Uncertainty quantification and sensitivity analysis of neuron models with ion concentration dynamics

Letizia Signorelli; Andrea Manzoni; Marte J. Sætra

doi:10.1371/journal.pone.0303822

Abstract

This paper provides a comprehensive and computationally efficient case study for uncertainty quantification (UQ) and global sensitivity analysis (GSA) in a neuron model incorporating ion concentration dynamics. We address how challenges with UQ and GSA in this context can be approached and solved, including challenges related to computational cost, parameters affecting the system’s resting state, and the presence of both fast and slow dynamics. Specifically, we analyze the electrodiffusive neuron-extracellular-glia (edNEG) model, which captures electrical potentials, ion concentrations (Na⁺, K⁺, Ca²⁺, and Cl⁻), and volume changes across six compartments. Our methodology includes a UQ procedure assessing the model’s reliability and susceptibility to input uncertainty and a variance-based GSA identifying the most influential input parameters. To mitigate computational costs, we employ surrogate modeling techniques, optimized using efficient numerical integration methods. We propose a strategy for isolating parameters affecting the resting state and analyze the edNEG model dynamics under both physiological and pathological conditions. The influence of uncertain parameters on model outputs, particularly during spiking dynamics, is systematically explored. Rapid dynamics of membrane potentials necessitate a focus on informative spiking features, while slower variations in ion concentrations allow a meaningful study at each time point. Our study offers valuable guidelines for future UQ and GSA investigations on neuron models with ion concentration dynamics, contributing to the broader application of such models in computational neuroscience.

Citation: Signorelli L, Manzoni A, Sætra MJ (2024) Uncertainty quantification and sensitivity analysis of neuron models with ion concentration dynamics. PLoS ONE 19(5): e0303822. https://doi.org/10.1371/journal.pone.0303822

Editor: Alexey Kuznetsov, Indiana University - Purdue University at Indidanapolis, UNITED STATES

Received: January 5, 2024; Accepted: May 1, 2024; Published: May 21, 2024

Copyright: © 2024 Signorelli et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting information files. The source codes for this study are available at https://github.com/CINPLA/edNEGmodel and https://github.com/letiziasignorelli/edNEGmodel_UQSA.

Funding: MJS acknowledges support from the Research Council of Norway (Norges Forskningsråd: https://www.forskningsradet.no) via FRIPRO grant no 324239 (EMIx). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

In the field of computational neuroscience, a multitude of models exists to describe neuronal activity, with ever-increasing biophysical intricacies [1]. Most of these models operate under the assumption of constant intra- and extracellular ion concentrations. This assumption is generally justified by the brain’s inherent mechanisms, such as pumps and cotransporters, that work to uphold ion concentration levels close to baseline values. However, to investigate scenarios where ion concentrations may change significantly—such as in epilepsy and spreading depression [2, 3]—certain models incorporate ion concentration dynamics and consider its influence on neuronal activity (see e.g., [4–21]).

This paper focuses on efficiently performing uncertainty quantification (UQ) and global sensitivity analysis (GSA) on neuron models incorporating ion concentration dynamics. Many model parameters inherently carry uncertainty arising from experimental measurement errors or physiological variations. For instance, the conductance g_x of a specific ion channel x is often uncertain. Despite the prevalence of such uncertainty, deterministic models assigning single numerical values to each parameter are common, and UQ and GSA have, to our knowledge, been overlooked or approached with simplistic one-at-a-time methods [19]. Understanding how uncertain parameters affect model outputs is crucial for extracting meaningful insights from the models [22–24]. Moreover, UQ and GSA serve as valuable tools to investigate channelopathies by revealing how modulations in specific ion channels or transporters influence neural activity. By conducting UQ and GSA on models with ion concentration dynamics, rather than more traditional models focused solely on membrane potentials, we can advance our understanding of such modulations. The conventional approach typically isolates the effects of channelopathies on neuronal membrane properties, overlooking their potential impact on the interstitial ion environment [25]. Neuron models with ion concentration dynamics enable us to examine the impact of channelopathies on ion concentrations, volume regulation, and neural activity patterns concurrently, providing a more comprehensive perspective. Furthermore, performing UQ and GSA under diverse conditions, such as normal spiking representing physiological states and activity indicating pathological states, would not only deepen our understanding of channelopathies, but also provide valuable insights into the underlying mechanisms associated with these varying conditions.

