Response times from ensembles of accumulators

Abstract

Decision-making is explained by psychologists through stochastic accumulator models and by neurophysiologists through the activity of neurons believed to instantiate these models. We investigated an overlooked scaling problem: How does a response time (RT) that can be explained by a single model accumulator arise from numerous, redundant accumulator neurons, each of which individually appears to explain the variability of RT? We explored this scaling problem by developing a unique ensemble model of RT, called e pluribus unum, which embodies the well-known dictum "out of many, one." We used the e pluribus unum model to analyze the RTs produced by ensembles of redundant, idiosyncratic stochastic accumulators under various termination mechanisms and accumulation rate correlations in computer simulations of ensembles of varying size. We found that predicted RT distributions are largely invariant to ensemble size if the accumulators share at least modestly correlated accumulation rates and RT is not governed by the most extreme accumulators. Under these regimes the termination times of individual accumulators was predictive of ensemble RT. We also found that the threshold measured on individual accumulators, corresponding to the firing rate of neurons measured at RT, can be invariant with RT but is equivalent to the specified model threshold only when the rate correlation is very high.

Keywords: computational model; diffusion model; mathematical psychology; neurophysiology; reaction time.

Fig. 1.

Response times predicted by ensembles of redundant stochastic accumulators. (A) Stochastic accumulator models describe RT in terms of an accumulation process (one trajectory per trial) that proceeds at a certain rate (v) to reach a fixed threshold (θ). Stochastic variation of RT arises from fluctuations of v between (η) and within trials (ξ). It is common to consider one accumulator associated with each of multiple responses; we considered instead the case of multiple accumulators associated with the same response (Inset). (B) RT can also be described by the time at which the evolving spike rates of certain neurons, averaged across bins of trials with common RTs (one trajectory per RT bin, replotted from ref. 49), reach an activation level that is invariant with RT (A_RT). These neurons have been argued to instantiate the process described by stochastic accumulator models. (C) Unless accumulators are perfectly correlated (Inset), it is unclear (i) how an ensemble of accumulators makes the transition from evidence accumulation to response execution, (ii) under what termination rules (p_N) and accumulation rate correlations (r_v) the dynamics of one accumulator (highlighted red) predicts RT distributions and the invariant relationship between A_RT and RT, as observed empirically, and (iii) how A_RT relates to the unobserved threshold of an accumulator (θ).

Fig. 2.

Predicted RT distributions as a function of ensemble size (N), termination rule (p_N), and accumulation rate correlation (r_v). Each panel shows the 0.1, 0.3, 0.5, 0.7, and 0.9 RT quantiles on a log-log scale (the x axis ranges from 10⁰ to 10³; the y axis ranges from 10² to 10³) as a function of N, p_N, and r_v vary across columns and rows, respectively. We identified conditions (p_N and r_v) under which RT distributions were (i) invariant over the entire interval of N (i.e., 1,1,000; white panels], (ii) invariant with N over the interval (10,1,000; light gray panels), (iii) invariant with N over the interval (100,1,000; medium gray panels), and (iv) not invariant with N (dark gray panels).

Fig. 3.

Relationship between A_RT and RT as a function of ensemble size (N), termination rule (p_N), and accumulation rate correlation (r_v). Each panel shows the linear regression slope of A_RT on RT, expressed as colored pixels, for three ensemble sizes (Left, n = 10; Center, n = 100; Right, n = 1,000) and all combinations of termination rules and accumulation rate correlations. Hatched pixels indicate parameter combinations for which A_RT varied systematically with RT. Thus, beige, nonhatched pixels represent parameter combinations for which the slope of the linear relationship between A_RT and RT was zero and nonsignificant.

Fig. 4.

Distribution of measured activation level around RT (A_RT) between trials in a randomly selected accumulator as a function ensemble size (N), termination rule (p_N), and accumulation rate correlation (r_v). The x axis ranges from 10⁰ to 10³, and the y axis ranges from 10² to 10³. Other conventions as in Fig. 2. Individual threshold (θ, red line) was identical across accumulators. Thus, correspondence between A_RT and θ is indicated by overlap of distributions (black lines) and threshold (red line).