Parameter identifiability and identifiable combinations in generalized Hodgkin–Huxley models

Parameter and coupling estimation in small networks of Izhikevich's neurons

Nowadays, experimental techniques allow scientists to have access to large amounts of data. In order to obtain reliable information from the complex systems that produce these data, appropriate analysis tools are needed. The Kalman filter is a frequently used technique to infer, assuming a model of the system, the parameters of the model from uncertain observations. A well-known implementation of the Kalman filter, the unscented Kalman filter (UKF), was recently shown to be able to infer the connectivity of a set of coupled chaotic oscillators. In this work, we test whether the UKF can also reconstruct the connectivity of small groups of coupled neurons when their links are either electrical or chemical synapses. In particular, we consider Izhikevich neurons and aim to infer which neurons influence each other, considering simulated spike trains as the experimental observations used by the UKF. First, we verify that the UKF can recover the parameters of a single neuron, even when the parameters vary in time. Second, we analyze small neural ensembles and demonstrate that the UKF allows inferring the connectivity between the neurons, even for heterogeneous, directed, and temporally evolving networks. Our results show that time-dependent parameter and coupling estimation is possible in this nonlinearly coupled system.

Parameter Identification Problem in the Hodgkin-Huxley Model

The Hodgkin-Huxley (H-H) landmark model is described by a system of four nonlinear differential equations that describes how action potentials in neurons are initiated and propagated. However, obtaining some of the parameters of the model requires a tedious combination of experiments and data tuning. In this letter, we propose the use of a minimal error iteration method to estimate some of the parameters in the H-H model, given the measurements of membrane potential. We provide numerical results showing that the approach approximates well some of the model's parameters, using the measured voltage as data, even in the presence of noise.

Calibration of ionic and cellular cardiac electrophysiology models

Cardiac electrophysiology models are among the most mature and well-studied mathematical models of biological systems. This maturity is bringing new challenges as models are being used increasingly to make quantitative rather than qualitative predictions. As such, calibrating the parameters within ion current and action potential (AP) models to experimental data sets is a crucial step in constructing a predictive model. This review highlights some of the fundamental concepts in cardiac model calibration and is intended to be readily understood by computational and mathematical modelers working in other fields of biology. We discuss the classic and latest approaches to calibration in the electrophysiology field, at both the ion channel and cellular AP scales. We end with a discussion of the many challenges that work to date has raised and the need for reproducible descriptions of the calibration process to enable models to be recalibrated to new data sets and built upon for new studies. This article is categorized under: Analytical and Computational Methods > Computational Methods Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Reducing complexity and unidentifiability when modelling human atrial cells

Mathematical models of a cellular action potential (AP) in cardiac modelling have become increasingly complex, particularly in gating kinetics, which control the opening and closing of individual ion channel currents. As cardiac models advance towards use in personalized medicine to inform clinical decision-making, it is critical to understand the uncertainty hidden in parameter estimates from their calibration to experimental data. This study applies approximate Bayesian computation to re-calibrate the gating kinetics of four ion channels in two existing human atrial cell models to their original datasets, providing a measure of uncertainty and indication of potential issues with selecting a single unique value given the available experimental data. Two approaches are investigated to reduce the uncertainty present: re-calibrating the models to a more complete dataset and using a less complex formulation with fewer parameters to constrain. The re-calibrated models are inserted back into the full cell model to study the overall effect on the AP. The use of more complete datasets does not eliminate uncertainty present in parameter estimates. The less complex model, particularly for the fast sodium current, gave a better fit to experimental data alongside lower parameter uncertainty and improved computational speed. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Reducing complexity and unidentifiability when modelling human atrial cells

Mathematical models of a cellular action potential (AP) in cardiac modelling have become increasingly complex, particularly in gating kinetics, which control the opening and closing of individual ion channel currents. As cardiac models advance towards use in personalized medicine to inform clinical decision-making, it is critical to understand the uncertainty hidden in parameter estimates from their calibration to experimental data. This study applies approximate Bayesian computation to re-calibrate the gating kinetics of four ion channels in two existing human atrial cell models to their original datasets, providing a measure of uncertainty and indication of potential issues with selecting a single unique value given the available experimental data. Two approaches are investigated to reduce the uncertainty present: re-calibrating the models to a more complete dataset and using a less complex formulation with fewer parameters to constrain. The re-calibrated models are inserted back into the full cell model to study the overall effect on the AP. The use of more complete datasets does not eliminate uncertainty present in parameter estimates. The less complex model, particularly for the fast sodium current, gave a better fit to experimental data alongside lower parameter uncertainty and improved computational speed. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

The parameter Houlihan: A solution to high-throughput identifiability indeterminacy for brutally ill-posed problems

One way to interject knowledge into clinically impactful forecasting is to use data assimilation, a nonlinear regression that projects data onto a mechanistic physiologic model, instead of a set of functions, such as neural networks. Such regressions have an advantage of being useful with particularly sparse, non-stationary clinical data. However, physiological models are often nonlinear and can have many parameters, leading to potential problems with parameter identifiability, or the ability to find a unique set of parameters that minimize forecasting error. The identifiability problems can be minimized or eliminated by reducing the number of parameters estimated, but reducing the number of estimated parameters also reduces the flexibility of the model and hence increases forecasting error. We propose a method, the parameter Houlihan, that combines traditional machine learning techniques with data assimilation, to select the right set of model parameters to minimize forecasting error while reducing identifiability problems. The method worked well: the data assimilation-based glucose forecasts and estimates for our cohort using the Houlihan-selected parameter sets generally also minimize forecasting errors compared to other parameter selection methods such as by-hand parameter selection. Nevertheless, the forecast with the lowest forecast error does not always accurately represent physiology, but further advancements of the algorithm provide a path for improving physiologic fidelity as well. Our hope is that this methodology represents a first step toward combining machine learning with data assimilation and provides a lower-threshold entry point for using data assimilation with clinical data by helping select the right parameters to estimate.

