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Giampaolo D'Alessandro is at the School of Mathematics, University of Southampton, Southampton, United Kingdom Robert C. Cannon is at the Department of Psychology Center for Memory and Brain, Boston, Massachusetts, United States of America, and at Textensor, Edinburgh, United Kingdom.

The authors have declared that no competing interests exist.

Ion channels are the building blocks of the information processing capability of neurons: any realistic computational model of a neuron must include reliable and effective ion channel components. Sophisticated statistical and computational tools have been developed to study the ion channel structure–function relationship, but this work is rarely incorporated into the models used for single neurons or small networks. The disjunction is partly a matter of convention. Structure–function studies typically use a single Markov model for the whole channel whereas until recently whole-cell modeling software has focused on serial, independent, two-state subunits that can be represented by the Hodgkin–Huxley equations. More fundamentally, there is a difference in purpose that prevents models being easily reused. Biophysical models are typically developed to study one particular aspect of channel gating in detail, whereas neural modelers require broad coverage of the entire range of channel behavior that is often best achieved with approximate representations that omit structural features that cannot be adequately constrained. To bridge the gap so that more recent channel data can be used in neural models requires new computational infrastructure for bringing together diverse sources of data to arrive at best-fit models for whole-cell modeling. We review the current state of channel modeling and explore the developments needed for its conclusions to be integrated into whole-cell modeling.

Ion channels drive voltage-based signal processing within neurons and convert chemical signals into voltage changes at the synapses between cells. They can be distinguished by the ions that they allow to cross the membrane and by their response to chemical signals or changes in the membrane potential. More than 140 types of voltage-gated channels have been identified so far [

The development of patch-clamp techniques [

The reason that ion channel research is so rarely used by neural modelers is that there is currently no mechanism to incorporate it in a whole-cell model. A model requires good coverage of the whole-channel dynamics in a relatively small parameter space; but channel studies often focus on details of particular aspects of behavior and may leave other areas relatively ill-defined. For example, the complex gating of T-type calcium channels has important functional implications [

Improving on this situation requires two complementary developments: reliable storage, dissemination, and reuse of experimental data [

The inverse approach differs from traditional data analysis and parameter estimation in two ways. First, its main objective is not to measure specific quantities believed to be part of the underlying system, but to deliver a computational equivalent with behavior that is as close as possible to that of the underlying system. Second, it avoids any form of model-specific analysis by comparing the model with data in the space of the raw data, with the result that the procedure can be scaled up to handle new preparations more readily. Naturally, if a model can be completely characterized, then the inverse approach ends up providing estimates for distinct physical properties of the underlying system, but in most cases the best approximation will be a more limited model that aggregates properties into a smaller set of parameters.

In this review we survey the mathematical methods and computational tools available for studying ion channels and map out the additional components needed to bridge the gap between what is available from studies of ion channels and what is needed to construct reliable models for use in computing neural activity. As increasing numbers of researchers turn to quantitative models of neurons to help refine and interpret their observations, it is vital that these models should be built from the best information available. Conversely, exploiting the wealth of ion channel studies to produce reliable channel models should facilitate new studies and make the characterization of neural activity dramatically more efficient. The next sections, respectively, introduce a) the channel-modeling problem, and discuss b) the potential for dedicated experimental work to facilitate model development, c) how diverse information from independent studies can be incorporated, and d) the greatest hurdles to be overcome in fully exploiting the wealth of empirical data that is collected.

With the growing availability of computational resources, numerical inverse approaches are increasingly used across a range of disciplines. Together with three dedicated journals,

Astronomers and geophysicists were among the first to exploit computational inverse methods, partly because, unlike other scientists, they have no option of interfering with the system under study, and partly because the forward processes giving rise to observational data can often be well-characterized. Even where direct inverses exist in the idealized case, such as deconvolving an image by the point spread function of the optics, it has long been recognized that better results can usually be obtained in practical problems by ignoring the direct inverse and using an iterative approach that incorporates other factors such as the power spectrum of the noise [

In its purest form, the approach specifically avoids any form of data processing, such as calculating an activation curve from voltage-clamp recordings of an ion channel, since these introduce unnecessary assumptions and reduce the dimension of the space in which models are compared. In effect, the motivation for this type of analysis is to reduce the dimension of the data so it can be handled more easily. But the inverse problem approach does not need this reduction and instead pushes the processing burden onto the computer.

In comparison with astronomical applications, the study of ion channels is characterized by relatively simple base models for the channels themselves, but by an enormous diversity of complex forward processes that give rise to distinct types of data. Historically, neuron models have tended to use the formulation of voltage-dependent ion channel gating first presented by Hodgkin and Huxley [

(A) Three-state scheme considered in the text.

