}~B with dots in black box in panel C).
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Changes in gating half activation potential \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance \(g\) affected the AUC (Figure \ref{fig:AUC_correlation}), but how exactly the AUC was affected usually depended on the specific neuronal model. Increasing the slope factor of the \Kv activation curve for example increased the AUC in all models (\( \text{Kendall} \ \tau \approx +1\)), but with different slopes (Figure \ref{fig:AUC_correlation}~D,E,F). Similar consistent positive correlations could be found for shifts in A-current activation \(V_{1/2}\). Changes in \Kv half activation \(V_{1/2}\) and in maximal A-current conductance resulted in negative correlations with the AUC in all models (\( \text{Kendall} \ \tau \approx -1\)).
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Qualitative differences could be found, for example, when increasing the maximal conductance of the delayed rectifier (Figure \ref{fig:AUC_correlation}~G,H,I). In some model neurons this increased AUC (\( \text{Kendall} \ \tau \approx +1\)), whereas in others AUC was decreased (\( \text{Kendall} \ \tau \approx -1\)). In model I, AUC depended in a non-linear way on the maximal conductance of the delayed rectifier, resulting in a \( \text{Kendall} \ \tau \) close to zero. Even more dramatic qualitative differences between models resulted from shifts of the activation curve of the delayed rectifier, as discussed already above (Figure \ref{fig:AUC_correlation}~A,B,C). Some model neurons did almost not depend on changes in K-current half activation \(V_{1/2}\) or showed strong non-linear dependencies, both resulting in \( \text{Kendall} \ \tau\) close to zero. Many model neurons showed strongly negative correlations, and a few displayed positive correlations with shifting the activation curve of the delayed rectifier.
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Changes in gating half activation potential \(V_{1/2}\) and slope factor \(k\) as well as the maximum conductance \(g\) affected rheobase (Figure \ref{fig:rheobase_correlation}). However, in contrast to AUC, qualitatively consistent effects on rheobase across models could be observed. An increasing of the maximal conductance of the leak current in the model A increased the rheobase (Figure \ref{fig:rheobase_correlation}~G). When these changes were plotted against the change in maximal conductance a monotonically increasing relationship was evident (thick teal line in Figure \ref{fig:rheobase_correlation}~H). This monotonically increasing relationship was evident in all models (\( \text{Kendall} \ \tau \approx +1\)), but with different slopes (thin lines in Figure \ref{fig:rheobase_correlation}~H). Similarly, positive correlations were consistently found across models for maximal conductances of delayed rectifier K, \Kv, and A type currents, whereas the maximal conductance of the sodium current was consistently associated with negative correlations (\( \text{Kendall} \ \tau \approx -1\); Figure \ref{fig:rheobase_correlation}~I), i.e. rheobase decreased with increasing maximum conductance in all models.
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Although changes in half maximal potential \(V_{1/2}\) and slope factor \(k\) generally correlated with rheobase similarly across models there were some exceptions. Rheobase was affected with both with positive and negative correlations in different models as a result of changing slope factor of Na\(^+\)-current inactivation (positive: models A--H and J; negative: models I, K and L), \Kv-current inactivation (positive: models I and K; negative: models E--G, J, H), and A-current activation (positive: models A, F and L; negative: model I; Figure \ref{fig:rheobase_correlation}~F). Departures from monotonic relationships also occurred in some models as a result of K\(^+\)-current activation \(V_{1/2}\) (e.g. model J) and slope factor \(k\) (models F and G), \Kv-current inactivation slope factor \(k\) (model K), and A-current activation slope factor \(k\) (model L). Thus, identical changes in current gating properties such as the half maximal potential \(V_{1/2}\) or slope factor \(k\) can have differing effects on firing depending on the model in which they occur.