Performing UQ and GSA on neuron models with ion concentration dynamics is not straightforward. The focus of this paper is to provide general guidelines on how UQ and GSA can be performed on these types of models, and address the following main challenges: 1) the high computational cost associated with these models, 2) the influence of certain parameters on the resting state, and 3) the models’ exhibition of both fast and slow dynamics. We will elaborate on these challenges in the following paragraphs.

The first challenge in performing UQ and GSA on models with ion concentration dynamics lies in the substantial computational cost involved. Neuron models with ion concentration dynamics are described by time-dependent, nonlinear, and strongly coupled ordinary differential equations (ODEs), or sometimes partial differential equations (PDEs). Their multi-scale nature introduces stiffness, leading to significant variations in time scales [26]. Simulating such models poses several numerical challenges arising from their inherent complexity. Capturing their dynamics precisely at different time scales is crucial for accurate simulations, but striking the right balance between accuracy and computational efficiency is challenging [27]. Moreover, calculating sensitivity indices within high-dimensional parameter spaces requires a considerable number of input-output evaluations. This challenge becomes particularly pronounced when dealing with models encompassing numerous state variables spanning several orders of magnitude and an expansive parameter space, as is often the case with neuron models incorporating ion concentration dynamics. To mitigate this challenge, the use of appropriate surrogate modeling techniques, such as polynomial chaos expansions (PCE) or reduced order models, becomes imperative for computational efficiency [22, 28–33]. These surrogate models enable the replacement of the original, computationally intensive model with a surrogate constructed from a relatively small experimental design. Subsequently, the surrogate can be utilized to compute sensitivity indices with minimal computational cost.

Another main challenge related to UQ and GSA on neuron models with ion concentration dynamics is that altering a membrane mechanism parameter may change the resting state of the system [19]. When the resting membrane potential is altered, it affects the driving force of all ion channels, subsequently impacting all active and passive currents and, consequently, the overall system dynamics. This inherent characteristic makes it difficult to distinguish the effect a specific parameter has on its associated membrane mechanism from the broader influence it exerts on all ion channels by altering the resting state.

The final challenge addressed in this paper pertains to the different temporal dynamics exhibited by the models’ outputs, spanning both fast and slow time scales. These differences require careful consideration when determining which outputs to examine and in what manner. For instance, membrane potentials often display rapid dynamics characterized by multiple action potentials (AP). In such cases, it proves more meaningful to focus on informative spiking features, such as the number of APs, rather than the time-dependent membrane potential. Conversely, ion concentrations exhibit variations over slower time scales, allowing for a meaningful study at each point in time. Temporal factors influence the evolution of uncertainty over time, requiring an in-depth examination of how time interacts with uncertainty and sensitivity in time-dependent outputs [34]. Moreover, this challenge is linked to the first one concerning computational costs, as slow dynamics processes necessitate running longer simulations to observe their long-term effects. Additionally, evaluating sensitivity indices for a time-dependent output (i.e., one for each time point) is computationally demanding, especially when aiming at ensuring sufficient accuracy.

In this paper, we present a comprehensive and computationally efficient case study for UQ and GSA on a neuron model with ion concentration dynamics. Specifically, we delve into the electrodiffusive neuron-extracellular-glia (edNEG) model presented in Sætra et al. 2021 [20]. Electrodiffusive neuron models represent a subgroup of models with ion concentration dynamics that carefully account for ion concentrations by factoring in both diffusion and electric drift effects on ionic movement [19, 20]. In doing so, they maintain a consistent relationship between ion concentrations and electrical potentials, fostering a comprehensive understanding of the intricate interplay between ion dynamics and electric potentials. Our primary aim is to precisely assess the model’s susceptibility to uncertainty, particularly during spiking dynamics, and compare it across physiological and pathological conditions. Thus, we specifically focus on the parameters of active ion channels, as they play a key role in the spiking activity of neurons. First, by considering effective implementation strategies and solver selection, we were able to run simulations that were 15 times more time-efficient than our original implementation presented in Sætra et al. 2021 [20]. This increased efficiency demonstrates the feasibility of conducting sensitivity analysis on complex neuroscience models. Second, to address the challenge associated with parameters potentially influencing the resting state of the system, we propose a method to isolate those parameters specifically affecting the resting state. This approach distinguishes parameters directly impacting system dynamics through active ion channels from those primarily affecting the resting state, subsequently shaping the system’s overall dynamics. Finally, we investigate how model parameters influence ion dynamics and membrane potentials, considering both scalar quantities of interest and time-dependent outputs. This is achieved through a variance-based GSA, accounting for multiple parameters’ simultaneous variation and interactions. We conduct this analysis on the edNEG model under both physiological and pathological conditions: first, when the neuron is subjected to a moderate stimulus current resulting in low-frequency firing, and second, when given a strong stimulus current inducing depolarization block. We believe our investigations can provide valuable guidelines for future studies on UQ and GSA applied to neuron models with ion concentration dynamics, thereby expanding the influence and application of such models in computational neuroscience.

