# 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.