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\subsection{\textit{KCNA1} Mutations}
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Mutations in \textit{KCNA1} are associated with episodic ataxia type~1 (EA1) and have been characterized biophysically (as reviewed by \citet{lauxmann_therapeutic_2021}). Here they were used as a test case in the effects of various ionic current environments on neuronal firing and on the outcomes of channelopathies. The changes in AUC and rheobase from wild type values for reported EA1 associated \textit{KCNA1} mutations were heterogeneous across models containing \Kv, but generally showed decreases in rheobase (Figure \ref{fig:simulation_model_comparision}A--I). Pairwise non-parametric Kendall \(\tau\) rank correlations between the simulated effects of these \Kv mutations on rheobase were highly correlated across models (Figure \ref{fig:simulation_model_comparision}J) indicating that EA1 associated \textit{KCNA1} mutations generally decrease rheobase across diverse cell-types. However, the effects of the \Kv mutations on AUC were more heterogenous as reflected by both weak and strong positive and negative pairwise correlations between models (Figure \ref{fig:simulation_model_comparision}K), suggesting that the effects of ion-channel variant on super-threshold neuronal firing depend both quantitatively and qualitatively on the specific composition of ionic currents in a given neuron.
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\section{Discussion}
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To compare the effects of ion channel mutations on neuronal firing of different neuron types, a diverse set of conductance-based models was used and the effect of changes in individual channel properties across conductance-based neuronal models. Additionally, the effects of episodic ataxia type~1 associated (EA1) \textit{KCNA1} mutations were simulated. Changes to single ionic current properties, as well as known EA1 associated \textit{KCNA1} mutations showed consistent effects on the rheobase across cell types, whereas the effects on AUC of the steady-state fI-curve depended on the cell type. Our results demonstrate that loss of function (LOF) and gain of function (GOF) on the biophysical level cannot be uniquely transferred to the level of neuronal firing. Thus, the effects caused by different mutations depend on the properties of the other ion channels expressed in a cell and are therefore depend on the channel ensemble of a specific cell type.
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\subsection{Firing Frequency Analysis}
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Although, firing differences can be characterized by an area under the curve of the fI curve for fixed current steps this approach characterizes firing as a mixture of key features: rheobase and the initial slope of the fI curve. By probing rheobase directly and using an AUC relative to rheobase, we disambiguate these features and enable insights into the effects on rheobase and initial fI curve steepness. This increases the specificity of our understanding of how ion channel mutations alter firing across cells types and enable classification as described in Figure \ref{fig:firing_characterization}. Importantly, in cases when ion channel mutations alter rheobase and initial fI curve steepness in ways that oppose effects on firing (upper left and bottom right quadrants of Figure \ref{fig:firing_characterization}~B) this disamgibuation is important for understanding the outcome of the mutation. In these cases, the regime the neuron is operating in is vital in determining the cell's firing outcome. If it is in its excitable regime and only occasionally generates an action potential, then the effect on the rheobase is more important. If it is firing periodically with high rates, then the change in AUC might be more relevant.
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\subsection{Modelling Limitations}
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The models used here are simple and while they all capture key aspects of the firing dynamics for their respective cell, they fall short of capturing the complex physiology and biophysics of real cells. However, for the purpose of understanding how different cell-types, or current environments, contribute to diversity in firing outcomes of ion channel mutations, the fidelity of the models to the physiological cells they represent is of a minor concern. For exploring possible cell-type specific effects, variety in currents and dynamics across models is of utmost importance. With this context in mind, the collection of models used here are labelled as models A-L to highlight that the physiological cells they represent is not of chief concern, but rather that the collection of models with different attributes respond heterogeneously to the same perturbation. Additionally, the development of more realistic models is a high priority and will enable cell-type specific predictions that may aid in precision medicine approaches. Thus, weight should not be put on any single predicted firing outcome here in a specific model, but rather on the differences in outcomes that occur across the cell-type spectrum the models used here represent.
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\subsection{Neuronal Diversity}
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The nervous system consists of a vastly diverse and heterogenous collection of neurons with variable properties and characteristics including diverse combinations and expression levels of ion channels which are vital for neuronal firing dynamics.