(B) Best-fit model derived by Vandenberg and Bezanilla [

(C) T-type calcium model by Frazier et al. [

A Markov model can be solved either as a stochastic process or using a mean field approach. In the first case, it represents the state of a single channel: one assumes that the model is in a given state, e.g., _{2},

A Markov model of a channel can be designed starting from its molecular representation, with each state of the Markov model corresponding to a different configuration of the molecule, e.g., [

The inverse channel–fitting problem, i.e., how to derive the values of the parameters of a Markov channel model from measured electrophysiological currents, has been studied in detail for many years. First of all, equilibrium distributions of currents are insufficient to distinguish between channel models [

The primary functional data for any model of an ion channel are in the form of the channel current in terms of the membrane potential and other environmental conditions as a function of time. The local conditions include the temperature, ionic concentrations on each side, and the concentrations of any ligands that act on the channel. The data may come from one or a few channels as in a cell-attached patch-clamp recording [

A wide range of ingenious voltage protocols have been developed for studying ion channels, beginning with steps to different voltages and extending to ramps, spikes, and multiple pulses (e.g., [

Once freed from any constraint to gather data that can be easily analyzed, one natural approach would be to subject channels to conditions that closely match those they normally experience. For example, a sequence of action potentials with a range of interspike intervals could be used for studying somatic channels in neurons. Alternatively, a sequence of steps to fixed potentials held for random time periods also exhibits a greater range of dynamics than conventional protocols. A good estimate of the discriminatory power for a given protocol can be readily obtained in simulations by measuring the sensitivity of the results to changes in the model parameters. One such case is shown in the last panel of

Sections of each command profile are shown in the first row. The strength of the constraints are shown in the second row for each of the twelve free parameters grouped into three voltage-dependent transitions. For each parameter, the symbols show the curvature of the error function around the exact model. Filled squares correspond to standard voltage steps, open triangles to a naturalistic spike-based waveform, and open squares to random steps.

Parameters for the Example Model

The second command profile is a sequence of spikes with a Poisson-distributed interspike interval (mean 50 ms) and a steady ramp to −40 mV between spikes. The third profile has random jumps to potentials between −80 mV and +20 mV in multiples of 10 mV. Each value is held for a Poisson-distributed period, again with mean 50 ms. These are taken as canonical examples of the two styles and have not been adjusted in any way to fit this channel. All the command waveforms have a total duration of ten seconds.

The lower row shows the strength of the parameter constraints in the vicinity of the correct model for each of the twelve parameters (four parameters for each of three transitions) for the three different voltage commands indicated by filled squares, open triangles, and open squares, respectively. The quantity being displayed is the second derivative of the likelihood function with respect to the corresponding parameter. The scale is logarithmic so a difference of one unit indicates a constraint that is ten times stronger.

The two-complex waveforms provide constraints that are typically at least ten times tighter than the step sequence. This is not surprising since the step sequence has a long, quiet holding period occupying almost 90% of the stimulation. But it does demonstrate that the use of complex command waveforms that do not allow any direct analysis is not a problem for the method. More interestingly, some of the constraints, such as the one on the gating asymmetry

Perhaps the most intriguing aspect of this style of model construction is that the optimal protocol inevitably depends strongly on the channel being studied. Given the limited time available during an experiment, there are therefore significant advantages, in terms of confidence in the resulting model, in adjusting the protocol in real time in the light of the results. Given sufficient computing power, this could involve performing the full inversion during the experiment and iteratively refining the best-fit models as more data becomes available. Or it could involve a less computationally costly decision process as the experiment proceeds to determine the regions of most active dynamics and to concentrate on them.

Another significant feature of the approach is that the feasibility of the inversion varies nonlinearly with the volume of data: at low volumes of data, inversion is highly degenerate with no clear optimal model. But increasing the breadth of data can resolve the degeneracy, turning an underspecified problem into one which is easily solved. For example, models that cannot be distinguished under equilibrium conditions [

The examples in

Although in the example the spiking command produces the tightest constraints, it is not necessarily the most appropriate command to use. The choice of command waveform should itself be treated as an optimization problem balancing a range of factors including: a) computational cost—channel models are easier to integrate if the stimulation has a step profile; b) screening efficiency—commands that can rapidly reject bad models could be very different from those that can refine good ones; c) off-minimum convergence—the example focuses on the final approach to a minimum, but it is equally important to help the fitting algorithm converge from more distant models.

