Transfer Entropy for solving the challenge.

We are going to virtually map the connectivity of neurons in our brain. We are using calcium imaging data. We are going to build a algorithm after predicting the connections between neurons. Transfer entropy is a key for understand the connectivity between neurons. We should understand it first.

Transfer entropy is a measuring way of transfer of information between two processes which are random. By knowing the past values we can measure the amount of uncertainty in future values. It is called as transfer entropy. This is valid only for two variables. To compare more than two variables, we have to consider two by two.

You can see the transfer entropy formula below

 

Here, X_n is the value of time series X at time n, Y_n is the value of time series Y at time n. P indicates the transition probabilities.

Transfer entropy is working with both linear and non linear interactions. By using the history of both two variables x and y, this method incorporate directional and dynamical information. Transfer is directly connected with the generalized Markov condition. Measure of deviation from generalized Markov condition is the Transfer entropy of two variables. Transfer entropy measure the differences between the distributions of the next value of sequence X given its own history and distribution of the next value of X given its own history and the history of Y. Simply, if X does not depend on Y, then transfer entropy is consider as zero.

The resolution of the measurements is lower than the synaptic time constants. Therefore, separate firing events can fall within the same measuring window, and the firing patterns of the neurons reflect the true connectivity of the network only during inter-bursting mode. So we have to consider about special transfer entropy. It is called as modified transfer entropy. Below you can see the modified transfer entropy.

Here, X_n is the differential fluorescence level of neuron X at time n, Y_n is the differential fluorescence level of neuron Y at time n and g_n is the average differential fluorescence level of the network at time t.

Transfer entropy is used for find the connectivity between neurons. Applying transfer entropy to calcium imaging data is challenging. Those researches derived generalized transfer entropy to overcome the difficulties of the transfer entropy.

Generalized transfer entropy for build the connectivity matrix of the transfer entropy values. Those values are directed functional connectivity network of neuron. Calcium imaging techniques can be used to get those values. We can take the fluorescence measured at each time point as time bin, because calcium imaging acquisition rate is much slower than synaptic activity. During burst, the network is excitable. So detecting directed functional connectivity is hard. But during non-burst phase it is easy to detect the connectivity. So, when the network is in non-bursting phase, we take the data for the transfer entropy calculations. Though fluorescence is continuous data, it is needed to be quantized before applying.  

Using those generalized transfer entropy and calcium imaging data this challenge can be solved and we can take those neurons connectivity map for further understanding of the neurons system.