From eee883459d13f43494a67279fae05695996d656e Mon Sep 17 00:00:00 2001 From: jovo Date: Tue, 11 Aug 2009 08:35:58 -0700 Subject: [PATCH] took liams revision, and revised --- .DS_Store | Bin 12292 -> 12292 bytes paper/Archive.zip | Bin 2196829 -> 2494246 bytes paper/abstract.tex | 0 paper/algorithmicx.sty | 0 paper/algpseudocode.sty | 0 paper/discussion.tex | 26 +- paper/hypernat.sty | 0 paper/intro.tex | 10 +- paper/main.aux | 174 +++++++------ paper/main.bbl | 102 ++++++++ paper/main.blg | 100 +++----- paper/main.log | 304 ++++++++-------------- paper/main.pdf | Bin 2216501 -> 2218365 bytes paper/main.tex | 5 +- paper/main.toc | 31 ++- paper/methods.tex | 551 +++++++++++++++++++++++++++++----------- paper/results.tex | 55 ++-- 17 files changed, 796 insertions(+), 562 deletions(-) mode change 100644 => 100755 paper/abstract.tex mode change 100644 => 100755 paper/algorithmicx.sty mode change 100644 => 100755 paper/algpseudocode.sty mode change 100644 => 100755 paper/discussion.tex mode 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$h_j$ vs. $h_{ij}$ reduces dimensionality of hidden space to $O(N)$ vs. $O(N^2)$, leading to our approach scaling well as $N \rightarrow$ big -\item $P(A|BC)\neq P(A|B)P(A|C)$ unless $B$ and $C$ are uncorrelated. we use this property, correlation coefficient doesn't, cross-correlations, etc., do not. -\item for the same reason, hidden neurons might be a problem for us. faster imaging, etc., should help alleviate that. -\item we can measure uncertainty in estimates, and use photo-stimulation to activate/deactivate small groups of neurons efficiently to help reduce variance in uncertainty. -\item certainly the issue of common input and unobserved neurons should be addressed a bit more in the discussion. -\item we can also talk a bit more about future directions involving fully-bayesian inference of the parameters (ie, MCMC over the parameter space, not just over the hidden variables X) instead of the MAP approach we took here; i tried to set this up a bit in the methods. -\item other important future work to mention: real data, faster mcmc sampling methods, dealing with nonstationarities (e.g. bleaching). -\item comparison with other work (e.g., duane's, Garofalo, Vakorina, transfer entropy, granger causality, etc.) -\end{itemize} +% \paragraph{j's outline of discussion} +% +% \begin{itemize} +% \item summary of main point: using sparse prior, we can recover a large fraction of variance of connection weights, assuming reasonable SNR, parameters, and imaging rate. +% \item $h_j$ vs. $h_{ij}$ reduces dimensionality of hidden space to $O(N)$ vs. $O(N^2)$, leading to our approach scaling well as $N \rightarrow$ big +% \item $P(A|BC)\neq P(A|B)P(A|C)$ unless $B$ and $C$ are uncorrelated. we use this property, correlation coefficient doesn't, cross-correlations, etc., do not. +% \item for the same reason, hidden neurons might be a problem for us. faster imaging, etc., should help alleviate that. +% \item we can measure uncertainty in estimates, and use photo-stimulation to activate/deactivate small groups of neurons efficiently to help reduce variance in uncertainty. +% \item certainly the issue of common input and unobserved neurons should be addressed a bit more in the discussion. +% \item we can also talk a bit more about future directions involving fully-bayesian inference of the parameters (ie, MCMC over the parameter space, not just over the hidden variables X) instead of the MAP approach we took here; i tried to set this up a bit in the methods. +% \item other important future work to mention: real data, faster mcmc sampling methods, dealing with nonstationarities (e.g. bleaching). +% \item comparison with other work (e.g., duane's, Garofalo, Vakorina, transfer entropy, granger causality, etc.) +% \end{itemize} % \paragraph{scale bias discussion} % diff --git a/paper/hypernat.sty b/paper/hypernat.sty old mode 100644 new mode 100755 diff --git a/paper/intro.tex b/paper/intro.tex index c5ecb66..6b5dae9 100755 --- a/paper/intro.tex +++ b/paper/intro.tex @@ -1,11 +1,5 @@ Since Ramon y Cajal discovered that the brain is a rich and dense \emph{network} of neurons \cite{RamonyCajal04,RamonyCajal23}, neuroscientists have been intensely curious about the details of these networks, which are believed to be the biological substrate for memory, cognition, and perception. While we have learned a great deal in the last century about ``macro-circuits'' --- the connectivity between coarsely-defined brain areas --- relatively little is known about ``micro-circuits,'' i.e., the connectivity within populations of neurons at a fine-grained cellular level. Broadly, one can imagine two complementary strategies for inferring microcircuit connectivity: anatomical and functional. Anatomical approaches to inferring circuitry include any strategy that does not consider physiological measurements of neural activity: for example, recently developed technologies including array tomography \cite{MichevaSmith07}, ``brainbow'' mice \cite{Brainbow07}, and serial electron microscopy \cite{Briggman2006} are rapidly improving and show great promise. Our work, on the other hand, takes a functional approach: our aim is to infer microcircuit properties by observing the activity of a population of neurons. -Experimental tools that enable approximately simultaneous observations of the activity of many (e.g., $O(10^3)$) neurons are now widely available. While arrays of extracellular electrodes have been exploited for this purpose, the arrays most often used \emph{in vivo} are inadequate for inferring monosynaptic connectivity, as the inter-electrode spacing is typically too large to record from all of the neurons in a given volume \cite{HATS98,HARR03,Stein04,Santhanam06,Harris07} \footnote{It is worth noting, however, that multielectrode arrays which have been recently developed for use in the retina are capable of much denser sampling \cite{Berry2004,Litke2004,Petrusca07,PILL07}.}. Alternately, calcium-sensitive fluorescent indicators allow us to observe the spiking activity of many neighboring neurons \cite{Tsien89}, which are more likely to be connected \cite{Abeles91, Braitenberg1998}. Some organic dyes achieve sufficiently high signal-to-noise ratios (SNR) that individual action potentials (spikes) may be resolved \cite{ImagingManual}, and bulk-loading techniques enable experimentalists to simultaneously fill populations of neurons with such dyes \cite{StosiekKonnerth03}. In addition, genetically encoded calcium indicators are under rapid development in a number of groups, and are approaching SNR levels of nearly single spike accuracy as well \cite{WallaceHasan08}. Microscopy technologies for collecting fluorescence signals are also rapidly developing. Cooled CCDs for wide-field imaging (either epifluorscence or confocal) now achieve a quantum efficiency of $\approx 90 \%$ with frame rates up to $60$ Hz or greater, depending on the width of the field of view \cite{Djurisic04}. For in vivo work, 2-photon laser scanning microscopy can achieve similar frame rates, using acoustic-optical deflectors to focus light at arbitrary locations in three-dimensional space \cite{ReddySaggau05, Iyer06, SalomeBourdieu06, ReddySaggau08}, or using resonant scanners \cite{NguyenParker01}. Together, these experimental tools can provide movies of calcium fluorescence transients for small populations of neurons (e.g., $O(10^2)$), with ``reasonable'' SNR, at imaging frequencies of 30 Hz or greater, in both the in vitro and in vivo settings. +Experimental tools that enable approximately simultaneous observations of the activity of many (e.g., $O(10^3)$) neurons are now widely available. While arrays of extracellular electrodes have been exploited for this purpose, the arrays most often used \emph{in vivo} are inadequate for inferring monosynaptic connectivity, as the inter-electrode spacing is typically too large to record from all of the neurons in a given volume \cite{HATS98,HARR03,Stein04,Santhanam06,Harris07} \footnote{It is worth noting, however, that multielectrode arrays which have been recently developed for use in the retina are capable of much denser sampling \cite{Berry2004,Litke2004,Petrusca07,PILL07}.}. Alternately, calcium-sensitive fluorescent indicators allow us to observe the spiking activity of many neighboring neurons \cite{Tsien89}, which are more likely to be connected \cite{Abeles91,Braitenberg1998}. Some organic dyes achieve sufficiently high signal-to-noise ratios (SNR) that individual action potentials (spikes) may be resolved \cite{ImagingManual}, and bulk-loading techniques enable experimentalists to simultaneously fill populations of neurons with such dyes \cite{StosiekKonnerth03}. In addition, genetically encoded calcium indicators are under rapid development in a number of groups, and are approaching SNR levels of nearly single spike accuracy as well \cite{WallaceHasan08}. Microscopy technologies for collecting fluorescence signals are also rapidly developing. Cooled CCDs for wide-field imaging (either epifluorscence or confocal) now achieve a quantum efficiency of $\approx 90 \%$ with frame rates up to $60$ Hz or greater, depending on the width of the field of view \cite{Djurisic04}. For in vivo work, 2-photon laser scanning microscopy can achieve similar frame rates, using acoustic-optical deflectors to focus light at arbitrary locations in three-dimensional space \cite{ReddySaggau05,Iyer06,SalomeBourdieu06,ReddySaggau08}, or using resonant scanners \cite{NguyenParker01}. Together, these experimental tools can provide movies of calcium fluorescence transients for small populations of neurons (e.g., $O(10^2)$), with ``reasonable'' SNR, at imaging frequencies of 30 Hz or greater, in both the in vitro and in vivo settings. -Given these experimental advances in functional neural imaging, our goal is to develop efficient computational and statistical methods to exploit this data for the analysis of neural connectivity. See Fig.~\ref{fig:data_schematic} for a schematic overview. One major challenge is that calcium transients due to action potentials decay about an order of magnitude more slowly than the time course of the underlying neural activity \cite{ImagingManual}. Thus, to properly analyze the functional network connectivity in this setting, we must incorporate methods for effectively deconvolving the observed noisy fluorescence signal to obtain estimates of the underlying spiking rates \cite{YaksiFriedrich06,GreenbergKerr08,Vogelstein2009}. To this end we introduce a coupled Markovian state-space model that relates the observed variables (fluorescence traces from the neurons in the microscope's field of view) to the hidden variables (spike trains and intracellular calcium concentrations of these neurons), as governed by a set of biophysical parameters including the network connectivity matrix. Given this model, we derive a Monte Carlo Expectation Maximization (MCEM) algorithm for approximating the maximum a posteriori estimates of the parameters of interest. Standard Monte Carlo sampling procedures (e.g., Gibbs sampling or sequential Monte Carlo) are inadequate in this setting, due to the high dimensionality and non-linear, non-Gaussian dynamics of the hidden variables in our model; we therefore develop a specialized blockwise-Gibbs approach to overcome these obstacles. This strategy enables us to accurately infer the functional connection matrix from moderately-sized simulated neural populations, under realistic assumptions about the dynamics and observation parameters. We describe our approach below, along with several methods for improving its computational speed and statistical efficiency. - - -\begin{figure}[t!] -\centering \includegraphics[width=\hsize]{../figs/yuri-paper-schematic} -\caption{Schematic overview. The raw observed data is a large-scale calcium fluorescence movie, which is pre-processed to correct for movement artifacts (in the in vivo setting) and find regions-of-interest (i.e., putative neurons); note that we have omitted details of these important preprocessing steps in this paper. Given the fluorescence traces from each neuron, we estimate the underlying spike trains (i.e., time series of neural activity) using statistical deconvolution methods. Then we estimate the parameters of a network model, given the observed data. Our major goal is to obtain an accurate estimate of the network connectivity matrix, which summarizes the information we are able to infer about the local neuronal microcircuit.} -\label{fig:data_schematic} \end{figure} +Given these experimental advances in functional neural imaging, our goal is to develop efficient computational and statistical methods to exploit this data for the analysis of neural connectivity. See Fig.~\ref{fig:data_schematic} for a schematic overview. One major challenge is that calcium transients due to action potentials decay about an order of magnitude more slowly than the time course of the underlying neural activity \cite{ImagingManual}. Thus, to properly analyze the functional network connectivity in this setting, we must incorporate methods for effectively deconvolving the observed noisy fluorescence signal to obtain estimates of the underlying spiking rates \cite{YaksiFriedrich06,GreenbergKerr08,Vogelstein2009}. To this end we introduce a coupled Markovian state-space model that relates the observed variables (fluorescence traces from the neurons in the microscope's field of view) to the hidden variables (spike trains and intracellular calcium concentrations of these neurons), as governed by a set of biophysical parameters including the network connectivity matrix. Given this model, we derive a Monte Carlo Expectation Maximization (MCEM) algorithm for approximating the maximum a posteriori estimates of the parameters of interest. Standard Monte Carlo sampling procedures (e.g., Gibbs sampling or sequential Monte Carlo) are inadequate in this setting, due to the high dimensionality and non-linear, non-Gaussian dynamics of the hidden variables in our model; we therefore develop a specialized blockwise-Gibbs approach to overcome these obstacles. This strategy enables us to accurately infer the functional connection matrix from moderately-sized simulated neural populations, under realistic assumptions about the dynamics and observation parameters. We describe our approach below, along with several methods for improving its computational speed and statistical efficiency. \ No newline at end of file diff --git a/paper/main.aux b/paper/main.aux index 1751419..e30cfb7 100644 --- a/paper/main.aux +++ b/paper/main.aux @@ -6,140 +6,150 @@ \citation{HATS98,HARR03,Stein04,Santhanam06,Harris07} \citation{Berry2004,Litke2004,Petrusca07,PILL07} \citation{Tsien89} -\citation{Abeles91, Braitenberg1998} +\citation{Abeles91,Braitenberg1998} \citation{ImagingManual} \citation{StosiekKonnerth03} \citation{WallaceHasan08} \citation{Djurisic04} -\citation{ReddySaggau05, Iyer06, SalomeBourdieu06, ReddySaggau08} +\citation{ReddySaggau05,Iyer06,SalomeBourdieu06,ReddySaggau08} \citation{NguyenParker01} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}} \newlabel{intro}{{1}{2}} \citation{ImagingManual} \citation{YaksiFriedrich06,GreenbergKerr08,Vogelstein2009} -\citation{BRIL88, CSK88, BRIL92, PG00, PILL07, PAN03d, PAN04c, Rigat06, TRUC05,NYK06,KP06,Vidne08,Stevenson2009} +\citation{BRIL88,CSK88,BRIL92,PG00,PILL07,PAN03d,PAN04c,Rigat06,TRUC05,NYK06,KP06,Vidne08,Stevenson2009} \@writefile{toc}{\contentsline {section}{\numberline {2}Methods}{3}} \newlabel{sec:methods}{{2}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Model}{3}} \newlabel{sec:methods:markov-setup}{{2.1}{3}} -\citation{PAN04c} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Schematic overview. 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Our major goal is to obtain an accurate estimate of the network connectivity matrix, which summarizes the information we are able to infer about the local neuronal microcircuit.}}{4}} -\newlabel{fig:data_schematic}{{1}{4}} -\newlabel{eqn:glm:definition}{{1}{4}} -\citation{LD89,PAN04c} +\citation{PAN04c,Escola07} \citation{Escola07} +\citation{LD89,PAN04c} \citation{Vogelstein2009} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A plot of the firing rate nonlinearity $f(J)$ used in our simulations; see Eq.\nobreakspace {}1\hbox {}. Note that the firing rate saturates at $1/\Delta $, because of our Bernoulli assumption (i.e., the spike count per bin is at most one); $\Delta = (60$ Hz$)^{-1}$ here.}}{4}} +\newlabel{fig:egfluor}{{1}{4}} +\newlabel{eqn:glm:definition}{{1}{4}} +\newlabel{eqn:h:definition}{{3}{4}} \citation{ImagingManual} \citation{Vogelstein2009} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Given our model, Eq. 1\hbox {}, the relationship between firing rate and the magnitude of the input to a neuron is highly nonlinear. 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XXX CAN WE INDENT THE BELOW PROPERLY? WOULD MAKE IT MORE LEGIBLE. ALSO, IT IS NOT EXACTLY CORRECT NOW XXX}}{12}} -\newlabel{eqn:pseudocode}{{1}{12}} -\@writefile{toc}{\contentsline {section}{\numberline {3}Results}{12}} -\newlabel{sec:results}{{3}{12}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulating neural activity in a neural population}{12}} -\newlabel{sec:results:simulations}{{3.1}{12}} +\@writefile{toc}{\contentsline {section}{\numberline {3}Results}{11}} +\newlabel{sec:results}{{3}{11}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Simulating neural activity in a neural population}{11}} +\newlabel{sec:results:simulations}{{3.1}{11}} +\citation{Braitenberg1998,Urquijo2000} +\citation{Braitenberg1998,Lefort2009} +\citation{Lefort2009,Sayer1990} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Schematic overview. The raw observed data is a large-scale calcium fluorescence movie, which is pre-processed to correct for movement artifacts (in the in vivo setting) and find regions-of-interest (i.e., putative neurons); note that we have omitted details of these important preprocessing steps in this paper. Given the fluorescence traces from each neuron, we estimate the underlying spike trains (i.e., time series of neural activity) using statistical deconvolution methods. Then we estimate the parameters of a network model, given the observed data. Our major goal is to obtain an accurate estimate of the network connectivity matrix, which summarizes the information we are able to infer about the local neuronal microcircuit. This figure adapted from personal communications with Rafael Yuste, Brendon O.