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Advances in high-throughput techniques have enabled large-scale investigation into single-cell properties across the CNS \citep{Poulin2016} that have revealed large diversity in neuronal gene expression, morphology and neuronal types in the motor cortex \citep{Scala2021}, neocortex \cite{Cadwell2016, Cadwell2020}, GABAergic neurons in the cortex and retina \citep{Huang2019, Laturnus2020}, cerebellum \citep{Kozareva2021}, spinal cord \citep{Alkaslasi2021}, visual cortex \citep{Gouwens2019} as well as the retina \citep{Baden2016, Voigt2019, Berens2017, Yan2020a, Yan2020b}.
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Diversity across neurons is not limited to gene expression and can also be seen electrophysiologically \citep{Tripathy2017, Gouwens2018, Tripathy2015, Scala2021, Cadwell2020, Gouwens2019, Baden2016, Berens2017} with correlations existing between gene expression and electrophysiological properties \citep{Tripathy2017}. At the ion channel level, diversity exists not only between the specific ion channels the different cell types express but heterogeneity also exists in ion channel expression levels within cell types \citep{marder_multiple_2011, goaillard_ion_2021,barreiro_-current_2012}. As ion channel properties and expression levels are key determinents of neuronal dynamics and firing \citep{Balachandar2018, Gu2014, Zeberg2015, Aarhem2007, Qi2013, Gu2014a, Zeberg2010, Zhou2020, Kispersky2012} neurons with different ion channel properties and expression levels display different firing properties.
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To capture the diversity in neuronal ion channel expression and its relevance in the outcome of ion channel mutations, we used multiple neuronal models with different ionic currents and underlying firing dynamics here.
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\subsection{Ionic Current Environments Determine the Effect of Ion Channel Mutations}
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To our knowledge, no comprehensive evaluation of how ionic current environment and cell type affect the outcome of ion channel mutations have been reported. However, comparisons between the effects of such mutations between certain cell types were described. For instance, the R1648H mutation in \textit{SCN1A} does not alter the excitability of cortical pyramidal neurons, but causes hypoexcitability of adjacent inhibitory GABAergic neurons \citep{Hedrich14874}. In the CA3 region of the hippocampus, the equivalent mutation in \textit{SCN8A}, R1627H, increases the excitability of pyramidal neurons and decreases the excitability of parvalbumin positive interneurons \cite{makinson_scn1a_2016}. Additionally, the L858H mutation in \(\text{Na}_\text{V}\text{1.7}\), associated with erythermyalgia, has been shown to cause hypoexcitability in sympathetic ganglion neurons and hyperexcitability in dorsal root ganglion neurons \citep{Waxman2007, Rush2006}. The differential effects of L858H \(\text{Na}_\text{V}\text{1.7}\) on firing is dependent on the presence or absence of another sodium channel, namely the \(\text{Na}_\text{V}\text{1.8}\) subunit \citep{Waxman2007, Rush2006}. These findings, in concert with our findings emphasize that the ionic current environment in which a channelopathy occurs is vital in determining the outcomes of the channelopathy on firing.
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Cell type specific differences in ionic current properties are important in the effects of ion channel mutations. However, within a cell type heterogeneity in channel expression levels exists and it is often desirable to generate a population of neuronal models and to screen them for plausibility to biological data in order to capture neuronal population diversity \citep{marder_multiple_2011,OLeary2016}. The models we used here are originally generated by characterization of current gating properties and by fitting of maximal conductances to experimental data \citep{pospischil_minimal_2008, ranjan_kinetic_2019, alexander_cerebellar_2019, otsuka_conductance-based_2004}. This practice of fixing maximal conductances based on experimental data is limiting as it does not reproduce the variability in channel expression and neuronal firing behavior of a heterogeneous neuron population \citep{verma_computational_2020}. For example, a model derived from the mean conductances in a neuronal sub-population within the stomatogastric ganglion, the so-called "one-spike bursting" neurons fire three spikes instead of one per burst due to an L-shaped distribution of sodium and potassium conductances \citep{golowasch_failure_2002}. Multiple sets of conductances can give rise to the same patterns of activity also termed degeneracy and differences in neuronal dynamics may only be evident with perturbations \citep{marder_multiple_2011, goaillard_ion_2021}. The variability in ion channel expression often correlates with the expression of other ion channels \citep{goaillard_ion_2021} and neurons whose behavior is similar may possess correlated variability across different ion channels resulting in stability in the neuronal phenotype \citep{lamb_correlated_2013, soofi_co-variation_2012, taylor_how_2009}. The variability of ionic currents and degeneracy of neurons may account, at least in part, for the observation that the effect of toxins within a neuronal type is frequently not constant \citep{khaliq_relative_2006, puopolo_roles_2007, ransdell_neurons_2013}.
