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<div class="section" id="model-description">
<h1>Model description<a class="headerlink" href="#model-description" title="Permalink to this headline">¶</a></h1>
<div class="section" id="blind-deconvolution-analysis-bda">
<h2>Blind Deconvolution Analysis (BDA)<a class="headerlink" href="#blind-deconvolution-analysis-bda" title="Permalink to this headline">¶</a></h2>
<p><strong>HemoLearn</strong> is a Python module offering a new algorithm that aims to fit a
rich multivariate decomposition of the BOLD data using a semi-blind
deconvolution and low-rank sparse decomposition. The model distinguishes two
major parts in the BOLD signal: the neurovascular coupling and the neural
activity signal.</p>
<p>Mathematically, if we have a single subject with <span class="math notranslate nohighlight">\(P\)</span> fMRI time series of length
<span class="math notranslate nohighlight">\(\widetilde{T}\)</span>, if we share the spatial maps, the considered data model is:</p>
<div class="math notranslate nohighlight">
\[\begin{align}
\boldsymbol{Y}_i &= \left( \sum_{m=1}^{M} \boldsymbol{\Theta}_m ^\top \boldsymbol{v}_{\delta_{im}} \right)
~\dot{*}~ \left( \sum_{k=1}^{K} \boldsymbol{u_k}^\top \boldsymbol{z_{ik}} \right)
+ \boldsymbol{E}_i
\enspace .
\end{align}\]</div>
<p>We aim to distangle the neurovascular coupling modelled by
<span class="math notranslate nohighlight">\(\sum_{m=1}^{M} \boldsymbol{\Theta}_m ^\top \boldsymbol{v}_{\delta_{im}}\)</span>
from the neural activation signals modelled by
<span class="math notranslate nohighlight">\(\sum_{k=1}^{K} \boldsymbol{u_k}^\top \boldsymbol{z_{ik}}\)</span> by minimizing
the following cost-function:</p>
<div class="math notranslate nohighlight">
\[\begin{split}\begin{equation}
\begin{split}
&\min_{(\boldsymbol{U}, \boldsymbol{Z}_i, \boldsymbol{\delta}_i)} ~
\frac{1}{2n} \sum_{i=1}^{n} \left\Vert \boldsymbol{Y}_i - \left( \sum_{m=1}^{M} \boldsymbol{\Theta}_m^\top \boldsymbol{v}_{\delta_{im}} \right) ~\dot{*}~ \left( \sum_{k=1}^{K} \boldsymbol{u}_k^\top \boldsymbol{z}_{ik} \right) \right\Vert_F^2 + \frac{1}{n} \sum_{i=1}^{n} \lambda_i \sum_{k=1}^{K} \| \nabla \boldsymbol{z}_{ik} \|_1 \enspace, \\
& \text{subject to} \quad \forall k \quad \|\boldsymbol{u_k}\|_1 = \eta, \quad \forall k, j \quad u_{kj} \geq 0, \quad \forall i, m \quad \delta_{im} \in [0.5, 2.0] \enspace . %\\
\end{split}
\end{equation}\end{split}\]</div>
<p>With <span class="math notranslate nohighlight">\(\lambda_i\)</span> being the temporal regularization parameter for the i-th subject, <span class="math notranslate nohighlight">\(\eta\)</span> the
spatial sparcity parameter, <span class="math notranslate nohighlight">\(K\)</span> the number of neural components and
<span class="math notranslate nohighlight">\(M\)</span> the number of vascular regions considered.</p>
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