\ Watson, and Adam Packer.}}{12}} +\newlabel{fig:data_schematic}{{2}{12}} +\citation{} +\citation{Abeles01} +\citation{Koch99} \citation{Sayer1990} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Schematic overview. First we obtain large-scale calcium recordings of spontaneous and evoked neural responses. These movies are then pre-processed to correct for movement artifacts, find regions-of-interest, and extract spike trains. Second, given the fluorescence traces from each neuron, we infer spike trains. Third, we use our population model to infer the most likely functional connectivity matrix, which is our approximation of the microcircuit.}}{13}} -\newlabel{fig:data_schematic}{{3}{13}} -\newlabel{eqn:convert}{{15}{13}} +\citation{Koch99} \citation{Vogelstein2009} -\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Table of simulation parameters. $\@mathcal {E}(\lambda )$ indicates an exponential distribution with $\lambda $, $\@mathcal {U}[a,b]$ indicates a uniform distribution between $a$ and $b$, and $\@mathcal {TN}[x]$ indicates a truncated normal with mean $x$ and variance and lower bound of $30 \%$ the mean. XXX Y: can you fix this to look nice? I think the line for Ca jump is good now, assuming we explain that the gaussian is truncated at 2 std below mean (or something like that) XXX}}{14}} +\citation{ImagingManual,HelmchenSakmann96,BrenowitzRegehr07} +\newlabel{eqn:convert}{{15}{13}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Inferring functional connectivity from the simulated calcium imaging data}{13}} +\newlabel{sec:results:inference}{{3.2}{13}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Scale bias in inferred connection weights due to coarse time discretization of calcium imaging data}{14}} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Table of simulation parameters. $\@mathcal {E}(\lambda )$ indicates an exponential distribution with $\lambda $, and $\@mathcal {N}(\mu ,\sigma ^2)$ indicates a truncated normal with mean $\mu $ and variance $\sigma ^2$, and lower bound of one standard deviation below the mean.}}{14}} \newlabel{table:caparm}{{1}{14}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Inferring functional connectivity from the simulated calcium imaging data}{14}} -\newlabel{sec:results:inference}{{3.2}{14}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Two in vivo fluorescence traces (top) and their corresponding inferred spike trains (bottom). The left panels show a high SNR case, that is sufficient quality for the factorization approximation to work nearly as well as the embedded-chain-within-Gibbs approach. The right panels show a low SNR case, in which the factorization approach is insufficient.}}{15}} -\newlabel{fig:example_traces}{{4}{15}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Scale bias in inferred connection weights due to coarse time discretization of calcium imaging data}{15}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Functional connectivity matrix can be reconstructed from calcium imaging data. In the upper panels inferred connection weights are shown in a scatter plot versus real connection weights, with inference performed using factorized approximation algorithm, exact embedded-chain-within-blockwise-Gibbs approach, and original spike trains observed at the frame rate of the calcium imaging. Network of $N=25$ neurons was used, firing at $\approx 5$ Hz, and imaged for T=600 sec at intermediate SNR (photon budget 10Kph/neuron/frame, see below). $r^2=0.47$ for factorized approximation algorithm was found, $r^2=0.48$ for embedded-chain-within-blockwise-Gibbs approach, and $r^2=0.57$ for the original spike trains. Thus, factorized approximation produced results almost as accurate as the exact embedded-chain-within-blockwise-Gibbs approach, and almost as accurate as the original spikes. Inferred connectivity weights (upper left) were scaled with respect to true connectivity by a constant amount due to time discretization bias (see below); other than scale, inferred connectivity represented the true connectivity matrix very well (upper right). Thus, calcium imaging is sufficient to identify connected pairs of neurons reliably.}}{16}} -\newlabel{fig:scatters}{{5}{16}} \citation{VogPan09} -\citation{NguyenParker01, ReddySaggau05, Iyer06, SalomeBourdieu06, ReddySaggau08} +\citation{NguyenParker01,ReddySaggau05,Iyer06,SalomeBourdieu06,ReddySaggau08} \citation{Vogelstein2009} -\newlabel{eqn:bias}{{17}{17}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The low-frame rate of calcium imaging can explain the observed scale bias from Figure 5\hbox {}. Our theoretical scale bias is determined by considering what fraction of spikes would fall within a single image frame, which is a function of $\Delta $ (c.f. Eq. 17\hbox {}). The center of each square indicates the mean scale bias of 5 simulations, the extent of each square indicates one standard deviation, and the errorbars indicate the $95$th and $5$th percentiles. XXX Y: can you correct these details? XXX}}{17}} -\newlabel{fig:bias}{{6}{17}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Impact of different imaging frame rates, noise levels, and durations on the estimator accuracy}{17}} -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Accuracy of the inferred connectivity weights as function of the frame rate of calcium imaging. Connectivity matrix here was inferred from the original spike trains observed at corresponding frame rates, thus establishing the upper performance bound for inference using calcium imaging data. A network of $N=25$ neurons, firing at $\approx 5$ Hz and imaged for $T=600$ sec was used.}}{18}} -\newlabel{fig:recvar}{{7}{18}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Accuracy of the estimates and Fisher information matrix}{18}} -\newlabel{sec:methods:accuracy_Fisher}{{3.5}{18}} -\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Accuracy of inferred connectivity weights as function of the noise amount in the calcium imaging data, as quantified by experimental photon budget per neuron-frame, for frame rates of 15 Hz, 33 Hz and 66 Hz. Photon counts on the order of 20-40 Kph/frame/neuron are required to achieve the upper bound due by the frame rate. Connectivity matrix here was inferred from simulated fluorescence data using factorized approximation algorithm. Simulation conditions are the same as in Figure 7\hbox {}.}}{19}} -\newlabel{fig:recvar-SNR}{{8}{19}} -\newlabel{eqn:fisher}{{19}{19}} -\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Accuracy of inferred connectivity weights as function of the imaging time and neural population size. Incorporating simple priors such as exponential prior on the connectivity weights allows to boost reconstruction accuracy dramatically (dashed lines). In this latter case, $T=300$ sec is already sufficient to recover 70\% of the variance in the connection weights. Incorporating Dale's prior leads to only marginal improvement (dotted line). As shown in the methods, reconstruction accuracy does not depend on the neural population size $N$. Here, neural population from $N=10$ to $N=200$ were simulated for different $T$, where $N=200$ (gray) and $N=100$ (black) are shown. All networks were prepared in similar state by adjusting strength of inhibitory connections to achieve similar mean firing rate $\approx 5$ Hz, although actual firing rate in these networks could vary. In all cases, $T=5$ min - 0.5 hour is sufficient to produce accurate reconstructions. }}{20}} -\newlabel{fig:recvar-NT}{{9}{20}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.6}Impact of using priors on the inference}{20}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.7}Impact of strong correlations and deviations from generative model on the inference}{20}} -\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Incorporating simple priors on the distribution of connectivity weights in the Bayesian inference algorithm, such as exponential sparseness prior, is essential to achieve much more accurate reconstructions than using simple GLM from a smaller amount of calcium imaging data. Here, connection weights reconstructed using simple GLM (left panel) or sparse-prior GLM (right panel) are shown in a scatter plot for a network of $N=50$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ sec. $r^2=0.64$ for simple GLM solution and $r^2=0.85$ for sparse-GLM solution.}}{21}} -\newlabel{fig:sparse}{{10}{21}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Two in vivo fluorescence traces (top) and their corresponding inferred spike trains (bottom). The left panels show a high SNR case, that is sufficient quality for the factorized approximation to work nearly as well as the embedded-chain-within-Gibbs approach. The right panels show a low SNR case, in which the factorized approach is insufficient. The ordinate is in arbitrary units. Data from the laboratory of Tom Mrsic-Flogel.}}{15}} +\newlabel{fig:example_traces}{{3}{15}} +\newlabel{eqn:bias}{{17}{15}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Functional connectivity matrix can be reconstructed from calcium imaging data. In the upper panels inferred connection weights are shown in a scatter plot versus real connection weights, with inference performed using factorized approximation algorithm, exact embedded-chain-within-blockwise-Gibbs approach, and original spike trains observed at the frame rate of the calcium imaging. Network of $N=25$ neurons was used, firing at $\approx 5$ Hz, and imaged for T=600 sec at intermediate SNR (photon budget 10Kph/neuron/frame, see below). $r^2=0.47$ for factorized approximation algorithm was found, $r^2=0.48$ for embedded-chain-within-blockwise-Gibbs approach, and $r^2=0.57$ for the original spike trains. Thus, factorized approximation produced results almost as accurate as the exact embedded-chain-within-blockwise-Gibbs approach, and almost as accurate as the original spikes. Inferred connectivity weights (upper left) were scaled with respect to true connectivity by a constant amount due to time discretization bias (see below); other than scale, inferred connectivity represented the true connectivity matrix very well (upper right). Thus, calcium imaging is sufficient to identify connected pairs of neurons reliably.}}{16}} +\newlabel{fig:scatters}{{4}{16}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Impact of different imaging frame rates, noise levels, and durations on the estimator accuracy}{16}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The low-frame rate of calcium imaging can explain the observed scale bias from Figure 4\hbox {}. Our theoretical scale bias is determined by considering what fraction of spikes would fall within a single image frame, which is a function of $\Delta $ (c.f. Eq. 17\hbox {}). The center of each square indicates the mean scale bias of 5 simulations, the extent of each square indicates one standard deviation, and the errorbars indicate the $95$th and $5$th percentiles. XXX Y: can you correct these details? XXX}}{17}} +\newlabel{fig:bias}{{5}{17}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Accuracy of the estimates and Fisher information matrix}{17}} +\newlabel{sec:methods:accuracy_Fisher}{{3.5}{17}} +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Accuracy of the inferred connectivity weights as function of the frame rate of calcium imaging. Connectivity matrix here was inferred from the original spike trains observed at corresponding frame rates, thus establishing the upper performance bound for inference using calcium imaging data. A network of $N=25$ neurons, firing at $\approx 5$ Hz and imaged for $T=600$ sec was used.}}{18}} +\newlabel{fig:recvar}{{6}{18}} +\newlabel{eqn:fisher}{{19}{18}} +\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Accuracy of inferred connectivity weights as function of the noise amount in the calcium imaging data, as quantified by experimental photon budget per neuron-frame, for frame rates of 15 Hz, 33 Hz and 66 Hz. Photon counts on the order of 20-40 Kph/frame/neuron are required to achieve the upper bound due by the frame rate. Connectivity matrix here was inferred from simulated fluorescence data using factorized approximation algorithm. Simulation conditions are the same as in Figure 6\hbox {}. Vertical black lines indicate noise levels for the two example traces shown in Figure 3\hbox {}, as determined using the SMC approach to estimate $\theta _i$ as described in Section 2.3\hbox {}.}}{19}} +\newlabel{fig:recvar-SNR}{{7}{19}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.6}Impact of using priors on the inference}{19}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.7}Impact of strong correlations and deviations from generative model on the inference}{19}} +\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Accuracy of inferred connectivity weights as function of the imaging time and neural population size. Incorporating simple priors such as exponential prior on the connectivity weights allows to boost reconstruction accuracy dramatically (dashed lines). In this latter case, $T=300$ sec is already sufficient to recover 70\% of the variance in the connection weights. Incorporating Dale's prior leads to only marginal improvement (dotted line). As shown in the methods, reconstruction accuracy does not depend on the neural population size $N$. Here, neural population from $N=10$ to $N=200$ were simulated for different $T$, where $N=200$ (gray) and $N=100$ (black) are shown. All networks were prepared in similar state by adjusting strength of inhibitory connections to achieve similar mean firing rate $\approx 5$ Hz, although actual firing rate in these networks could vary. In all cases, $T=5$ min - 0.5 hour is sufficient to produce accurate reconstructions. }}{20}} +\newlabel{fig:recvar-NT}{{8}{20}} +\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Incorporating simple priors on the distribution of connectivity weights in the Bayesian inference algorithm, such as exponential sparseness prior, is essential to achieve much more accurate reconstructions than using simple GLM from a smaller amount of calcium imaging data. Here, connection weights reconstructed using simple GLM (left panel) or sparse-prior GLM (right panel) are shown in a scatter plot for a network of $N=50$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ sec. $r^2=0.64$ for simple GLM solution and $r^2=0.85$ for sparse-GLM solution.}}{21}} +\newlabel{fig:sparse}{{9}{21}} +\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Distribution of inferred connection weights using simple GLM (left) and sparse GLM (right) vs true distributions. When sparse exponential prior on the distribution of connection weights is enacted, dispersion in inferred connection weights is substantially reduced and, in particular, it becomes possible to reliably determine which neural pairs are connected. Distributions are shown for a network of $N=200$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ s was used here.}}{22}} +\newlabel{fig:distros}{{10}{22}} +\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Diverseness of observed neural activity patterns is required for functional connectivity to give access to the actual ``anatomical'' structure of the neural circuit. Here, 15 sec of simulated spike trains for a weakly coupled network (upper-left) and a network with strongly coupled component (upper-right) are shown. In weakly coupled network spikes are sufficiently uncorrelated to give access to all different neural activity patterns needed to properly estimate true weights ${\bf w}_i$. In strongly coupled case, many instances of highly synchronous locked firings are evident, thus preventing observation of sufficiently rich ensemble of activity patterns. Accordingly, GLM solution for the strongly coupled neural network (lower-right) does not represent the true connectivity of the circuit, even for the weakly coupled circuit's component. This is contrary to the weakly-coupled network (lower-left) where true connectivity is successfully estimated. Networks of $N=50$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ sec were used to produce this figure.}}{23}} +\newlabel{fig:rasters}{{11}{23}} +\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Bayesian inference algorithm is robust to distortions of the underlying generative model. One distortion that should be expected is variability of the EPSP time courses from neuron to neuron, and possibly synapse to synapse. With up to 25\% variability allowed in EPSP time scales $\tau _w$ (right panel) our algorithm provided reconstructions of the same quality as when all $\tau _w$ were the same (left panel). Simulation conditions are the same as in Figure 6\hbox {}.}}{24}} +\newlabel{fig:vartau}{{12}{24}} \citation{Chlovksii} \citation{emory, liam, Garofalo, Vakorina, etc.} -\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Distribution of inferred connection weights using simple GLM (left) and sparse GLM (right) vs true distributions. When sparse exponential prior on the distribution of connection weights is enacted, dispersion in inferred connection weights is substantially reduced and, in particular, it becomes possible to reliably determine which neural pairs are connected. Distributions are shown for a network of $N=200$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ s was used here.