Biophysically interpretable inference of single neuron dynamics

Identification of key ionic channel contributors to the overall dynamics of a neuron is an important problem in experimental neuroscience. Such a problem is challenging since even in the best cases, identification relies on noisy recordings of membrane potential only, and strict inversion to the constituent channel dynamics is mathematically ill-posed. In this work, we develop a biophysically interpretable, learning-based strategy for data-driven inference of neuronal dynamics. In particular, we propose two optimization frameworks to learn and approximate neural dynamics from an observed voltage trajectory. In both the proposed strategies, the membrane potential dynamics are approximated as a weighted sum of ionic currents. In the first strategy, the ionic currents are represented using voltage dependent channel conductances and membrane potential in a parametric form, while in the second strategy, the currents are represented as a linear combination of generic basis functions. A library of channel activation/inactivation and time-constant curves describing prototypical channel kinetics are used to provide estimates of the channel variables to approximate the ionic currents. Finally, a linear optimization problem is solved to infer the weights/scaling variables in the membrane-potential dynamics. In the first strategy, the weights can be used to recover the channel conductances, and the reversal potentials while in the second strategy, using the estimated weights, active channels can be inferred and the trajectory of the gating variables are recovered, allowing for biophysically salient inference. Our results suggest that the complex nonlinear behavior of the neural dynamics over a range of temporal scales can be efficiently inferred in a data-driven manner from noisy membrane potential recordings.

Four Ways to Fit an Ion Channel Model

Mathematical models of ionic currents are used to study the electrophysiology of the heart, brain, gut, and several other organs. Increasingly, these models are being used predictively in the clinic, for example, to predict the risks and results of genetic mutations, pharmacological treatments, or surgical procedures. These safety-critical applications depend on accurate characterization of the underlying ionic currents. Four different methods can be found in the literature to fit voltage-sensitive ion channel models to whole-cell current measurements: method 1, fitting model equations directly to time-constant, steady-state, and I-V summary curves; method 2, fitting by comparing simulated versions of these summary curves to their experimental counterparts; method 3, fitting to the current traces themselves from a range of protocols; and method 4, fitting to a single current trace from a short and rapidly fluctuating voltage-clamp protocol. We compare these methods using a set of experiments in which hERG1a current was measured in nine Chinese hamster ovary cells. In each cell, the same sequence of fitting protocols was applied, as well as an independent validation protocol. We show that methods 3 and 4 provide the best predictions on the independent validation set and that short, rapidly fluctuating protocols like that used in method 4 can replace much longer conventional protocols without loss of predictive ability. Although data for method 2 are most readily available from the literature, we find it performs poorly compared to methods 3 and 4 both in accuracy of predictions and computational efficiency. Our results demonstrate how novel experimental and computational approaches can improve the quality of model predictions in safety-critical applications.

Identifiability from a Few Species for a Class of Biochemical Reaction Networks

Under mass-action kinetics, biochemical reaction networks give rise to polynomial autonomous dynamical systems whose parameters are often difficult to estimate. We deal in this paper with the problem of identifying the kinetic parameters of a class of biochemical networks which are abundant, such as multisite phosphorylation systems and phosphorylation cascades (for example, MAPK cascades). For any system of this class, we explicitly exhibit a single species for each connected component of the associated digraph such that the successive total derivatives of its concentration allow us to identify all the parameters occurring in the component. The number of derivatives needed is bounded essentially by the length of the corresponding connected component of the digraph. Moreover, in the particular case of the cascades, we show that the parameters can be identified from a bounded number of successive derivatives of the last product of the last layer. This theoretical result induces also a heuristic interpolation-based identifiability procedure to recover the values of the rate constants from exact measurements.

A systematic approach for doing an a priori identifiability study of dynamical nonlinear models

This paper presents a method for investigating, through an automatic procedure, the (lack of) structural identifiability of dynamical models parameters. This method takes into account constraints on parameters and returns parameters whose estimations turn unidentifiable parameters into identifiable ones. It is based on (i) an equivalence between an extension of the notion of identifiability and the existence of solutions of algebraic systems, (ii) the use of symbolic computations for testing their existence. This method is described in details and is applied to two examples, the last one involving 12 parameters.

Data Assimilation Methods for Neuronal State and Parameter Estimation

This tutorial illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. The two main classes of DA techniques are sequential methods and variational methods. We provide computer code implementing basic versions of a method from each class, the Unscented Kalman Filter and 4D-Var, and demonstrate how to use these algorithms to infer several parameters of the Morris-Lecar model from a single voltage trace. Depending on parameters, the Morris-Lecar model exhibits qualitatively different types of neuronal excitability due to changes in the underlying bifurcation structure. We show that when presented with voltage traces from each of the various excitability regimes, the DA methods can identify parameter sets that produce the correct bifurcation structure even with initial parameter guesses that correspond to a different excitability regime. This demonstrates the ability of DA techniques to perform nonlinear state and parameter estimation and introduces the geometric structure of inferred models as a novel qualitative measure of estimation success. We conclude by discussing extensions of these DA algorithms that have appeared in the neuroscience literature.

Computational Neuroscience: Mathematical and Statistical Perspectives

Mathematical and statistical models have played important roles in neuroscience, especially by describing the electrical activity of neurons recorded individually, or collectively across large networks. As the field moves forward rapidly, new challenges are emerging. For maximal effectiveness, those working to advance computational neuroscience will need to appreciate and exploit the complementary strengths of mechanistic theory and the statistical paradigm.