Inferring meaningful predictive relationships between events across time is a problem commonly faced by neural systems — such as the hippocampal system in learning cognitive maps (Behrens et al., 2018, Momennejad, 2020, Peer et al., 2021) and the mesolimbic dopamine system in learning cue-reward relationships (Gershman et al., 2024, Lerner et al., 2021). An effective approach to this problem is to maintain a memory trace of the temporal history and to strengthen associations between stimuli that regularly co-occur close together in time.

Learning such temporal contiguities forms the basis of the successor representation, in which an event is represented as a temporally weighted sum of its expected successor events (Dayan, 1993, Momennejad et al., 2017, Stachenfeld et al., 2017). The successor representation and its temporal difference (TD) learning rule have been investigated in both hippocampal and striatal systems. Notably, it has long been suggested that mesolimbic dopaminergic dynamics implement the reward prediction error necessary for TD learning (Eshel et al., 2015, Montague et al., 1996, Schultz et al., 1997).

At the same time, temporal contiguities do not allow inference of causal relations between events. If some events have higher marginal probabilities, for example, those events will tend to have a higher expectation as successors — they will temporally co-occur with other events more often merely due to being more frequent a priori. Causal inference requires information about temporal contingencies between events, namely the extent to which co-occurrences exceed what would be expected based on events’ marginal probabilities. For events to be causally related, they should not just co-occur often but should show statistical dependencies across time.

Several distinct but related measures of temporal contingency have been proposed, such as the adjusted net contingency for causal relations (Jeong et al., 2022), mutual information (Gallistel et al., 2019, Gershman, 2025), and pointwise mutual information (Haga et al., 2025, Williams, 2022). In various ways, these measures all rely on comparing the joint probability of two events against the background co-occurrence rate expressed as the product of marginal probabilities.

Here, we formalize a definition of temporal contingency that maps onto an efficient, neurally plausible learning rule. We begin with a measure of contingency derived from cognitive maps and the successor representation (Namboodiri & Stuber, 2021) that has been found to predict mesolimbic dopamine responses (Garr et al., 2024, Jeong et al., 2022). Work on the adjusted net contingency for causal relations (ANCCR) raises the possibility that mesolimbic regions widely seen as substrates of TD learning also show non-TD-like characteristics consistent with performing causal inference. So far, that work has used prospective and retrospective contingencies based on forward and backward conditional probabilities, combining the forward and backward representations to produce a measure of ”net contingency”.

In contrast, we first define temporal contingency in terms of the core computation underlying both prospective and the retrospective contingencies: the temporally weighted joint probability corrected by the background co-occurrence rate. We show that this quantity can be directly estimated through a simple associative learning rule, providing an efficient algorithm to support the type of causal inference suggested by recent neural studies. The key to making an associative learning rule compute temporal contingency rather than contiguity is to use an activation function with suppressed responses to more frequent events. The resulting measure of contingency can then be converted to the forward contingency, backward contingency, or net contingency by normalizing.

It further follows that learning temporal contingency is equivalent to learning the cross-covariance between a memory trace and a stimulus, whereas learning contiguity is equivalent to learning the cross-correlation. This relationship indicates a generalized from of associative temporal learning that accounts for both temporal contingencies and temporal contiguities.