The present study is organized as follows: First, we give a general introduction to the edNEG model, followed by a description of the numerical integration methods employed and the techniques applied for uncertainty quantification and sensitivity analysis. Next, we present the results from UQ and GSA. Finally, we provide discussions and future perspectives.

Materials and methods

The edNEG model: Mathematical and computational framework

The edNEG model [20] considers three domains—a neuron, extracellular space, and glia—each divided into two subdomains, resulting in a total of six compartments (Fig 1). Specifically, within the neuronal domain, the two compartments represent the somatic and dendritic layers, while the glial domain corresponds to a segment of astrocyte syncytium. Utilizing the electrodiffusive Kirchhoff-Nernst-Planck (KNP) framework, the model predicts the evolution in time of ion concentrations for four ion species (Na⁺, K⁺, Ca²⁺, and Cl⁻), the electrical potentials ϕ, and the volumes V in all compartments.

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Fig 1. edNEG model schematics.

The edNEG model comprises three domains representing a neuron, ECS, and glia, each subdivided into two compartments representing the somatic (bottom) and dendritic (top) layers. The model is embedded in the KNP framework and predicts the temporal evolution of the ion concentrations [k] (k = {Na⁺, K⁺, Ca²⁺, Cl⁻}), the electrical potential ϕ, and the volume V in each compartment. Both neuronal membranes feature Na⁺, K⁺, and Cl⁻ leak channels (j_leak,k), a 3Na⁺/2K⁺ pump (j_pump), K⁺/Cl⁻ (j_KCC2) and Na⁺/K⁺/2Cl⁻ (j_NKCC1) cotransporters, and a Ca²⁺/2Na⁺ exchanger (j_Ca-dec). Additionally, the soma contains Na⁺ (j_Na) and K⁺ delayed rectifier (j_K-DR) channels, while the dendrite includes a voltage-dependent Ca²⁺ channel (j_Ca), a voltage-dependent K⁺ afterhyperpolarization channel (j_K-AHP), and a Ca²⁺-dependent K⁺ channel (j_K-C). Both glial compartments feature Na⁺ and Cl⁻ leak channels (j_leak,k), an inward rectifying K⁺ channel (j_K-IR), and a 3Na⁺/2K⁺ pump (j_pump). Intra- and extracellular fluxes (j_n, j_e and j_g) are driven by diffusion and electric drift. The model also accounts for transmembrane water flow (w_m) due to changes in the osmotic pressure across the membranes. The figure is adapted from Sætra et al. 2021 [20].

https://doi.org/10.1371/journal.pone.0303822.g001

Interactions between the three domains are modeled by ionic exchange across the neuron-extracellular and glia-extracellular membranes, taking into account the distinct ion channels present in neuronal soma and dendrites, and in glial membrane. Specifically, in order to describe the transmembrane ion movements across the neuron-extracellular membrane, the following channels are considered: Na⁺, K⁺ and Cl⁻ leak channels, a 3Na⁺/2K⁺ pump, K⁺/Cl⁻ and Na⁺/K⁺/2Cl⁻ cotransporters, and a Ca²⁺/2Na⁺ exchanger. Additionally, the soma compartment contains Na⁺ and K⁺ delayed rectifier channels, while the dendrite encloses a voltage-dependent Ca²⁺ channel, a K⁺ afterhyperpolarization channel, and a Ca²⁺-dependent K⁺ channel. Conversely, the glia-extracellular membrane includes Na⁺ and Cl⁻ leak channels, an inward rectifying K⁺ channel, and a 3Na⁺/2K⁺ pump. Movement of ions may also occur within domains, driven by diffusion and electric drift. The active ion channels are modeled through a Hodgkin-Huxley formalism for the voltage-dependent conductances, with six differential equations for the gating variables. Finally, volume dynamics induced by osmotic changes are described by six differential equations, each corresponding to a specific compartment. The ultimate model comprises d = 34 ODEs, with d_N = 22 addressing ion dynamics (six for each ion species, except for Ca²⁺, which requires only four as it does not enter the glial compartments), d_x = 6 concerning gating variables, and d_V = 6 related to volume dynamics. This model can be represented by an ODE system in the form: (1) Here, is the ion dynamics state vector, where N are the number of ions, is the gating variables state vector, and is the volume dynamics state vector. is a known system input vector, specifically a stimulus current in our case, is a vector of known parameters, is a set of coupled nonlinear functions, and represent the initial conditions of the system. The six electrical potentials are derived algebraically at each time step, while the four membrane potentials are defined as the difference between intra- and extracellular electrical potentials, both for neuronal and glial cells in each of their two compartments. Subsequently, we can establish the set of output equations as follows: (2) where are the outputs. The detailed mathematical formulation is reported in Sætra et al. 2021 [20].

