In this series, we are observing the semantic errors of a hippocampal simulation of neurointerfaces, and the sampling grid approach used to model its unsupervised feature maps. This section will get into the linear algebra and calculus behind the sampling grids and how they relate to a variate error in the final system.

### Parametrized Sampling Grid

A sampling grid, neuroanatomically a receptive network, will be parameterized to allow the mutation of various neurobiological parameters, such as dopamine, oxytocin, or adrenaline, and produce a synthetic, reactionary response in the neurointerface stack. A modulation of the initial sampling grid will be used to classify the transformations to their respective location in the comprehensive memory field. In order to perform the spatial transform of the normalized input feature map, a sampler must sample a set of parameters from ${ \tau }_{ \theta }({ G }_{ i })$ where $G$ represents a static translational grid of the applied transforms. The input feature map $U$, the raw equivalent of the receptive fields, along with its primed resultant of the ${ f }_{ loc }(x) = V$ function will be accounted for as well in the translational grid. Each coordinate in $G$ represented as ${ \left( { x }_{ j }^{ s },{ y }_{ j }^{ s } \right) }_{ j }$, giving a gradient dimensionality $j$ to the spatial grid input. A gradient dimensionality allows the sparse network to have an infinite number of spatial perspectives as I will soon be posting about concentric bias simulation for mental illnesses.

Each coordinate in the ${ \tau }_{ \theta }({ G }_{ i })$ represents a spatial location in the input where the sampling kernel can concentrically be applied to get a projected and subsequent value in $V$ \. This, for stimuli transforms, can be written as:

${ V }_{ i }^{ c }(j)=\frac { \sum _{ n }^{ H }{ \sum _{ m }^{ W }{ { U }_{ nm }^{ c } } k\left( { x }_{ i }^{ s }-{ m };{ \Phi }_{ x } \right) k\left( { y }_{ i }^{ s }-n;{ \Phi }_{ x } \right) { :\quad \forall }_{ i }\in \left[ 1\dots { H }^{ ' }{ W }^{ ' } \right] } { :\quad \forall }_{ c }\in \left[ 1\dots C \right] }{ \left< { j }|{ { H }^{ ' } }|{ { W }^{ ' } } \right> }$

Here, $\Phi$ represents the parameterized potential of the sampling kernel of the spatial transformer which will be used to forward neuroanatomical equivalences through recall gradients.

The use of kernel sampling can be varied as long as all levels of gradients can be simplified to functions of ${ \left( { x }_{ j }^{ s },{ y }_{ j }^{ s } \right) }_{ j }$. For the purposes of our experimentation, a bilinear sampling kernel will be used to co-parallely process inputs, allowing for a larger parametrization of learning transforms. To allow backpropagation of loss through this sampling mechanism, the gradient functions must be with respect to $U$ and $G$. This observation was initially established as a means to allow sub-differentiable sampling in a similar bilinear sampling method:

$\frac { \delta { V }_{ i }^{ c } }{ \delta { U }_{ nm }^{ c } } =\sum _{ n }^{ H }{ \sum _{ m }^{ W }{ \max _{ j }{ (0,1-\left| { x }_{ i }^{ s }-m \right| ) } \max _{ j }{ (0,1-\left| { y }_{ i }^{ s }-n \right| ) } } }$

$\frac { \delta { V }_{ i }^{ c } }{ \delta { x }_{ i }^{ s } } =\sum _{ n }^{ H }{ \sum _{ m }^{ W }{ { U }_{ nm }^{ c }\max _{ j }{ (0,1-\left| { y }_{ i }^{ s }-n \right| ) } \begin{cases} 0 & if\left| m-{ x }_{ i }^{ s } \right| \ge 1 \\ 1 & if\quad m\ge { x }_{ i }^{ s } \\ -1 & if\quad m<{ x }_{ i }^{ s } \end{cases} } }$

Therefore, loss gradients can be attributed not only to the spatial transformers, but also to the input feature map, sampling grid, and, finally, back to the parameters, $\Phi$ & $\theta$. The bilinear sampler has been slightly modified in this case to allow for concentric recall functions to be applied to its resultant fields. It is worth noting that due to this feature, the spatial networks representation of the learned behavior is unique in the rate and method of preservation, much like how each person is unique in his ability to learn and process information. The observable synthetic activation complexes can also be modeled through the monitoring of these parameters as they elastically adapt to the stimulus. The knowledge of how to transform is encoded in localization networks, which fundamentally are non-static as well.