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\subsection{Effects of \textit{KCNA1} Mutations}
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Changes in delayed rectifier potassium currents, analogous to those seen in LOF \textit{KCNA1} mutations, change the underlying firing dynamics of the Hodgkin Huxley model result in reduced thresholds for repetitive firing and thus contribute to increased excitability \citep{hafez_altered_2020}. Although the Hodgkin Huxley delayed rectifier lacks inactivation, the increases in excitability observed by \citet{hafez_altered_2020} are in line with our simulation-based predictions of the outcomes of \textit{KCNA1} mutations. LOF \textit{KCNA1} mutations generally increase neuronal excitability, however the varying susceptibility on rheobase and different effects on AUC of the fI-curve of \textit{KCNA1} mutations across models are indicative that a certain cell type specific complexity exists. Increased excitability is seen experimentally with \Kv null mice \citep{smart_deletion_1998, zhou_temperature-sensitive_1998}, with pharmacological \Kv block \citep{chi_manipulation_2007, morales-villagran_protection_1996} and by \citet{hafez_altered_2020} with simulation-based predictions of \textit{KCNA1} mutations. Contrary to these results, \citet{zhao_common_2020} predicted \textit{in silico} that the depolarizing shifts seen as a result of \textit{KCNA1} mutations broaden action potentials and interfere negatively with high frequency action potential firing. However, they varied stimulus duration between different models and therefore comparability of firing rates is lacking in this study.
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In our simulations, different current properties alter the impact of \textit{KCNA1} mutations on firing as evident in the differences seen in the impact of \(\text{I}_\text{A}\) and \IKv in the Cb stellate and STN model families on \textit{KCNA1} mutation firing. This highlights that not only knowledge of the biophysical properties of a channel but also its neuronal expression and other neuronal channels present is vital for the holistic understanding of the effects of a given ion channel mutation both at the single cell and network level.
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\subsection{Loss or Gain of Function Characterizations Do Not Fully Capture Ion Channel Mutation Effects on Firing}
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The effects of changes in channel properties depend in part on the neuronal model in which they occur and can be seen in the variance of correlations (especially in AUC of the fI-curve) across models for a given current property change. Therefore, relative conductances and gating properties of currents in the ionic current environment in which an alteration in current properties occurs plays an important role in determining the outcome on firing. The use of LOF and GOF is useful at the level of ion channels to indicate whether a mutation results in more or less ionic current. However, the extension of this thinking onto whether mutations induce LOF or GOF at the level of neuronal firing based on the ionic current LOF/GOF is problematic due to the dependency of neuronal firing changes on the ionic channel environment. Thus, the direct leap from current level LOF/GOF characterizations to effects on firing without experimental or modelling-based evidence, although tempting, should be refrained from and viewed with caution when reported. This is especially relevant in the recent development of personalized medicine for channelopathies, where a patient's specific channelopathy is identified and used to tailor treatments \citep{Weber2017, Ackerman2013, Helbig2020, Gnecchi2021, Musto2020, Brunklaus2022, Hedrich2021}. However, in these cases the effects of specific ion channel mutations are often characterized based on ionic currents in expression systems and classified as LOF or GOF to aid in treatment decisions \citep{johannesen_genotype-phenotype_2021, Brunklaus2022, Musto2020}. Although positive treatment outcomes occur with sodium channel blockers in patients with GOF \(\text{Na}_{\text{V}}\text{1.6}\) mutations, patients with both LOF and GOF \(\text{Na}_{\text{V}}\text{1.6}\) mutations can benefit from treatment with sodium channel blockers \citep{johannesen_genotype-phenotype_2021}. This example suggests that the relationship between effects at the level of ion channels and effects at the level of firing and therapeutics is not linear or evident without further contextual information.
