^{*}

Conceived and designed the experiments: AR AL. Performed the experiments: AR AL. Analyzed the data: AR AL. Contributed reagents/materials/analysis tools: AR AL. Wrote the paper: AR AL.

The authors have declared that no competing interests exist.

The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons.

The brain holds a central position in scientific theories of rational behavior. For example, brain activity is thought to stand in a causal relation to the decision making behavior observed in two-choice perceptual discrimination tasks. Although a lot is known about both the brain activity and the response behavior during these tasks, the relationships between the two are not fully understood. In particular, how can one relate the high-dimensional dynamic activity of the brain to the low-dimensional descriptions of response behavior such as performance and reaction-times? Our approach to this question is to relate existing neurobiological models of brain activity to existing models of response behavior. In this paper we establish a formal link between standard, winner-take-all models of brain activity during two-choice tasks and a family of one-dimensional behavioral models known as diffusion models. Our analysis demonstrates a universal functional dependence between the external inputs to the neural populations in the neurobiological model on the one hand, and reaction times and performance in the one-dimensional model on the other. Importantly, we show that experimentally measured performance and reaction-times can be predicted through changes in these external inputs alone.

In perceptual two-choice decision making experiments one studies how sensory information influences response behavior. In each trial the experimental subject is presented with a stimulus and must use the information thus provided to choose one of two possible responses. The response behavior in these tasks, as defined by reaction times and performance, has been studied for over a hundred years

The aim of the work presented here is to account for the response behavior in two-choice decision making tasks in terms of the underlying neurobiology. In the remaining part of this section we will first describe one prominent family of behavioral models of response behavior, the sequential sampling models. Subsequently we will describe some neurophysiological findings, and models thereof, pertinent to our modeling framework.

The response behavior in two-choice decision making tasks is well described by so-called sequential sampling models _{L}_{L}_{L}

The Diffusion model can be related to other behavioral models of decision making. It can be conceived of as a continuous-time version of random walk models (e.g.,

Recently, neuroscientists have begun to investigate the single-cell neurophysiology of the decision making process in two-choice tasks (e.g.,

Assuming that the brain regions involved in the decision making process implement a winner-take-all strategy, as suggested by computational models, it remains unclear how this might lead to a response behavior best described by a one-dimensional diffusion processes. In other words, what is the relationship between the neuronal activity in putative decision making circuits and decision variables in behavioral models such as

Here we show how one can go beyond linearizations to take into account nonlinear effects in neural winner-take-all models. In particular, we will show how models of neuronal dynamics in two-choice decision making tasks can be formally reduced to a one-dimensional nonlinear diffusion equation. Instead of focusing on a particular model system we consider a generic model of the neuronal dynamics with two key features: nonlinearity and competition. One obvious advantage of using such a general framework is that the extent of validity of the reduction is potentially very large. Moreover, most detailed models of the neuronal underpinnings of decision making do include nonlinearity and competition as important ingredients (e.g.,

Here we show how a winner-take-all model can be reduced to a nonlinear diffusion equation through general symmetry arguments alone. Despite the generality of the derivation, the resulting coefficients are directly related to biological meaningful parameters. We furthermore illustrate the correspondence of the nonlinear diffusion equation with neural winner-take-all models by deriving it directly from a system of coupled rate equations. Further examples, including a network of spiking neurons, are provided in the supporting material (

We begin on a fairly technical note in order to provide some sense of the generality of the result. We will then make use of, for illustrative purposes, a simple model which nonetheless retains some biophysical plausibility . While the method we use may seem complicated and the algebra is, in general, involved, the idea behind the reduction is simple. We take advantage of the dramatic reduction in dimensionality which occurs spontaneously in dynamical systems near a point where the qualitative behavior of the system changes, i.e. stationary states appear, disappear or change in nature. Such transition points or bifurcations, are ubiquitous in physical and biological systems, e.g. see

