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commit 9a18c160185713d1011d66c527c277e95198d9f3 2 parents 743a52f + 888ef4d
@shababo shababo authored
Showing with 34 additions and 1 deletion.
  1. BIN  proposal/main.pdf
  2. +34 −1 proposal/main.tex
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35 proposal/main.tex
@@ -128,10 +128,42 @@ \section{Introduction}
\subsection{Formal Model}
+We extend and synthesize a parametric generative model proposed by
+\citep{mishchencko2011} of joint spike trains on $N$ neurons in
+discrete time, together with a method to infer the structure of
+dynamic bayesian networks proposed by \citep{patnaik2011}. Mischencko
+et al. propose a model to infer the connectivity matrix $W$, where
+each entry $w_{ij}$ encodes the influence of neuron $i$ on subsequent
+firing of neuron $j$, given the history of directly from calcium
+flourescence data. Their model can be decomposed into one part
+inferring neural spike train data from flourescence imaging, and
+another part inferring $W$ from spike train data. We focus on the
+Denote by $ n_i(t) $ whether neuron $i$ fired at time $t$. We observe
+fired neurons, so we set $n_i(t) = 1$ when we observe in the spike
+train that neuron $i$ fires at time $t$. We model $n_i(T)$ as a
+binary random variable with parameter $f(J_i(t))$, where
+\begin{equation}\label{J} J_i(t) = b_i + I_i(t) + \sum_{j=1}^{N}
+w_{ij}h_{ij}(t) \end{equation} where $b_i$ is a baseline, $I_i(t)$
+accounts for indirect contributions to neuron $i$ from a fixed window of
+past timeslices (discussed below). The history term, $h_{ij}$, encodes
+the influence of neuron $i$ on neuron $j$ from one earlier timeslice.
+From \citep{mishchencko2011}, we model $h$ as an autoregressive function of
+$n_j(t)$: \begin{equation}\label{h} h_{ij}(t) = (1-\Delta/\tau_{ij}^h)h_{ij}(t-\Delta)
+ + n_j(t-\Delta)+\sigma_{ij}^h\sqrt{\Delta}\epsilon_{ij}^h(t) \end{equation}
+where $ \tau_{ij}^h $ is the decay time constant, $\sigma_{ij}^h$ is the
+standard deviation of the noise and $\epsilon_{ij}^h$ is a standard
+normal random variable representing noise.
+Following \citep{mishchencko2011}, we define $f$ to ensure that log likelihood of
+spiking and correct scaling with respect to timeslice granularity: \begin{equation}
+\label{f} f(J) = P\left(n>0 | n \sim \text{Poiss}(e^J\Delta)\right) = 1 - \exp(-e^J\Delta) \end{equation}
\subsection{Testing Method}
+We will test on spike trains of 87 PFC neurons that were simultaneously recorded during a single 40-minute behaving session.
+<<<<<<< HEAD
The model as presented thus far only takes into account first-order interactions, that is, it only infers connectivity between two neurons over the course of one discrete time step. Given that the data only represents only a subset of the microcircuit, there are many inputs being ignored. We propose modeling these indirect inputs simply by creating and indirect input to each neuron at each time step which is a weighted sum of inputs from all neurons at each time step from $t-2$ to $t-S$, $S$ being the maximum number of time steps within which $n_j(t)$ can have an indirect influence on $n_i(t)$.
These higher order interactions are incorporated into the summed input of neuron $n_i(t)$ by adding the term,
@@ -145,6 +177,7 @@ \subsection{Testing Method}
to the input function $J_i(t)$. Here, $s$ is the number of time steps back, $\Delta$ is the temporal resolution of the model, and $\beta_{ijs}$ is the weight of the indirect influence of $n_j(t)$ on $n_i$.
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