CoTETE.jl

Continuous-Time Event-based Transfer Entropy

Contains an implementation of the estimator proposed in this paper

It is easy to call this package from Python. See this tutorial for a quick guide on how to do this.

Documentation

Introduction

Transfer entropy (TE) is a measure of information flow between time series. It can be used to infer functional and effective networks.

This package allows one to estimate the TE between event-based time series (such as spike trains or social media post times) in continuous time (that is, without discretising time into bins). The advantages of this approach over the discrete-time approach include:

• The continuous-time approach is provably consistent - it is guaranteed to converge to the true value of the TE in the limit of infinite data. The discrete-time estimator is not consistent. It is easy to create examples where it does not converge to the true value of the TE.
• The discrete-time approach is thwarted by having an effective limit on the total number of bins that can be used for history embeddings. This means that the user of this approach must choose between capturing relationships occurring over long time intervals, or those that occurr with fine time precision (by choosing either a large or small bin size). They can never capture both simultaneously. By contrast, the continuous-time approach can capture relationships occurring over relatively long time intervals with no loss of precision.
• On synthetic examples studied, the continuous-time approach converges orders of magnitude faster than the discrete-time approach and exhibits substantially lower bias.
• In the inference of structural and functional connectivity, the discrete-time approach was typically coupled with a surrogate generation method which utilised an incorrect null hypothesis. The use of this method can be demonstrated to lead to high false-positive rates. CoTETE.jl contains an implementation of a method for generating surrogates which conform to the correct null hypothesis of conditional independence. See our paper for more details on all of these points.

Transfer entropy has already been widely applied to the spiking activity of neurons. Notable work on the application of TE to spike trains include:

• The reconstruction of the structural connectivity of neurons from simulated calcium imaging data. See here for an extension to this work.
• The inference of structural connectivity from models of spiking neural networks (1, 2).
• Investigation of the energy efficiency of synaptic information transfer.
• The inference of functional and effective networks ( 1, 2, 3, 4 )

Getting Started

Install Julia

Clone this repo (make sure to include the --recurse-submodules flag so that the modified nearest neighbours package gets included as well as the --branch flag to avoid recent potentially unstable changes).

david@home:~$ git clone --recurse-submodules --branch v0.2.3 https://github.com/dpshorten/CoTETE.jl.git

make sure that CoTETE.jl/src/ is on your JULIA_LOAD_PATH. eg:

david@home:~$ export JULIA_LOAD_PATH=:/home/david/CoTETE.jl/src/

Fire up the Julia REPL

david@home:~$ julia

You will need to add three prerequisite packages.

Note for new Julia users: The Julia REPL has a nifty feature called prompt pasting, which means that it will automatically remove the julia> prompt when you paste. You can, therefore, just copy and paste the entire block below without worrying about these prompts.

julia> import Pkg
julia> Pkg.add("Distances")
julia> Pkg.add("StaticArrays")
julia> Pkg.add("SpecialFunctions")
julia> Pkg.add("Parameters")
julia> Pkg.add("StatsBase")

For the first example, lets estimate the TE between uncoupled homogeneous Poisson processes. This is covered in section II A of [1]. We first create the source and target processes, each with 1 000 events and with rate 1.

julia> source = 1e3*rand(Int(1e3));
julia> sort!(source);
julia> target = 1e3*rand(Int(1e3));
julia> sort!(target);

We can now estimate the TE between these processes, with history embeddings of length 1.

julia> import CoTETE
julia> parameters = CoTETE.CoTETEParameters(l_x = 1, l_y = 1);
julia> CoTETE.estimate_TE_from_event_times(parameters, target, source)

The answer should be close to 0.

Let's apply the estimator to a more complex problem. We shall simulate the process described as example B in [2]. The application of the estimator to this example is covered in section II B of [1]. We create the source process as before

julia> source = 1e3*rand(Int(1e3));
julia> sort!(source);

We create the target as an homogeneous Poisson process with rate 10. This rate has to be larger than the largest rate achieved by the rate function in our example, so that the thinning algorithm can operate.

julia> target = 1e3*rand(Int(1e4));
julia> sort!(target);

We then run the thinning algorithm on the target.

julia> function thin_target(source, target, target_rate)
           # Remove target events occurring before first source
    	   start_index = 1
    	   while target[start_index] < source[1]
           	 start_index += 1
    	   end
    	   target = target[start_index:end]

	   new_target = Float64[]
    	   index_of_last_source = 1
    	   for event in target
               while index_of_last_source < length(source) && source[index_of_last_source + 1] < event
               	     index_of_last_source += 1
               end
               distance_to_last_source = event - source[index_of_last_source]
               λ = 0.5 + 5exp(-50(distance_to_last_source - 0.5)^2) - 5exp(-50(-0.5)^2)
               if rand() < λ/target_rate
               	  push!(new_target, event)
               end
           end
    	   return new_target
       end
julia> target = thin_target(source, target, 10);

We can now estimate the TE

julia> CoTETE.estimate_TE_from_event_times(parameters, target, source)

The answer should be close to 0.5.

For both of these examples, increasing the number of events in the processes will give estimates closer to the true value.

Note on negative values

Kraskov-style estimators of information-theoretic quantities (such as this one) can produce negative values. In TE estimation this is most commonly encountered when the target present state has a strong dependence on the target history but is only weakly dependent (or conditionally independent) of the source history. This leads to a violation of the assumption of local uniformity and a negative bias. This issue is discussed in detail in the 9th paragraph of the Discussion section of the paper proposing this estimator. A good discussion of it can also be found in the JIDT documentation. As mentioned in these resources, the issue can be easily resolved by debiasing the estimator by subtracting the mean of the surrogate TE estimates from the estimated value. This debiasing procedure should be incorporated as an option in this library in the near future.

Note on short event sequences

When using the estimator on very short event sequences, it is recommended to set the variable use_exclusion_windows in the parameters struct to false. In very short event sequences, it is possible for all history embedding vectors to overlap. This could result in searches excluding all candidate neighbours, resulting in the estimator returning an error. See the Methods subsection `Handling dynamic correlations' in [1] for more details on the use of these windows.

Assistance

If you have any issues using this software, please add an issue here on github, or email me at [email protected]

References

[1] Shorten, D. P., Spinney, R. E., Lizier, J.T. (2021). Estimating Transfer Entropy in Continuous Time Between Neural Spike Trains or Other Event-Based Data. PLoS Computaional Biology.

[2] Spinney, R. E., Prokopenko, M., & Lizier, J. T. (2017). Transfer entropy in continuous time, with applications to jump and neural spiking processes. Physical Review E, 95(3), 032319.