Causal influence in linear Langevin networks without feedback

The intuition of causation is so fundamental that almost every research study in life sciences refers to this concept. However, a widely accepted formal definition of causal influence between observables is still missing. In the framework of linear Langevin networks without feedback (linear response models) we propose a measure of causal influence based on a new decomposition of information flows over time. We discuss its main properties and we compare it with other information measures like the transfer entropy. We are currently unable to extend the definition of causal influence to systems with a general feedback structure and nonlinearities.

Figure 1

Stochastic dynamics of the basic linear response model. The parameters are $\alpha =0.1, \beta =0.2, t_{\mathrm{rel}}=10$, and $D=10$. All graphs are produced using R [11].

Figure 2

Conditional probability distributions over time. Given a particular condition (input) at time $t=0, x\left(0\right)\equiv x_0=28$, we plot the conditioned expectation values $\langle y\left(t\right)\left|x\right(0)\rangle$ and $\langle x\left(t\right)\left|x\right(0)\rangle$ with the relative standard deviations $\pm {\sigma }_{y\left(t\right)\left|x\right(0)}$ and $\pm {\sigma }_{x\left(t\right)\left|x\right(0)}$ (thinner lines) as a function of the time shift $t$. The parameters are $\alpha =0.1, \beta =0.2, t_{\mathrm{rel}}=10, D=10$.

Figure 3

Previously proposed PIDs where $R\left(\tau \right)=I_{\min }$. Information measures are expressed in natural units, $\mathrm{Nats}=\frac{\mathrm{bits}}{ln2}$. The $\tau$ axis is on the logarithmic scale. The parameters are $\beta =0.2, t_{\mathrm{rel}}=10$.

Figure 4

Linear information decomposition $x⟶y$. The thick black curve represents the unique information that $x\left(t\right)$ gives on $y(t+\tau )$, which is our measure of causal influence $C_{x\to y}\left(\tau \right)$. The parameters are $\beta =0.2, t_{\mathrm{rel}}=10$.

Figure 5

Linear information decomposition $y⟶x$. The redundant information is equal to the mutual information, meaning that there is no causal influence. The parameters are $\beta =0.2, t_{\mathrm{rel}}=10$.

Figure 6

Linear information decomposition $x⟶y$. High-information scenario: $\beta =1000, t_{\mathrm{rel}}=1000$.

Figure 7

Linear information decomposition $x⟶y$. Low-information scenario: $\beta =0.02, t_{\mathrm{rel}}=0.02$.

Figure 8

The coefficient ${\gamma }_{xy}$ of the vector autoregressive model compared with the information measures. ${\gamma }_{xy}$ and $\sqrt{\langle {\xi }^2\left(\tau \right)\rangle }$ are adimensional. The parameters are $\beta =0.2, t_{\mathrm{rel}}=10, \alpha =0.3, D=0.03$.

Figure 9

Feed-forward loop, the three-dimensional general case. Causal influence $x⟶y$ (numerical simulation). The parameters are $t_{\mathrm{rel}}=10, \gamma ={\alpha }_x={\alpha }_y=1, {\beta }_x={\beta }_y=0.2, D_z=10, D_x=D_y=0.1$.