Associative learning of events’ co-occurrence rates across time can generate useful predictive representations (e.g., the successor representation), but temporal contiguities alone are not enough to infer causal relations. Recent work suggests that neural substrates long thought to implement temporal difference learning may perform causal inference by tracking temporal contingences — coincidences between events corrected by background co-occurrence rate. We show that changing the activation function enables simple associative updates to directly compute temporal contingencies. Temporal contiguities can be learned as the cross-correlation between a stimulus and a memory trace, and temporal contingencies can be learned as the cross-covariance. An implication is that neurally plausible causal learning algorithms can be implemented through simple associative updates. These results highlight a family of learning rules for incremental computation of forward, backward, and joint temporal contiguities and contingencies.
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