|
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Therefore, the transferring of LOF or GOF from the current to the firing level should be used with caution; the cell type in which the mutant ion channel is expressed may provide valuable insight into the functional consequences of an ion channel mutation. Experimental assessment of the effects of a patient's specific ion channel mutation \textit{in vivo} is not generally feasible at a large scale. Therefore, modelling approaches investigating the effects of patient specific channelopathies provide an alternative bridge between characterization of changes in biophysical properties of ionic currents and the firing consequences of these effects. In both experimental and modelling investigation into the effects of ion channel mutations on neuronal firing the specific cell-type dependency should be considered.
|
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The effects of altered ion channel properties on firing is generally influenced by the other ionic currents present in the cell. In channelopathies the effect of a given ion channel mutation on neuronal firing therefore depends on the cell type in which those changes occur \citep{Hedrich14874, makinson_scn1a_2016, Waxman2007, Rush2006}. Although certain complexities of neurons such as differences in cell-type sensitivities to current property changes, interactions between ionic currents, cell morphology and subcellular ion channel distribution are neglected here, it is likely that this increased complexity \textit{in vivo} would contribute to the cell-type dependent effects on neuronal firing. The complexity and nuances of the nervous system, including cell-type dependent firing effects of channelopathies explored here, likely underlie shortcomings in treatment approaches in patients with channelopathies. Accounting for cell-type dependent firing effects provides an opportunity to further the efficacy and precision in personalized medicine approaches.
|
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With this study we suggest that cell-type specific effects are vital to a full understanding of the effects of channelopathies at the level of neuronal firing. Furthermore, we highlight the use of modelling approaches to enable relatively fast and efficient insight into channelopathies.
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\notenk{did this for Python in the methods}
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\notenk{Not sure if we need to do this, or what to do?}
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\subsection*{\textcolor{red}{Contribution to the field}}
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\textit{Significance statement from eNeuro}\\
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\textit{Although the genetic nature of ion channel mutations as well as their effects on the biophysical properties of an ion channel are routinely assessed experimentally, determination of their role in altering neuronal firing is more difficult. In particular, cell-type dependency of ion channel mutations on firing has been observed experimentally, and should be accounted for. In this context, computational modelling bridges this gap and demonstrates that the cell type in which a mutation occurs is an important determinant in the effects of neuronal firing. As a result, classification of ion channel mutations as loss or gain of function is useful to describe the ionic current but should not be blindly extend to classification at the level of neuronal firing.}
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\section*{Conflict of Interest Statement}
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%All financial, commercial or other relationships that might be perceived by the academic community as representing a potential conflict of interest must be disclosed. If no such relationship exists, authors will be asked to confirm the following statement:
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The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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\section*{Author Contributions}
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NK, LS, UBSH, SL, JB contributed to conception and design of the study. NK performed simulation and wrote the first draft of the manuscript. NK, LS analyzed simulation data. All authors wrote sections of the manuscript and contributed to manuscript revision, read and approved the submitted version.
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\section*{Funding}
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%Details of all funding sources should be provided, including grant numbers if applicable. Please ensure to add all necessary funding information, as after publication this is no longer possible.
