A recurrent network mechanism of time integration in perceptual decisions

Abstract

Recent physiological studies using behaving monkeys revealed that, in a two-alternative forced-choice visual motion discrimination task, reaction time was correlated with ramping of spike activity of lateral intraparietal cortical neurons. The ramping activity appears to reflect temporal accumulation, on a timescale of hundreds of milliseconds, of sensory evidence before a decision is reached. To elucidate the cellular and circuit basis of such integration times, we developed and investigated a simplified two-variable version of a biophysically realistic cortical network model of decision making. In this model, slow time integration can be achieved robustly if excitatory reverberation is primarily mediated by NMDA receptors; our model with only fast AMPA receptors at recurrent synapses produces decision times that are not comparable with experimental observations. Moreover, we found two distinct modes of network behavior, in which decision computation by winner-take-all competition is instantiated with or without attractor states for working memory. Decision process is closely linked to the local dynamics, in the "decision space" of the system, in the vicinity of an unstable saddle steady state that separates the basins of attraction for the two alternative choices. This picture provides a rigorous and quantitative explanation for the dependence of performance and response time on the degree of task difficulty, and the reason for which reaction times are longer in error trials than in correct trials as observed in the monkey experiment. Our reduced two-variable neural model offers a simple yet biophysically plausible framework for studying perceptual decision making in general.

Figure 1.

Reduction of a biophysical neuronal decision-making model. The original model (top) is endowed with strong recurrent excitation between neurons with similar stimulus selectivity, and effective inhibition between them via shared inhibition. NS and I denote the nonselective excitatory (black) and inhibitory (green) pools of cells, respectively. Arrows, Excitatory connections; circles, inhibitory connections. I₁ and I₂ are inputs from external stimulus to selective neural populations 1 (blue) and 2 (red). Brown arrows, Background noisy inputs. w₊ denotes enhanced excitatory connections within each selective neural pool. The numbers on the right displays the total number of dynamical equations involved in the model. First step, Mean-field approach reduces 2000 spiking neurons into four neural units (with a total of 11 dynamical variables). Second step, Simplify the linear input–output relation (F–I curves) of the cells: (1) fit the input–output relation (F–I curve) of the spiking neuronal model with a simple function (Abbott and Chance, 2005); (2) linearize F–I curve for I cells; and (3) assume constant activity of NS cells. The final step involves the assumption that all fast variables of the system reach steady states earlier than that of NMDAR. The final reduced two-variable model (bottom) consists of two neural units, endowed with self-excitation and effective mutual inhibition.

Figure 2.

Time course with two different motion strengths. Motion coherence of 0% (black traces) and 51.2% (light gray traces) each with 10 sample trials. Firing rates that ramp upward (bold traces) are for saccades made toward the RF of the neuron, whereas downward (dashed traces) are for saccades away from RF. Ramping is steeper for higher coherence level. The prescribed threshold is fixed at 15 Hz. Once the firing rate crosses the threshold, a decision is made, and the decision time is the time it takes from stimulus onset (0 ms) until the threshold is crossed. The reaction time is defined as the decision time plus a nondecision latency of 100 ms. The bold horizontal line at the top of the figure denotes the duration, at zero coherence, where the firing rates toward and away from RF are indistinguishable.

Figure 3.

Performance and reaction time of models and the experiment of Roitman and Shadlen (2002). First column, Psychometric data from experiment and the models (data are fit with a Weibull function). Second column, Reaction time from experiment and the models. Open circles joined by dashed lines, Mean reaction of error trials; filled circles, correct trials. σ_noise = 0.008 nA. Experimental data are adapted from Mazurek et al. (2003).

Figure 4.

Random choice with stimulus at zero coherence. A, Phase-plane without stimulus. Black circles, Stable steady states (attractors); gray circles, saddle-type unstable steady states. The green and orange lines are the nullclines for the synaptic dynamical variables S₁ and S₂. Using Equation 8, a threshold at 15 Hz would correspond to S = 0.49 in phase space. B, With an unbiased stimulus of 30 Hz, the two unstable steady states together with the low stable steady state disappear, and a new symmetric unstable steady state is formed. The black line with arrows toward (away) from the saddle point is the stable (unstable) manifold of this saddle point. The stable manifold is exactly the boundary between the two basins of attraction of the two choice attractors (when there is no noise). Superimposed are two typical single-trial trajectories (blue and red lines) of the state of the system from simulations. Color labeling is the same as in Figure 1. C, Schematic diagram of a generic saddle-like steady state (gray circle) and the local flows (arrows) around it. The lines directly toward (magenta) and away (brown) the steady state (gray) are its stable and unstable eigenvectors, respectively, with an exponential temporal dynamics determined by τ_stable (brown) and τ_unstable (magenta). D, A diagram of how a one-dimensional decision “landscape” changes with stimulus inputs in a single trial, illustrating decision computation and working memory by the same network. See Results for detailed description.

Figure 5.

Basins of attraction with stimulus at nonzero coherence (c′ > 0%). A, Phase-plane without stimulus as in Figure 4A. B, The stable manifold is tilted away from the spontaneous state and toward the less favored attractor when c′ is nonzero (6.4%). As a result, at the onset of stimulus, the system starts in a resting state that has a higher chance of falling in the basin of attraction of the favored attractor state. The blue and red lines are typical single-trial trajectories for correct and error choices, respectively. C, Stronger bias between the basins of the two competing attractor states with a larger c′ (=51.2%). D, When c′ is sufficiently large, the saddle steady state annihilates with the less favored attractor, leaving only one choice attractor. c′ = 100%.

Figure 6.

