Fig 1.
Diagrams highlighting the differences in the embeddings used by the discrete and continuous-time estimators.
The discrete-time estimator (A) divides the time series into time bins. A binary value is assigned to each bin denoting the presence or absence of a (spiking) event–alternatively, this could be a natural number to represent the occurrence of multiple events. The process is thus recast as a sequence of binary values and the history embeddings (xt−4:t−1 and yt−4:t−1) for each point are binary vectors. The probability of an event occurring in a bin, conditioned on its associated history embeddings, is estimated via the plugin (histogram) [38] estimator. Conversely, the continuous-time estimator (B) performs no time binning. History embeddings and
for events or
and
for arbitrary points in time (not shown in this figure, see Fig 10) are constructed from the raw interspike intervals. This approach estimates the TE by comparing the probabilities of the history embeddings of the target processes’ history as well as the joint history of the target and source processes at both the (spiking) events and arbitrary points in time.
Fig 2.
Evaluation of the continuous-time estimator on independent homogeneous Poisson processes.
The solid line shows the average TE rate across multiple runs and the shaded area spans from one standard deviation below the mean to one standard deviation above it. Plots are shown for two different values of k nearest neighbours, and four different values of the ratio of the number of sample points to the number of events NU/NX (See Methods).
Fig 3.
Result of the discrete-time estimator applied to independent homogeneous Poisson processes.
The solid line shows the average TE rate across multiple runs and the shaded area spans from one standard deviation below the mean to one standard deviation above it. Plots are shown for four different values of the bin width Δt as well as different source and target embedding lengths, l and m.
Fig 4.
The discrete-time and continuous-time estimators were run on coupled point processes for which the ground-truth value of the TE is known.
(A) shows the firing rate of the target process as a function of the history of the source. (B) and (C) show the estimates of the TE provided by the two estimators. The solid blue line shows the average TE rate across multiple runs and the shaded area spans from one standard deviation below the mean to one standard deviation above it. The black line shows the true value of the TE. For the continuous-time estimator the parameter values of NU/NX = 1 and k = 4 were used along with the ℓ1 (Manhattan) norm. Plots are shown for three different values of the length of the target history component lX. For the discrete-time estimator, plots are shown for four different values of the bin width Δt. The source and target history embedding lengths are chosen such that they extend back one time unit (the known length of the history dependence).
Fig 5.
Diagram of the noisy copy process.
Events in the mother process M occur periodically with intervals T + ξM (ξM and are noise terms). Events in the daughter processes D1 and D2 occur after each event in the mother process, at a distance of
(with
). (A) shows a graph of the dependencies with the labels on the edges representing delays. (B) shows a diagram of a representative spike raster.
Fig 6.
Results of the continuous-time estimator run on a noisy copy process , where conditioning on a strong common driver M should lead to zero information flow being inferred.
The translation ω of the source, relative to the target and common driver, controls the strength of the correlation between the source and target (maximal at zero translation). For each translation, the estimator is run on both the original process as well as embeddings generated via two surrogate generation methods: our proposed local permutation method and a traditional source time-shift method. The solid lines show the average TE rate across multiple runs and the shaded areas span from one standard deviation below the mean to one standard deviation above it. The bias of the estimator changes with the translation ω, and we expect the estimates to be consistent with appropriately generated surrogates reflecting the same strong common driver effect. This is the case for our local permutation surrogates, as shown in (A). This leads to the correct bias-corrected TE value of 0, as shown in (B).
Fig 7.
The p-values obtained when using continuous and discrete-time estimators to infer non-zero information flow in the noisy copy process.
The estimators are applied to both (expected to have zero flow) and
(expected to have non-zero flow and therefore be indicated as statistically significant). Only the results from the continuous-time estimator match these expectations. Ticks represent the particular combination of estimator and surrogate generation scheme making the correct inference in the majority of cases when a cutoff value of p = 0.05 is used. The dotted line shows p = 0.05.
Fig 8.
The scaling of the combination of the continuous-time estimator and the local permutation surrogate generation scheme on correctly identifying conditional independence relationships with increasing dimension and data size (A).
This is compared with the performance of the combination of the discrete-time estimator with the traditional time-shift surrogate generation procedure (B). The y axis represents the number of background processes being conditioned on. Above a certain moderate threshold of data size, the discrete-time approach infers a non-zero TE in all of the runs, including those where the source was in fact not connected to the target. This renders it impractical for this task.
Fig 9.
Results of both estimator and surrogate generation combinations being applied to data from simulations of a biophysical model of a neural circuit inspired by the pyloric circuit of the crustacean stomatogastric ganglion.
The circuit, shown in (A), is fully connected apart from the missing connection between the PY neuron and the AB/PD complex, and generates membrane potential traces which are bursty and highly-periodic with cross-correlated activity. The distribution of p values from the combination of the continuous-time estimator and local permutation surrogate generation scheme are shown in (C). They demonstrate that this combination is capable of correctly identifying the conditional dependence and independence relationships in this circuit in all runs, apart from two false negatives. By contrast, the distribution of p values produced by the combination of the discrete-time estimator and the traditional source time-shift surrogate generation method shown in (D) mis-specified the relationship from the PY to the ABPD in every run. Ticks represent the particular combination of estimator and surrogate generation scheme making the correct inference of dependence or independence in the majority of cases when a cutoff value of p = 0.05 is used.
Fig 10.
Examples of history embeddings.
(A) shows an example of a joint embedding constructed at a target event (). (B) shows an example of a joint embedding constructed at a sample event (
).
Fig 11.
Diagrammatic representation of the local permutation surrogate generation scheme.
For our chosen sample we find a
where we have that the
component of
is similar to the
component of
and
component of
is similar to the
component of
. We then form a single surrogate sample by combining the
and
components of
with the
component of
. Corresponding colours of the dotted interval lines indicates corresponding length. The grey boxes indicate a small delta.