Characteristics of the neural coding of causality

While causality processing is an essential cognitive capacity of the neural system, a systematic understanding of the neural coding of causality is still elusive. We propose a physically fundamental analysis of this issue and demonstrate that the neural dynamics encodes the original causality between external events near homomorphically. The causality coding is memory robust for the amount of historical information and features high precision but low recall. This coding process creates a sparser representation for the external causality. Finally, we propose a statistic characterization for the neural coding mapping from the original causality to the coded causality in neural dynamics.

Figure 1

Neural dynamics characterization. We set a stimulus sequence in $\left[0,200\right]$ and a neural population with 500 neurons (the ratio of input to intermediary neurons is $7:3$). (a) The tuning curve of a randomly picked input neuron in this neural population. (b) For this input neuron, the Poisson process of its stimulus-triggered neural activities defined by (1) as well as its neural response arrival time sequence is shown. (c) For a randomly picked intermediary neuron, its presynaptic inputs in (4) and the estimated stimulus-triggered neural activities in (5) with $h=100$ is shown. (d) The response train and population codes of the whole neural population in (6).

Figure 2

A case study for the causality matrices and graphs before and after neural coding. (a) The matrices and graphs of Lasso-Granger causality (LG). (b) The matrices and graphs of copula-Lasso-Granger causality (CG). (c) The matrices and graphs of the transfer entropy (TE). Note that the color of a node in each graph is determined by the averaged causality between it and other nodes.

Figure 3

A case study for the neural coding effects and neural coding capacity of causality. The analysis is implemented based on the data in Fig. 2. (a) The linear regression relations between the coding effects and the original causality. (b) The coding capacity of causality by neural dynamics. (c) The comparison between the causality distributions before and after the coding process.

Figure 4

The causality coding capacity with different $\left({\tau }_{\mathrm{origin}},{\tau }_{\mathrm{coded}}\right)$ settings. (a, b) The precision and recall measurements with different $\left({\tau }_{\mathrm{origin}},{\tau }_{\mathrm{coded}}\right)$. (c) The frequency distributions of precision and recall. (d) The causality graphs vary with the threshold $\alpha$ ($\alpha \in \left[0.5,0.99\right]$) while the trend that precision is higher and recall is lower always holds ($\text{Precision}-\text{Recall}>0$).

Figure 5

The linear regression coefficients with different $\left({\tau }_{\mathrm{origin}},{\tau }_{\mathrm{coded}}\right)$ settings. (a, b) The measurements of $\kappa$ and $\omega$ with different $\left({\tau }_{\mathrm{origin}},{\tau }_{\mathrm{coded}}\right)$. (c) The frequency distributions of $\kappa$ and $\omega$.

Figure 6

The estimation of the coded causality based on the original causality. (a, b) The estimations for the Granger-type causality (GT) and the transfer entropy (TE) data sets. Note that all results are obtained in a causality coding experiment for a stimulus sequence set with 100 stimulus sequences, where each sequence lasts for a duration of $\left[0,1000\right]$. The neural population has 500 neurons (the ratio of input to intermediary neurons is $6:4$). The settings of causality measurement are the same as those in Sec. 4.