Complex cognition can be predicted by remarkably simple tasks. For example, the speed with which you choose one of two possible responses can reliably predict IQ. Some theories propose that this relationship is due to differences in something called “processing speed,” but more recent work has shown the effect is really due to the slowness of your slowest reaction times on such simple tasks. Known as the “worst performance rule,” this can be revealed through various RT distribution decomposition techniques (e.g., “binning” of reaction times or ex-gaussian analysis).

A particular class of computational models provides an elegant explanation for the cognitive processes that generate this “worst performance rule.” As described by Usher & McClelland, Ratcliff’s diffusion model posits that information processing occurs gradually and unfolds continuously over time, that a response is provided once information has accumulated past a threshold, and that information cannot be accumulated prior to some startup time. Almost all parts of this model are subject to variability, including the rate and direction of information accumulation (e.g., “drift”), and the starting point of information accumulation. This variability is critical to the predictive power of these models – e.g., all source of variability are important for capturing asymmetries in the relationship of stimulus discriminability to reaction time, and variability in drift or information accumulation rate is particularly important for capturing skew in reaction time. It is this “skew” which makes up the slowest portion of reaction times, and which lead to the worst performance rule.

Simen et al describe how this model is fundamentally equivalent to the sequential probability ratio test, which is the most efficient procedure known in a sequential sampling model (in which evidence is gradually accumulated) for ensuring a given level of accuracy in the minimum amount of time.

Importantly, new work by Schmiedek et al. shows that the drift rate parameter of diffusion models – the rate of information accumulation – appears to explain the relationship between high level cognition (in this case, IQ) and elementary cognitive tasks like the 2-choice reaction time.

This simple model is not only useful at a cognitive level of analysis, but is increasingly proving its worth as a way of analyzing physiological data, both in fMRI and ERP.

For example, Forstmann et al shows that subjects who have undergone a behavioral speed-accuracy manipulation may accomplish this by resetting their response thresholds (the “absorbing boundaries” in diffusion model parlance), and that this change in threshold is monotonically related to the fMRI BOLD response in the anterior striatum and pre-supplementary motor area – both of them regions of the brain which are important for thresholding or initiating motor commands.

In ERPs, Philiastides et al. showed that a diffusion model can be used to interpret the results of a mathematical classifier of ERP components in a face/car discrimination task (where the faces and cars were embedded in noisy backgrounds and varied in terms of discriminability [manipulated via differences in phase coherence]). This classifier showed two dissociable ERP components, an early N170-like component with consistent onset across subjects, and a later ERP complex whose onset varied between subjects – and predicted their behavioral performance on the task! This latter complex was interpreted to reflect difficulty with the task, and was also predictive of reaction time variability. The strength of this component for each subject was correlated (with R>.85) with the drift rate in a diffusion model decomposition of their reaction times. This confirms that drift rate indexes a psychologically and physiologically important variable, consistent with the diffusion model’s specific claim that it reflects the rate of quality information accumulation.

As described by Usher & McClelland, there are potentially some shortcomings of this model, including that information is not lost over time (although the fact that drift can be positive or negative seems to contradict this), and that the model can only account for 2-choice tasks (although it may be possible to extend the diffusion model into multiple dimensions).

Although the Usher & McClelland neural network implementation of the diffusion model (the leaky competing accumulator model) apparently does not capture as many phenomena of reaction times as the diffusion model, a new contender from Simen et al. may.

Simen et al implemented the diffusion model in a neural network by using two decision units, which accumulate evidence in favor of one option or another. These units mutually compete (through lateral inhibition), and are subject to decay of information over time (via a leak current). Stochastic differential equations are used to calculate changes in activation of each unit as a function of time, net input from other units, and the leak and inhibition currents. This process is fundamentally equivalent to the diffusion model when leak and inhibition are perfectly balanced, and is similar to the Usher & McClelland model. Their unique addition to this system is a set of two units which detect when the accumulators have gone beyond threshold, and a final unit which “biases” those threshold detectors as a function of “reward rate.” It is this latter term which allows the network to learn, by adjusting the decision thresholds to provide the maximum reward (number of trials correct) in the fastest time.

It may be possible with this new implementation to fit behavioral reaction times by training the network on those reaction times themselves. This capacity was not demonstrated by the authors, so it’s not yet clear that the model will surpass the classic diffusion model in utility for understanding behavioral data (at least at the individual differences level). In fact, no one to my knowledge has demonstrated what empirical failures of the classic diffusion model warrant the inclusion of leak and mutual inhibition – it is true that these features characterize neuronal processing, but it is far from clear that they are not already captured by the classic “drift rate” parameter.

In case you wish to explore the diffusion model in your own data, you can look at the MatLab diffusion model toolbox .

As noted by Wagenmakers et al., even a simplified version of the diffusion model requires solving for an infinite sum and a triple integral in terms of trial-to-trial variability. Therefore Wagenmakers et al have developed the Robust-EZ-Diffusion model (downloadable here; previous iterations included EZ-Diffusion and EZ2)