Existing Methods to Reconstruct Neural Connections.

There are several existing methods that are published. In this post I’m going to describe some of them.

1. Neural​ ​connectivity​ ​analyzing​ ​using​ ​Connectivity​ ​Matrix​ ​construction [1]

As I said in the previous post, we gather the pairwise neural connections using calcium florescence imaging [2]. And using this method we can discover and analyze the neural connectivity. At the end of the post I will mention the references and if you interested in this method you can refer them.

This method uses Matrix completion methods and local thresholding to reconstruct neural connections. Basically this method has three steps.

• Connectivity​ ​Matrix​ ​Semi-Construction

Suppose there are N neurons are in the sample, then we use NxN dimension matrix to denote the pairwise connections that identified from the Calcium Florescence Imaging. But in this step we enter only fraction of data to the connectivity matrix. The completion of other half of the matrix will done in the next step.

• Connectivity​ ​Matrix​ ​Completion

To complete the rest of the matrix we use matrix completion techniques. There is a fact that we need to consider, many times neurons are connected in same way. As an example two different neurons may connect to the same neuron. Using this fact and matrix completion techniques we can deduce an equation to complete the rest of the matrix. 

X_M-calculated entries in the connectivity matrix

Z_M-corresponding entries of the connectivity matrix approximation Z

||.||* Used to illustrate nuclear form [3]

And we can use convex optimization solver to solve this equation [4]. Note that solving this equation is much cheaper than calculating pairwise connectivity scores. So that if we complete the other half of the matrix by using this equation we reduce the half of our computation.

Local Thresholding

• Local Thresholding

Using Local thresholding we can analyze the connectivity matrix [5], [6]. We apply the following equation to each neuron we can identify the connections of each this will help us to reconstruct the neural network.

     2. Transfer Entropy Method [7]

This method also gather data to analyze from Calcium Imaging. This method basically have three steps.

• Time Series Chart

From the data that gather from the Calcium Imaging we can draw the time series diagram for each neuron. The time series chart is the illumination inside the neuron with respect to time. We use this charts to analyze each neurons with other neurons.

• Transfer Entropy

We use following equation to calculate the transfer entropy. Consider two neurons as x and y.

Where,

X_t – Value of the time series of x at time t

Y_{t -} Value of the time series of y at time t

This equation calculates the next value of sequence of x with respect to its own history and get the deference between the next values of x with respect to history of y. If x do not depend on y, then TE = 0 and otherwise TE>0. After some modifications we can construct the neural network.

References,

[1]Fast algorithm for neural network reconstruction, Sean Bittner, Siheng Chen & Jelena Kovacevic, Available; http://repository.cmu.edu/cgi/viewcontent.cgi?article=1309&context=ece

[2]Imaging Calcium in Neurons, Christine Grienberger & Arthur Konnerth, Available;
http://www.sciencedirect.com/science/article/pii/S0896627312001729

[3] Nuclear norm. G.L. Litvinov (originator), Encyclopedia of Mathematics. Available;
http://www.encyclopediaofmath.org/index.php?title=Nuclear_norm&oldid=19242

[4] E. J. Cand`es and B. Recht, “Exact matrix completion via convex optimization,” Journal Foundations of Computational Mathematics, vol. 9, no. 2, pp. 717–772, Dec. 2009.

[5] Normalized Iterative Hard Thresholding for Matrix Completion, Jared Tanner & Ke Wei, Available;https://people.maths.ox.ac.uk/tanner/papers/TaWei_NIHT.pdf

[6] A Singular Value Thresholding Algorithm for Matrix Completion, Jian-Feng Cai, Emmanuel J. Cand`Es &Zuowei Shen, Available; https://statweb.stanford.edu/~candes/papers/SVT.pdf

[7] Reconstructing Neuronal Connectivity from Calcium Imaging Data Using Generalized Transfer Entropy, Jina Li, Available; http://elischolar.library.yale.edu/cgi/viewcontent.cgiarticle=1178&context=ysphtdl