theory.tex

\input{preamble}
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%\addbibresource{neuro_theory.bib}

\title{Information-Theoretic, Probabilistic, and Algorithmic Considerations for the Design of Brain Activity Imaging Experiments}

%\author[1,2]{\ \lift{$\jointfirst\,$}Adam~H.~Marblestone\rlap{,}}
\author[1,2]{David~A.~Dalrymple}

%\affil[$\jointfirst$]{Joint first authors}
%\affil[$\jointlast$]{Joint last authors}

\newcommand\et{{\em \&}}

\affil[1]{Nemaload, San Francisco, CA{~94107, USA}}
\affil[2]{Media Lab{oratory,} Massachusetts Institute of Technology{, Cambridge,~MA~02139, USA}}

\renewcommand{\maketitlehookc}{{\small\raggedright Correspondence to: \texttt{david\,\textnormal{(at)}\,\,dalrymple.co}}}

\begin{document}
\maketitle
\pagestyle{plain}
\thispagestyle{empty}

\section{Introduction}

\textsc{Imaging} can be defined as the estimation, from sensor readings, of a physical quantity that varies over some particular region of space (and possibly time). In many cases (including almost all medical imaging, except for the humble X-ray), this is a nontrivial inference problem.

In the case of ``brain activity mapping,'' imaging is only one piece of the overall puzzle, which is illustrated in \autoref{fig:puzzle}.

%The neural computation should be estimated as a stochastic differential equation, of the type $\mbox{Phase Space} \rightarrow \left(\mbox{$\sigma$-Algebra of Tangent Space} \rightarrow \mathbb{R}\right)$.

%Neural Computation: $X_{\mbox{ph}} \rightarrow \left( \sigma(X_{\mbox{ph}}^*) \rightarrow \mathbb{R} \right)$ \\

\begin{landscape}
\begin{figure}[p]
\caption{An overview of variables involved in an electro-optical measurement of neural computation.}
\label{fig:puzzle}
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\end{figure}
\end{landscape}


Let us focus on the imaging problem for now.

\subsection{Forward Model}

\subsubsection{Emission: from Discrete Signals to Fluorescence in Cartesian Space}

Here we begin with an
``{\color{blue!60!green}Observable State History},'' presumed to consist of one
time-varying signal for each neuron. We define $\mathcal{S} \in
\left[C(\Re_{\ge 0})\right]^n$ such that $\mathcal{S}_i(t)$ are these
time-varying signals. Next, the Physical Indicator associates each point in
$(0,1)^3$ (the specimen coordinate system) with a single such time series, multiplied by a factor (proportional to the local density of emitted photon flux) and convolved
with characteristic time behavior: $\mathcal{P} \in
C\!\left[(0,1)^3,\bigvee_{i=0}^{n-1} \Re_{\ge 0}\right] \times
C^\infty_C(\Re_{\ge 0})$, such that $f_0(u,v,w) = \left(\mathcal{S} \cdot \mathcal{P}_0(u,v,w) \right) \ast \mathcal{P}_1$, i.e. $f_0 = \left(\mathcal{P}_1 \,\ast\right) \circ \left(\mathcal{S} \, \cdot\right) \circ \mathcal{P}_0$. We then add in time-invariant background, $\mathcal{B}_0 \in C\left[(0,1)^3,\Re_{\ge 0}\right]$, such that $f_b(u,v,w)(t)=f_0(u,v,w)(t)+\mathcal{B}_0(u,v,w)$, or simply $f_b = f_0 + \mathcal{B}_0$.
Next, the Specimen Movement induces a time-dependent embedding of the unit $(u,v,w)$ cube (considered to parameterize the anterior-posterior, dorsal-ventral, and left-right axes of the specimen) into ``relative'' coordinates (i.e. a Cartesian coordinate system referenced to the specimen-holder). We define $\mathcal{Q} \in C\left[\Re_{\geq 0}, C^1\left[(0,1)^3,\Re^3\right]\right]$. Now, for any bounded
subset $S \subset (0,1)^3$, we want $\int_S f_b = \int_{\mathcal{Q}(t)(S)} f_1$. By the change of variables theorem,
\[\int_{\mathcal{Q}(t)(S)} f_1 = \int_S \left(f_1 \circ \mathcal{Q}(t)\right) \left|\det D\mathcal{Q}(t)\right|\]
\[\frac{1}{\left|\det D\mathcal{Q}(t)\right|} f_b = f_1 \circ \mathcal{Q}(t) \]
\[f_1(x,y,z) = \begin{dcases} \frac{1}{\left|\det D\mathcal{Q}(t)\right|} f_b \circ \mathcal{Q}(t)^{-1} & \mbox{if } (x,y,z) \in \mathcal{Q}(t)\left((0,1)^3\right) \\ 0 & \mbox{otherwise} \end{dcases}\]
Finally, we add in another time-invariant background, this one fixed in $(x,y,z)$ space, $\mathcal{B}_1 \in C\left[\Re^3,\Re_{\geq 0}\right]$ with $f_2 = f_1 + \mathcal{B}_1$. The expanded equation for fluorescence in specimen-holder coordinates is:
\[f_2(x,y,z)(t) = \mathcal{B}_1(x,y,z) + \begin{dcases} \dfrac{\mathcal{B}_0(u,v,w) + \int_{-\infty}^{\infty} d\tau \mathcal{P}_1(t-\tau) \left( \mathcal{S}(\tau) \cdot \mathcal{P}_0(u,v,w)\right) }{\left|\det D\mathcal{Q}(t)(u,v,w)\right|} & \mbox{if } (x,y,z) = \mathcal{Q}(t)(u,v,w) \mbox{, } (u,v,w)\in(0,1)^3 \\ 0 & \mbox{otherwise}\end{dcases}\]