Decision time and local dynamics in the vicinity of a saddle point. Zero coherence level from A to C. A, Longer reaction time for smaller stimulus strength μ₀. Error bars indicate SD. B, Typical time courses: ramping is faster for larger stimulus strength, μ₀. C, Time constants of saddle-like unstable steady state with different μ₀. For μ₀ > 17 Hz, τ_stable is larger than τ_unstable, whereas the opposite is true for μ₀ < 17 Hz. D, Time constants of the unstable saddle as function of coherence level c′ (μ₀ fixed at 30 Hz). The unstable time constant is essentially constant up to c′ ∼ 70%. The sudden increase in τ_unstable happens just before the bifurcation point at which the saddle coalesces with the less favored attractor and disappears (see Fig. 5).

Figure 7.

Dependence of decision-making behavior on the AMPA:NMDA ratio at recurrent synapses. A, Typical time courses: faster ramping neural activity at larger AMPA:NMDA ratios. Top black (gray) horizontal bar denotes the duration where the firing rates are not distinguishable [i.e., the trajectory lies along the stable manifold of the saddle point, when AMPA:NMDA is 35:65 (0:100)]. B, Reaction time is shorter with a higher AMPA:NMDA ratio. C, The performance, however, becomes less accurate. Accuracy data are fitted by a Weibull function. D, For c′ = 0%, a higher AMPA:NMDA ratio decreases the reaction time for the entire range of stimulus strengths μ₀. x-axis, Difference between μ₀ and μ*₀, which is the bifurcation point at which the saddle steady state appears and whose value depends on the AMPA:NMDA ratio. C and D have the same symbolic notations as in B. Error bars indicate SD.

Figure 8.

Decision time with only AMPA at recurrent synapses (c′ = 0%). A, A sample time course with very fast dynamics (t = 0 is stimulus onset). μ₀ = μ*₀ + 15 Hz, where μ*₀ is the value of μ₀ when the saddle point is just created. Bottom horizontal bar denotes the duration where the firing rates are indistinguishable; trajectory lies near the stable manifold of the saddle point. B, Reaction time as a function of μ₀ − μ*₀. Note that reaction times longer than 200 ms can hardly be realized even with parameter fine-tuning. Error bars indicate SD.

Figure 9.

Neural firing activity in delayed-response task in which a motion stimulus presentation is followed by a memory period. A, Sample time courses within a trial with different coherence levels. Black and dashed lines are for saccades moving toward and away from the response field of the neuron, respectively. The coherence level is shown at the top of each panel. Shaded regions, Motion viewing period. The black horizontal line at the bottom indicates the time epoch when the two firing rates are indistinguishable. B, Dependence of neural activity on motion strength in different epochs. Opened and filled circles, Saccades toward and away from the response field of the neuron. The largest dependence on the motion strength, as well as the greatest difference in the two neural responses, correspond to the late phase (0.5–1 s epoch) of stimulus presentation. In the third epoch (early delay period), there is a still a residual effect of the dependence of neural response on motion strength. Figures are calculated using correct trials only and averaged over 2000 trials.

Figure 10.

Bifurcation diagram of a selective population with stimulus strength μ₀ as parameter (c′ = 0%). Bold lines, Stable steady states; dashed lines, saddle steady states. Spontaneous state before stimulus presentation is denoted by the filled square. With a μ₀ = 30 Hz stimulus, the spontaneous stable state loses stability, and a saddle steady state appears (open square). The state either goes toward the upper or lower stable state (filled circles). The population wins the competition if the upper branch is chosen, and loses otherwise. When stimulus is removed, hysteresis of the upper stable branch allows the activity to persist (memory storage of a decision choice). Arrow with an asterisk, Point where spontaneous state loses stability. Arrow with double asterisks, Saddle point turns into an attractor.

Figure 11.

Dependence of integration time on the relative strength of recurrent excitation. A, Sample time courses: faster ramping activity with stronger recurrent strengths, w₊. B, The unstable time constant of a saddle point dominates the dynamics when recurrent strength w₊ is weak. C, Reaction time decreases with increasing w₊. Error bars indicate SD. D, Accuracy of performance decreases with increasing w₊. Data are fit with a Weibull function.

Figure 12.

Distinct modes of operation in the two parameter space with zero coherence. In general, there are three types of regions. Bistable (red) region, A symmetric and two asymmetric attractors coexist; blue competition region, one saddle with two asymmetric competing attractors; monostable region, only one attractor. Depending on the strength of recurrent excitation w₊, the network responds to a stimulus (of suitable intensity μ₀) in four different ways, shown as regimes I, II, III, IV in insets. Regime I and II do not support working memory (of decision). Regime I, No decision making nor memory. Regime II, The network can produce a binary decision during stimulation but cannot store it in working memory. Regime III, The network is capable of both decision-making computation and working memory (our standard parameter set). Regime IV, For any μ₀, there is always a stable symmetric stable state. Dark and dashed branches denote loci of stable and unstable steady states, respectively. A and AS are labels for branches with symmetric and asymmetric steady states, respectively.

Figure 13.

Decision making without short-term memory in a network with low recurrent strength (w₊ = 1.59). A, A typical trial showing slow ramping up activity for both neural populations. Note that the two firing rates are indistinguishable for many hundreds of milliseconds (indicated by the black horizontal bar), before they separate by a modest difference. Stimulus is applied from time 0–2 s (gray region). B, Phase-plane without stimulus has only one low stable attractor. C, The response of the system to a stimulus with zero coherence, in two trials, plotted as trajectories in the decision space (red and blue). Stimulus intensity is μ₀ = 45 Hz lasting for 2 s. D, Comparison between stable and unstable time constants of the saddle-type unstable steady state (μ₀ = 45 Hz).