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This work was supported by the German Research Foundation in the Frame of the Research Unit FOR-2715 (DFG, grants Le1030/15-1/2 and HE 8155/1-2) and the Network for Rare Ion Channel Disorders Treat-ION of the Federal Ministry for Education and Research (BMBF, grants 01GM1907A and 01GM2210A). SL was supported with an Otfrid-Foerster stipend from the German Society for Epileptology (DGfE).
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\section*{Acknowledgments}
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\section*{Supplemental Data}
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\notenk{See ./Koch\_SupplementaryMaterial.tex}
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\section*{Data Availability Statement}
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%The datasets [GENERATED/ANALYZED] for this study can be found in the [NAME OF REPOSITORY] [LINK].
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The datasets generated for this study as well as the code/software described in this study are publicly available. This data and software can be found here: \href{https://github.com/nkoch1/LOFGOF2023}{https://github.com/nkoch1/LOFGOF2023}.
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%The original contributions presented in the study are publicly available. This data can be found here: [link/accession number].
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%\bibliographystyle{Frontiers-Vancouver} % Many Frontiers journals use the numbered referencing system, to find the style and resources for the journal you are submitting to: https://zendesk.frontiersin.org/hc/en-us/articles/360017860337-Frontiers-Reference-Styles-by-Journal
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\newpage
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\section*{Figures}
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%%% Please be aware that for original research articles we only permit a combined number of 15 figures and tables, one figure with multiple subfigures will count as only one figure.
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.83\linewidth]{Figures/diversity_in_firing.pdf}
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\linespread{1.}\selectfont
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\caption[]{Diversity in Neuronal Model Firing. Spike trains (left), frequency-current (fI) curves (right) for Cb stellate \textbf{(A)}, RS inhibitory \textbf{(B)}, FS \textbf{(C)}, RS pyramidal \textbf{(D)}, RS inhibitory +\Kv \textbf{(E)}, Cb stellate +\Kv \textbf{(F)}, FS +\Kv \textbf{(G)}, RS pyramidal +\Kv \textbf{(H)}, STN +\Kv \textbf{(I)}, Cb stellate \(\Delta\)\Kv \textbf{(J)}, STN \(\Delta\)\Kv \textbf{(K)}, and STN \textbf{(L)} neuron models. Models are sorted qualitatively based on their fI curves. Black markers on the fI curves indicate the current step at which the spike train occurs. The green marker indicates the current at which firing begins in response to an ascending current ramp, whereas the red marker indicates the current at which firing ceases in response to a descending current ramp \textcolor{red}{(see Figure \ref{ramp_firing}).}}
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\label{fig:diversity_in_firing}
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\end{figure}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=0.5\linewidth]{Figures/firing_characterization_arrows.pdf}
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\linespread{1.}\selectfont
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\caption[]{Characterization of firing with AUC and rheobase. \textbf{(A)} The area under the curve (AUC) of the repetitive firing frequency-current (fI) curve. \textbf{(B)}
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Changes in firing as characterized by \(\Delta\)AUC and \(\Delta\)rheobase occupy four quadrants separated by no changes in AUC and rheobase. Representative schematic fI curves in red with respect to a reference (or wild type) fI curve (blue) depict the general changes associated with each quadrant.}
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\label{fig:firing_characterization}
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\end{figure}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=\linewidth]{Figures/AUC_correlation.pdf}
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\linespread{1.}\selectfont