\subsubsection{Imaging: from Fluorescence to Sensor Intensity}

The position and orientation of the specimen-holder with respect to the optical system are characterized by $\mathcal{T} \in C\!\left[\Re_{\ge 0}, \Re^3 \times SO(3)\right]$, such that $f_r\left(\mathcal{T}_1(t) \vec{x} - \mathcal{T}_0(t)\right)(t) = f_2(\vec{x})(t) $. We may write $\mathcal{T}_1(t) \vec{x} - \mathcal{T}_0(t)$ as $\mathcal{T}(t)(\vec{x})$ or simply $\vec{r}$, and the inverse $\mathcal{T}_1(t)^{-1} ( \vec{r} + \mathcal{T}_0(t) )$ as $\mathcal{T}^{-1}(t)(\vec{r})$, so that $f_r(\vec{r})(t) = f_2(\mathcal{T}^{-1}(t)(\vec{r}))(t)$.

The input and output intensities of any passive optical system are related by a Fredholm integral equation:
\[f_s(\vec{s})(t) = \int d\vec{r}\,K(\vec{r},\vec{s})\,f_r(\vec{r})(t)\]
We also consider active optical systems, which we model using a time-dependent Fredholm kernel:
\[f_s(\vec{s})(t) = \int d\vec{r}\,K(\vec{r},\vec{s},t)\,f_r(\vec{r})(t)\]

The expanded equation for sensor intensity in a generic active optical system is:
\[f_s(\vec{s})(t) = \int_{\Re^3} K(\mathcal{T}^{-1}(t)(\vec{x}),\vec{s},t)\,\mathcal{B}_1\!\left(\vec{x}\right) d\vec{x} + \int_{(0,1)^3} K\left(\left(\mathcal{T}\circ\mathcal{Q}\right)(t)(\vec{u}),\vec{s},t\right) \frac{\mathcal{B}_0(\vec{u}) + \int_{-\infty}^{\infty} d\tau \mathcal{P}_1(t-\tau) \left( \mathcal{S}(\tau) \cdot \mathcal{P}_0(\vec{u})\right) }{\left|\det D\mathcal{Q}(t)(\vec{u})\right|} d\vec{u} \]

\subsubsection{Measurement: from Sensor Intensity to Data}

The sensor comprises a finite grid of \textit{pixels}, each of which induces shot noise, read noise, and discretization
(in space, time, and value) on the local sensor intensity. Specifically,
\[D_{ijk} = \max 0, \min 2^{16}\!-1, \left\lfloor\mathcal{L}_0+\mathcal{L}_1 \cdot \left(\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot \left(t_{k,1}-t_{k,0}\right) + \mathcal{E}\cdot\mathrm{Poisson}^{-1}\left[\mathcal{U}_{ijk},\int_{t_{k,0}}^{t_{k,1}} \int_{\alpha_{ij}} f_s(\vec{s})(t)\, d\vec{s}\,dt\right]\right)\right\rfloor\]
where $\alpha_{ij}$ is the area of $\vec{s}$-space occupied by pixel $i,j$, $\mathcal{U}_{ijk}\in(0,1)$ represents the shot noise, $\mathcal{E} \in (0,1)$ represents the quantum efficiency (average electrons per photon), $\mathcal{C}_{ij}\in\Re_{\ge 0}$ is the dark current of pixel $i,j$ (electrons per second), $\mathcal{R}_{ijk} \in \Re$ represents the read noise, $\mathcal{L} \in \Re_{\ge 0}^2$ is a linear model of the amplifier and A/D, and $D_{ijk} \in 2^{16}$ is a datum (the value of pixel $i,j$ in frame $k$). Further, we assume the priors $\mathcal{R}_{ijk} \sim \mathrm{Gaussian}(0, \mathcal{R}_{ij})$, $\mathcal{R}_{ij} \sim \mbox{L\'evy}(\mathcal{R}_\mu,\mathcal{R}_c)$ ($\mathcal{R}_\mu\approx 0.8$, $\mathcal{R}_c\approx 0.4$ estimated from Andor Neo datasheet), and $\mathcal{U}_{ijk} \sim \mathrm{Uniform}(0,1)$.