}}{22}} -\newlabel{fig:distros}{{11}{22}} -\@writefile{toc}{\contentsline {section}{\numberline {4}Discussion}{22}} -\newlabel{sec:discussion}{{4}{22}} \citation{Duane 1 and 2, liam} \bibdata{mybib} -\@writefile{toc}{\contentsline {paragraph}{j's outline of discussion}{23}} +\@writefile{toc}{\contentsline {section}{\numberline {4}Discussion}{25}} +\newlabel{sec:discussion}{{4}{25}} \bibcite{Abeles91}{Abeles, 1991} \bibcite{Andrieu2007}{Andrieu et\nobreakspace {}al., 2007} \bibcite{BickelBengtsson08}{Bickel et\nobreakspace {}al., 2008} \bibcite{Binzegger04}{Binzegger et\nobreakspace {}al., 2004} \bibcite{CONV04}{Boyd and Vandenberghe, 2004} \bibcite{Braitenberg1998}{Braitenberg and Schuz, 1998} +\bibcite{BrenowitzRegehr07}{Brenowitz and Regehr, 2007} \bibcite{Briggman2006}{Briggman and Denk, 2006} \bibcite{BRIL88}{Brillinger, 1988} +\bibcite{BRIL92}{Brillinger, 1992} \bibcite{Buhl94}{Buhl et\nobreakspace {}al., 1994} \bibcite{Candes2005}{Candes and Romberg, 2005} +\bibcite{CSK88}{Chornoboy et\nobreakspace {}al., 1988} \bibcite{DLR77}{Dempster et\nobreakspace {}al., 1977} \bibcite{Djurisic04}{Djurisic et\nobreakspace {}al., 2004} \bibcite{DE03}{Donoho and Elad, 2003} @@ -149,10 +159,15 @@ \bibcite{Feldmeyer99}{Feldmeyer et\nobreakspace {}al., 1999} \bibcite{FeldmeyerSakmann00}{Feldmeyer and Sakmann, 2000} \bibcite{GDW04}{Godsill et\nobreakspace {}al., 2004} +\bibcite{Urquijo2000}{Gomez-Urquijo et\nobreakspace {}al., 2000} \bibcite{GreenbergKerr08}{Greenberg et\nobreakspace {}al., 2008} \bibcite{Gupta00}{Gupta et\nobreakspace {}al., 2000} \bibcite{HARR03}{Harris et\nobreakspace {}al., 2003} \bibcite{HATS98}{Hatsopoulos et\nobreakspace {}al., 1998} +\bibcite{HelmchenSakmann96}{Helmchen et\nobreakspace {}al., 1996} +\bibcite{Iyer06}{Iyer et\nobreakspace {}al., 2006} +\bibcite{Koch99}{Koch, 1999} +\bibcite{KP06}{Kulkarni and Paninski, 2007} \bibcite{Lefort2009}{Lefort et\nobreakspace {}al., 2009} \bibcite{LD89}{Li and Duan, 1989} \bibcite{Litke2004}{Litke et\nobreakspace {}al., 2004} @@ -165,16 +180,22 @@ \bibcite{NBR03}{Neal et\nobreakspace {}al., 2003} \bibcite{NG04}{Ng, 2004} \bibcite{NguyenParker01}{Nguyen et\nobreakspace {}al., 2001} +\bibcite{NYK06}{Nykamp, 2007} \bibcite{PAN04c}{Paninski, 2004} +\bibcite{PAN03d}{Paninski et\nobreakspace {}al., 2004} \bibcite{PetersenSakmann00}{Petersen and Sakmann, 2000} \bibcite{Petrusca07}{Petrusca et\nobreakspace {}al., 2007} \bibcite{PILL07}{Pillow et\nobreakspace {}al., 2008} +\bibcite{PG00}{Plesser and Gerstner, 2000} \bibcite{RAB89}{Rabiner, 1989} \bibcite{RamonyCajal04}{Ramon\nobreakspace {}y Cajal, 1904} \bibcite{RamonyCajal23}{Ramon\nobreakspace {}y Cajal, 1923} +\bibcite{ReddySaggau08}{Reddy et\nobreakspace {}al., 2008} \bibcite{ReddySaggau05}{Reddy and Saggau, 2005} \bibcite{Reyes98}{Reyes et\nobreakspace {}al., 1998} \bibcite{Rigat06}{Rigat et\nobreakspace {}al., 2006} +\bibcite{RC05}{Robert and Casella, 2005} +\bibcite{SalomeBourdieu06}{Salome et\nobreakspace {}al., 2006} \bibcite{Santhanam06}{Santhanam et\nobreakspace {}al., 2006} \bibcite{Sayer1990}{Sayer et\nobreakspace {}al., 1990} \bibcite{Berry2004}{Segev et\nobreakspace {}al., 2004} @@ -182,18 +203,17 @@ \bibcite{Song2005}{Song et\nobreakspace {}al., 2005} \bibcite{Stein04}{Stein et\nobreakspace {}al., 2004} \bibcite{Stevenson08}{Stevenson et\nobreakspace {}al., 2008} +\bibcite{Stevenson2009}{Stevenson et\nobreakspace {}al., 2009} \bibcite{StosiekKonnerth03}{Stosiek et\nobreakspace {}al., 2003} \bibcite{Thompson88}{Thompson et\nobreakspace {}al., 1988} \bibcite{Tibs96}{Tibshirani, 1996} \bibcite{TIP01}{Tipping, 2001} +\bibcite{TRUC05}{Truccolo et\nobreakspace {}al., 2005} \bibcite{Tsien89}{Tsien, 1989} +\bibcite{Vidne08}{Vidne et\nobreakspace {}al., 2009} \bibcite{Vogelstein2009}{Vogelstein et\nobreakspace {}al., 2009} \bibcite{WallaceHasan08}{Wallace et\nobreakspace {}al., 2008} \bibcite{YaksiFriedrich06}{Yaksi and Friedrich, 2006} \bibcite{Yasuda2004}{Yasuda et\nobreakspace {}al., 2004} \bibcite{ImagingManual}{Yuste et\nobreakspace {}al., 2006} \bibstyle{apalike} -\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Diverseness of observed neural activity patterns is required for functional connectivity to give access to the actual ``anatomical'' structure of the neural circuit. Here, 15 sec of simulated spike trains for a weakly coupled network (upper-left) and a network with strongly coupled component (upper-right) are shown. In weakly coupled network spikes are sufficiently uncorrelated to give access to all different neural activity patterns needed to properly estimate true weights ${\bf w}_i$. In strongly coupled case, many instances of highly synchronous locked firings are evident, thus preventing observation of sufficiently rich ensemble of activity patterns. Accordingly, GLM solution for the strongly coupled neural network (lower-right) does not represent the true connectivity of the circuit, even for the weakly coupled circuit's component. This is contrary to the weakly-coupled network (lower-left) where true connectivity is successfully estimated. Networks of $N=50$ neurons firing at $\approx 5$ Hz and imaged for $T=600$ sec were used to produce this figure.}}{28}} -\newlabel{fig:rasters}{{12}{28}} -\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Bayesian inference algorithm is robust to distortions of the underlying generative model. One distortion that should be expected is variability of the EPSP time courses from neuron to neuron, and possibly synapse to synapse. With up to 25\% variability allowed in EPSP time scales $\tau _w$ (right panel) our algorithm provided reconstructions of the same quality as when all $\tau _w$ were the same (left panel). Simulation conditions are the same as in Figure 7\hbox {}.}}{29}} -\newlabel{fig:vartau}{{13}{29}} diff --git a/paper/main.bbl b/paper/main.bbl old mode 100644 new mode 100755 index c3a3f33..a552e31 --- a/paper/main.bbl +++ b/paper/main.bbl @@ -33,6 +33,12 @@ Braitenberg, V. and Schuz, A. (1998). \newblock {\em Cortex: statistics and geometry of neuronal connectivity.} \newblock Springer, Berlin. +\bibitem[Brenowitz and Regehr, 2007]{BrenowitzRegehr07} +Brenowitz, S.~D. and Regehr, W.~G. (2007). +\newblock Reliability and heterogeneity of calcium signaling at single + presynaptic boutons of cerebellar granule cells. +\newblock {\em J Neurosci}, 27(30):7888--7898. + \bibitem[Briggman and Denk, 2006]{Briggman2006} Briggman, K.~L. and Denk, W. (2006). \newblock Towards neural circuit reconstruction with volume electron microscopy @@ -45,6 +51,11 @@ Brillinger, D. (1988). cells. \newblock {\em Biological Cyberkinetics}, 59:189--200. +\bibitem[Brillinger, 1992]{BRIL92} +Brillinger, D. (1992). +\newblock Nerve cell spike train data analysis: a progression of technique. +\newblock {\em Journal of the American Statistical Association}, 87:260--271. + \bibitem[Buhl et~al., 1994]{Buhl94} Buhl, E., Halasy, K., and Somogyi, P. (1994). \newblock Diverse sources of hippocampal unitary inhibitory postynaptic @@ -55,6 +66,11 @@ Buhl, E., Halasy, K., and Somogyi, P. (1994). Candes, E.~J. and Romberg, J. (2005). \newblock Practical signal recovery from random projections. +\bibitem[Chornoboy et~al., 1988]{CSK88} +Chornoboy, E., Schramm, L., and Karr, A. (1988). +\newblock Maximum likelihood identification of neural point process systems. +\newblock {\em Biological Cybernetics}, 59:265--275. + \bibitem[Dempster et~al., 1977]{DLR77} Dempster, A., Laird, N., and Rubin, D. (1977). \newblock Maximum likelihood from incomplete data via the {EM} algorithm. @@ -106,6 +122,13 @@ Godsill, S., Doucet, A., and West, M. (2004). \newblock Monte {C}arlo smoothing for non-linear time series. \newblock {\em Journal of the American Statistical Association}, 99:156--168. +\bibitem[Gomez-Urquijo et~al., 2000]{Urquijo2000} +Gomez-Urquijo, S.~M., Reblet, C., Bueno-Lopez, J.~L., and Gutierrez-Ibarluzea, + I. (2000). +\newblock Gabaergic neurons in the rabbit visual cortex: percentage, + distribution and cortical projections. +\newblock {\em Brain Res}, 862:171--9. + \bibitem[Greenberg et~al., 2008]{GreenbergKerr08} Greenberg, D.~S., Houweling, A.~R., and Kerr, J. N.~D. (2008). \newblock Population imaging of ongoing neuronal activity in the visual cortex @@ -129,6 +152,28 @@ Hatsopoulos, N., Ojakangas, C., Paninski, L., and Donoghue, J. (1998). of motor cortical neurons. \newblock {\em PNAS}, 95:15706--15711. +\bibitem[Helmchen et~al., 1996]{HelmchenSakmann96} +Helmchen, F., Imoto, K., and Sakmann, B. (1996). +\newblock {Ca}$^{2+}$ buffering and action potential-evoked {Ca}$^{2+}$ + signaling in dendrites of pyramidal neurons. +\newblock {\em Biophys J}, 70(2):1069--1081. + +\bibitem[Iyer et~al., 2006]{Iyer06} +Iyer, V., Hoogland, T.~M., and Saggau, P. (2006). +\newblock Fast functional imaging of single neurons using random-access + multiphoton ({RAMP}) microscopy. +\newblock {\em J Neurophysiol}, 95(1):535--545. + +\bibitem[Koch, 1999]{Koch99} +Koch, C. (1999). +\newblock {\em Biophysics of Computation}. +\newblock Oxford University Press. + +\bibitem[Kulkarni and Paninski, 2007]{KP06} +Kulkarni, J. and Paninski, L. (2007). +\newblock Common-input models for multiple neural spike-train data. +\newblock {\em Network: Computation in Neural Systems}, 18:375--407. + \bibitem[Lefort et~al., 2009]{Lefort2009} Lefort, S., Tomm, C., Floyd~Sarria, J.-C., and Petersen, C. C.~H. (2009). \newblock The excitatory neuronal network of the c2 barrel column in mouse @@ -205,12 +250,25 @@ Nguyen, Q.~T., Callamaras, N., Hsieh, C., and Parker, I. (2001). imaging. \newblock {\em Cell Calcium}, 30(6):383--393. +\bibitem[Nykamp, 2007]{NYK06} +Nykamp, D. (2007). +\newblock A mathematical framework for inferring connectivity in probabilistic + neuronal networks. +\newblock {\em Mathematical Biosciences}, 205:204--251. + \bibitem[Paninski, 2004]{PAN04c} Paninski, L. (2004). \newblock Maximum likelihood estimation of cascade point-process neural encoding models. \newblock {\em Network: Computation in Neural Systems}, 15:243--262. +\bibitem[Paninski et~al., 2004]{PAN03d} +Paninski, L., Fellows, M., Shoham, S., Hatsopoulos, N., and Donoghue, J. + (2004). +\newblock Superlinear population encoding of dynamic hand trajectory in primary + motor cortex. +\newblock {\em J. Neurosci.}, 24:8551--8561. + \bibitem[Petersen and Sakmann, 2000]{PetersenSakmann00} Petersen, C.~C. and Sakmann, B. (2000). \newblock The excitatory neuronal network of rat layer 4 barrel cortex. @@ -230,6 +288,12 @@ Pillow, J., Shlens, J., Paninski, L., Sher, A., Litke, A., Chichilnisky, E., neuronal population. \newblock {\em Nature}, 454:995--999. +\bibitem[Plesser and Gerstner, 2000]{PG00} +Plesser, H. and Gerstner, W. (2000). +\newblock Noise in integrate-and-fire neurons: From stochastic input to escape + rates. +\newblock {\em Neural Computation}, 12:367--384. + \bibitem[Rabiner, 1989]{RAB89} Rabiner, L. (1989). \newblock A tutorial on hidden {M}arkov models and selected applications in @@ -246,6 +310,12 @@ Ramon~y Cajal, S. (1923). \newblock {\em Recuerdos de mi vida: Historia de mi labor cientifica}. \newblock Alianza Editorial. +\bibitem[Reddy et~al., 2008]{ReddySaggau08} +Reddy, G., Kelleher, K., Fink, R., and Saggau, P. (2008). +\newblock {Three-dimensional random access multiphoton microscopy for + functional imaging of neuronal activity}. +\newblock {\em Nature Neuroscience}, 11(6):713--720. + \bibitem[Reddy and Saggau, 2005]{ReddySaggau05} Reddy, G.~D. and Saggau, P. (2005). \newblock Fast three-dimensional laser scanning scheme using acousto-optic @@ -264,6 +334,18 @@ Rigat, F., de~Gunst, M., and van Pelt, J. (2006). \newblock Bayesian modelling and analysis of spatio-temporal neuronal networks. \newblock {\em Bayesian Analysis}, 1:733--764. +\bibitem[Robert and Casella, 2005]{RC05} +Robert, C. and Casella, G. (2005). +\newblock {\em Monte {C}arlo Statistical Methods}. +\newblock Springer. + +\bibitem[Salome et~al., 2006]{SalomeBourdieu06} +Salome, R., Kremer, Y., Dieudonne, S., Leger, J.-F., Krichevsky, O., Wyart, C., + Chatenay, D., and Bourdieu, L. (2006). +\newblock Ultrafast random-access scanning in two-photon microscopy using + acousto-optic deflectors. +\newblock {\em Journal of Neuroscience Methods}, 154(1-2):161--174. + \bibitem[Santhanam et~al., 2006]{Santhanam06} Santhanam, G., Ryu, S.~I., Yu, B.~M., Afshar, A., and Shenoy, K.~V. (2006). \newblock A high-performance brain-computer interface. @@ -305,6 +387,13 @@ Stevenson, I., Rebesco, J., Hatsopoulos, N., Haga, Z., Miller, L., and \newblock Inferring network structure from spikes. \newblock {\em Statistical Analysis of Neural Data meeting}. +\bibitem[Stevenson et~al., 2009]{Stevenson2009} +Stevenson, I.~H., Rebesco, J.~M., Hatsopoulos, N.~G., Haga, Z., Miller, L.~E., + and Kording, K.~P. (2009). +\newblock Bayesian inference of functional connectivity and network structure + from spikes. +\newblock {\em IEEE Trans. Neural Systems and Rehab.}, 17:203--13. + \bibitem[Stosiek et~al., 2003]{StosiekKonnerth03} Stosiek, C., Garaschuk, O., Holthoff, K., and Konnerth, A. (2003). \newblock In vivo two-photon calcium imaging of neuronal networks. @@ -328,11 +417,24 @@ Tipping, M. (2001). \newblock Sparse {B}ayesian learning and the relevance vector machine. \newblock {\em Journal of Machine Learning Research}, 1:211--244. +\bibitem[Truccolo et~al., 2005]{TRUC05} +Truccolo, W., Eden, U., Fellows, M., Donoghue, J., and Brown, E. (2005). +\newblock A point process framework for relating neural spiking activity to + spiking history, neural ensemble and extrinsic covariate effects. +\newblock {\em Journal of Neurophysiology}, 93:1074--1089. + \bibitem[Tsien, 1989]{Tsien89} Tsien, R.~Y. (1989). \newblock Fluorescent probes of cell signaling. \newblock {\em Ann. Rev. Neurosci.}, 12:227--253. +\bibitem[Vidne et~al., 2009]{Vidne08} +Vidne, M., Kulkarni, J., Ahmadian, Y., Pillow, J., Shlens, J., Chichilnisky, + E., Simoncelli, E., and Paninski, L. (2009). +\newblock Inferring functional connectivity in an ensemble of retinal ganglion + cells sharing a common input. +\newblock {\em COSYNE}. + \bibitem[Vogelstein et~al., 2009]{Vogelstein2009} Vogelstein, J.~T., Watson, B.~O., Packer, A.~M., Yuste, R., Jedynak, B., and Paninski, L. (2009). diff --git a/paper/main.blg b/paper/main.blg index f911ab2..333831c 100644 --- a/paper/main.blg +++ b/paper/main.blg @@ -1,42 +1,10 @@ This is BibTeX, Version 0.99c (Web2C 7.5.4) The top-level auxiliary file: main.aux -White space in argument---line 9 of file main.aux - : \citation{Abeles91, - : Braitenberg1998} -I'm skipping whatever remains of this command -White space in argument---line 14 of file main.aux - : \citation{ReddySaggau05, - : Iyer06, SalomeBourdieu06, ReddySaggau08} -I'm skipping whatever remains of this command -White space in argument---line 20 of file main.aux - : \citation{BRIL88, - : CSK88, BRIL92, PG00, PILL07, PAN03d, PAN04c, Rigat06, TRUC05,NYK06,KP06,Vidne08,Stevenson2009} -I'm skipping whatever remains of this command -White space in argument---line 80 of file main.aux - : \citation{Braitenberg1998, - : Urquijo2000, Lefort2009, Sayer1990} -I'm skipping whatever remains of this command -White space in argument---line 81 of file main.aux - : \citation{Braitenberg1998, - : Urquijo2000} -I'm skipping whatever remains of this command -White space in argument---line 82 of file main.aux - : \citation{Braitenberg1998, - : Lefort2009} -I'm skipping whatever remains of this command -White space in argument---line 83 of file main.aux - : \citation{Lefort2009, - : Sayer1990} -I'm skipping whatever remains of this command -White space in argument---line 105 of file main.aux - : \citation{NguyenParker01, - : ReddySaggau05, Iyer06, SalomeBourdieu06, ReddySaggau08} -I'm skipping whatever remains of this command -White space in argument---line 125 of file main.aux +White space in argument---line 135 of file main.aux : \citation{emory, : liam, Garofalo, Vakorina, etc.} I'm skipping whatever remains of this command -White space in argument---line 130 of file main.aux +White space in argument---line 136 of file main.aux : \citation{Duane : 1 and 2, liam} I'm skipping whatever remains of this command @@ -62,49 +30,51 @@ Repeated entry---line 22006 of file mybib.bib : @book{ShumwayStoffer06 : , I'm skipping whatever remains of this entry +Warning--I didn't find a database entry for "" +Warning--I didn't find a database entry for "Abeles01" Warning--I didn't find a database entry for "VogPan09" Warning--I didn't find a database entry for "Chlovksii" Warning--I didn't find a database entry for "emory" Warning--can't use both author and editor fields in DFG01 -You've used 62 entries, +You've used 79 entries, 1935 wiz_defined-function locations, - 878 strings with 14775 characters, -and the built_in function-call counts, 25952 in all, are: -= -- 2483 -> -- 1288 -< -- 27 -+ -- 456 -- -- 444 -* -- 2288 -:= -- 4610 -add.period$ -- 185 -call.type$ -- 62 -change.case$ -- 521 -chr.to.int$ -- 62 -cite$ -- 63 -duplicate$ -- 878 -empty$ -- 1678 -format.name$ -- 534 -if$ -- 4954 + 979 strings with 17707 characters, +and the built_in function-call counts, 33245 in all, are: += -- 3191 +> -- 1636 +< -- 34 ++ -- 580 +- -- 564 +* -- 2948 +:= -- 5892 +add.period$ -- 236 +call.type$ -- 79 +change.case$ -- 664 +chr.to.int$ -- 79 +cite$ -- 80 +duplicate$ -- 1115 +empty$ -- 2160 +format.name$ -- 679 +if$ -- 6345 int.to.chr$ -- 1 int.to.str$ -- 0 -missing$ -- 68 -newline$ -- 311 -num.names$ -- 188 -pop$ -- 444 +missing$ -- 87 +newline$ -- 396 +num.names$ -- 239 +pop$ -- 560 preamble$ -- 1 -purify$ -- 530 +purify$ -- 675 quote$ -- 0 -skip$ -- 666 +skip$ -- 841 stack$ -- 0 -substring$ -- 1781 -swap$ -- 83 +substring$ -- 2342 +swap$ -- 100 text.length$ -- 2 text.prefix$ -- 0 top$ -- 0 -type$ -- 356 +type$ -- 454 warning$ -- 1 -while$ -- 198 +while$ -- 257 width$ -- 0 -write$ -- 789 -(There were 15 error messages) +write$ -- 1007 +(There were 7 error messages) diff --git a/paper/main.log b/paper/main.log index db20091..2a82a97 100644 --- a/paper/main.log +++ b/paper/main.log @@ -1,4 +1,4 @@ -This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4) (format=pdflatex 2009.4.16) 10 AUG 2009 11:15 +This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4) (format=pdflatex 2009.4.16) 11 AUG 2009 08:35 entering extended mode **main (./main.tex @@ -171,11 +171,7 @@ Package: amsthm 2004/08/06 v2.20 \thm@postskip=\skip51 \thm@headsep=\skip52 \dth@everypar=\toks28 -) (./main.aux - -LaTeX Warning: Label `fig:data_schematic' multiply defined. - -) +) (./main.aux) \openout1 = `main.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 44. @@ -190,7 +186,8 @@ LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 44. LaTeX Font Info: ... okay on input line 44. