## 3 Methodology
### Bio-Inspired Embedding Fusion  
Integration of multi-modal information is a considerably complex task characterized by deeply convoluted data correlations. In the animal kingdom, such a task is unconsciously performed on certain brain regions, allowing for beings and equally humans to derive multi-sensory conclusions. Inspired by the work of [PERSON] et al. in [29] and their model of the brain's superior colliculus (SC) for multi-sensory integration, we present a novel bio-inspired neural network layer for audio and visual embedding fusion. This section describes its structure and behavior. The proposed SC fusion layer, henceforth abbreviated to SCF, is composed by two upstream uni-modal neural areas corresponding to the audio and visual modalities, plus the additional downstream multi-modal area in charge of audiovisual integration. Each upstream area comprises \(N\times M\) basic neuron units interlinked within that same area, but not between upstream areas, and topologically organized to communicate inputs to corresponding downstream neurons whose receptive fields (RF)--kernels--are in the same spatial positions. The downstream area in turn, also made up of \(N\times M\) neurons, feeds back to each upstream area in the same neuron spatial fashion. This serves to emulate how proximal biological neurons respond to stimuli in proximal position of space, and allows for visual area neurons to indirectly excite audio area neurons or vice-versa through the multi-modal area. Lateral synapses between intra-area neurons are modelled after a mexican hat (Ricker) wavelet, so they may be both excitatory (central region) and inhibitory (surrounding region). The overall described structure is depicted in Figure 2.  
Considering a neural area \(s\in S=\{a,v,m\}\), pertaining to the auditory \(a\), visual \(v\) and multi-modal \(m\) factors. Asserting \(s\) to be composed of \(N\times M\) neurons as previously described, each neuron \(ij\) or \(hk\) receives a composite input \(u_{ij}^{s}[n]\), which is described by Equation (1). Here \(i,j,h\) and \(k\) denote iterators over \(N\times M\) as such: \(i,h=1,\ldots,N;j,k=1,\ldots,M\). \[u_{ij}^{s}[n]=\left\{ \begin{array}{l}r_{ij}^{s}[n]+l_{ij}^{s}[n]+f_{ij}^{s}[ n],\ s=\{a,v\}\\ \\ k^{a}\cdot z_{ij}^{a}[n]+k^{c}\cdot z_{ij}^{v}[n]+l_{ij}^{s}[n],\ s=m\end{array} \right. \tag{1}\]  
where \(r_{ij}^{s}[n]\) denotes the input at neuron \(ij\) triggered by an external stimulus, \(l_{ij}^{s}[n]\) represents the input caused by lateral synapses between intra-area neurons, \(f_{ij}^{s}[n]\) is the feedback input from the multi-modal SC neurons up to the uni-modal neurons, and \(z_{ij}^{s}[n]\) describes an arbitrary neuron's activity whose continuous differential model is discretized in our implementation as the signal in Equation (2) for any \(s\in S\). \[\tau_{s}z_{ij}^{s}[n+1]=(\tau_{s}-1)z_{ij}^{s}[n]+\phi(p^{s}(u_{ij}^{s}[n]- \theta^{s})) \tag{2}\]  
For such a response \(\tau_{s}\) refers to the speed with which a neuron responds to some stimulus, and \(\phi\) is the sigmoidal activation of some input \(u_{ij}^{s}[n]\) normalized by \(\theta^{s}\) and \(p^{s}\)--respectively the input value at the central point of neuron activity and its slope. As a consequence, the input to a multi-modal neuron can be seen as the weighted sum of the input activities of each of its spatial uni-sensory counterparts (\(k^{a}\) and \(k^{v}\) being inter-area synaptic connection intensity) plus the area's own lateral synapses. The laterally synaptic input received by some neuron \(ij\) of an area \(s\) is defined simply as the weighted sum of the neighboring neurons activities, following (3). Here \(L_{ij,hk}^{s}\) represents the strength of the synaptic connection from neuron \(hk\) to neuron \(ij\), which follows the mexican hat wavelet in (4) as previously explained.