The thermodynamics of human reaction times

I present a new approach for the interpretation of reaction time (RT) data from behavioral experiments. From a physical perspective, the entropy of the RT distribution provides a model-free estimate of the amount of processing performed by the cognitive system. In this way, the focus is shifted from the conventional interpretation of individual RTs being either long or short, into their distribution being more or less complex in terms of entropy. The new approach enables the estimation of the cognitive processing load without reference to the informational content of the stimuli themselves, thus providing a more appropriate estimate of the cognitive impact of different sources of information that are carried by experimental stimuli or tasks. The paper introduces the formulation of the theory, followed by an empirical validation using a database of human RTs in lexical tasks (visual lexical decision and word naming). The results show that this new interpretation of RTs is more powerful than the traditional one. The method provides theoretical estimates of the processing loads elicited by individual stimuli. These loads sharply distinguish the responses from different tasks. In addition, it provides upper-bound estimates for the speed at which the system processes information. Finally, I argue that the theoretical proposal, and the associated empirical evidence, provide strong arguments for an adaptive system that systematically adjusts its operational processing speed to the particular demands of each stimulus. This finding is in contradiction with Hick’s law, which posits a relatively constant processing speed within an experimental context. keywords: cognition | entropy | lexical decision | reaction time | word naming Ever since its introduction by Donders [1] in the very early days of experimental psychology, reaction time (RT) has been among the most widely used measures of cognitive processing in human and animal behavioral experiments. Very generally speaking, following Donders’ seminal work, the logic underlying the analysis of data in RT experiments is that, information processing takes time, thus the average time taken to initiate or complete a task reflects the duration of the process(es) that are involved in the task. Therefore, if certain types of stimuli, tasks, or groups of subjects elicit longer RTs than others, it is generally inferred that the former involve more cognitive processing than the latter. In this study, I propose a qualitatively different perspective on the understanding of RT data: Rather than focusing on whether some experimental conditions elicit shorter or longer RTs than others, I investigate whether different conditions elicit RT distributions with different degrees of complexity. As I will argue, an increase in the complexity of the RT distribution constitutes an indirect measure of the amount of information processing that has been performed by the system. For this, I take a psychologically naive, model-free, approach: Instead of guiding the RT analysis using knowledge about the relevant neural and/or psychological processes that give rise to RTs, I intend to draw inferences on the former by studying only the properties of the latter. The cognitive system can be considered a system in the thermodynamical sense of the word. In particular, it is an open system that exchanges energy (and information) with its environment. Performing an experimental task involves an exchange of information with the environment. The experimental instructions and the presentation of stimuli are a source of information. Performing the experimental task requires the processing of this external information, and information processing is costly in energy terms. As discussed by Brillouin [2], the acquisition of information by any part of a system must be offset with a decrease of information somewhere else. In Brillouin’s terms there is a balance between gained and lost ‘negentropy’, 1 ar X iv :0 90 8. 31 70 v1 [ qbi o. N C ] 2 1 A ug 2 00 9 that is, information. Having received energy and information, the stimulus is processed and a response is initiated. Once more, this process involves a further exchange of negentropy and energy with the environment. An ideal system with perfect efficiency could perhaps achieve a perfect balance between the received, and the spent negentropy. However, as the efficiency is never perfect, some negentropy will be lost in the process. Eventually, in the case of the cognitive system, this loss of negentropy can be compensated for by a supply of energy, normally by metabolic means, that would enable the system to return to its ‘resting’ state. In short, the processing of experimental stimuli should temporarily increase the entropy of the cognitive system by an amount directly proportional to the amount of information that has been processed, corresponding to the negentropy that was wasted in the process. In essence, a measure of the increases in the entropy of the cognitive system elicited by different experimental conditions or stimuli would provide an estimate of the amount of information that has been processed (see [3] for a detailed physical description of this type of processes). Measuring the overall state of entropy of the cognitive system might not be an easy task, as it would involve a quantification of the uncertainty in the state of all the microscopic units in the system. However, collateral measures of the ‘noise’ emitted by the system should reflect increases in its state of complexity. This is to say, if the system is in a higher state of complexity, the noises it emits will also increase in their complexity. The random variability of times at which responses happen in a particular experimental condition can be considered as part of this emitted ‘noise’. Therefore the uncertainty (i.e., entropy, in its statistical sense [4]) of this distribution can be taken to reflect the state of the system that generated them. My working assumption is that one can measure the entropy of the distribution of RTs in a particular condition (i.e., the temporal entropy), and make inferences about variations in the entropy (in its physical sense) of the underlying system. In short, an increase in the informational entropy of an RT distribution is directly proportional to the amount of information that has been processed. In a typical repeated measures RT experiment, the differential entropy [4] of the RT distribution can be expressed as a mixed effect model (MEM; see [5, 6, 7] for recent introductions to this technique) with meaningful (and thus very constrained) parameter values: E [− log p (t)] = h0 + k N ∑ i=1 [ θiIi(S, P ) ] + ε + ES [− log p (t)] + EP [− log p (t)] . (1) In this model, the independent variable is the self-information of the RTs (i.e., − log p(t)), whose expected value is – by definition – the entropy. The intercept of the model (h0) corresponds to the baseline entropy of the RT distribution, which must always be positive and provides an indication of task complexity. The fixed effect coefficients (kθi) indicate the relative contribution of the i-th known source of information in the stimuli (Ii), and must all be positive and smaller than or equal to one. In this product, the θi represent the proportion of the i-th source of information that is processed. On the other hand, k is constant for all sources of information representing the proportion of the wasted negentropy that is reflected in the RT variability. Therefore both k and the θi must also lie within the (0, 1] interval. This has the additional implication that k is bound to be larger than or equal to the largest observed fixed effect value, least the estimated value for some of the θi would be greater than one. The last two terms on the right-hand side of Eqn. 1 correspond to random effects of the individual stimulus S and participant P . These correspond to other unknown sources of information linked to the identity of the stimulus or participant that are not accounted for by the Ii. Finally, ε accounts for the error in the estimations. If estimates of p(t) and of Ii(S, P ) can somehow be obtained, this relationship can be tested directly. Information Theory has a long history in the study of behavior, particularly so in the study of RTs. Very soon after Shannon’s development of information theory in telecommunications [4], psychologists were applying it to the study of human RTs. This produced one of the few standing laws of experimental psychology: The time it takes to make a choice is linearly related to the entropy of the possible alternatives; 1In this study, I follow [2]’s interpretation equating negentropy and information. 2See supplemental materials for the derivation of this equation.