To emphasize the reduction in dimensionality we first consider a system of _{i}^{th}_{1} and _{2} are populations whose activity correlates with the two possible developing choices while the remaining populations are non-selective given the particular task. We note that for _{1} = _{2} the equations are invariant under the transformation (_{1},_{2})→(_{2},_{1}), a property known as reflection symmetry. We see from this symmetry that the existence of the fixed point (_{1},_{2},…) = (_{high}_{low}_{1}, _{2},…) = (_{low}_{high}_{SS} are the steady state values and perturbations with growth rate λ have the form _{1} in Equation 2 and _{2} in Equation 3, β is the derivative of _{2} in Equation 2 and _{1} in Equation 3, the γs are derivatives with respect to _{1} and _{2} and all derivatives are evaluated at the fixed point. It is clear that for α = β, which will occur only for a special parameter set, the first two rows cease to be linearly independent implying a zero eigenvalue with eigenvector _{cr} = (1,−1,0,…,0) which corresponds to a mode for which either _{1} or _{2} increases while the other decreases, i.e. the winner-take-all dynamics we are interested in. If we wish our system to exhibit winner-take-all behavior then it must also be that the real part of the remaining

We now wish to derive an equation for the dynamics of the ‘winner-take-all’ instability. To do so we express the dynamical variables as _{SS}+_{cr}_{1} and _{2}. We assume that the increase in input, _{1} and _{2} leads to the developing decision in the winner-take-all system and is thus the bifurcation parameter. This means that the linear growth rate of the spontaneous state must be proportional to the difference between the presynaptic input and the value of the input at the bifurcation although with an unknown prefactor, i.e. μ(_{cr}). The difference in inputs, _{1}−_{2}, breaks the reflection symmetry thereby introducing a constant term which, to first approximation, must be proportional to that difference although with an unknown prefactor, i.e. _{cr} only when α = β identically, i.e. at point of instability, and ∂_{T}_{1}−_{2} the equation is invariant under ^{3} is the lowest order nonlinearity which obeys reflection symmetry). The coefficients

The framework we chose above for illustration, Equations 2–5, is a system of coupled nonlinear ordinary differential equations. However, the derivation of the nonlinear diffusion equation Equation 8 is not contingent on the original system having this particular form. For this reason we have chosen to illustrate the derivation of the nonlinear diffusion equation from three distinct model systems. Below we study a system of three coupled rate equations. In the supporting material (

We consider a simple model describing the activity of two excitatory populations of neurons which compete via a population of inhibitory interneurons. The equations are_{I}_{1} and _{2} are the activity of the inhibitory and two excitatory populations respectively. The input to each population consists of a combination of recurrent and external inputs. For each excitatory population there is a recurrent excitatory coupling of strength _{i}_{I}_{E}_{I}_{cr}_{cr}

Generically, two qualitatively different scenarios for winner-take-all dynamics can occur in nonlinear systems.

In the first scenario, the two ‘decision’ fixed points bifurcate continuously from the spontaneous state and therefore with small amplitude. Such a supercritical bifurcation can be seen in _{1} and _{2} (black) during a typical simulation in the supercritical case, as well as the rates predicted by Equation 8 (red), see the figure caption for parameter values. In comparing the original system and the nonlinear diffusion equation we must choose appropriate initial conditions. The derivation of the nonlinear diffusion equation itself assumes strongly attracting dynamics in all directions except along the slow manifold whose dynamics are described precisely by Equation 8. Assuming symmetric initial conditions in the original system (pre-stimulus), an infinitely fast approach to the slow manifold with stimulus onset would lead to _{1} = _{2} = _{I} = 0) after which the dynamics converges rapidly to that predicted by the nonlinear diffusion equation (red). If we wish to treat the system as a model of decision making in the supercritical regime we must choose the appropriate placement for the thresholds. This has been studied elsewhere, e.g.

