Causal connectivity of evolved neural networks during behavior

Abstract

To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and ‘Granger causality,’ for characterizing causal interactions generated within intact neural mechanisms. This method, called ‘causal connectivity analysis’ is illustrated via model neural networks optimized for controlling target fixation in a simulated head–eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of ‘causal sources’ and ‘causal sinks’: nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics.

Keywords:

• Granger causality
• graph theory
• causality
• embodiment

Notes

^*‘Causal flow’ is distinct from a previous definition of ‘flow’ in graph theory, which refers to the problem of assigning non-negative values to directed edges such that total inflow is equal to total outflow for all nodes except two (Bollobás 1985).

^†This required estimating a 32-dimensional p = 4 VAR. The mean R2 adj was 0.9, and the residuals were uncorrelated (P < 0.01, Ljung-Box ‘Q’ statistic).

^‡As was remarked in the Introduction, if common inputs or intermediate variables (e.g., C and D in the above example) are part of the observed system, a multivariate Granger causality analysis will reveal the causal interactions mediated by these variables, instead of the indirect causal interactions that they support.)