The problem definition

Consider a model \(U\) that depends on space \(\boldsymbol{x}\) and time \(t\), has \(D\) uncertain input parameters \(\boldsymbol{Q} = \left[Q_1, Q_2, \ldots, Q_D \right]\), and gives the output \(Y\):

\[Y = U(\boldsymbol{x}, t, \boldsymbol{Q}).\]

The output \(Y\) can be any value within the output space \(\Omega_Y\) and has an unknown probability density function \(\rho_Y\). The goal of an uncertainty quantification is to describe the unknown \(\rho_Y\) through statistical metrics. We are only interested in the input and output of the model, and we ignore all details on how the model works. The model \(U\) is thus considered a black box, and may represent any model, for example a spiking neuron model that returns a voltage trace, or a network model that return a spike train.

We assume the model includes uncertain parameters that can be described by a multivariate probability density function \(\rho_{\boldsymbol{Q}}\). Examples of parameters that can be uncertain in neuroscience are the conductance of a single ion channel, or the synaptic weight between two species of neurons in a network. If the uncertain parameters are independent, the multivariate probability density function \(\rho_{\boldsymbol{Q}}\) can be given as separate univariate probability density functions \(\rho_{Q_i}\), one for each uncertain parameter \(Q_i\). The joint multivariate probability density function for the independent uncertain parameters is then:

\[\rho_{\boldsymbol{Q}} = \prod_{i=1}^D \rho_{Q_i}.\]

In cases where the uncertain input parameters are dependent, the multivariate probability density function \(\rho_{\boldsymbol{Q}}\) must be defined directly. We assume the probability density functions are known, and are not here concerned with how they are determined. They may be the product of a series of measurements, a parameter estimation, or educated guesses made by experts.