For all panels shown, τ = _{I}_{1} = _{2} = 0. (B) Typical dynamics for a single trial given a supercritical bifurcation. Shown are the time-dependent variables _{1} and _{2} from integrating Equations 11–13 in black and the approximation obtained by integrating the nonlinear diffusion equation Equation 8 with coefficients Equations 14–17, in red. Left inset: the same trial shown for a longer time. Right inset: The energy function given the parameter values used for this trial. Here _{cr}_{1} = 0.0025, _{2} = −0.0025, σ_{E}_{I}_{1}(0) = _{2}(0) = _{I}_{1} = _{2} = 0. (D) Typical dynamics for a single trial given a subcritical bifurcation. Lines and insets are as in (C). Here _{cr}_{1} = 0.001, _{2} = −0.001. Initial conditions as for (B). (E) A comparison of the fraction of ‘correct decisions’ and mean reaction-times calculated by conducting simulations in the full system, Equations 11–13 (symbols), and with the nonlinear diffusion equation, Equation 8, with coefficients, Equations 14–17 (lines). The parameter values correspond to the bifurcation structure shown in (C). Different symbols indicate different values for the common input _{cr}

A positive cubic coefficient in Equation 8 indicates a subcritical bifurcation. In this case, the fixed points corresponding to a decision appear in a saddle-node bifurcation already below the critical input at the pitchfork bifurcation. These solution branches therefore already have finite amplitude at this point. Such a situation is shown in

The explicit dependence of the coefficients in Equation 8 on the inputs to the two populations allows us to directly relate modulations in these inputs to changes in reaction-times and performance. In doing so we will make use of the formulation of a nonlinear diffusion equation as the motion of a particle in a potential, Equations 9–10, see

The populations are shown schematically as circles and the respective inputs as arrows. The relative level of input to the populations is represented by the spike trains “recorded” from the input arrows. (A) The energy

Below, we discuss these effects in greater detail, making use of exact expressions for reaction times _{0}) and performance _{0}) as a function of the initial condition _{0}. See supporting material (

When both populations receive the same mean input, i.e. Δν = 0, the energy function, Equation 10, is symmetric, leading to an equal probability of escape through either boundary, i.e. performance

Changes in the input common to both populations affect the quadratic term in _{L}X

We note that increasing the noise amplitude σ in Equation 8 leads to decreasing performance. Increasing noise amplitude also tends to reduce reaction-times given initial conditions in the vicinity of the spontaneous state.

To reiterate,

We now show that such changes in input are sufficient to describe behavioral data in two-choice decision making tasks. Specifically, we consider data from two separate studies using the so-called random moving dots task, namely from Roitman and Shadlen

In light of these experimental observations, it is reasonable to assume that the difference in inputs from MT cells to the putative neuronal populations in LIP which encode the two possible directions, increases linearly with increasing coherence. Therefore, we assume a linear dependence of Δ

Black symbols: experimental data. Error bars represent approximate 95% confidence intervals. Lines: solution of Equation 8. Red symbols: simulated data from a system of rate equations (Equations 11–13). (A) Reaction times on correct and error trials as a function of coherence. Circles (solid line) and diamonds (dotted line) are for correct and error trials respectively. (B) Fraction of correct responses as a function of coherence. Experimental data are shown as crosses. Parameter values for Equation 8: ηΔν =

Filled circles: experimental data. Lines: solution of nonlinear diffusion equation, Equation 8. Open squares: system of rate equations. Error bars represents approximate 95% confidence intervals. Data are from three sets of experiments in which subjects are instructed to respond with 0.5 s (blue) 1 s (red) and 2 s (black). (A) Reaction times on correct trials as a function of coherence for subject 1. (B): Fraction of correct responses as a function of coherence for subject 1. Parameter values from Equation 8 for subject 1: ηΔν =

We now show that the trade-off between speed and accuracy, commonly observed in reaction-time experiments

Since Equation 8 can be derived analytically from more complex model systems, we can map the values of the coefficients obtained from fits to behavioral data back to more physiologically meaningful parameters. An example of this is shown in

The fits of the nonlinear diffusion equation Equation 8 to behavioral data suggest not only that the putative decision making circuit behaves in a way consistent with a winner-take-all framework, but that changes in inputs to this circuit alone are sufficient to account for performance and mean reaction-times. In addition, the best fits to the data were found for |ηΔν|,

One-dimensional diffusion equations have long been used to model behavior in two-choice reaction-time tasks. Recently, researchers discovered that the trial-averaged single-unit activity recorded in areas of the brain which are implicated in generating this behavior closely resemble the dynamics of a linear diffusion process

