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Edited by: Ken Miller, Columbia University, USA

Reviewed by: Surya Ganguli, University of California, USA; Philip Holmes, Princeton University, USA

*Correspondence: Dominic Standage, Queen's University, Botterell Hall, Room 453, Kingston, ON, Canada K7L 3N6. e-mail:

This is an open-access article subject to an exclusive license agreement between the authors and Frontiers Media SA, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

The speed–accuracy trade-off (SAT) is ubiquitous in decision tasks. While the neural mechanisms underlying decisions are generally well characterized, the application of decision-theoretic methods to the SAT has been difficult to reconcile with experimental data suggesting that decision thresholds are inflexible. Using a network model of a cortical decision circuit, we demonstrate the SAT in a manner consistent with neural and behavioral data and with mathematical models that optimize speed and accuracy with respect to one another. In simulations of a reaction time task, we modulate the gain of the network with a signal encoding the urgency to respond. As the urgency signal builds up, the network progresses through a series of processing stages supporting noise filtering, integration of evidence, amplification of integrated evidence, and choice selection. Analysis of the network's dynamics formally characterizes this progression. Slower buildup of urgency increases accuracy by slowing down the progression. Faster buildup has the opposite effect. Because the network always progresses through the same stages, decision-selective firing rates are stereotyped at decision time.

Subjects in decision making experiments trade speed and accuracy at will (van Veen et al.,

Experimental and theoretical work indicates that decisions result from mutual inhibition between neural populations selective for each option of a decision (see Gold and Shadlen,

A convergence of evidence offers an answer. In neuronal networks with extensive intrinsic connectivity, integration times are determined by network dynamics (Usher and McClelland,

We model a decision circuit in the lateral intraparietal area (LIP) of posterior parietal cortex with a recurrent network model. We choose LIP because this area is extensively correlated with decision making (Roitman and Shadlen,

A cortical decision circuit was simulated with a network from a class of models widely used in population and firing rate simulations of cortical circuits (Wilson and Cowan,

_{T} and r_{D} refer to firing rates of the target- and distractor-selective populations. Lighter shades correspond to higher-rate activity. The state of the network at the end of the trial is shown on the right side of the figure.

The model is constrained by signature characteristics of neural and behavioral data from visuospatial decision making experiments. The neural data we consider were recorded in LIP, but similar activity is seen in other decision-correlated cortical areas, e.g., the frontal eye fields (see Schall,

We simulated a two-choice visual discrimination task by providing two noisy inputs to the model for 1000 ms. The task was to distinguish the stronger input (the target) from the weaker input (the distractor). While the spatial and temporal profiles of the inputs were constrained by the above data, the task clearly generalizes to other decision tasks, just as the model generalizes to other cortical regions. The neural coding of elapsed time (urgency) was simulated with a piecewise linear function (Figure

We ran 1000 trials across a range of task difficulties and rates of buildup of urgency (Section

The network is a fully connected recurrent rate model with

where the phenomenological state variable _{v}_{h}_{h}

The population rate

where β determines the slope of the function, scaled by urgency signal

The intercolumnar interaction structure

where _{w}

To simulate a reaction time version of a two-choice visual discrimination task, Gaussian response fields (RF) were defined for all columns

where γ_{s}_{div}_{μ} = 25 ms determines the rate of input decay, and _{vrd}_{μ} was set to 1 for the target and to γ_{ext}_{ext}_{i}

The urgency signal was simulated with a piecewise linear function

where _{max}_{μ} ∈ {500, 750, 1000} ms determines time over which the signal increases toward _{max}

Signal detection theory (Green and Swets,

where

We begin the Section _{eff}_{lin}_{eff}

For all rates of buildup of urgency

Consistent with neural data (Hanes and Schall,

The time over which a recurrent network can accumulate evidence (its effective time constant of integration τ_{eff}_{eff}_{eff}

We approximated τ_{eff}_{eff}^{−6}). Longer buildup of _{eff}

_{eff}_{eff}_{eff}_{ext}_{eff}

The above calculation of the network's effective time constant τ_{eff}_{eff}_{eff}_{eff}_{eff}

Under gain modulation, the network progresses through all these processing stages on each trial. As such, the decision variable is dominated by leakage early in the trial when urgency is low and is amplified later in the trial when urgency is high (see Figure _{eff}

Note that our method of calculating τ_{eff}_{eff}_{eff}_{eff}

The progression from leakage to amplification of the decision variable with the buildup of urgency can be further understood by observing the activity in the fixed-gain network during the decision task for different values of

Whereas the fixed-gain network implements a single regime for each value of

The above simulations demonstrate the SAT with a fixed neural threshold and provide an intuitive picture of network dynamics with growing urgency. In this section, we take a non-linear dynamics approach to formally characterize this picture.