These considerations demonstrate that before it is worth attempting the inverse problem on real channels, a simulation study should be used to work out what data should be collected to make the inversion feasible. Such a study can also indicate the level of confidence likely to be achievable in the resulting models.

In practical terms, a natural starting point for the search process is the set of previously published schemes and parameter sets for channels similar to the one under study as well as the previous results of the inverse process itself. The careful choice of this set is a one-off task with a significant effect on the success and speed of the inverse process. Occasional tests against a sequence of systematically generated possible schemes could be used to adjust the set of models for initial screening. Computational performance is heavily influenced by the type of optimization methods that can be applied. Gradients on the likelihood functions are easily computed and can be used in the final approach to optimal models, but local minima prevent the exclusive use of downhill methods. As

The final test of a model is how well it can replace the real channel in its contribution to the activity of a neuron. Fitting to spiking protocols ensures the model sees the full dynamic range of the natural environment, but whole-cell models are also influenced by numerous factors that are still unquantified such as channel densities, localization, and even possible cooperation between channels [

The examples in

There are also many other sources of data about the structure and kinetics of ion channels. A few channels have been crystallized and their three-dimensional structure is known [

Once these questions are satisfactorily answered, the technical process of incorporating the data is usually straightforward: an extra term is added to the error function with a weighting reflecting the combination of possible errors and uncertainties. For a biophysical model, the example above with four voltage-sensing groups translates directly to a prior, saying that schemes with other than four closed states are extremely unlikely. For a cell-level model, where the object is to derive a “semi–black-box” system that provides the best performance when computing whole-cell behavior, the prior will be much more complicated. It should take account of the tradeoff between an extensive state diagram and very uncertain parameters, or a small-state diagram and well-constrained parameters. The value of such models in neuroinformatics derives from their connection of two levels of description. They package up knowledge about protein conformation and dynamics in a form that can be used in studying whole cells. This is just one of many such connections needed in whole-cell modeling. It will be equally important to produce models of how cells use nuclear processes such as post-translation modification and channel transport and insertion to regulate which channels are actually present in the membrane.

Although many statistical methods exist to help address the question of how to choose the right level of complexity of the model to fit under these questions [

When treated as an inverse problem, the construction of channel models is most productive when there is a wide variety of different raw data available to be fitted. Ideally this should include recordings with complex waveforms in a range of different temperatures and chemical environments for the channel. For recordings to be of use in the fitting process they require extensive metadata detailing the preparation and recording conditions. The publication of such data and metadata is fully in line with emerging policies on data sharing [

In practice, however, it is difficult to provide and validate metadata for which there is no market: why take the time and trouble to package up data when there is no apparent further use for it? And how can one be sure the metadata is sufficient for the data to be useful, when the application that may eventually use it does not currently exist? Some of these concerns may be overcome by negotiated pairwise collaborations [

It should be stressed here that the model evaluation and optimization algorithms will form only a small part of any practical system for large-scale channel modeling. Much of the work will concern essential processes related to data and metadata management, data formats, data provenance, and user interaction with the system. From this perspective, the output models are not so much research products themselves, as the transient outputs of two more fundamental research efforts: first, the experimental recording themselves, and second, the software systems and additional input data (weights, priors, algorithm choices) that implement solutions to the inverse problem. Naturally, it is sure to be useful to fix and archive particular models, but this should not preclude their revision when more data is available or when new algorithms come online. Shifting the focus from building databases of model parameters to databases of original recordings with best-fit models as a transient product should also overcome objections that databases can become filled with unreliable information. Rather than offering a single set of parameters, it allows the user to pick the model most suited to their application, and to see how it was derived and how well it reproduces the original data.

Once sufficient data is available for largely automated harvesting and processing, the computational procedures outlined here can be seen in the role of an ongoing compression process. The resulting models provide a condensed version of the raw data: an investigator could use the model to find out what a particular channel is expected to do in any given circumstance without having to refer back to the original experiments. This type of interaction is increasingly important with the growth of multidisciplinary studies where the user of the channel model can benefit from incorporating the latest data but is unlikely to have the relevant expertise to work directly with the data. It is the software equivalent of routine developments in hardware where a new machine suddenly makes widely accessible a set of processes that used to require very specific training and expertise. And just as with hardware, although functional prototypes can be built that can be successfully operated by their creators, for the technology to achieve its full potential and gain widespread use, substantial investment is required in productizing the ideas and algorithms. The bioinformatics community has led the way in doing this in their own domain. Perhaps the ion channel inverse problem can be the first instance of this philosophy spreading across the boundary into neuroinformatics.

Hodgkin–Huxley

^{2+}channel

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