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 44. LaTeX Font Info: ... okay on input line 44. - (/opt/local/share/texmf-dist/tex/context/base/supp-pdf.tex + +(/opt/local/share/texmf-dist/tex/context/base/supp-pdf.tex (/opt/local/share/texmf-dist/tex/context/base/supp-mis.tex loading : Context Support Macros / Miscellaneous (2004.10.26) \protectiondepth=\count109 @@ -239,131 +236,55 @@ File: umsb.fd 2002/01/19 v2.2g AMS font definitions (./abstract.tex) [1 -{/opt/local/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./intro.tex - -LaTeX Warning: Citation ` Braitenberg1998' on page 2 undefined on input line 3. - - - -LaTeX Warning: Citation ` Iyer06' on page 2 undefined on input line 3. - - -LaTeX Warning: Citation ` SalomeBourdieu06' on page 2 undefined on input line 3 -. - - -LaTeX Warning: Citation ` ReddySaggau08' on page 2 undefined on input line 3. - -[2] <../figs/yuri-paper-schematic.pdf, id=41, 794.97pt x 614.295pt> -File: ../figs/yuri-paper-schematic.pdf Graphic file (type pdf) - -) (./methods.tex - -LaTeX Warning: Citation ` CSK88' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` BRIL92' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` PG00' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` PILL07' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` PAN03d' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` PAN04c' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` Rigat06' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation ` TRUC05' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation `NYK06' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation `KP06' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation `Vidne08' on page 3 undefined on input line 10. - - -LaTeX Warning: Citation `Stevenson2009' on page 3 undefined on input line 10. - -[3] [4 <../figs/yuri-paper-schematic.pdf>] -<../figs/fr_vs_J.pdf, id=106, 393.47pt x 277.035pt> +{/opt/local/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./intro.tex [ +2]) (./methods.tex [3] +<../figs/fr_vs_J.pdf, id=47, 393.47pt x 277.035pt> File: ../figs/fr_vs_J.pdf Graphic file (type pdf) -[5 <../figs/fr_vs_J.pdf>] -Overfull \hbox (38.28987pt too wide) in paragraph at lines 56--56 -[] +[4 <../figs/fr_vs_J.pdf>] [5] +Overfull \hbox (7.05049pt too wide) in paragraph at lines 195--195 +[]\OT1/cmr/bx/n/12 Initialization of ``in-trin-sic'' pa-ram-e-ters via se-quen- +tial Monte Carlo meth- [] [6] -Overfull \hbox (4.6977pt too wide) in paragraph at lines 74--74 -[]\OT1/cmr/bx/n/12 Initialization of ``in-ter-nal'' pa-ram-e-ters via se-quen-t -ial Monte Carlo meth- - [] - - -Overfull \hbox (25.76118pt too wide) in paragraph at lines 86--86 +Overfull \hbox (64.72421pt too wide) in paragraph at lines 225--225 [] [] -[7] -Overfull \hbox (50.0147pt too wide) in paragraph at lines 95--99 -\OT1/cmr/m/n/10.95 where $\OML/cmm/m/it/10.95 r[]\OT1/cmr/m/n/10.95 (\OML/cmm/m -/it/10.95 t; t \OMS/cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 ^^A)$ de-notes the we -ight at-tached to the par-ti-cle pair with po-si-tions $[]$. + +Overfull \hbox (26.71463pt too wide) detected at line 240 +\OML/cmm/m/it/10.95 P\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 X[]\OT1/cmr/m/n/10 +.95 (\OML/cmm/m/it/10.95 t\OT1/cmr/m/n/10.95 )\OML/cmm/m/it/10.95 ; X[]\OT1/cmr +/m/n/10.95 (\OML/cmm/m/it/10.95 t \OMS/cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 ^^ +A)\OMS/cmsy/m/n/10.95 j\OT1/cmr/bx/n/10.95 F[]\OML/cmm/m/it/10.95 ; ^^R[]\OT1/c +mr/m/n/10.95 ] \OMS/cmsy/m/n/10.95 ^^Y [] \OML/cmm/m/it/10.95 r[]\OT1/cmr/m/n/1 +0.95 (\OML/cmm/m/it/10.95 t; t \OMS/cmsy/m/n/10.95 ^^@ \OT1/cmr/m/n/10.95 ^^A)\ +OML/cmm/m/it/10.95 ^^N [] \OMS/cmsy/m/n/10.95 ^^B \OML/cmm/m/it/10.95 ^^N [] ; [] -[8] -Overfull \hbox (11.18362pt too wide) in paragraph at lines 171--172 +[7] [8] +Overfull \hbox (11.6322pt too wide) in paragraph at lines 347--359 []\OT1/cmr/m/n/10.95 We can ap-ply this embedded-chain method quite di-rectly h -ere to sam-ple from $\OML/cmm/m/it/10.95 P\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10. +ere to sam-ple from $\OML/cmm/m/it/10.95 P\OT1/cmr/m/n/10.95 [\OT1/cmr/bx/n/10. 95 X[]\OMS/cmsy/m/n/10.95 j\OT1/cmr/bx/n/10.95 X[]\OML/cmm/m/it/10.95 ; \OT1/cm r/bx/n/10.95 F\OT1/cmr/m/n/10.95 ; \OML/cmm/m/it/10.95 ^^R\OT1/cmr/m/n/10.95 ]$ . [] -[9] -Overfull \hbox (3.83298pt too wide) in paragraph at lines 211--225 -\OT1/cmr/m/n/10.95 Enforcing sparse-ness for sig-nal re-cov-ered with a se-ries - of lin-ear mea-sure-ments via $\OML/cmm/m/it/10.95 L\OT1/cmr/m/n/10.95 1$-regu -larizer - [] - -[10] [11] +[9] [10] <../figs/yuri-paper-schematic.pdf, id=98, 794.97pt x 614.295pt> File: ../figs/yuri-paper-schematic.pdf Graphic file (type pdf) - - -LaTeX Warning: `h' float specifier changed to `ht'. - -) (./results.tex - -LaTeX Warning: Citation ` Urquijo2000' on page 12 undefined on input line 3. - - -LaTeX Warning: Citation ` Lefort2009' on page 12 undefined on input line 3. - - -LaTeX Warning: Citation ` Sayer1990' on page 12 undefined on input line 3. - - -LaTeX Warning: Citation ` Urquijo2000' on page 12 undefined on input line 5. +) (./results.tex [11] [12 <../figs/yuri-p +aper-schematic.pdf>] -LaTeX Warning: Citation ` Lefort2009' on page 12 undefined on input line 5. +LaTeX Warning: Citation `Abeles01' on page 13 undefined on input line 11. +<../figs/example_traces.pdf, id=157, 418.56375pt x 214.8025pt> +File: ../figs/example_traces.pdf Graphic file (type pdf) -LaTeX Warning: Citation ` Sayer1990' on page 12 undefined on input line 5. - -[12] [13] -Overfull \hbox (35.78934pt too wide) in paragraph at lines 46--47 + +Overfull \hbox (35.78934pt too wide) in paragraph at lines 47--55 \OT1/cmr/m/n/10.95 cence data. Specif-i-cally, we es-ti-mated the con-nec-tiv-i ty ma-trix by max-i-miz-ing $\OML/cmm/m/it/10.95 E\OT1/cmr/m/n/10.95 [[] \OML/c mm/m/it/10.95 P\OT1/cmr/m/n/10.95 [\OML/cmm/m/it/10.95 n[]\OT1/cmr/m/n/10.95 (\ @@ -372,11 +293,10 @@ OML/cmm/m/it/10.95 t\OT1/cmr/m/n/10.95 )\OMS/cmsy/m/n/10.95 j\OT1/cmr/bx/n/10.9 /10.95 b[]; \OT1/cmr/bx/n/10.95 w[]\OT1/cmr/m/n/10.95 ]]$ [] -<../figs/example_traces.pdf, id=158, 418.56375pt x 214.8025pt> -File: ../figs/example_traces.pdf Graphic file (type pdf) - [14] -<../figs/FigureA3_scatter_three.pdf, id=162, 420.57124pt x 316.18124pt> +LaTeX Warning: `h' float specifier changed to `ht'. + +[13] <../figs/FigureA3_scatter_three.pdf, id=162, 420.57124pt x 316.18124pt> File: ../figs/FigureA3_scatter_three.pdf Graphic file (type pdf) @@ -387,192 +307,168 @@ File: ../figs/FigureA3_scatter_three_corrected.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -[15 <../figs/example_traces.pdf>] [16 <../figs/FigureA3_scatter_three.pdf> <../ -figs/FigureA3_scatter_three_corrected.pdf>] +[14] -LaTeX Warning: Citation `VogPan09' on page 17 undefined on input line 99. +LaTeX Warning: Citation `VogPan09' on page 15 undefined on input line 100. -<../figs/FigureA4_scale_bias.pdf, id=188, 420.57124pt x 316.18124pt> +<../figs/FigureA4_scale_bias.pdf, id=167, 420.57124pt x 316.18124pt> File: ../figs/FigureA4_scale_bias.pdf Graphic file (type pdf) -<../figs/FigureA5_recvar.pdf, id=189, 420.57124pt x 316.18124pt> + +LaTeX Warning: `h' float specifier changed to `ht'. + +<../figs/FigureA5_recvar.pdf, id=168, 420.57124pt x 316.18124pt> File: ../figs/FigureA5_recvar.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -<../figs/FigureA6_recvar_SNR.pdf, id=190, 422.57875pt x 343.2825pt> +<../figs/FigureA6_recvar_SNR.pdf, id=169, 422.57875pt x 343.2825pt> File: ../figs/FigureA6_recvar_SNR.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -<../figs/FigureA7_recvar_NT.pdf, id=191, 420.57124pt x 316.18124pt> +<../figs/FigureA7_recvar_NT.pdf, id=170, 420.57124pt x 316.18124pt> File: ../figs/FigureA7_recvar_NT.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. - -LaTeX Warning: Citation ` ReddySaggau05' on page 17 undefined on input line 136 -. - - -LaTeX Warning: Citation ` Iyer06' on page 17 undefined on input line 136. - - -LaTeX Warning: Citation ` SalomeBourdieu06' on page 17 undefined on input line -136. - - -LaTeX Warning: Citation ` ReddySaggau08' on page 17 undefined on input line 136 -. - -[17 <../figs/FigureA4_scale_bias.pdf>] -Overfull \hbox (60.19464pt too wide) detected at line 163 +[15 <../figs/example_traces.pdf>] [16 <../figs/FigureA3_scatter_three.pdf> <../ +figs/FigureA3_scatter_three_corrected.pdf>] +Overfull \hbox (60.19464pt too wide) detected at line 164 [] [] -[18 <../figs/FigureA5_recvar.pdf>] -<../figs/FigureA10_regular_sol.pdf, id=209, 420.57124pt x 316.18124pt> +[17 <../figs/FigureA4_scale_bias.pdf>] +<../figs/FigureA10_regular_sol.pdf, id=202, 420.57124pt x 316.18124pt> File: ../figs/FigureA10_regular_sol.pdf Graphic file (type pdf) -<../figs/FigureA10_sparse_sol.pdf, id=210, 420.57124pt x 316.18124pt> +<../figs/FigureA10_sparse_sol.pdf, id=203, 420.57124pt x 316.18124pt> File: ../figs/FigureA10_sparse_sol.pdf Graphic file (type pdf) - [19 <../figs/FigureA6_recvar_SNR.pdf>] + LaTeX Warning: `h' float specifier changed to `ht'. -<../figs/FigureA3_hist_glm200.pdf, id=218, 420.57124pt x 316.18124pt> +<../figs/FigureA3_hist_glm200.pdf, id=204, 420.57124pt x 316.18124pt> File: ../figs/FigureA3_hist_glm200.pdf Graphic file (type pdf) -<../figs/FigureA3_hist_spa200.pdf, id=219, 420.57124pt x 316.18124pt> +<../figs/FigureA3_hist_spa200.pdf, id=205, 420.57124pt x 316.18124pt> File: ../figs/FigureA3_hist_spa200.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -<../figs/Figure7b_raster_weak.pdf, id=220, 420.57124pt x 313.17pt> +[18 <../figs/FigureA5_recvar.pdf>] +<../figs/Figure7b_raster_weak.pdf, id=213, 420.57124pt x 313.17pt> File: ../figs/Figure7b_raster_weak.pdf Graphic file (type pdf) -<../figs/Figure7a_raster_strong.pdf, id=221, 420.57124pt x 294.09875pt> +<../figs/Figure7a_raster_strong.pdf, id=214, 420.57124pt x 294.09875pt> File: ../figs/Figure7a_raster_strong.pdf Graphic file (type pdf) -<../figs/FigureA8_weak_corr.pdf, id=222, 420.57124pt x 316.18124pt> +<../figs/FigureA8_weak_corr.pdf, id=215, 420.57124pt x 316.18124pt> File: ../figs/FigureA8_weak_corr.pdf Graphic file (type pdf) -<../figs/FigureA8_strong_corr.pdf, id=223, 420.57124pt x 316.18124pt> +<../figs/FigureA8_strong_corr.pdf, id=216, 420.57124pt x 316.18124pt> File: ../figs/FigureA8_strong_corr.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -<../figs/FigureA9_all_same_sol.pdf, id=224, 420.57124pt x 316.18124pt> +<../figs/FigureA9_all_same_sol.pdf, id=217, 420.57124pt x 316.18124pt> File: ../figs/FigureA9_all_same_sol.pdf Graphic file (type pdf) -<../figs/FigureA9_variable_25.pdf, id=225, 420.57124pt x 316.18124pt> +<../figs/FigureA9_variable_25.pdf, id=218, 420.57124pt x 316.18124pt> File: ../figs/FigureA9_variable_25.pdf Graphic file (type pdf) LaTeX Warning: `h' float specifier changed to `ht'. -[20 <../figs/FigureA7_recvar_NT.pdf>]) [21 <../figs/FigureA10_regular_sol.pdf> -<../figs/FigureA10_sparse_sol.pdf>] (./discussion.tex +[19 <../figs/FigureA6_recvar_SNR.pdf>] [20 <../figs/FigureA7_recvar_NT.pdf>]) +[21 <../figs/FigureA10_regular_sol.pdf> <../figs/FigureA10_sparse_sol.pdf>] +[22 <../figs/FigureA3_hist_glm200.pdf> <../figs/FigureA3_hist_spa200.pdf>] +[23 <../figs/Figure7b_raster_weak.pdf> <../figs/Figure7a_raster_strong.pdf> <.. +/figs/FigureA8_weak_corr.pdf> <../figs/FigureA8_strong_corr.pdf>] [24 <../figs/ +FigureA9_all_same_sol.pdf> <../figs/FigureA9_variable_25.pdf>] +(./discussion.tex -LaTeX Warning: Citation `Chlovksii' on page 22 undefined on input line 1. +LaTeX Warning: Citation `Chlovksii' on page 25 undefined on input line 1. -LaTeX Warning: Citation `emory' on page 22 undefined on input line 3. +LaTeX Warning: Citation `emory' on page 25 undefined on input line 3. -LaTeX Warning: Citation ` liam' on page 22 undefined on input line 3. +LaTeX Warning: Citation ` liam' on page 25 undefined on input line 3. -LaTeX Warning: Citation ` Garofalo' on page 22 undefined on input line 3. +LaTeX Warning: Citation ` Garofalo' on page 25 undefined on input line 3. -LaTeX Warning: Citation ` Vakorina' on page 22 undefined on input line 3. +LaTeX Warning: Citation ` Vakorina' on page 25 undefined on input line 3. -LaTeX Warning: Citation ` etc.' on page 22 undefined on input line 3. +LaTeX Warning: Citation ` etc.' on page 25 undefined on input line 3. -[22 <../figs/FigureA3_hist_glm200.pdf> <../figs/FigureA3_hist_spa200.pdf>] -LaTeX Warning: Citation `Duane 1 and 2' on page 23 undefined on input line 5. +LaTeX Warning: Citation `Duane 1 and 2' on page 25 undefined on input line 5. -LaTeX Warning: Citation ` liam' on page 23 undefined on input line 5. +LaTeX Warning: Citation ` liam' on page 25 undefined on input line 5. -LaTeX Font Info: Try loading font information for OMS+cmr on input line 15. -(/opt/local/share/texmf-dist/tex/latex/base/omscmr.fd -File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions -) -LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available -(Font) Font shape `OMS/cmsy/m/n' tried instead on input line 15. - -Overfull \hbox (7.20935pt too wide) in paragraph at lines 19--20 -[]\OT1/cmr/m/n/10.95 we can mea-sure un-cer-tainty in es-ti-mates, and use phot -o-stimulation to ac-ti-vate/deactivate - [] +) (./main.bbl [25 -) (./main.bbl [23] [24] -Underfull \hbox (badness 10000) in paragraph at lines 175--180 +] [26] [27] +Underfull \hbox (badness 10000) in paragraph at lines 220--225 []\OT1/cmr/m/n/10.95 Mishchenko, Y. (2009). Strate-gies for iden-ti-fy-ing ex- act struc-ture of neu- [] -Underfull \hbox (badness 10000) in paragraph at lines 175--180 +Underfull \hbox (badness 10000) in paragraph at lines 220--225 \OT1/cmr/m/n/10.95 ral cir-cuits with broad light mi-croscopy con-nec-tiv-ity p robes. \OT1/cmr/m/it/10.95 Preprint: [] -[25] [26]) [27] [28 <../figs/Figure7b_raster_weak.pdf> <../figs/Figure7a_raster -_strong.pdf> <../figs/FigureA8_weak_corr.pdf> <../figs/FigureA8_strong_corr.pdf ->] [29 <../figs/FigureA9_all_same_sol.pdf> <../figs/FigureA9_variable_25.pdf>] -(./main.aux) - -LaTeX Warning: There were multiply-defined labels. - - ) +[28] [29] [30]) [31] (./main.aux) ) Here is how much of TeX's memory you used: - 2605 strings out of 94500 - 34789 string characters out of 1155413 - 89057 words of memory out of 1000000 - 5697 multiletter control sequences out of 10000+50000 + 2586 strings out of 94500 + 34464 string characters out of 1155413 + 94873 words of memory out of 1000000 + 5681 multiletter 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z0Y?IB;~eJ!h(MLa*%AN|_pB1Y1OA#C#}}|L;0oOE4=y&~n2}6O+tn3&9TBh}_5!T3 zi>)j04RG*3-xN5FCFRgpuOeY`5R{_)Rgj{B0#Xrn6@E<-B8QZN1BV%KY0m$D6zqjU dvU~0pcip}3+Srm|1p}Hb5;8tMMa^qu{~vEhMLz%l diff --git a/paper/main.tex b/paper/main.tex index 613590e..e9dfd0d 100755 --- a/paper/main.tex +++ b/paper/main.tex @@ -87,17 +87,18 @@ \section{Results} \label{sec:results} \input{results.tex} +\clearpage \section{Discussion} \label{sec:discussion} \input{discussion.tex} % \begin{acknowledgments} \section*{Acknowledgements} -Thank everyone for their help and support [Bows, Bows, Bows] !!! +The authors would like to acknowledge Rafael Yuste, Brendon O.\ Watson, Adam Packer, Tanya Sippy, Tom Mrsic-Flogel and Vincent Bonin for data and helpful discussions. % \end{acknowledgments} \bibliography{mybib} -%\bibliographystyle{amsplain} +% \bibliographystyle{amsplain} \bibliographystyle{apalike} \end{document} \ No newline at end of file diff --git a/paper/main.toc b/paper/main.toc index c6280bf..b38b439 100644 --- a/paper/main.toc +++ b/paper/main.toc @@ -1,21 +1,20 @@ \contentsline {section}{\numberline {1}Introduction}{2} \contentsline {section}{\numberline {2}Methods}{3} \contentsline {subsection}{\numberline {2.1}Model}{3} -\contentsline {subsection}{\numberline {2.2}Goal and general strategy}{6} -\contentsline {subsection}{\numberline {2.3}Initialization of ``internal'' parameters via sequential Monte Carlo methods}{7} +\contentsline {subsection}{\numberline {2.2}Goal and general strategy}{5} +\contentsline {subsection}{\numberline {2.3}Initialization of ``intrinsic'' parameters via sequential Monte Carlo methods}{6} \contentsline {subsection}{\numberline {2.4}Estimating joint posteriors over weakly coupled neurons}{8} -\contentsline {subsubsection}{\numberline {2.4.1}A cheaper high-SNR approximation of the joint posteriors}{10} -\contentsline {subsection}{\numberline {2.5}Estimating the functional connectivity matrix}{10} -\contentsline {subsubsection}{\numberline {2.5.1}Imposing a sparse prior on the functional connectivity}{11} -\contentsline {subsubsection}{\numberline {2.5.2}Imposing Dale's law on the functional connectivity}{11} +\contentsline {subsubsection}{\numberline {2.4.1}A factorized approximation of the joint posteriors}{9} +\contentsline {subsection}{\numberline {2.5}Estimating the functional connectivity matrix}{9} +\contentsline {subsubsection}{\numberline {2.5.1}Imposing a sparse prior on the functional connectivity}{10} +\contentsline {subsubsection}{\numberline {2.5.2}Imposing Dale's law on the functional connectivity}{10} \contentsline {subsection}{\numberline {2.6}Specific implementation notes}{11} -\contentsline {section}{\numberline {3}Results}{12} -\contentsline {subsection}{\numberline {3.1}Simulating neural activity in a neural population}{12} -\contentsline {subsection}{\numberline {3.2}Inferring functional connectivity from the simulated calcium imaging data}{14} -\contentsline {subsection}{\numberline {3.3}Scale bias in inferred connection weights due to coarse time discretization of calcium imaging data}{15} -\contentsline {subsection}{\numberline {3.4}Impact of different imaging frame rates, noise levels, and durations on the estimator accuracy}{17} -\contentsline {subsection}{\numberline {3.5}Accuracy of the estimates and Fisher information matrix}{18} -\contentsline {subsection}{\numberline {3.6}Impact of using priors on the inference}{20} -\contentsline {subsection}{\numberline {3.7}Impact of strong correlations and deviations from generative model on the inference}{20} -\contentsline {section}{\numberline {4}Discussion}{22} -\contentsline {paragraph}{j's outline of discussion}{23} +\contentsline {section}{\numberline {3}Results}{11} +\contentsline {subsection}{\numberline {3.1}Simulating neural activity in a neural