The dependence of the coefficients in Equation 8 on external inputs is explicit and independent of the details of the underlying model. This suggests that the functional dependence of behavioral measures in two-choice decision making on changes in inputs is universal. In particular, we predict that modulations of the input common to both populations can account for the speed-accuracy trade-off. This mechanism differs from that evoked by others previously, which consists of varying the threshold for detection of the decision (e.g. a higher threshold increases reaction times and increases performance),

While Equation 8 appears similar in form to other diffusion models which have been used to describe behavior in two-choice decision making

As in the linear diffusion equations, bias in external inputs in the nonlinear diffusion equation appears to leading order as a constant drift term. In contrast, while reductions of linear connectionist models to the linear diffusion equation lead to a linear (Ornstein-Uhlenbeck) term proportional to the difference between intrinsic ‘leak’ and the effective cross inhibition, this is not the case in nonlinear systems. Rather, this term reflects the linear growth rate of the spontaneous state which, given that the input is the bifurcation parameter, is simply proportional to the distance of the common external input from the critical value at the bifurcation. Thus this term varies with modulations of the external input, unlike in the linear case. Finally, the cubic nonlinearity, which is the lowest order nonlinearity consistent with the reflection symmetry of the original system, leads to an inverted-U potential. This drives the activity to infinity in finite time, reflecting the escape from the spontaneous state to the ‘decision’ state. As illustrated in

As it turns out, Equation 8 can account for behavioral data for the random moving dot task in monkeys and humans, c.f.

Soltani and Wang

The reduction to Equation 8 is strictly valid only in the immediate vicinity of the bifurcation. For this reason it might be argued that the current scenario is tantamount to fine-tuning and may not be biologically relevant. Three facts indicate this is not the case. (I) As we have shown here Equation 8 can be rigorously derived from model systems and can provide a

Here we derive the nonlinear diffusion equation (noise driven amplitude equation for an imperfect pitchfork bifurcation) from Equations 11–13. We first study the linear stability of the spontaneous fixed point, analogously to Equation 6 and then extend this analysis to take into account nonlinear effects in a so-called

We assume that _{A}_{B}_{A}_{B}_{I}_{I}_{A}_{B}_{I}^{λt}, where _{I}_{I}_{I}_{I}

The eigenvalue corresponding to the eigenvector (1,−1,0) is equal to zero for

We expand the input current and the rates around the steady instability found above. We take^{2}

We recover the linear stability problem_{1} = (1,−1,0)

_{2}. A solution therefore only exists if the vector _{2} is in the left-null eigenspace of the linear operator. This can be expressed as 〈^{†},_{2}^{†} = (1,−1,0) and the ^{T}_{2} can then be found by projecting onto the eigenspace orthogonal to the left-null eigenvector, i.e. 〈(1,1,1),_{2}−_{2}〉 = 0 and 〈(1,1,−2),_{2}−_{2}〉 = 0. Doing so yields

We have^{†},_{2}_{1}−_{3}〉 = 0. This leads to the equation_{A}_{B}^{1/2}

We solve for the performance and reaction time in Equation 8 (solid lines in

Once the fits have been made using the nonlinear diffusion equation, we must choose parameters in the rate equations which give the proper values for the coefficients, using the expressions Equations 14–17. Various parameter combinations are possible, indicative of the reduction in dimensionality of the system and a potential mechanism for robustness in functionality.

For the simulations in _{0} = 1, _{I}_{E} = σ_{I}_{1}−_{2} = 2.168

For the simulations for subject 1 in _{0} = 1, _{I}_{E} = σ_{I}_{1}−_{2} = 4.066

For the simulations for subject 2 in _{0} = 1, _{I}_{E} = σ_{I}_{1}−_{2} = 4.228

In all cases, a trial ends once one of the rates crosses a fixed threshold of 0.7. Initial condition was _{A}_{B}_{I}_{A}_{B}_{I}

Detailed description of nonlinear diffusion equation: closed-form expressions for RT and P and derivations from three model systems. Comparison between nonlinear and linear diffusion models.

(0.51 MB PDF)

We thank John Palmer for providing us with the data used in