Estimating Temporal Causal Interaction between Spike Trains with Permutation and Transfer Entropy

<div><p>Estimating the causal interaction between neurons is very important for better understanding the functional connectivity in neuronal networks. We propose a method called normalized permutation transfer entropy (NPTE) to evaluate the temporal causal interaction between spike trains, which quantifies the fraction of ordinal information in a neuron that has presented in another one. The performance of this method is evaluated with the spike trains generated by an Izhikevich’s neuronal model. Results show that the NPTE method can effectively estimate the causal interaction between two neurons without influence of data length. Considering both the precision of time delay estimated and the robustness of information flow estimated against neuronal firing rate, the NPTE method is superior to other information theoretic method including normalized transfer entropy, symbolic transfer entropy and permutation conditional mutual information. To test the performance of NPTE on analyzing simulated biophysically realistic synapses, an Izhikevich’s cortical network that based on the neuronal model is employed. It is found that the NPTE method is able to characterize mutual interactions and identify spurious causality in a network of three neurons exactly. We conclude that the proposed method can obtain more reliable comparison of interactions between different pairs of neurons and is a promising tool to uncover more details on the neural coding.</p></div

Removal of the bias and normalization of the PTE.

<p>(A) Permutation transfer entropy. (B) Unbiased permutation transfer entropy. (C) Conditional entropy. (D) Normalized permutation transfer entropy.</p

Extracting motifs from a simulated spike train.

<p>(A) A simulated spike train . (B) Discretizing the spike train by counting spikes in each bin and the generation of and . (C) Some motifs contained in the discretized sequence. (D) All motifs for the order (3! = 6 different motifs), which are identical to those defined in Ref. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070894#pone.0070894-Olofsen1" target="_blank">[23]</a>.</p

NPTE estimate in a network with three neurons.

<p>(A) Three neurons sampled from a simulated network. (B), (C) and (D) The pairwise NPTE estimate of connections on two directions.</p

Estimated values for varying spike trains duration.

<p>(A) NPTE. (B) NTE. (C) PCMI. (D) STE.</p

NPTE estimate in mutual coupling.

<p>(A) Two mutual coupled neurons sampled from a simulated network. (B) The NPTE estimate of connections on two directions.</p

NPTE values for different and .

<p>NPTE values for different and .</p

Variation of estimated values with coupling strength between spike trains.

<p>(A) NPTE. (B) NTE. (C) PCMI. (D) STE.</p

Effect of firing rate on the values estimated by different methods.

<p>(A) NPTE and NTE under . (B) PCMI and STE under . (C) NPTE and NTE under . (D) PCMI and STE under . (E) Coefficients of variation for the four methods under different coupling strength.</p

The error of estimation methods for different time delays.

<p>(A) NTE. (B) NPTE.</p