population}{11} +\contentsline {subsection}{\numberline {3.2}Inferring functional connectivity from the simulated calcium imaging data}{13} +\contentsline {subsection}{\numberline {3.3}Scale bias in inferred connection weights due to coarse time discretization of calcium imaging data}{14} +\contentsline {subsection}{\numberline {3.4}Impact of different imaging frame rates, noise levels, and durations on the estimator accuracy}{16} +\contentsline {subsection}{\numberline {3.5}Accuracy of the estimates and Fisher information matrix}{17} +\contentsline {subsection}{\numberline {3.6}Impact of using priors on the inference}{19} +\contentsline {subsection}{\numberline {3.7}Impact of strong correlations and deviations from generative model on the inference}{19} +\contentsline {section}{\numberline {4}Discussion}{25} diff --git a/paper/methods.tex b/paper/methods.tex old mode 100644 new mode 100755 index 2fd5b7c..fbe4ff3 --- a/paper/methods.tex +++ b/paper/methods.tex @@ -1,201 +1,411 @@ \long\def\comment#1{} -\comment{Our goal is to estimate the connection matrix from a population of observable neurons, given only their calcium fluorescence observations. We take a model based approach, meaning that we first describe a parametric generative model that completely characterizes the statistics of the data, and then we derive algorithms to learn the parameters, given the data. - -We use the following conventions throughout the paper, unless indicated otherwise. Time is discrete, taking values $t=1,\ldots,T$. We let $X_i(t)$ indicate the state of neuron $i$ at time $t$, $X_i=\{X_i(t), t=1,\ldots, T\}$, and ${\bf X}= \{X_1, \ldots, X_N\}$. Conditional probability distributions will be written $P[\bf F | \bf X; \bth]$, where $\bf X$ indicates some random variables, $\bth$ indicates some parameters, and a semicolon separates the two. To indicate that a random variable, $X$, is independently and identically distributed according to some distribution $P$, we have $X \overset{iid}{\sim} P$. } - \subsection{Model} \label{sec:methods:markov-setup} -We first describe a parametric generative model that characterizes the statistics of the (unobserved) joint spike trains of all $N$ observable neurons, along with the observed calcium fluorescence data. Each neuron is modeled as a generalized linear model (GLM); this class of model is known to capture the statistical firing properties of the of individual neurons fairly accurately \cite{BRIL88, CSK88, BRIL92, PG00, PILL07, PAN03d, PAN04c, Rigat06, TRUC05,NYK06,KP06,Vidne08,Stevenson2009}. We denote the $i$-th neuron's activity at time $t$ as $n_i(t)$: in continuous time, $n_i(t)$ could be modeled as an unmarked point process, but we will take a discrete-time approach here, and so $n_i(t)$ will be a binary random variable. We model the spiking probability of neuron $i$ via an instantaneous nonlinear function, $f(\cdot)$, of the filtered and summed input to that neuron at that time, $J_i(t)$. The input is composed of: (i) some baseline value, $b_i$; (ii) some external stimulus, $S(t)$, that is linearly filtered by $k_i$; and (iii) spike history terms, $h_j(t)$, from each neuron $j$, weighted by $\w_{ij}$: - +We begin by detailing a parametric generative model for the +(unobserved) joint spike trains of all $N$ observable neurons, along +with the observed calcium fluorescence data. Each neuron is modeled as +a generalized linear model (GLM); this class of model is known to +capture the statistical firing properties of individual neurons fairly +accurately \cite{BRIL88,CSK88,BRIL92,PG00,PILL07,PAN03d,PAN04c,Rigat06,TRUC05,NYK06,KP06,Vidne08,Stevenson2009}. We denote the +$i$-th neuron's activity at time $t$ as $n_i(t)$: in continuous time, +$n_i(t)$ could be modeled as an unmarked point process, but we will +take a discrete-time approach here, with each $n_i(t)$ taken to be a +binary random variable. We model the spiking probability of neuron +$i$ via an instantaneous nonlinear function, $f(\cdot)$, of the +filtered and summed input to that neuron at that time, $J_i(t)$. This +input is composed of: (i) some baseline value, $b_i$; (ii) some +external vector stimulus, $S(t)$, that is linearly filtered by $k_i$; +and (iii) spike history terms, $h_{ij}(t)$, from each neuron $j$, +weighted by $\w_{ij}$: \begin{equation} \label{eqn:glm:definition} \begin{array}{l} -n_i(t)\overset{iid}{\sim} \text{Bernoulli}\big(f\big(J_i(t) \big) -\big), \qquad J_i(t)=b_i+k_i\cdot S(t)+\sum \limits_{j=1}^{N} \w_{ij} -h_{j}(t). +n_i(t) \sim \text{Bernoulli}\big(f\big(J_i(t) \big) \big), \qquad +J_i(t)=b_i+k_i\cdot S(t)+\sum \limits_{j=1}^{N} \w_{ij} h_{ij}(t). \end{array} \end{equation} - -To ensure computational tractability, we must impose some reasonable constraints on the instantaneous nonlinearity $f(\cdot)$ (which plays the role of the inverse of the link function in the standard GLM setting) and on the dynamics of the spike-history effects $h_j(t)$. More specifically, first, we restrict our attention to functions $f(\cdot)$ which ensure the concavity of the spiking loglikelihood in this model \cite{PAN04c}, as we will discuss at more length below. In this paper, we use \begin{equation} f(J) = P[X>0 | X \sim Poiss(e^J \Delta)] = 1 - \exp[-e^J \Delta] \end{equation} (where the inclusion of $\Delta$, the time step size, ensures that the firing rate scales properly with respect to the time discretization), though in our experience the results depend only weakly on the details of $f(\cdot)$ within the class of log-concave models \cite{LD89,PAN04c}; see \cite{Escola07} for a proof that this $f(\cdot)$ satisfies the required concavity constraints. - -\begin{figure}[h] -\centering -\includegraphics[width=3in]{../figs/fr_vs_J} -\caption{Given our model, Eq. \ref{eqn:glm:definition}, the relationship between firing rate and the magnitude of the input to a neuron is highly nonlinear. Note that firing rate saturates at $1/\Delta$, because of our Bernoulli assumption.} +To ensure computational tractability, we must impose some reasonable +constraints on the instantaneous nonlinearity $f(\cdot)$ (which plays +the role of the inverse of the link function in the standard GLM +setting) and on the dynamics of the spike-history effects $h_{ij}(t)$. +First, we restrict our attention to functions $f(\cdot)$ which ensure +the concavity of the spiking loglikelihood in this model +\cite{PAN04c,Escola07}, as we will discuss at more length below. In +this paper, we use \begin{equation} f(J) = P[n>0 ~|~ n \sim Poiss(e^J +\Delta)] = 1 - \exp[-e^J \Delta] \end{equation} +(Fig.~\ref{fig:egfluor}), where the inclusion of $\Delta$, the time +step size, ensures that the firing rate scales properly with respect +to the time discretization; see \cite{Escola07} for a proof that this +$f(\cdot)$ satisfies the required concavity constraints. However, we +should note that in our experience the results depend only weakly on +the details of $f(\cdot)$ within the class of log-concave models +\cite{LD89,PAN04c}. + +\begin{figure}[t!] +\centering \includegraphics[width=3in]{../figs/fr_vs_J} +\caption{A plot of the firing rate nonlinearity $f(J)$ used in our + simulations; see Eq.~\ref{eqn:glm:definition}. Note that the firing + rate saturates at $1/\Delta$, because of our Bernoulli assumption + (i.e., the spike count per bin is at most one); $\Delta = (60$ + Hz$)^{-1}$ here.} \label{fig:egfluor} \end{figure} -Second, because the algorithms we develop below assume Markovian dynamics, we model the spike history terms as: - -\begin{equation} \label{eqn:h:definition} h_j(t) = (1- \Delta/\tau_h) h_j(t- \Delta) +n_j(t) + \sigma_h \sqrt{\Delta} \epsilon^h_j(t), \end{equation} - -where $\tau_h$ is the decay time constant for spike history terms, $\sigma_h$ is a standard deviation parameter, $\sqrt{\Delta}$ ensures that the statistics of this Markov process have a proper Ornstein-Uhlenbeck limit as $\Delta \to 0$, and throughout this paper, $\epsilon$ denotes an independent standard normal random variable. Note that this model generalizes (via a simple augmentation of the state variable $h_j(t)$) to allow each neuron to have several spike history terms, each with a unique time constant, which when weighted and summed allow us to model a wide variety of possible post-synaptic effects, including bursting, facilitating, and depressing synapses; see \cite{Vogelstein2009} for further details. In this paper, for simplicity, we assume that $\tau_h$ and $\sigma_h$ are known synaptic parameters, and therefore our model spiking parameters $\bth^n$ are given by $\{\bth^n_i\}_{i=1}^N$, where $\bth^n_i=\{\bw_i,k_i,b_i\}$, with $\bw_i=(\w_{i1},\ldots, \w_{iN})$. - -The problem of estimating the connectivity parameters $\bw_i$ in this type of GLM, given a fully-observed ensemble of neural spike train $\{n_i(t)\}$, has recently received a great deal of attention; see the references above for a partial list. In the calcium fluorescent imaging setting, however, we do not directly observe spike trains; $\{n_i(t)\}$ must be considered a hidden variable here. Instead, each spike in a given neuron leads to a rapid increase in the intracellular calcium concentration, which then decays slowly due to various cellular buffering and extrusion mechanisms. We in turn make only noisy, indirect, and subsampled observations of this intracellular calcium concentration, via fluorescent imaging techniques \cite{ImagingManual}. To perform statistical inference in this setting, \cite{Vogelstein2009} proposed a simple conditional first-order hidden Markov model (HMM) for the intracellular calcium concentration $C_i(t)$ in cell $i$ at time $t$, along with the observed fluorescence $F_i(t)$: - -\begin{align} \label{eqn:ca:definition} C_i(t) &= C_i(t-\Delta) + (C_i^b-C_i(t-\Delta)) \Delta/\tau^c_i + A_i n_i(t)+\sigma^c_i \sqrt{\Delta} \epsilon^c_i(t), \\ F_i(t) &= \alpha_i S(C_i(t)) + \beta_i + \sqrt{\gamma_i S(C_i(t)) + \sigma^F_i} \epsilon^F_i(t). \label{eqn:F:definition} +Second, because the algorithms we develop below assume Markovian +dynamics, we model the spike history terms in terms of an +autoregressive process: +\begin{equation} \label{eqn:h:definition} +h_{ij}(t) = (1- \Delta/\tau_h) h_{ij}(t- \Delta) +n_j(t) + \sigma_h + \sqrt{\Delta} \epsilon_{ij}(t), +\end{equation} +where $\tau_h$ is the decay time constant for spike history terms, +$\sigma_h$ is a standard deviation parameter, $\sqrt{\Delta}$ ensures +that the statistics of this Markov process have a proper +Ornstein-Uhlenbeck limit as $\Delta \to 0$, and throughout this paper, +$\epsilon$ denotes an independent standard normal random variable. +Note that this model generalizes (via a simple augmentation of the +state variable $h_{ij}(t)$) to allow each neuron to have several spike +history terms, each with a unique time constant, which when weighted +and summed allow us to model a wide variety of possible post-synaptic +effects, including bursting, facilitating, and depressing synapses; +see \cite{Vogelstein2009} for further details. We restrict our +attention to the case of a single time constant $\tau_h$ here, both +for simplicity and also because we find that the detailed shape of the +coupling terms $h_{ij}(t)$ had a limited effect on the inference of +the connectivity matrix, as illustrated in Fig.~\ref{fig:vartau} +below. In addition, we treat $\tau_h$ and $\sigma_h$ as known +synaptic parameters; therefore our unknown model spiking parameters +are $\{\bw_i,k_i,b_i\}$, with $\bw_i=(\w_{i1},\ldots, \w_{iN})$. + +The problem of estimating the connectivity parameters $\bw_i$ in this +type of GLM, given a fully-observed ensemble of neural spike train +$\{n_i(t)\}$, has recently received a great deal of attention; see the +references above for a partial list. In the calcium fluorescent +imaging setting, however, we do not directly observe spike trains; +$\{n_i(t)\}$ must be considered a hidden variable here. Instead, each +spike in a given neuron leads to a rapid increase in the intracellular +calcium concentration, which then decays slowly due to various +cellular buffering and extrusion mechanisms. We in turn make only +noisy, indirect, and subsampled observations of this intracellular +calcium concentration, via fluorescent imaging techniques +\cite{ImagingManual}. To perform statistical inference in this +setting, \cite{Vogelstein2009} proposed a simple conditional +first-order hidden Markov model (HMM) for the intracellular calcium +concentration $C_i(t)$ in cell $i$ at time $t$, along with the +observed fluorescence $F_i(t)$: +\begin{align} +\label{eqn:ca:definition} +C_i(t) &= C_i(t-\Delta) + (C_i^b-C_i(t-\Delta)) \Delta/\tau^c_i + A_i +n_i(t)+\sigma^c_i \sqrt{\Delta} \epsilon^c_i(t), \\ F_i(t) &= \alpha_i +S(C_i(t)) + \beta_i + \sqrt{\gamma_i S(C_i(t)) + \sigma^F_i} +\epsilon^F_i(t). \label{eqn:F:definition} \end{align} - -This model can be interpreted as a simple driven autoregressive process: under nonspiking conditions, $C_i(t)$ fluctuates around the baseline level of $C_i^b$, driven by normally-distributed noise $\epsilon^c_i(t)$ with standard deviation $\sigma^c_i \sqrt{\Delta}$. Whenever the neuron fires a spike, $n_i(t)=1$, causing the calcium variable $C_i(t)$ to jump by a fixed amount $A_i$, and subsequently decay with time constant $\tau^c_i$. The fluorescence signal $F_i(t)$ corresponds to the count of photons collected at the detector per neuron per imaging frame. This photon count may be modeled with normal statistics, with the mean and variance given by generalized Hill functions, where $S(C)=C/(C+K_d)$ \cite{Yasuda2004}. Because the parameter $K_d$ effectively acts as a simple scale factor, and is a property of the fluorescent indicator, we assume throughout this work that it is known. - -To summarize, Eqs. \ref{eqn:glm:definition} -- \ref{eqn:F:definition} define a coupled HMM: the underlying spike trains $n_i(t)$ and spike history terms $h_i(t)$ evolve in a Markovian manner, driving the intracellular calcium concentrations $C_i(t)$, which are themselves Markovian, but evolving at a slower timescale $\tau_i^c$. Finally, we observe only the fluorescence signals $\{F_i(t)\}$, which are related in a simple Markovian fashion to the calcium variables $C_i(t)$. +This model can be interpreted as a simple driven autoregressive +process: under nonspiking conditions, $C_i(t)$ fluctuates around the +baseline level of $C_i^b$, driven by normally-distributed noise +$\epsilon^c_i(t)$ with standard deviation $\sigma^c_i +\sqrt{\Delta}$. Whenever the neuron fires a spike, $n_i(t)=1$, causing +the calcium variable $C_i(t)$ to jump by a fixed amount $A_i$, and +subsequently decay with time constant $\tau^c_i$. The fluorescence +signal $F_i(t)$ corresponds to the count of photons collected at the +detector per neuron per imaging frame. This photon count may be +modeled with normal statistics, with the mean and variance given by +generalized Hill functions, $S(\cdot)$, where $S(C)=C/(C+K_d)$ +\cite{Yasuda2004}. Because the parameter $K_d$ effectively acts as a +simple scale factor, and is a property of the fluorescent indicator, +we assume throughout this work that it is known. + +To summarize, Eqs. \ref{eqn:glm:definition} -- \ref{eqn:F:definition} +define a coupled HMM: the underlying spike trains $n_i(t)$ and spike +history terms $h_{ij}(t)$ evolve in a Markovian manner given the +stimulus $S(t)$. These spike trains in turn drive the intracellular +calcium concentrations $C_i(t)$, which are themselves Markovian, but +evolving at a slower timescale $\tau_i^c$. Finally, we observe only +the fluorescence signals $\{F_i(t)\}$, which are related in a simple +Markovian fashion to the calcium variables $C_i(t)$. \subsection{Goal and general strategy} \label{sec:methods:goal} -Our primary goal is to estimate the connectivity matrix, $\bw$, given the observed set of calcium fluorescence signals ${\bf F}$. We must also deal with a number of nuisance parameters: the spiking parameters $\{k_i, b_i\}$ and the calcium parameters $\{C^b_i, \tau^c_i, A_i, \sigma^c_i, \alpha_i, \beta_i, \gamma_i, \sigma^F_i\}$. We addressed the problem of estimating these latter parameters in earlier work \cite{Vogelstein2009}; thus our focus here will be on $\bw$. A Bayesian approach is natural here, since we have a good deal of prior information about neural connectivity; see \cite{Rigat06} for a related discussion. However, a fully-Bayesian approach, in which we numerically integrate over the very high-dimensional parameter $\theta = \{\bw, k_i, b_i, C^b_i, \tau^c_i, A_i, \sigma^c_i, \alpha_i, \beta_i, \gamma_i, \sigma^F_i\}$, is not particularly attractive here, from a computational point of view. Thus we take a compromise approach and compute \emph{maximum a posteriori} (MAP) estimates for the parameters via an expectation-maximization (EM) algorithm in which the sufficient statistics are computed by a hybrid blockwise Gibbs sampler and sequential Monte Carlo (SMC) method. +Our primary goal is to estimate the connectivity matrix, $\bw$, given +the observed set of calcium fluorescence signals ${\bf F}$. We must +also deal with a number of nuisance parameters: the spiking parameters +$\{k_i, b_i\}$ and the calcium parameters $\{C^b_i, \tau^c_i, A_i, +\sigma^c_i, \alpha_i, \beta_i, \gamma_i, \sigma^F_i\}$. We addressed +the problem of estimating these latter parameters in earlier work +\cite{Vogelstein2009}; thus our focus here will be on the connectivity +matrix $\bw$. A Bayesian approach is natural here, since we have a +good deal of prior information about neural connectivity; see +\cite{Rigat06} for a related discussion. However, a fully-Bayesian +approach, in which we numerically integrate over the very +high-dimensional parameter $\theta = \{\bw, k_i, b_i, C^b_i, \tau^c_i, +A_i, \sigma^c_i, \alpha_i, \beta_i, \gamma_i, \sigma^F_i\}$, is not +particularly attractive here, from a computational point of view. Thus +we take a compromise approach and compute \emph{maximum a posteriori} +(MAP) estimates for the parameters via an expectation-maximization +(EM) algorithm in which the sufficient statistics are computed by a +hybrid blockwise Gibbs sampler and sequential Monte Carlo (SMC) +method. More specifically, we iterate the steps: \begin{align*} -\textbf{E step} &\text{: