2.2. Example application: Selecting among disparate parameter sets of a biophysical model#
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To explain the construction of the EMD rejection rule, we use the dynamical model for a neuron membrane potential described by Prinz et al.~[6]. This choice was motivated by the fact that fitting a neuron model is a highly ill-posed problem, and therefore corresponds to the situation we set out in the Introduction: disparate sets of parameters for a structurally fixed model which nevertheless produce similar outputs. We will focus on the particular LP neuron type; Prinz et al. find five distinct parameter sets which reproduce its experimental characteristics. Throughout this work, we reserve LP 1 as the model that generates through simulation the true data \(\D_\test\), and use LP 2 to LP 5 to define the candidate models \(\M_A\) to \(\M_D\) which we compare against \(\D_\test\). We intentionally exclude LP 1 from the candidate parameters to emulate the typical situation where none of the candidate models fit the data perfectly. Visual inspection of model outputs suggests that two candidates (models \(\M_A\) and \(\M_B\)) are more similar to the true model output (Fig. 2.1). We will show that our method not only concords with those observations, but makes the similarity between models quantitative.
It is important to note that we treat the models \(\M_A\) to \(\M_D\) as simply given. We make no assumption as to how they were obtained, or whether they correspond to the maximum of a likelihood. We chose this neural example, where parameters are obtained with a multi-stage, semi-automated grid search, partly to illustrate that our method is agnostic to the fitting procedure.
The possibility of comparing models visually was another factor in choosing this example. Since numbers can be misleading, this allows us to confirm that the method works as expected. Especially for our target application where all candidate models share the same equation structure, a visual validation is key to establishing the soundness of the \(\Bemd{}\), since none of the established model selection like the Bayes factor, AIC or WAIC [32, 33] are applicable. Indeed, these other methods only compare alternative equation structures, not alternative parameter sets. We include a more in-depth comparison to these other methods at the end of the Results.
Fig. 2.1 Comparison of LP neuron responses
The goal is to estimate a good set of candidate models for neuron LP.
a) We consider a simple circuit with a known pacemaker neuron (AB) and a post-synaptic neuron of unknown response (LP).
b) Output of the AB neuron. This serves as input to the LP neuron.
In practice, these data would more likely be provided by an experimental recording of neuron AB, but here we assume it is a simulatable known model for convenience.
c) Response of neuron model LP 1 to the input in (b).
Together, (b) and © serve as our observed data \(\D_{\test}\).
d) Response of neuron models LP 2 to LP 5 to the input in (b). These are our four candidate models of neuron LP.
[source]#
The datasets in this example take the form of one-dimensional time series, with the time \(t \in \Tspace\) as the independent variable and the membrane potential \(V^{\LP{\!\!}} \in \Vspace\) as the dependent variable. We denote the space of all possible time-potential tuples \(\Tspace \!\times\! \Vspace\). Model specifics are given in the Methods; from the point of view of model selection, what matters is that we have ways of generating series of these time-potential tuples: either using the true data-generating process (\(\Mtrue\)) or one of the candidate models (\(\M_A\) to \(\M_D\)).
We will assume that the dataset used to evaluate models is composed of \(L\) samples, with each sample a \((t, \tilde{V}^{\LP{\!\!}})\) tuple:
The original model by Prinz et al. [6] produced deterministic traces \(V^{\LP{\!\!}}\). Experimental measurements however are variable, and our approach depends on that variability. For this work we therefore augment the model with two sources of stochasticity. First, the system as a whole receives a coloured noise input \(I_{\mathrm{ext}}\), representing an external current received from other neurons. (External currents may also be produced by the experimenter, to help model the underlying dynamics.) Second, we think of the system as the combination of a biophysical model — described by Eq. 4.1, Eq. 4.2 and Eq. 4.3 — and an observation model which adds Gaussian noise. The observation model represents components which don’t affect the biophysics — like noise in the recording equipment — and can be modeled to a first approximation as:
The parameters of the external input \(I_{\mathrm{ext}}\) are \(τ\) and \(σ_i\), which respectively determine its autocorrelation time and strength (i.e. its amplitude). The observation model has only one parameter, \(σ_o\), which determines the strength of the noise. The symbol \(\Mtrue\) is shorthand for everything defining the data-generating process, which includes the parameters \(τ\), \(σ_i\) and \(σ_o\). The indices \(i\) and \(o\) are used to distinguish input and output model parameters.
Since the candidate models in this example assume additive observational Gaussian noise, a natural choice for the loss function is the negative log likelihood:
However one should keep in mind that a) it is not necessary to define the loss in terms of the log likelihood, and b) the loss used to compare models need not be the same used to fit the models. For example, if the log likelihood of the assumed observation model is non-convex or non-differentiable, one might use a simpler objective for optimization. Alternatively, one might fit using the log likelihood, but compare models based on global metrics like the interspike interval. Notably, Prinz et al. [6] do not directly fit to potential traces \(\tilde{V}\), but rather use a set of data-specific heuristics and global statistics to select candidate models. Gonçalves et al. similarly find that working with specially crafted summary statistics — rather than the likelihoods — is more practical for inferring this type of model. In this work we nevertheless stick with the generic form of Eq. 2.6, which makes no assumptions on the type of data used to fit the models and is thus easier to generalise to different scenarios.
The definition of the risk then follows immediately:
Here \(a\) stands for one of the model labels \(A\), \(B\), \(C\) or \(D\). The notation \((t,\tilde{V}^{\LP{\!\!}}) \sim \Mtrue\) denotes that the distribution from which the samples \((t,\tilde{V}^{\LP{\!\!}})\) are drawn is \(\Mtrue\). We can think of \(\Mtrue\) as producing an infinite sequence of data points by integrating the model dynamics and appyling Eq. 2.5 multiple times, or by going to the laboratory and performing the experiment multiple times.
As alluded to above, the candidate model traces in Fig. 2.1d suggest two groups of models: the \(\M_A\) and \(\M_B\) models seem to better reproduce the data than \(\M_C\) or \(\M_D\). Within each group however, it is hard to say whether one model is better than the other; in terms of the risks, this means that we expect \(R_A, R_B < R_C, R_D\). This also means that we should be wary of a selection criterion which unequivocally ranks \(\M_A\) better than \(\M_B\), or \(\M_C\) better than \(\M_D\).
In other words, we expect that uncertainties on \(R_A\) and \(R_B\) should be at least commensurate with the difference \(\lvert R_A - R_B \rvert\), and similarly for the uncertainties on \(R_C\) and \(R_D\).