Albany Center for Collaborative Computational Neuroscience - Data Analysis

Home

 A particle move involving annihilation. Before the move: . 
 After the move: . The constraints  are 
 preserved by this move.

Maple Code

MEspheres.mw: Maple Worksheet with examples of how to use the code.
spread.mpl: This is the actual program that performs the Swarm Particle Optimization.
ivols.mpl: Quasi-Monte Carlo computation of the intersecting volumes of 7 spheres.
Circ5.mpl: Image of the intersection regions of 5 circles.

Imaging and Model Building from Nanoprobe Data

New algorithms are currently being developed for 1) the visualization of local neuronal activity using swarm particle optimization and Maximum Entropy, and 2) to model the raw data from the nanoprobes as a discrete dynamic bayesian network. We plan to use the newest methods of bayesian analysis for modeling the joint distribution of the data array by as simple as possible a collection of meaningful parameters. These likelihoods will of course depend on the neuronal circuit under study but we expect some common characteristics to underlie all of them. The raw data from the nanoprobes will be used to produce a temporal sequence of 3D maps of local neuronal activity. These maps will be further assembled into movies with a user specified point of view. The raw data will also be used for estimating the structure and the parameters of a dynamic graphical model This probabilistic network will be used for testing hypotheses about brain activity in the neighborhood of the sensors. A description of both computational aspects follows.

Imaging by Swarm Particle Optimization and Maximum Entropy

To simplify the presentation and illustrate the main ideas of our new algorithms we think of the array of electrodes as located on a regular unit 3D lattice. To each electrode in this lattice we associate a ``listening’‘ sphere of radius $1/2<\delta< 2$ , so that spheres that are nearest neighbors (nn) overlap. After collecting readings

$n_{0},n_1,\ldots,n_6$ from a central electrode and its 6 nn in the

lattice, we would like to estimate the distribution of activity within the central sphere. We assume the readings $n_i\ge 0$ to be integers. These integers are proportional to the number of charges inside a given listening sphere. Due to the overlapping of nn spheres the actual total number $N$ of charges observed by a detector and its nn, is at most $N_{\max}=n_0 + n_1 \ldots + n_6$ but the minimum value

$N_{\min}$ for $N$ depends on $\delta$ . Thus, we are uncertain about

the exact location and number of charges within the total volume $V$ of the 7 nn spheres. Now suppose the $N$ charges were distributed uniformly in $V$ . Then, the expected numbers falling within the $m$ different regions created by the intersection of the 7 nn spheres will be proportional to the volumes of those $m$ regions. However, these expectations must agree with the observations $n_{0},\ldots,n_6$ and that is in general not possible unless the distribution of charges $p=(p_1,\ldots,p_m)$ deviates from the uniform $q=(q_1,\ldots,q_m)$ of volume proportions. We therefore, arrive to a classic constrained maximum entropy problem:

$\displaystyle \min_{N,p} \sum_{j=1}^{m} p_j \log\ \frac{p_j}{q_j}$

subject to:

$\begin{eqnarray*}\n N \sum_{j\in J_0} p_j = n_0 \\\n N \sum_{j\in J_1} p_j = n_1 \\\n \vdots \\\n N \sum_{j\in J_6} p_j = n_6 \\\n \end{eqnarray*}$

where $J_i$ is the set of indices of the regions that intersect the $i$ -th sphere and $N_{\min}\le N \le N_{\max}$ .

For a given $\delta$ we can easily obtain $m$ and $q$ by quasi-monte carlo. The $m$ sequence is piecewise constant given by $7, 13, 25, 37, 45, 53, 59$ at $\delta$ values

$0.5, 0.53849, 0.70735, 0.73695, 0.81819, 0.91785, 1.020203$ .

For example if $\delta< 0.5$ there are only 7 regions (one per detector) but as soon as $\delta> 0.5$ the value for $m$ jumps to 13, staying there until $\delta > .53849$ when it becomes 25, etc. The same quasi-monte carlo algorithm that produced the $m$ sequence above, is used for estimating the proportions $q$ for any $\delta$ . We solved the constrained maximum entropy problem by simulating annealing. We made use of the assumption of discrete integral values $n_j$ to implement simple stochastic particle moves that always preserve the constrains. We start with

$N=N_{\max}$ charges at the centers. We then move pairs of particles

at random from one region to another adjacent region in such a way that the total numbers in each of the spheres remain constant. We also allow creation and annihilation of charges that change the value for $N$ (see the above Figure).

This solves the reconstruction problem within one listening sphere. To get a complete image it is sufficient to sweep across the lattice but our preliminary experiments suggest other alternatives. More experimentation will be needed to fully optimize the procedure. We emphasize again that the actual geometry of the nanoprobes is not expected to always fit the assumption of regular spacing in all directions. The basic ideas illustrated above for the special case of a regular 3D lattice still hold whenever the distances between the probes are either known apriori or there is the possibility of estimating them from anatomical information or from other available data.

New Geometric Bayesian Model Selection and Parameter Estimation

To be able to test hypotheses about neural mechanisms we will model the raw data from the nanoprobes as a discrete dynamic bayesian network (DBN). Thus, an observation consists of a list of non negative integers $\{n(x,y,z,t)\}$ . Each entry in the list has associated a spacetime location that is useful for adding prior information and for pruning the search for best models. Huge simplifications are obtained by assuming that the data consists of (approximately) independent identically distributed (iid) chunks of binary variables. For finding these chunks we are planning to use the newly discovered geometric model selection method. Once the chunks are identified we plan to further reduce the dimensionality of the hypothesis space by performing an stochastic search in model space guided by the available anatomical and physical constraints. We anticipate the need to evaluate the performance of variational bayes, expectation propagation as well as MCMC methods for computing posterior expectations of relevant functions of the parameters.

Classification of Pathogens and Behaviors

After fitting probabilistic models for the observations, in the form of a bayesian network, we can exploit those models to obtain characteristic brain activity patterns for the different conditions of our experiments. For example, in the study of the inflammatory response circuit (see below) we can perform a standard bayesian classification algorithm (see e.g. to obtain signatures for the different pathogens. These signatures could lead us to discover some relevant easy to observe variables highly correlated with the different pathogens. An obvious immediate application of this procedure could be the rapid identification of a pathogen from for example body temperatures and blood pressure measurements on an infected organism. The availability of such a procedure would be important in the event of a bio-terrorist attack.

Using Hidden Markov Models for Estimating Neural States

The nanoprobes do not measure the firings of individual neurons directly. They only sense the presence of electrical charges that enter their listening sphere. By design, the nanoprobes in an array are of sizes typically smaller than individual neurons so that a sudden increase of charge within the listening sphere of a probe is expected to be produced by the firing of one (or at most a few) nearby neuron. One of our goals is to be able to reconstruct the activity of the unobserved neural net from the data provided by the nanoprobes. This goal can be achieved by adding extra unobserved parent nodes to our original DBN. These extra nodes represent local brain states at different times and are assumed to follow a finite state Markov Chain. Standard neurophysiology will be used to guide our choice for the hidden Markov Chain (i.e., number of states, labels for the states and the Markov transition diagram). Standard DBN techniques (see for example and open software are available for estimating the parameters of these kinds of models.

Page last modified on April 24, 2009, at 01:09 PM