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\caption[]{Effects of altered channel kinetics on AUC in various neuron models. The fI curves corresponding to shifts in model G delayed rectifier K half activation \(V_{1/2}\) \textbf{(A)}, changes \Kv activation slope factor \(k\) in model G \textbf{(D)}, and changes in maximal conductance of delayed rectifier K current in the model I \textbf{(G)} are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for \textbf{(A,D,} and \textbf{G}) in accordance to the greyscale of the x-axis in B, E, and H. The \ndAUC of fI curves is plotted against delayed rectifier K half activation potential (\(\Delta V_{1/2}\); \textbf{B}), \Kv activation slope factor \(k\) (k/\(\text{k}_{\text{WT}}\); \textbf{E}) and maximal conductance \(g\) of the delayed rectifier K current (g/\(\text{g}_{\text{WT}}\); \textbf{H}) for all models (thin lines) with relationships from the fI curve examples ( \textbf{A, D, G} respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \ndAUC \textbf{(C)}, slope factor k and \ndAUC \textbf{(F)} as well as maximal current conductances and \ndAUC \textbf{(I)} for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\text{k}_{\text{WT}}\), and g/\(\text{g}_{\text{WT}}\) and \ndAUC for the Kendall rank correlations highlighted in the black boxes are depicted in \textbf{(B)}, \textbf{(E)} and \textbf{(H)} respectively. }
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\label{fig:AUC_correlation}
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\end{figure}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=\linewidth]{Figures/rheobase_correlation.pdf}
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\linespread{1.}\selectfont
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\caption[]{Effects of altered channel kinetics on rheobase. The fI curves corresponding to shifts in model G \Kv activation \(V_{1/2}\) \textbf{(A)}, changes \Kv inactivation slope factor \(k\) in model F \textbf{(D)}, and changes in maximal conductance of the leak current in model A \textbf{(G)} are shown. The fI curves from the smallest (grey) to the largest (black) alterations are seen for ( \textbf{A,D,} and \textbf{G}) in accordance to the greyscale of the x-axis in \textbf{B}, \textbf{E}, and \textbf{H}. The \drheo of fI curves is plotted against \Kv half activation potential (\(\Delta V_{1/2}\); \textbf{B}), \Kv inactivation slope factor \(k\) (k/\(\text{k}_{\text{WT}}\); \textbf{E}) and maximal conductance \(g\) of the leak current (g/\(\text{g}_{\text{WT}}\); \textbf{H}) for all models (thin lines) with relationships from the fI curve examples ( \textbf{A, D, G} respectively) highlighted by thick lines with colors corresponding to the box highlighting each set of fI curves. The Kendall rank correlation (Kendall \(\tau\)) coefficients between shifts in half maximal potential \(V_{1/2}\) and \drheo \textbf{(C)}, slope factor k and \drheo \textbf{(F)} as well as maximal current conductances and \drheo \textbf{(I)} for each model and current property is computed. The relationships between \(\Delta V_{1/2}\), k/\(\text{k}_{\text{WT}}\), and g/\(\text{g}_{\text{WT}}\) and \drheo for the Kendall rank correlations highlighted in the black boxes are depicted in \textbf{(B)}, \textbf{(E)} and \textbf{(H)} respectively.}
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\label{fig:rheobase_correlation}
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\end{figure}
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\begin{figure}[h!]
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\centering
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\includegraphics[width=\linewidth]{Figures/simulation_model_comparison.pdf}
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\linespread{1.}\selectfont
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\caption[]{Effects of episodic ataxia type~1 associated \textit{KCNA1} mutations on firing. Effects of \textit{KCNA1} mutations on AUC (percent change in normalized \(\Delta\)AUC) and rheobase (\(\Delta\)Rheobase) compared to wild type for model H \textbf{(A)}, model E \textbf{(B)}, model G \textbf{(C)}, model A \textbf{(D)}, model F \textbf{(E)}, model J \textbf{(F)}, model L \textbf{(G)}, model I \textbf{(H)} and model K \textbf{(I)}. All \textit{KCNA1} Mutations are marked in grey with the V174F, F414C, E283K, and V404I \textit{KCNA1} mutations highlighted in color for each model. Pairwise Kendall rank correlation coefficients (Kendall \(\tau\)) between the effects of \textit{KCNA1} mutations on rheobase and on AUC are shown in J and K respectively. Marker shape is indicative of model/firing type, and grey dashed lines denote the quadrants of firing characterization (see Figure \ref{fig:firing_characterization}).}
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\label{fig:simulation_model_comparision}
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\end{figure}
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\section*{Tables}
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\input{g_table}
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\input{gating_table}
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\end{document}
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