Numerical implementation and validation

Approximating, simulating, and validating multi-scale biophysical models such as the edNEG model poses several numerical challenges. Furthermore, it is crucial to minimize the computational cost needed for model execution to facilitate efficient UQ and GSA within a reasonable timeframe. In reference to the implementation proposed in Sætra et al. 2021 [20], where we utilized the solve_ivp function from the Python library SciPy [35] and its RK23 method, we successfully improved convergence by rescaling units and implementing an analytical Jacobian. The first strategy aimed at confining the range of orders of magnitude for the state variables. The second strategy sped up computations and improved result accuracy. Specifically, the analytical Jacobian prevented approximation errors resulting from SciPy’s default finite difference approximation. These changes made implicit solvers work effectively, and resulted in simulations that were up to fifteen times faster than the original ones, depending on the choice of integration method and maximum time step. We subsequently conducted a convergence analysis of the number of action potentials and time of the last action potential for ϕ_msn, the extracellular potassium concentration [K⁺]_se, and the extracellular volume V_se for solve_ivp’s different solvers. As a result of this analysis, for UQ and GSA, we opted for the implicit solver Radau with a maximum time-step length of Δt_max = 10 ms. Note that the solve_ivp function utilizes adaptive time stepping, and Δt_max indicates the maximum allowed time step. Our choice balanced computational efficiency with the necessity to effectively capture dynamics within a simulation time of T = 6 s. Our selection of the Radau solver was primarily due to its implicit nature, which allows accurate results to be obtained with larger time-step lengths. Additionally, it proved to be more accurate at large time-step lengths for our particular model compared to other SciPy implicit solvers.

The complete set of rescaled parameters along with initial conditions are summarized in S1 Appendix. The source codes for this study are available at https://github.com/CINPLA/edNEGmodel and https://github.com/letiziasignorelli/edNEGmodel_UQSA [36].

Variance-based global sensitivity analysis

Let us call a computational model , i.e. a six-compartmental neuron model, that depends on time t and has N_p uncertain input parameters , with output y: (3) where denotes the support of the set of N_p input parameters. To address the uncertainties associated with the input, the computational model can be analyzed using a probabilistic approach. Therefore, suppose that the uncertainty in the input parameters is modeled by a random vector with prescribed joint probability density function f_P(p). The resulting quantity of interest Y is now a random variable obtained by propagating the uncertainty in P through : (4) Variance-based GSA provides valuable information about which parameters have the most significant influence on the output variability and uncovers potential interactions among these parameters. One widely used approach within variance-based GSA is the computation of Sobol’ sensitivity indices [37, 38]. Sobol’ indices are a set of sensitivity indices that partition the total variance of the model output into contributions from individual parameters and their interactions. Considering the quantity of interest as defined in Eq 4, let us fix the factor P_i at a particular value . The resulting variance of the model output, with P_i fixed, will measure the relative importance of P_i and can be defined as follows: (5) Taking the average and according to the law of total variance, the variance of the whole model output can be decomposed as (6) The conditional variance is defined as the first-order effect of P_i on Y. Finally, the First Order Sobol’ sensitivity index of the input P_i on Y is given by (7) and is interpreted as the fraction of the output variance that can be associated to the variance of P_i. A high value of S_i indicates that changes in parameter P_i have a significant impact on the output, while a low value suggests that variations in P_i have a relatively minor effect on the overall output variance.