Evaluate } Q(\bth^{(l+1)},\bth^{(l)}) = -E_{P[\bX | \bF; \bth^{(l+1)}]} \ln P \left[ \bF, \bX | \bth^{(l)} -\right] = \int P[\bX | \bF; \bth^{(l+1)}] \ln P[\bF, \bX | \bth^{(l)}] -d \bX \\ \textbf{M step} &\text{: Solve } \bth^{(l+1)} = -\argmax_{\bth} \left\{ Q(\bth,\bth^{(l)}) + \ln P(\theta) \right\}, +\textbf{E step} &\text{: Evaluate } Q(\bth,\bth^{(l)}) = E_{P[\bX | +\bF; \bth^{(l)}]} \ln P \left[ \bF, \bX | \bth \right] = \int P[\bX | +\bF; \bth^{(l)}] \ln P[\bF, \bX | \bth] d \bX \\ \textbf{M step} +&\text{: Solve } \bth^{(l+1)} = \argmax_{\bth} \left\{ +Q(\bth,\bth^{(l)}) + \ln P(\theta) \right\}, \end{align*} - -where $\bX$ denotes the set of all hidden variables $\{ C_i(t), n_i(t), h_i(t) \}_{i \leq N, t \leq T}$ and $P(\theta)$ denotes a (possibly improper) prior on the parameter space $\theta$. According to standard EM theory \cite{DLR77,McLachlanKrishnan96}, each iteration of these two steps is guaranteed to increase the log-posterior $\ln P(\bth^{(l+1)} | \bF)$, and will therefore lead to at least a locally maximum a posteriori estimator. - -Now our major challenge is to evaluate the auxiliary function $Q(\bth^{(l+1)},\bth^{(l)})$ in the E-step. Because our model is a coupled HMM, as discussed in the previous section, $Q$ simplifies considerably \cite{RAB89}: - +where $\bX$ denotes the set of all hidden variables $\{ C_i(t), +n_i(t), h_{ij}(t) \}_{i,j \leq N, t \leq T}$ and $P(\theta)$ denotes a +(possibly improper) prior on the parameter space $\theta$. According +to standard EM theory \cite{DLR77,McLachlanKrishnan96}, each iteration +of these two steps is guaranteed to increase the log-posterior $\ln +P(\bth^{(l)} | \bF)$, and will therefore lead to at least a locally +maximum a posteriori estimator. + +Now our major challenge is to evaluate the auxiliary function +$Q(\bth,\bth^{(l)})$ in the E-step. Because our model is a coupled +HMM, as discussed in the previous section, $Q$ simplifies considerably +\cite{RAB89}: \begin{eqnarray} Q(\bth,\bth^{(l)}) &=& \sum_{it} P[C_i(t) | \bF; \bth] \times \ln P[F_i(t)|C_i(t); \alpha_i, \beta_i, \gamma_i, \sigma^F_i] \nonumber \\ &+& \sum_{it} P[C_i(t), C_i(t-\Delta), n_i(t) | \bF; \bth] \times \ln P[C_i(t)|C_i(t-\Delta), n_i(t); C^b_i, \tau^c_i, A_i, \sigma^c_i] -\nonumber \\ &+& \sum_{it} P[n_i(t), \bh(t) | \bF; \bth] \times \ln -P[n_i(t)| \bh(t); b_i, k_i, \bw_i, S(t)], +\nonumber \\ &+& \sum_{it} P[n_i(t), \bh_i(t) | \bF; \bth] \times \ln +P[n_i(t)| \bh_i(t); b_i, k_i, \bw_i, S(t)], \label{eqn:loglik:definition-expl} \end{eqnarray} - -\noindent where $\bh(t)=\{h_i(t)\}_{i=1}^N$. Thus we need only compute low-dimensional marginals of the full posterior distribution $P[\bX(t) | \bF; \bth]$; specifically, we need pairwise marginals, of the form $P[X_i(t), X_i(t-1) | \bF; \bth]$. The high dimensionality of the hidden variable $\bX$ necessitates the development of specialized blockwise Gibbs-SMC sampling methods, as we describe in sections \ref{sec:methods:indep} and \ref{sec:methods:joint} below. Once we have obtained these marginals, the M-step breaks up into a number of independent optimizations that may be computed in parallel and which are therefore relatively straightforward (section \ref{sec:methods:parameters HMM}); see section \ref{sec:methods:specific_implementation} for a pseudocode summary along with some specific implementation details. - -\subsection{Initialization of ``internal'' parameters via sequential Monte Carlo methods} \label{sec:methods:indep} - -We begin by constructing relatively cheap, approximate preliminary estimators for the nuisance parameters $\theta \backslash \bw_{ij}$, i.e., the observation parameters, $\{\alpha_i,\beta_i,\gamma_i,\sigma^F_i\}$, the calcium dynamics parameters $\{C^b_i,\tau^c_i, A_i, \sigma^c_i\}_i$, and the intrinsic dynamics parameters, $\{b_i,w_{ii}\}$ (note that we include all the parameters except $w_{ij}$ for $i\neq j$). The idea is to initialize our estimate $\theta^{(0)}$ by assuming that each neuron is observed independently. Thus we want to compute $P[X_i(t), X_i(t-\Delta) | \bF_i; \bth_i]$, and solve the M-step for each individual parameter $\theta_i$, with the connection matrix $\bw$ held fixed. This single-neuron case is much simpler, and has been discussed at length in \cite{Vogelstein2009}; therefore, we only provide a brief overview here. The standard forward and backward recursions provide these posteriors \cite{ShumwayStoffer06}: - -\begin{align} - P[X_i(t) | F_i(0:t)] &\propto P[F_i(t)| X_i(t)] \int P[X_i(t) -| X_i(t-\Delta)] P[X_i(t-\Delta) | F_i(0:t-\Delta)] dX_i(t-\Delta) -\label{eqn:forward} \\ P[X_i(t), X_i(t-\Delta) | F_i] &= P[X_i(t) | F_i] +where $\bh_i(t)=\{h_{ij}(t)\}_{j \leq N}$. Thus we need only compute +low-dimensional marginals of the full posterior distribution $P[\bX(t) +| \bF; \bth]$; specifically, we need pairwise marginals, of the form +$P[X_i(t), X_i(t- \Delta) | \bF; \bth]$. The high dimensionality of +the hidden variable $\bX$ necessitates the development of specialized +blockwise Gibbs-SMC sampling methods, as we describe in sections +\ref{sec:methods:indep} and \ref{sec:methods:joint} below. Once we +have obtained these marginals, the M-step breaks up into a number of +independent optimizations that may be computed in parallel and which +are therefore relatively straightforward (section +\ref{sec:methods:parameters HMM}); see section +\ref{sec:methods:specific_implementation} for a pseudocode summary +along with some specific implementation details. + +\subsection{Initialization of ``intrinsic'' parameters via sequential + Monte Carlo methods} \label{sec:methods:indep} + +We begin by constructing relatively cheap, approximate preliminary +estimators for the nuisance parameters $\theta \backslash +\{w_{ij}\}_{i \neq j}$, i.e., the observation parameters, +$\{\alpha_i,\beta_i,\gamma_i,\sigma^F_i\}$, the calcium dynamics +parameters $\{C^b_i,\tau^c_i, A_i, \sigma^c_i\}_i$, and the intrinsic +spiking parameters, $\{k_i,b_i,w_{ii}\}$. The idea is to initialize +our estimator by assuming that each neuron is observed independently. +Thus we want to compute $P[X_i(t), X_i(t-\Delta) | \bF_i; \bth_i]$ +(where $\bF_i$ denotes the $i$-th observed fluorescence signal), and +solve the M-step for each individual parameter $\theta_i$ (where +$\theta_i$ includes all of neuron $i$'s intrinsic parameters but +excludes the connectivity parameters $\{w_{ij}\}_{i \neq j}$), with +the connection matrix parameter held fixed. This single-neuron case +is much simpler, and has been discussed at length in +\cite{Vogelstein2009}; therefore, we only provide a brief overview +here. The standard forward and backward recursions provide these +posteriors \cite{ShumwayStoffer06}: +\begin{align} +\hspace{-25mm} P[X_i(t) | F_i(0:t)] &\propto P[F_i(t)| X_i(t)] \int P[X_i(t) | + X_i(t-\Delta)] P[X_i(t-\Delta) | F_i(0:t-\Delta)] dX_i(t-\Delta), + ~ for ~ t=\Delta, 2 \Delta, \ldots, T +\label{eqn:forward} \\ +\hspace{-25mm} P[X_i(t), X_i(t-\Delta) | \bF_i] &= P[X_i(t) | \bF_i] \frac{P[X_i(t) | X_i(t-\Delta)] P[X_i(t-\Delta) | F_i(0:t-\Delta)]}{\int P[X_i(t) | X_i(t-\Delta)] P[X_i(t-\Delta) | -F_i(0:t-\Delta)] dX_i(t-\Delta)}, +F_i(0:t-\Delta)] dX_i(t-\Delta)}, ~ for ~ t=T, T- \Delta, \ldots, +\Delta \label{eqn:backward} \end{align} - -where we have dropped the conditioning on the parameters $\theta$ for brevity's sake. Because these integrals cannot be analytically evaluated for our model, we approximate them using SMC (``marginal particle filtering'') methods \cite{DGA00,DFG01,GDW04}; see \cite{Vogelstein2009} for details on the proposal density and resampling methods used here. The output of these SMC techniques comprise an array of particle positions $\{X_i^{(l)}(t)\}$, where $l$ indexes the particle number, and a discrete approximation to the marginals $P[X_i(t), X_i(t-\Delta) | F_i]$, - +where we have dropped the conditioning on the parameters $\theta$ for +brevity's sake. Because these integrals cannot be analytically +evaluated for our model, we approximate them using SMC (``marginal +particle filtering'') methods \cite{DGA00,DFG01,GDW04}; see +\cite{Vogelstein2009} for details on the proposal density and +resampling methods used here. The output of these SMC techniques +comprise an array of particle positions $\{X_i^{(m)}(t)\}$, where $m$ +indexes the particle number, and a discrete approximation to the +marginals $P[X_i(t), X_i(t-\Delta) | \bF_i, \theta_i]$, \begin{equation} - P[X_i(t), X_i(t-\Delta) | F_i] \approx \sum_{j,l} - r_i^{(j,l)}(t,t-\Delta) \delta \left[ X_i(t) - X_i^{(l)}(t) \right] - \times \delta \left[ X_i(t-\Delta) - X_i^{(j)}(t-\Delta) \right], + P[X_i(t), X_i(t-\Delta) | \bF_i, \theta_i] \approx \sum_{m,m'} + r_i^{(m,m')}(t,t-\Delta) \delta \left[ X_i(t) - X_i^{(m)}(t) \right] + \times \delta \left[ X_i(t-\Delta) - X_i^{(m')}(t-\Delta) \right], \label{eq:particle-fb} \end{equation} -where $r_i^{(j,l)}(t,t-\Delta)$ denotes the weight attached to the -particle pair with positions $\left( X_i^{(l)}(t), X_i^{(j)}(t-\Delta) -\right)$. +where $r_i^{(m,m')}(t,t-\Delta)$ denotes the weight attached to the +pair of particles with positions $\left( X_i^{(m)}(t), +X_i^{(m')}(t-\Delta) \right)$, and $\delta(.)$ denotes a Dirac mass. As discussed above, the sufficient statistics for estimating the -parameters for each neuron, $\bth_i$, are exactly these marginal -posteriors. As shown in Eq. \ref{eqn:loglik:definition-expl}, the -M-step decouples into three independent subproblems. The first term -depends on only $\{\alpha_i, \beta_i, \gamma_i, \sigma_i\}$; since -$\ln P[F_i(t)|C_i(t); \theta_i]$ is quadratic (by our Gaussian -assumption on the fluorescent observation noise), we can estimate -these parameters by solving a weighted regression problem -(specifically, we use a coordinate-optimization approach: we solve a -quadratic problem for $\{\alpha_i, \beta_i\}$ while holding -$\{\gamma_i, \sigma_i\}$ fixed, then estimate $\{\gamma_i,\sigma_i\}$ -by the usual residual error formulas while holding $\{\alpha_i, -\beta_i\}$ fixed). Similarly, the second term requires us to optimize -over $\{\tau_i^c, A_i, C_i^b\}$ using a quadratic solver, and then we -use the residuals to estimate $\sigma_i^c$. Note that all the -parameters mentioned so far are constrained to be non-negative, but -may be solved efficiently using standard quadratic program solvers. -Finally, the last term, assuming neurons are independent, may be -expanded: +intrinsic parameters for each neuron, $\bth_i$, are exactly these +marginal posteriors. As shown in +Eq. \ref{eqn:loglik:definition-expl}, the M-step decouples into three +independent subproblems. The first term depends on only $\{\alpha_i, +\beta_i, \gamma_i, \sigma_i\}$; since $\ln P[F_i(t)|C_i(t); \theta_i]$ +is quadratic (by our Gaussian assumption on the fluorescent +observation noise), we can estimate these parameters by solving a +weighted regression problem (specifically, we use a +coordinate-optimization approach: we solve a quadratic problem for +$\{\alpha_i, \beta_i\}$ while holding $\{\gamma_i, \sigma_i\}$ fixed, +then estimate $\{\gamma_i,\sigma_i\}$ by the usual residual error +formulas while holding $\{\alpha_i, \beta_i\}$ fixed). Similarly, the +second term requires us to optimize over $\{\tau_i^c, A_i, C_i^b\}$, +and then we use the residuals to estimate $\sigma_i^c$. Note that all +the parameters mentioned so far are constrained to be non-negative, +but may be solved efficiently using standard quadratic program solvers +if we use the simple reparameterization $\tau_i^c \to 1- \Delta / +\tau_i^c$. Finally, the last term, assuming neurons are independent, +may be expanded: \begin{equation} - E [\ln P[n_i(t), \bh_i(t) | \bF; \bth]] = P[n_i(t), h_i(t) | F_i] -\ln f (J_i(t)] + (1-P[n_i(t), h_i(t) | F_i]) \ln [1- f(J_i(t))]; + E [\ln P[n_i(t), \bh_i(t) | \bF; \bth]] = P[n_i(t), \bh_i(t) | +\bF] \ln f (J_i(t)] + (1-P[n_i(t), \bh_i(t) | \bF]) \ln [1- +f(J_i(t))]; \label{eqn:bw} \end{equation} - since $J_i(t)$ is a linear function of $(b_i,k_i,\bw_i)$, and the right-hand side of (\ref{eqn:bw}) is concave in $J_i(t)$, we see that -the third term in \ref{eqn:loglik:definition-expl} is a sum of terms -which are concave in $(b_i,k_i,\bw_i)$, and may therefore be solved -efficiently using any convex optimization method, e.g.\ Newton-Raphson -or conjugate gradient ascent. +the third term in Eq.~(\ref{eqn:loglik:definition-expl}) is a sum of +terms which are concave in $(b_i,k_i,\bw_i)$ --- and therefore also +concave in the linear subspace $(b_i,k_i, w_{ii})$ with $\{w_{ij}\}_{i +\neq j}$ held fixed --- and may thus be maximized efficiently using +any convex optimization method, e.g.\ Newton-Raphson or conjugate +gradient ascent. Our procedure therefore is to initialize the parameters for each -neuron using some default values that we have found to be effective in -practice, and then recursively (i) estimate the marginal posteriors (E -step), and (ii) maximize the parameters (M step), using the above -described approach. We iterate these two steps until the change in -parameters does not exceed some minimum threshold. We can then use +neuron using some default values that we have found to be practically +effective in analyzing real data, and then recursively (i) estimate +the marginal posteriors via Eq.~(\ref{eq:particle-fb}) (E step), and +(ii) maximize the intrinsic parameters $\theta_i$ (M step), using the +above described approach. We iterate these two steps until the change +in parameters does not exceed some minimum threshold. We can then use the marginal posteriors from the last iteration to seed the blockwise Gibbs sampling procedure described below, to obtain a rough estimate -of $P[\bh(t) | \bF]$. +of $P[\bh_i(t), n_i(t) | \bF]$. \subsection{Estimating joint posteriors over weakly coupled neurons} \label{sec:methods:joint} -Now we turn to the key problem: computing $P(\bh(t),n_i(t) | F,\theta)$, which encapsulates the sufficient statistics for estimating the connectivity matrix $\bw$ (recall equation (\ref{eqn:loglik:definition-expl})). The SMC methods described in the preceding section only provide the marginals over each neuron, $P[X_i(t) | F_i; \bth_i]$; these methods may in principle be extended to obtain the desired full posterior $P[\bX(t) | \bF; \bth]$, but since the SMC algorithm is fundamentally a sequential importance sampling method, these techniques scale poorly as the dimensionality of the hidden state $\bX(t)$ increases \cite{BickelBengtsson08}. Thus we need a different approach. +Now we turn to the key problem: computing $P(\bh_i(t),n_i(t) | +\bF,\theta)$, which encapsulates the sufficient statistics for +estimating the connectivity matrix $\bw$ (recall equation +(\ref{eqn:loglik:definition-expl})). The SMC methods described in the +preceding section only provide the marginals over each neuron, +$P[X_i(t), X_i(t-\Delta) | \bF_i; \bth_i]$; these methods may in +principle be extended to obtain the desired full posterior $P[\bX(t), +\bX(t-\Delta) | \bF; \bth]$, but the SMC algorithm is fundamentally a +sequential importance sampling method, and therefore scales poorly as +the dimensionality of the hidden state $\bX(t)$ increases +\cite{BickelBengtsson08}. Thus we need a different approach. One very simple idea is to use a Gibbs sampler: sample sequentially from - \begin{align} X_i(t) \sim P[X_i(t) | \bX_{\i}, X_i(0), \ldots, X_i(t-\Delta), X_i(t+\Delta), \ldots, X_i(T), \bF; \bth], \end{align} - -looping in some order over all cells $i$ and all time bins $t$. -Unfortunately, this approach is likely to mix very poorly, due to the -strong temporal dependence between $X_i(t)$ and $X_i(t+\Delta)$. -Instead, we propose to use a blockwise Gibbs strategy, sampling each -spike train as a block: - +looping over all cells $i$ and all time bins $t$. Unfortunately, this +approach is likely to mix poorly, due to the strong temporal +dependence between $X_i(t)$ and $X_i(t+\Delta)$. Instead, we propose +a blockwise Gibbs strategy, sampling each spike train as a block: \begin{align} X_i \sim P[X_i | \bX_{\i}, \bF; \bth]; \end{align} - -if we can draw these blockwise samples $X_i = \{X_i(t)\}$ efficiently for a large subset of timebins $t$ simultaneously, then we would expect the resulting Markov chain to mix much more quickly than the naive element-wise Gibbs chain, since by assumption the hidden variables $X_i,X_j$ are weakly dependent for different cells $i \neq j$, and Gibbs is most efficient for weakly-dependent variables. - -So, how can we efficiently sample from $P[X_i | \bX_{\i}, \bF; \bth]$? One attractive approach is to try to repurpose the SMC methods described above, which are quite effective for drawing approximate samples from $P[X_i | \bX_{\i}, \bF_i; \bth]$ for one neuron $i$ at a time. Recall that sampling from an HMM is in principle easy by the ``propagate forward, sample backward'' method: we first compute the forward probabilities $P[X_i(t) | \bX_{\i}(0:t), \bF(0:t); \bth]$ recursively for timesteps $0$ up to $T$, then sample backwards from $P[X_i(t) | \bX_{\i}(0:T), \bF(0:T), X_i(t+\Delta); \bth]$. This approach is powerful because each sample requires just linear time to compute (i.e., $O(T/\Delta)$ time, where $T/\Delta$ is the number of desired time steps). Unfortunately, in this case we can only compute the forward probabilities approximately (with the SMC forward recursion (\ref{eqn:forward})), and so therefore this attractive forward-backward approach only provides approximate samples from $P[X_i | \bX_{\i}, \bF; \bth]$, not the exact samples required to establish the validity of the Gibbs method. +if we can draw these blockwise samples $X_i = \{X_i(t)\}$ efficiently +for a large subset of adjacent timebins $t$ simultaneously, then we +would expect the resulting Markov chain to mix much more quickly than +the single-element Gibbs chain, since by assumption the hidden +variables $\bX_i,\bX_j$ are weakly dependent for different cells $i +\neq j$, and Gibbs is most efficient for weakly-dependent variables +\cite{RC05}. + +So, how can we efficiently sample from $P[\bX_i | \bX_{\i}, \bF; +\bth]$? One attractive approach is to try to repurpose the SMC +methods described above, which are quite effective for drawing +approximate samples from $P[\bX_i | \bX_{\i}, \bF_i; \bth]$ for one +neuron $i$ at a time. Recall that sampling from an HMM is in +principle easy by the ``propagate forward, sample backward'' method: +we first compute the forward probabilities $P[X_i(t) | \bX_{\i}(0:t), +F_i(0:t); \bth]$ recursively for timesteps $0$ up to $T$, then sample +backwards from $P[X_i(t) | \bX_{\i}(0:T), F_i(0:T), X_i(t+\Delta); +\bth]$. This approach is powerful because each sample requires just +linear time to compute (i.e., $O(T/\Delta)$ time, where $T/\Delta$ is +the number of desired time steps). Unfortunately, in this case we can +only compute the forward probabilities approximately (with the SMC +forward recursion (\ref{eqn:forward})), and so therefore this +attractive forward-backward approach only provides approximate samples +from $P[\bX_i | \bX_{\i}, \bF; \bth]$, not the exact samples required +to establish the validity of the Gibbs method. Of course, in principle we should be able to use the Metropolis-Hastings (M-H) algorithm to correct these approximate samples. The problem is that the M-H acceptance ratio in this setting involves a high-dimensional integral over the set of paths that the particle filter might possibly trace out, and is therefore difficult to compute directly. \cite{Andrieu2007} discuss this problem at more length, along with some proposed solutions. However, a slightly simpler approach was introduced by \cite{NBR03}. Their idea is to exploit the $O(T/\Delta)$ forward-backward sampling method by embedding a discrete Markov chain within the continuous state space $\mathcal{X}_t$; the state space of this discrete embedded chain is sampled randomly according to some distribution $\rho_t$ with support on $\mathcal{X}_t$. It turns out that an appropriate acceptance probability (defined in terms of the original state space model transition and observation probabilities, along with the auxiliary sampling distributions $\rho_t$) may be computed quite tractably, guaranteeing that the samples produced by this algorithm form a Markov chain with the desired equilibrium density. See \cite{NBR03} for details. -We can apply this embedded-chain method quite directly here to sample from $P[X_i | \bX_{\i}, \bF; \bth]$. The one remaining question is how to choose the auxiliary densities $\rho_t$. We would like to choose these densities to be close to the desired marginal densities $P[X_i(t) | \bX_{\i}, \bF; \bth]$, and conveniently, we have already computed a good (discrete) approximation to these densities, using the SMC methods described in the last section. The algorithm described in \cite{NBR03} requires that $\rho_t$ be continuous densities, so we simply convolve our discrete SMC-based approximation (specifically, the marginal of (\ref{eq:particle-fb})) with an appropriate normal density to arrive at a very tractable mixture-of-Gaussians representation for $\rho_t$. - -Thus, to summarize, our procedure for sampling from the desired joint state distributions $P(\bh(t),n_i(t) | F,\theta)$ has a Metropolis-within-blockwise-Gibbs flavor, where the internal Metropolis step is replaced by the $O(T/\Delta)$ embedded-chain method introduced by \cite{NBR03}, and the auxiliary densities $\rho_t$ necessary for implementing the embedded-chain sampler are obtained using the SMC methods from \cite{Vogelstein2009}. - -\subsubsection{A cheaper high-SNR approximation of the joint posteriors} +We can apply this embedded-chain method quite directly here to sample +from $P[\bX_i | \bX_{\i}, \bF; \bth]$. The one remaining question is +how to choose the auxiliary densities $\rho_t$. We would like to +choose these densities to be close to the desired marginal densities +$P[X_i(t) | \bX_{\i}, \bF; \bth]$, and conveniently, we have already +computed a good (discrete) approximation to these densities, using the +SMC methods described in the last section. The algorithm described in +\cite{NBR03} requires that $\rho_t$ be continuous densities, so we +simply convolve our discrete SMC-based approximation (specifically, +the $X_i(t)$-marginal of Eq.~(\ref{eq:particle-fb})) with an +appropriate normal density to arrive at a very tractable +mixture-of-Gaussians representation for $\rho_t$. + +Thus, to summarize, our procedure for sampling from the desired joint +state distributions $P(\bh_i(t),n_i(t) | F,\theta)$ has a +Metropolis-within-blockwise-Gibbs flavor, where the internal +Metropolis step is replaced by the $O(T/\Delta)$ embedded-chain method +introduced by \cite{NBR03}, and the auxiliary densities $\rho_t$ +necessary for implementing the embedded-chain sampler are obtained +using the SMC methods from \cite{Vogelstein2009}. + +\subsubsection{A factorized approximation of the joint posteriors} \label{sec:cheaper-high-snr} -If the SNR in the calcium imaging is sufficiently high, then by definition the observed fluorescence data $F_i$ will provide enough information to exactly determine the underlying hidden variables $X_i$. Thus, in this case the joint posterior approximately factorizes into a product of marginals for each neuron $i$: +If the SNR in the calcium imaging is sufficiently high, then by +definition the observed fluorescence data $F_i$ will provide enough +information to determine the underlying hidden variables $X_i$. Thus, +in this case the joint posterior approximately factorizes into a +product of marginals for each neuron $i$: \begin{equation} \label{eqn:indep_approx} - P[{\bf X}|{\bf F};\theta] \approx \prod_{i=1}^N P[X_i|F_i; \bth]. + P[{\bf X}|{\bf F};\theta] \approx \prod_{i=1}^N P[\bX_i|\bF_i; + \bth]. \end{equation} -We can take advantage of this representation because we have already estimated all the above marginals using the SMC methods described in section \ref{sec:methods:indep}. In particular, we can obtain the sufficient statistics $P(\bh(t),n_i(t) | \bF,\theta)$ by forming a product over the marginals $P(X_i(t) | \bF_i,\theta)$ obtained from (\ref{eq:particle-fb}). This approximation entails a very significant gain in efficiency for two reasons: first, it obviates the need to generate joint samples via the expensive blockwise-Gibbs approach described above; and second, because we can very easily parallelize the SMC step, inferring the marginals $P[X_i(t) | \bF_i; \bth_i]$ and estimating parameters $\bth_i$ for each neuron on a separate processor. We will discuss the empirical accuracy of this approximation in more depth in the Results section. +We can take advantage of this representation because we have already +estimated all the above marginals using the SMC methods described in +section \ref{sec:methods:indep}. In particular, we can obtain the +sufficient statistics $P(\bh_i(t),n_i(t) | \bF,\theta)$ by forming a +product over the marginals $P(X_i(t) | \bF_i,\theta)$ obtained from +(\ref{eq:particle-fb}). This factorized approximation entails a very +significant gain in efficiency for two reasons: first, it obviates the +need to generate joint samples via the expensive blockwise-Gibbs +approach described above; and second, because we can very easily +parallelize the SMC step, inferring the marginals $P[X_i(t) | \bF_i; +\bth]$ and estimating the parameters $(\bth_i, \bw_i)$ for each neuron +on a separate processor. We will discuss the empirical accuracy of +this approximation in more depth in the Results section. \subsection{Estimating the functional connectivity matrix} \label{sec:methods:parameters HMM} Computing the M-step for the connectivity matrix, $\bw$, is an -optimization problem with on the order of $N^2$ variables. By -construction, however, the auxiliary function -(\ref{eqn:loglik:definition-expl}) is concave in $\bw$, and decomposes -into $N$ terms which may be optimized independently using standard -ascent methods. To improve our estimates, we will incorporate two -sources of strong \emph{a priori} information via our prior $P(\bw)$: -first, prior anatomical studies have established that connectivity in -many neuroanatomical substrates is ``sparse,'' i.e., most neurons form -synapses with only a fraction of their neighbors +optimization problem with on the order of $N^2$ variables. The +auxiliary function (\ref{eqn:loglik:definition-expl}) is concave in +$\bw$, and decomposes into $N$ separable terms that may be optimized +independently using standard ascent methods. To improve our +estimates, we will incorporate two sources of strong \emph{a priori} +information via our prior $P(\bw)$: first, previous anatomical studies +have established that connectivity in many neuroanatomical substrates +is ``sparse,'' i.e., most neurons form synapses with only a fraction +of their neighbors \cite{Buhl94,Thompson88,Reyes98,Feldmeyer99,Gupta00,FeldmeyerSakmann00,PetersenSakmann00,Binzegger04,Song2005,Mishchenko2009b}, implying that many elements of the connectivity matrix $\bw$ are zero; see also \cite{PAN04c,Rigat06,PILL07,Stevenson08} for further @@ -208,9 +418,9 @@ \subsection{Estimating the functional connectivity matrix} \label{sec:methods:pa \subsubsection{Imposing a sparse prior on the functional connectivity} -Enforcing sparseness for signal recovered with a series of linear -measurements via $L1$-regularizer is known to dramatically reduce the -amount of data necessary to accurately reconstruct the signal +It is well-known that imposing sparseness via an $L1$-regularizer can +dramatically reduce the amount of data necessary to accurately +reconstruct high-dimensional parameters \cite{Tibs96,TIP01,DE03,NG04,Candes2005,Mishchenko2009}. We incorporate a prior of the form $\ln p(\bw) = const. - \lambda \sum_{i,j} |\w_{ij}|$, and additionally enforce the constraints @@ -225,42 +435,70 @@ \subsubsection{Imposing a sparse prior on the functional connectivity} \subsubsection{Imposing Dale's law on the functional connectivity} -Enforcing Dale's law requires us to solve a non-convex, non-separable problem: we need to optimize the concave function $Q(\bth,\bth^{(l)}) + \ln P(\theta)$ under the non-convex, non-separable constraint that all of the columns of the matrix $\bw$ are of a fixed sign (either nonpositive or nonnegative). It is difficult to solve this problem exactly, but we have found that simple greedy methods are quite efficient in finding good (possibly approximate) solutions. We begin with our original sparse solution, obtained as discussed in the previous subsection without enforcing Dale's law. Then we assign each neuron as either excitatory or inhibitory, based on the weights we have inferred in the previous step: i.e., neurons $i$ whose inferred postsynaptic connections $w_{ij}$ are largely positive are tentatively labeled excitatory, and neurons with largely inhibitory inferred postsynapic connections are labeled inhibitory. Neurons which are highly ambiguous may be unassigned in the early iterations, to avoid making mistakes from which it might be difficult to recover. Given the assignments $a_i$ ($a_i =1$ for putative excitatory cells, $-1$ for inhibitory, and $0$ for neurons which have not yet been assigned) we solve the convex, separable problem - +Enforcing Dale's law requires us to solve a non-convex, non-separable +problem: we need to optimize the concave function $Q(\bth,\bth^{(l)}) ++ \ln P(\theta)$ under the non-convex, non-separable constraint that +all of the elements in any column of the matrix $\bw$ are of a +definite sign (either nonpositive or nonnegative). It is difficult to +solve this nonconvex problem exactly, but we have found that simple +greedy methods are quite efficient in finding good approximate +solutions. We begin with our original sparse solution, obtained as +discussed in the previous subsection without enforcing Dale's +law. Then we assign each neuron as either excitatory or inhibitory, +based on the weights we have inferred in the previous step: i.e., +neurons $i$ whose inferred postsynaptic connections $w_{ij}$ are +largely positive are tentatively labeled excitatory, and neurons with +largely inhibitory inferred postsynapic connections are labeled +inhibitory. Neurons which are highly ambiguous may be unassigned in +the early iterations, to avoid making mistakes from which it might be +difficult to recover. Given the assignments $a_i$ ($a_i =1$ for +putative excitatory cells, $-1$ for inhibitory, and $0$ for neurons +which have not yet been assigned) we solve the convex, separable +problem \begin{equation} -\argmax_{\substack{a_i \w_{ij} \geq 0 ~ \forall i,j}} Q(\bth,\bth^{(l)}) + -\ln P(\theta), +\argmax_{\substack{a_i \w_{ij} \geq 0, |w_{ij}|\eta^w$} \ForAll{$i=1\ldots N$} \While{$|{\bth_i}^{(l)}-{\bth_i}^{(l-1)}|> \eta^F$} - \State Approximate $P[X_i(t)|F_i; \bth]$ using SMC (section + \State Approximate $P[X_i(t)|\bF_i; \bth]$ using SMC (section \ref{sec:methods:indep}) - \State Perform the M-step for the ``internal'' parameters - $\theta \backslash \bw_{ij}$ (section \ref{sec:methods:indep}) + \State Perform the M-step for the ``intrinsic'' parameters + $\theta \backslash \{w_{ij}\}_{i \neq j}$ (section + \ref{sec:methods:indep}) \EndWhile \EndFor \ForAll{$i=1\ldots N$} - \State Approximate $P[n_i(t), \bh(t) |{\bf F}; \bth]$ using + \State Approximate $P[n_i(t), \bh_i(t) |{\bf F}; \bth]$ using either the blockwise Gibbs method or the high-SNR factorized approximation (section \ref{sec:methods:joint}) \EndFor @@ -272,13 +510,26 @@ \subsection{Specific implementation notes} \label{sec:methods:specific_implement \end{algorithmic} \end{algorithm} -As emphasized above, the parallel nature of these EM steps is essential for making these computations tractable. We performed the bulk of our analysis on a high-performance cluster of Intel Xeon L5430 based computers (2.66 GHz). For 10 minutes of simulated fluorescence data, imaged at $30$ Hz, calculations typically took 10-20 minutes per neuron using the factorized approximation, with time split approximately equally between (i) estimating the internal parameters $\theta \backslash \bw_{ij}$, (ii) approximating the posteriors using the independent SMC method, and (iii) estimating the functional connectivity matrix, $\bw$. The hybrid embedded-chain-within-blockwise-Gibbs sampler was substantially slower, up to an hour per neuron per Gibbs pass, with the Gibbs sampler dominating the computation time (because the Gibbs process requires 5 Gibbs loops, each requiring about 5 MCMC forward-backward passes). - -\begin{figure}[h] +As emphasized above, the parallel nature of these EM steps is +essential for making these computations tractable. We performed the +bulk of our analysis on a 100-node cluster of Intel Xeon L5430 based +computers (2.66 GHz). For 10 minutes of simulated fluorescence data, +imaged at $30$ Hz, calculations using the factorized approximation +typically took 10-20 minutes per neuron (divided by the number of +available processing nodes on the cluster), with time split +approximately equally between (i) estimating the intrinsic parameters +$\theta \backslash \bw_{ij}$, (ii) approximating the posteriors using +the independent SMC method, and (iii) estimating the functional +connectivity matrix, $\bw$. The hybrid +embedded-chain-within-blockwise-Gibbs sampler was substantially +slower, up to an hour per neuron per Gibbs pass, with the Gibbs +sampler dominating the computation time, because we thinned the chain +by a factor of five (since we found empirically that the +autocorrelation of the Gibbs chain had a scale of about five time +steps). + +\begin{figure}[t!] \centering \includegraphics[width=\hsize]{../figs/yuri-paper-schematic} -\caption{Schematic overview. First we obtain large-scale calcium recordings of spontaneous -and evoked neural responses. These movies are then pre-processed to correct for movement artifacts, find regions-of-interest, and extract spike trains. Second, given the fluorescence traces from each neuron, we infer spike trains. Third, we use our population model to infer the most likely functional connectivity matrix, which is our approximation of the microcircuit.} -\label{fig:data_schematic} -\end{figure} - +\caption{Schematic overview. The raw observed data is a large-scale calcium fluorescence movie, which is pre-processed to correct for movement artifacts (in the in vivo setting) and find regions-of-interest (i.e., putative neurons); note that we have omitted details of these important preprocessing steps in this paper. Given the fluorescence traces from each neuron, we estimate the underlying spike trains (i.e., time series of neural activity) using statistical deconvolution methods. Then we estimate the parameters of a network model, given the observed data. Our major goal is to obtain an accurate estimate of the network connectivity matrix, which summarizes the information we are able to infer about the local neuronal microcircuit. This figure adapted from personal communications with Rafael Yuste, Brendon O.\ Watson, and Adam Packer.