\subsection{Induced Constraints}

Given a full complement of $\{D_{ijk}\}$ (where $i \in W$, $j \in H$, and $k \in T$), what conclusions can we draw? Let us begin by introducing new variables $I_{ijk} \in \Re_{\ge 0}$ (the intensity at pixel $(i,j)$ integrated over frame $k$) and $G_{ijk}\in\mathbb{N}$ (the number of photons actually detected within pixel $i,j$ during the interval of frame $k$) to eliminate the special function $\mathrm{Poisson}^{-1}$:
\[D_{ijk} = \max 0, \min 2^{16}\!-1, \left\lfloor\mathcal{L}_0+\mathcal{L}_1 \cdot \left(\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot \left(t_{k,1}-t_{k,0}\right) + \mathcal{E}\cdot G_{ijk}\right)\right\rfloor\]
%\[\frac{1}{G_{ijk}!} \int_{I_{ijk}}^{\infty} t^{G_{ijk}} e^{-t} dt \leq \mathcal{U}_{ijk} < \frac{1}{(G_{ijk}+1)!} \int_{I_{ijk}}^{\infty} t^{G_{ijk} + 1} e^{-t} dt \]
\[e^{-I_{ijk}} \sum_{z=0}^{G_{ijk} - 1} \frac{\left(I_{ijk}\right)^z}{z!} \leq \mathcal{U}_{ijk} < e^{-I_{ijk}} \sum_{z=0}^{G_{ijk}} \frac{\left(I_{ijk}\right)^z}{z!}\]
\[\int_{t_{k,0}}^{t_{k,1}} \int_{\alpha_{ij}} f_s(\vec{s})(t)\, d\vec{s}\,dt=I_{ijk}\]

The first constraint can be solved for the unknowns in $ijk$, $G_{ijk}$ and $\mathcal{R}_{ijk}$ (here assuming $D_{ijk}$ is neither 0 or its maximum value):
\[D_{ijk} \leq \mathcal{L}_0+\mathcal{L}_1 \cdot \left(\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot \left(t_{k,1}-t_{k,0}\right) + \mathcal{E}\cdot G_{ijk}\right) < D_{ijk} + 1\]
\[\frac{D_{ijk}-\mathcal{L}_0}{\mathcal{L}_1} \leq\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot \left(t_{k,1}-t_{k,0}\right) + \mathcal{E}\cdot G_{ijk} < \frac{D_{ijk} - \mathcal{L}_0}{\mathcal{L}_1} + \frac{1}{\mathcal{L}_1}\]
%\[\frac{D_{ijk}-\mathcal{L}_0}{\mathcal{E}\mathcal{L}_1} - \frac{\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot (t_{k,1} - t_{k,0})}{\mathcal{E}} \leq G_{ijk} < \frac{D_{ijk} - \mathcal{L}_0}{\mathcal{E}\mathcal{L}_1} - \frac{\mathcal{R}_{ijk} + \mathcal{C}_{ij} \cdot (t_{k,1} - t_{k,0})}{\mathcal{E}} + \frac{1}{\mathcal{E}\mathcal{L}_1}\]
\[\frac{D_{ijk}-\mathcal{L}_0 - \mathcal{L}_1 \mathcal{C}_{ij} \cdot (t_{k,1} - t_{k,0})}{\mathcal{E}\mathcal{L}_1} \leq G_{ijk} + \frac{\mathcal{R}_{ijk}}{\mathcal{E}} < \frac{D_{ijk}-\mathcal{L}_0 - \mathcal{L}_1 \mathcal{C}_{ij} \cdot (t_{k,1} - t_{k,0})}{\mathcal{E}\mathcal{L}_1} + \frac{1}{\mathcal{E}\mathcal{L}_1}\]

\end{document}