From a dynamic systems point of view, the accumulation of evidence in the network is the evolution of a dynamic system from an initial state to an attractor corresponding to the target or the distractor. The evolution is determined by the structure of the steady states of the system (Strogatz, _{ext}

_{ext}_{ext}_{ext}_{ext}_{ext}

The two regimes have several notable features. Firstly, the similarity of the target and the distractor plays a different role in each regime. In regime 1, the height of the bump related to the target is higher than the bump related to the distractor for distinguishable tasks (γ_{ext}_{ext}

Secondly, by variation of the target-distractor similarity γ_{ext}_{ext}

Thirdly, the evolution from the initial state to the stable steady state is determined by the structure of the steady states. In regime 1, the system evolves directly from the initial state to the single stable steady state of the target. The dynamics in the vicinity of this stable state determine the characteristics of the decision. In regime 2, the initial firing rate of the target population equals that of the distractor population (see Figure

As described above, the decision ensues with the evolution of the system to the stable steady state. The dynamics in the vicinity of the steady state (the stable steady state in regime 1 and the unstable steady state in regime 2) thus determine task performance, such as speed, accuracy, and decisiveness (whether or not a decision is made). We have used signal detection theory to determine the decision on each trial, where the AUROC is used to estimate the separation of the distributions of target and distractor-selective activation. On any given trial, the activation of the noisy system fluctuates around the rates of the noise-free system. Therefore, the overlap of the two distributions decreases (and thus the AUROC increases) with the increase of the difference between the target and distractor-selective activation in the noise-free system. In regime 1, if the difference between the two bumps of the stable steady state is small [i.e., a low value of (1 + _{ext}

In regime 2, the system is driven away from the unstable steady state to one of the two stable steady states, where the difference between target and distractor-selective activation is large enough to make a decision (see

In the vicinity of the steady state, the dynamic system (Equation

where _{i}_{i}

In regime 1, the eigenvalues for the stable steady state are negative, and the evolution of the system along the invariant manifold tangent to the eigenvector corresponding to the largest eigenvalue (i.e., the one closest to 0) determines how slowly the system reaches the stable steady state. Therefore, the absolute value of the reciprocal of the largest eigenvalue can be used to approximate the time it takes for the system to reach the stable steady state and is defined as the time constant. In regime 2, we consider the time it takes for the system to evolve away from the unstable steady state. The reciprocal of the largest positive eigenvalue approximates the time over which the system departs from the unstable steady state of the invariant manifold tangent to the eigenvector corresponding to the largest eigenvalue. Therefore, the absolute reciprocal of the largest eigenvalue of the stable steady state in regime 1 and the unstable steady state in regime 2 depicts the time over which the system makes a decision and is denoted τ_{lin}

For a given target-distractor similarity γ_{ext}_{lin}_{ext}

_{lin}_{ext}_{ext}_{ext}_{ext}

With growing urgency, decision processing under gain modulation can be described according to the above two regimes. In the early stages of a decision, the system operates in regime 1 because the urgency signal is low. The system has only one stable steady state with activation bumps centered at the target and distractor columns. As the urgency signal grows, the target and distractor-selective activation in the steady state gradually increases and decreases respectively (see Figures

_{ext}

In the fixed-gain network, the decision process occurs either in regime 1 or in regime 2, depending on the slope parameter _{ext}_{ext}_{lin}_{lin}

The above analysis shows that the bifurcation between regime 1 and regime 2 under gain modulation puts the network in a state closer to the target attractor when it enters regime 2 than would be the case in the fixed-gain network with slope parameter _{ext}_{u}_{u}_{u}_{u}_{max}

_{u}_{u}_{u}

Because mean decision time on error trials was longer than on correct trials under gain modulation (see Section

This finding is instructive in several regards. Earlier work showed that a spatially non-selective signal, variable between blocks of trials, but constant within each trial, could potentially produce the SAT with a fixed threshold (Bogacz et al.,

Our model offers a candidate neural mechanism for the SAT. We propose that gain modulation by the encoding of urgency controls the time constant of cortical decision circuits “on the fly.” The rate of buildup of urgency determines how long the circuit integrates evidence before the decision variable is amplified. Longer (shorter) estimates of the time available to respond result in slower (faster) buildup of the signal, so the circuit spends more (less) time integrating evidence. Importantly, decision-correlated neural activation reaches a fixed level at decision time, consistent with neural data (see Schall,

Our neural model is grounded in abstract, mathematical models that have been instrumental in characterizing decision processes. Sequential sampling models are based on the premise that evidence is integrated until it reaches a threshold level (see Smith and Ratcliff,

The DDM has been augmented with a time-variant mechanism similar in principle to our use of urgency (differences between the models are described below). In the model by Ditterich (

Neural models have addressed the possible mechanisms underlying the above mathematical models, several of which have been shown to be equivalent to the DDM under biophysical constraints. In these models, the subtractive operation is implemented by mutual inhibition between neural populations selective for each option of a decision (Bogacz et al.,

Our model trades speed and accuracy with a fixed threshold by exploiting the time constant of recurrent networks (Sections

We have used a network belonging to a class of local-circuit models (Wilson and Cowan,

We have modeled subjects’ estimates of the passage of time with a piecewise linear function, where different slopes correspond to different urgency conditions (Figure

Neural mechanisms that may underlie gain modulation include recurrent processing of spatially non-selective input (Salinas and Abbott,

While conceptually similar to Ditterich's (^{t}^{d}^{t}^{d}

In the time-variant DDM (Ditterich, ^{t}^{d}

Like Equation

where α is a (positive) scale factor. The practical difference between the two models is the time of arrival of the evidence subject to strong amplification. In Equation

The implications of the difference between the two models can be seen in Figure _{max}

While the difference between these time-variant DDMs is clear from Equations

Our model also bears conceptual similarities with the “urgency gating” models of Cisek et al. (

Ultimately, the modulation of decision circuitry is likely to occur on more than one timescale. Two such timescales are captured by the mathematical models of Gold and Shadlen (

Models of decision circuits have demonstrated that network dynamics determine time constants of integration (Wang,

The notion of urgency in decision making is not new (Reddi and Carpenter,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the Canadian Institutes of Health Research. Da-Hui Wang was supported by NSFC under Grant 60974075. Dominic Standage thanks Martin Paré and Thomas Trappenberg for helpful discussions.