} +\label{fig:data_schematic} \end{figure} diff --git a/paper/results.tex b/paper/results.tex old mode 100644 new mode 100755 index c423492..ab13486 --- a/paper/results.tex +++ b/paper/results.tex @@ -1,41 +1,42 @@ \subsection{Simulating neural activity in a neural population} \label{sec:results:simulations} -To test the described method for inferring functional connectivity from calcium imaging data, we simulated networks (according to our model, Eqs. \ref{eqn:glm:definition} -- \ref{eqn:F:definition}) of stochastically connected neurons. Although simulations ran at $1$ msec time discretization, imaging rate was assumed to be much slower. Simulations lasted anywhere between 5 minutes and 1 hours (of simulated time). Model parameters were chosen based on experimental data available from the literature \cite{Braitenberg1998, Urquijo2000, Lefort2009, Sayer1990}. +To test the described method for inferring functional connectivity from calcium imaging data, we simulated networks (according to our model, Eqs. \ref{eqn:glm:definition} -- \ref{eqn:F:definition}) of stochastically connected neurons. Although simulations ran at $1$ msec time discretization, imaging rate was assumed to be much slower. Simulations lasted anywhere between 5 minutes and 1 hours (of simulated time). Model parameters were chosen based on experimental data available from the literature \cite{Braitenberg1998,Urquijo2000,Lefort2009,Sayer1990}. -More specifically, the network was divided into excitatory (80\%) and inhibitory (20\%) neurons \cite{Braitenberg1998, Urquijo2000}, each respecting Dale's law, i.e., each neurons was either excitatory or inhibitory (corresponding to all positive or all negative columns in our functional connection weight matrix, $\bw$). Neurons were randomly connected to each other with probability $0.1$ \cite{Braitenberg1998, Lefort2009}. Synaptic weights for excitatory connections, as defined by EPSP peak amplitude, were randomly drawn from exponential distribution with the mean of $0.5 \mu V$ \cite{Lefort2009, Sayer1990}. To convert these into functional connectivity weights, we note that $w_{ij}=1$ corresponds to a jump in $J_i=1$, immediately following a spike from neuron $j$. Defining $V_\delta$ as the difference between threshold and rest for a neuron, and $V_E$ as the peak amplitude of an EPSP (and similarly for IPSP's), we can say the change in probability after a spike is $\Delta P=V_E/V_\delta$. Because Eq. \ref{eqn:glm:definition} provides the same quantity in our model parameter, $w_{ij}$, we have: +More specifically, the network was divided into excitatory (80\%) and inhibitory (20\%) neurons \cite{Braitenberg1998,Urquijo2000}, each respecting Dale's law, i.e., each neurons was either excitatory or inhibitory (corresponding to all positive or all negative columns in our functional connection weight matrix, $\bw$). Neurons were randomly connected to each other with probability $0.1$ \cite{Braitenberg1998,Lefort2009}. Synaptic weights for excitatory connections, as defined by EPSP peak amplitude, were randomly drawn from exponential distribution with the mean of $0.5 \mu V$ \cite{Lefort2009,Sayer1990}. To convert these into functional connectivity weights, we note that $w_{ij}=1$ corresponds to a jump in $J_i=1$, immediately following a spike from neuron $j$. Defining $V_\delta$ as the difference between threshold and rest for a neuron, and $V_E$ as the peak amplitude of an EPSP (and similarly for IPSP's), we can say the change in probability after a spike is $\Delta P=V_E/V_\delta$. Because Eq. \ref{eqn:glm:definition} provides the same quantity in our model parameter, $w_{ij}$, we have: \begin{equation}\label{eqn:convert} \w_{ij}=\ln(-\ln(e^{-r_i\tau_w}-V_E/V_{\delta})/r_i\tau_w), \end{equation} -\noindent where $r_i=\exp(b_i)$ is the base firing rate of neuron $i$ and $\tau_w=10$ msec was the typical EPSP/IPSP scale over which single EPSP affects the firing probability of the neuron $i$. Inhibitory connections were also drawn from exponential distribution; their strengths chosen so as to balance excitatory and inhibitory currents in the network, and achieve an average firing rate of $\approx 5 $ Hz. Practically, the mean strength of inhibitory connections was about 10 times larger than that of the excitatory connections. PSP shapes were modeled as the difference of two exponentials, corresponding to a sharp rise and relatively slow decay \cite{Sayer1990}. +\noindent where $r_i=\exp(b_i)$ is the base firing rate of neuron $i$ and $\tau_w=10$ msec was the typical EPSP/IPSP scale over which single EPSP affects the firing probability of the neuron $i$ \cite{}. Inhibitory connections were also drawn from exponential distribution; their strengths chosen so as to balance excitatory and inhibitory currents in the network, and achieve an average firing rate of $\approx 5 $ Hz \cite{Abeles01}. Practically, the mean strength of inhibitory connections was about 10 times larger than that of the excitatory connections. PSP shapes were modeled as an alpha function \cite{Koch99}, by differencing of two exponentials, corresponding to a sharp rise and relatively slow decay \cite{Sayer1990}. %The time course of functional connectivity weights $\w_{ij}(t)$ was modeled as the difference of two exponentials, resulting in a rise time of $\tau_r=1$ msec and decay time of $\tau_{E_d}=10$ msec for excitatory and $\tau_{I_d}=20$ msec for inhibitory currents . In practice, we sampled time constants from uniform distributions with means as stated above and support of $\pm 25 \%$ of the means. -We neglected conduction delays, given that the time delay below $\sim 1$ msec expected in local cortical circuit was smaller than the time step of our computer simulation. Each neuron also had an exponential refractory current with a $10$ msec time constant. XXX Y: did that not get sampled from some distribution? XXX +We neglected conduction delays, given that the time delay below $\sim 1$ msec expected in local cortical circuit was smaller than the time step of our computer simulation. Each neuron also had an exponential refractory current with a $10$ msec time constant \cite{Koch99}. XXX Y: did that not get sampled from some distribution? XXX -Parameters for the calcium dynamics and observation statistics were chosen according to our experience with several cells analyzed using algorithm of \cite{Vogelstein2009}. Each parameter was generated from a normal distribution with specified mean and variance at about 30\% of the mean, truncated at the lower bound at about 30\% of the mean value. Table \ref{table:caparm} provides details for each of the parameters in our model. +Parameters for the calcium dynamics and observation statistics were chosen according to our experience with several cells analyzed using algorithm of \cite{Vogelstein2009}, and conformed to other experimental observations \cite{ImagingManual,HelmchenSakmann96,BrenowitzRegehr07}. Each parameter was generated from a normal distribution with specified mean and variance at about 30\% of the mean, truncated at the lower bound at about 30\% of the mean value. Table \ref{table:caparm} provides details for each of the parameters in our model. \begin{table}[h!b!p!] -\caption{Table of simulation parameters. $\mathcal{E}(\lambda)$ indicates an exponential distribution with $\lambda$, $\mathcal{U}[a,b]$ indicates a uniform distribution between $a$ and $b$, and $\mathcal{TN}[x]$ indicates a truncated normal with mean $x$ and variance and lower bound of $30 \%$ the mean. XXX Y: can you fix this to look nice? I think the line for Ca jump is good now, assuming we explain that the gaussian is truncated at 2 std below mean (or something like that) XXX}\label{table:caparm} -\begin{tabular}{ll} +\caption{Table of simulation parameters. $\mathcal{E}(\lambda)$ indicates an exponential distribution with $\lambda$, and $\mathcal{N}(\mu,\sigma^2)$ indicates a truncated normal with mean $\mu$ and variance $\sigma^2$, and lower bound of one standard deviation below the mean.}\label{table:caparm} + +\begin{tabular}{lll} \hline Total neurons & 10-500 \\ -Excitatory neurons & 80\% \\ -Connections sparseness & 10\% \\ -Baseline firing rate & 5 Hz\\ +Excitatory neurons & $80\%$ \\ +Connections sparseness & $10\%$ \\ +Baseline firing rate & $5$ & Hz\\ \hline -Mean EPSP strength & $\sim \mathcal{E}[0.5] \mu$V \\ -Mean IPSP strength & $\sim -\mathcal{E}[2.3] \mu$V \\ -EPSP rise time & $\sim \mathcal{U}[0.75, 1.25]$ msec \\ -EPSP decay time & $\sim \mathcal{U}[7.5, 12.5]$ msec \\ -IPSP rise time & $\sim \mathcal{U}[0.75, 1.25]$ msec \\ -IPSP decay time & $\sim \mathcal{U}[15, 25]$ msec \\ +EPSP peak height & $\sim \mathcal{E}(0.5)$ & $\mu$V \\ +IPSP peak height & $\sim -\mathcal{E}(2.3)$ & $\mu$V \\ +EPSP rise time & $\sim \mathcal{N}(1,0.3)$ & msec \\ +EPSP decay time & $\sim \mathcal{N}(10,2.5)$& msec \\ +IPSP rise time & $\sim \mathcal{N}(1,0.3)$ & msec \\ +IPSP decay time & $\sim \mathcal{N}(20,7)$ & msec \\ \hline -Mean Ca noise $\sigma_c$ & $\sim \mathcal{TN} [28] \mu$M \\ -Calcium jump after spike, $A_c$ & $\sim \mathcal{N} (80, 20) \mu$M \\ -Mean Ca background $C_b$ & $\sim \mathcal{TN} [24] \mu$M \\ -Mean Ca decay time $\tau_c$ & $\sim \mathcal{TN} [0.25]$ sec \\ -Mean photon budget $\alpha_c$ & 1-80 Kph/neuron/frame \\ -$K_d$ & 200 $\mu$M \\ +Calcium std. $\sigma_c$ & $\sim \mathcal{N}(28,10)$ & $\mu$M \\ +Calcium jump after spike, $A_c$ & $\sim \mathcal{N}(80,20)$ & $\mu$M \\ +Calcium baseline, $C_b$ & $\sim \mathcal{N}(24,8)$ & $\mu$M \\ +Calcium decay time, $\tau_c$ & $\sim \mathcal{N}(500, 170)$ & msec \\ +Mean photon budget $\alpha_c$ & $1$--$80$ & Kph/neuron/frame \\ +Dissociation constant, $K_d$ & $200$ & $\mu$M \\ \hline \end{tabular} \end{table} @@ -43,12 +44,12 @@ \subsection{Simulating neural activity in a neural population} \label{sec:result \subsection{Inferring functional connectivity from the simulated calcium imaging data} \label{sec:results:inference} -With neural population activity prepared as described in the previous section, we used our inference algorithms to reconstruct the functional connectivity matrix from simulated fluorescence data. Specifically, we estimated the connectivity matrix by maximizing $E[\ln P[n_i(t)| \bh(t); b_i, \bw_i]]$ with respect to $\{b_i,\bw_i\}$, for each neuron (c.f. Eq. \ref{eqn:loglik:definition-expl}), using both the embedded-chain-within-blockwise-Gibbs approach as well as using factorized approximation, Figure \ref{fig:scatters}. We found that factorized approximation algorithm was able to provide reconstructions almost as accurate as the exact embedded-chain-within-blockwise-Gibbs approach --- $r^2=0.47$ versus $r^2=0.48$ --- when parameters corresponded to a realistic (but relatively high quality) preparation, given $10$ minutes simulation time, in a population of $N=25$ neurons. Figure \ref{fig:example_traces} depicts a couple different fluorescence traces, of varying SNR. In practice, we found that for data looking like the left panel, the factorization approach is sufficient, but for data like the right panel, too much information is lost by the factorization. - +With neural population activity prepared as described in the previous section, we used our inference algorithms to reconstruct the functional connectivity matrix from simulated fluorescence data. Specifically, we estimated the connectivity matrix by maximizing $E[\ln P[n_i(t)| \bh(t); b_i, \bw_i]]$ with respect to $\{b_i,\bw_i\}$, for each neuron (c.f. Eq. \ref{eqn:loglik:definition-expl}), using both the embedded-chain-within-blockwise-Gibbs approach as well as using factorized approximation, Figure \ref{fig:scatters}. We found that factorized approximation algorithm was able to provide reconstructions almost as accurate as the exact embedded-chain-within-blockwise-Gibbs approach --- $r^2=0.47$ versus $r^2=0.48$ --- when parameters corresponded to a realistic (but relatively high quality) preparation, given $10$ minutes simulation time, in a population of $N=25$ neurons. Figure \ref{fig:example_traces} depicts a couple different fluorescence traces, of varying SNR. +factorized \begin{figure}[h] \centering \includegraphics[width=\hsize]{../figs/example_traces} -\caption{Two in vivo fluorescence traces (top) and their corresponding inferred spike trains (bottom). The left panels show a high SNR case, that is sufficient quality for the factorization approximation to work nearly as well as the embedded-chain-within-Gibbs approach. The right panels show a low SNR case, in which the factorization approach is insufficient.} +\caption{Two in vivo fluorescence traces (top) and their corresponding inferred spike trains (bottom). The left panels show a high SNR case, that is sufficient quality for the factorized approximation to work nearly as well as the embedded-chain-within-Gibbs approach. The right panels show a low SNR case, in which the factorized approach is insufficient. The ordinate is in arbitrary units. Data from the laboratory of Tom Mrsic-Flogel.} \label{fig:example_traces} \end{figure} @@ -118,7 +119,7 @@ \subsection{Impact of different imaging frame rates, noise levels, and durations \begin{figure}[h] \centering \includegraphics[width=3in]{../figs/FigureA6_recvar_SNR} -\caption{Accuracy of inferred connectivity weights as function of the noise amount in the calcium imaging data, as quantified by experimental photon budget per neuron-frame, for frame rates of 15 Hz, 33 Hz and 66 Hz. Photon counts on the order of 20-40 Kph/frame/neuron are required to achieve the upper bound due by the frame rate. Connectivity matrix here was inferred from simulated fluorescence data using factorized approximation algorithm. Simulation conditions are the same as in Figure \ref{fig:recvar}.} +\caption{Accuracy of inferred connectivity weights as function of the noise amount in the calcium imaging data, as quantified by experimental photon budget per neuron-frame, for frame rates of 15 Hz, 33 Hz and 66 Hz. Photon counts on the order of 20-40 Kph/frame/neuron are required to achieve the upper bound due by the frame rate. Connectivity matrix here was inferred from simulated fluorescence data using factorized approximation algorithm. Simulation conditions are the same as in Figure \ref{fig:recvar}. Vertical black lines indicate noise levels for the two example traces shown in Figure \ref{fig:example_traces}, as determined using the SMC approach to estimate $\theta_i$ as described in Section \ref{sec:methods:indep}.} \label{fig:recvar-SNR} \end{figure} @@ -133,7 +134,7 @@ \subsection{Impact of different imaging frame rates, noise levels, and durations \end{figure} -What minimal conditions for the experimental setup should be met to allow for successful reconstruction of the connectivity from calcium imaging data? In Figures \ref{fig:recvar} -- \ref{fig:recvar-NT} we address this question. Figure \ref{fig:recvar} shows the quality of the inferred connectivity matrix as function of the imaging frame rate; imaging frame rates 30-60 Hz are needed to achieve meaningful reconstruction results. These imaging frame rates are feasible for already existing experimental setups \cite{NguyenParker01, ReddySaggau05, Iyer06, SalomeBourdieu06, ReddySaggau08}. Figure \ref{fig:recvar-SNR} shows the quality of the inferred connectivity matrix as function of effective SNR and photon budget. Operationally, we define effective SNR here as $eSNR=$ XXX and photon budget as $1/\gamma$XXX. Note, therefore, that the relationship between $eSNR$ and $1/\gamma$ is XXX. From our experience with the analysis of real cells \cite{Vogelstein2009}, the eSNR in real data was $\sim XXX$ for in vivo data collected at 15 Hz and $\sim XXX$ for in vitro data at the same frame rate. As can be seen from Figure \ref{fig:recvar-SNR}, the effective SNR necessary for accurate reconstructions was $XXX$. For lower photon counts, the amount of noise in calcium imaging data degraded inferred connectivity matrices significantly. Finally, Figure \ref{fig:recvar-NT} shows the quality of the inferred connectivity matrix as function of the experimental duration. The minimal duration for a particular $r^2$ depended substantially on whether prior information about the distribution of connectivity weights was incorporated into the M-step (as described in detail below). In particular, for the M-step lacking a sparse prior, the calcium imaging duration necessary to achieve $r^2=0.5$ for the reconstructed connectivity matrix was $T\approx 10$ min, whereas, for the same data, when incorporating a sparse prior, $r^2>0.5$ was achieved already at $T\sim 5$ min. These numbers appear to be well within limitations of the existing experimental preparations. Furthermore, in agreement with theoretical analysis in Section \ref{sec:methods}, the accuracy of the reconstruction did not depend on the size of imaged neural population, with the same reconstruction quality observed for the same amount of data for $N=50-200$ neurons. In all cases, good reconstructions were obtained with $T\sim 5--30$ min of calcium imaging data. +What minimal conditions for the experimental setup should be met to allow for successful reconstruction of the connectivity from calcium imaging data? In Figures \ref{fig:recvar} -- \ref{fig:recvar-NT} we address this question. Figure \ref{fig:recvar} shows the quality of the inferred connectivity matrix as function of the imaging frame rate; imaging frame rates 30-60 Hz are needed to achieve meaningful reconstruction results. These imaging frame rates are feasible for already existing experimental setups \cite{NguyenParker01,ReddySaggau05,Iyer06,SalomeBourdieu06,ReddySaggau08}. Figure \ref{fig:recvar-SNR} shows the quality of the inferred connectivity matrix as function of effective SNR and photon budget. Operationally, we define effective SNR here as $eSNR=$ XXX and photon budget as $1/\gamma$XXX. Note, therefore, that the relationship between $eSNR$ and $1/\gamma$ is XXX. From our experience with the analysis of real cells \cite{Vogelstein2009}, the eSNR in real data was $\sim XXX$ for in vivo data collected at 15 Hz and $\sim XXX$ for in vitro data at the same frame rate. As can be seen from Figure \ref{fig:recvar-SNR}, the effective SNR necessary for accurate reconstructions was $XXX$. For lower photon counts, the amount of noise in calcium imaging data degraded inferred connectivity matrices significantly. Finally, Figure \ref{fig:recvar-NT} shows the quality of the inferred connectivity matrix as function of the experimental duration. The minimal duration for a particular $r^2$ depended substantially on whether prior information about the distribution of connectivity weights was incorporated into the M-step (as described in detail below). In particular, for the M-step lacking a sparse prior, the calcium imaging duration necessary to achieve $r^2=0.5$ for the reconstructed connectivity matrix was $T\approx 10$ min, whereas, for the same data, when incorporating a sparse prior, $r^2>0.5$ was achieved already at $T\sim 5$ min. These numbers appear to be well within limitations of the existing experimental preparations. Furthermore, in agreement with theoretical analysis in Section \ref{sec:methods}, the accuracy of the reconstruction did not depend on the size of imaged neural population, with the same reconstruction quality observed for the same amount of data for $N=50-200$ neurons. In all cases, good reconstructions were obtained with $T\sim 5$--$30$ min of calcium imaging data. \subsection{Accuracy of the estimates and Fisher information matrix} \label{sec:methods:accuracy_Fisher}

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