Estimating temporal causal interaction between spike trains with permutation and transfer entropy.
Bottom Line:
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.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.
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PubMed Central - PubMed
Affiliation: Institute of Information Science and Engineering, Yanshan University, Qinhuangdao, China.
ABSTRACT
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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. |
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Mentions: We select two parts of a simulated network to investigate the performance of NPTE for characterizing mutual coupling and interactions between three neurons. As shown in Figure 8(A), two neurons N13 and N14 are connected through a mutual coupling with different strengths (denoted by s) and time delays (denoted by D) on two directions. In Figure 8, it can be seen that the maximum NPTE estimation from N13 to N14 occurs at the delay of 5 ms, while the maximum NPTE estimation appears at the delay of 29 ms in the opposite direction. This means that the NPTE method is able to describe the mutual coupling, with correct reflection of the strength but small error for the time delay estimate (1 ms for both directions). Next, we discuss how the NPTE measure performs in a sub-network of three neurons in which N12 is coupled via a synapse to N4, who is then connected to N1, as illustrated in Figure 9 (A). The NPTE estimate of these three pairwise coupling is plotted in Figure 9 (B–D). The causal relationships from N12 to N4 and from N4 to N1 are described effectively. At the same time, it can be also seen that there is a large value for the NPTE estimate from N12 to N1. In other words, the NPTE method shows an interaction raised by a third neuron. However, it should be noted that the time delay from N12 to N1 indicated by the NPTE method is 36 ms, which approximates to the summation of the delays from N12 to N4 and from N4 to N1 (19+5 = 34 ms). In addition, the maximum NPTE value from N12 to N1 is less than the values from N12 to N4 or from N4 to N1. Considering these two aspects, the causality appeared in the direction from N12 to N1 can be reckoned as a spurious one. Thus, it is reasonable to conclude that the NPTE method is able to distinguish real causalities from spurious ones. The underlying idea is explained as follows. For example, if there is a causal interaction from neuron X to neuron Y, then a portion of spikes of neuron Y will occur after some spikes of neuron X with a certain time delay. Suppose that this is hold for neuron Y and neuron Z. That is to say, some spikes of neuron Z will fire after certain spikes of neuron X, resulting in a large NPTE estimation that implies a spurious connection from neuron X to neuron Z. In fact, only the spikes of neuron Y that contribute to the causal relationship from neuron Y to neuron Z that also participate in the generation of causal connection from neuron X to neuron Y, can lead to a NPTE estimate about this spurious causal relationship. Consequently, this spurious NPTE estimate is less than the real ones, and the spurious delay is related to the two real delays. Thus, if this is observed in a network of three neurons, a spurious causality can be identified. To summarize, although the NPTE method cannot eliminate spurious interactions directly in the computation, it can tell whether interactions are direct or indirect by comparing the results of three neurons. |
View Article: PubMed Central - PubMed
Affiliation: Institute of Information Science and Engineering, Yanshan University, Qinhuangdao, China.