First Order Sobol’ indices do not consider potential interactions among parameters, a common occurrence in neuroscience models. To address this limitation, a possible approach is to compute the total effects. We define the variance of the expected value when all parameters except P_i are fixed as (8) Therefore, due to the law of total variance, the Total Order Sobol’ sensitivity of P_i on Y is given by (9) provides a measure of the output variance attributed to P_i considering all possible interactions of any order with any other parameter.

The case of time-dependent processes.

Many neuroscience models, such as the edNEG model examined in this study, exhibit time-dependent outputs, yielding additional challenges for sensitivity analysis. While it is possible to apply Sobol’s approach pointwise in time, for instance on a set of grid points, 0 = t₀ < t₁ < … < t_n−1 < t_n = T, this approach has some limitations. First, the variance of the process itself varies in time, leading to distortions in the evaluation of the relative importance of uncertain parameters over different time periods. Second, considering the model outputs at different time points as independent variables ignores the temporal correlation structure of the process. To overcome the first issue, one possible option is to compute a weighted Sobol’ indices pointwise in time, where at each time t_k the Sobol’ index is multiplied by the standard deviation of the output : (10) A further improvement could involve incorporating a generalized version of Sobol’ indices considering potential time correlations [34]. The generalized First Order Sobol’ indices read as (11) while the generalized Total Order Sobol’ indices are: (12)

Factor fixing

Procedure description.

Based on the edNEG model’s biophysics, and informed by a preliminary and previously done sensitivity analysis of the analogous edPR model [39], we made an educated guess to categorize the parameters of interest into two distinct groups. The first group influences the model’s dynamics by altering the resting state, while the second group directly impacts the dynamics themselves. We have labeled these groups as “non-dynamic” parameters and “dynamic” parameters, respectively. The nominal values and the detailed description for these parameters are listed in Tables 1 and 2. To ensure that only the “non-dynamic” parameters influence the resting state and subsequently modify the dynamics, we carried out a sensitivity analysis by treating the two parameter groups as two inputs of uncertainty. Consequently, we computed the Total Order Sobol’ indices with respect to both groups of parameters. The aim of this factor fixing procedure was to ensure that the “dynamic” parameters did not affect the resting state, allowing us to subsequently concentrate on examining how uncertainty in these parameters uniquely influences firing dynamics.

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Table 1. “Non-dynamic” parameters: Conductances and strengths of leak channels and homeostatic mechanisms.

https://doi.org/10.1371/journal.pone.0303822.t001

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Table 2. “Dynamic” parameters: Maximum conductances of active channels.

https://doi.org/10.1371/journal.pone.0303822.t002

Parameter distributions.

We adopted the assumption that all parameters under consideration for the sensitivity analysis followed a uniform distribution [23] within a predefined percentage range centered around their nominal values. For this reason a ‘hyper-parameter’ was introduced to control the uncertainty across all parameters, as in Pathmanathan et al. 2019 [24]: (13) where P_i,nom is the nominal value of P_i. In the factor fixing analysis, was established at a value of 15%.

Simulation protocol.

To evaluate the parameters’ impact on the resting state, we conducted the GSA on simulations that we ran for 240 s without applying any stimulus.

Quantities of interest.

We opted to analyze six representative variables:

ϕ_msn: neuronal membrane potential (soma layer),
ϕ_mdn: neuronal membrane potential (dendrite layer),
ϕ_msg: glial membrane potential (soma layer),
ϕ_mdg: glial membrane potential (dendrite layer),
[K⁺]_se: extracellular potassium concentration (soma layer),
[K⁺]_de: extracellular potassium concentration (dendrite layer),

and designated as the quantities of interest (QoI) their values at the simulation’s final time. Subsequently, we computed the Total Order Sobol’ indices for each of them. We selected these QoIs because temporally constant membrane potentials and ion concentrations characterize the system’s resting state. If a parameter is altered, the membrane potentials and ion concentrations may deviate from an initial resting state until the system stabilizes in a new resting state.

Implementation details.

We employed the Python library SaLib [40, 41] to create a set of parameter configurations using the Saltelli method. This approach extends the Sobol’ sequence, a widely utilized quasi-random low-discrepancy sequence, to generate uniform samples across the parameter space. This extension aims to minimize error rates in the subsequent sensitivity index computations [42]. Our study employed 512 samples, which translated to 3072 parameter sets. Since the factor-fixing analysis was a preliminary analysis conducted with the model in resting conditions and with scalar outputs, it did not require the use of surrogate models.