Alex Roxin; Anders Ledberg

“ 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 [1] – [2] leading to a wealth of data and modeling results [3] . ”

“ 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 P (0) = 0.5, see Figure 2A green. If one population receives more input than the other, i.e. its activity encodes the ‘correct’ choice, Δν≠0 and the probability of making the corresponding choice will be greater than chance, P (0) >0.5. This is reflected in the asymmetry of the energy function, which is now tilted towards the correct choice, see Figure 2A black. ”

“ That is, if there is state in which population 1 exhibits a high level of activity and population 2 a low level of activity, then we are assured the existence of the opposite state. If these are the only possible stable states at long times then Equations 2–5 constitute a so-called ‘winner-take-all’ system. Since we want the system to behave in a winner-take-all fashion only when provided with sufficient input, we furthermore assume the existence of a fixed point , representing the spontaneous state. ”

“ The inhibitory population receives excitatory drive from both populations with a strength g and input II and we have neglected any self-inhibition term. The Φs are nonlinear transformations of the input. Fluctuations in the activity variables are expressed via unit variance Gaussian white noise terms ξ (t) with strength σE and σI for the excitatory and inhibitory populations respectively. Additionally, we have normalized time by the time constant of the excitatory populations and τ thus represents the ratio of the inhibitory to the excitatory time constant. ”

“ The average firing rate of cells in these areas is correlated to the response behavior of the animals. This indicates that the firing rate “represents [...] the information on which the developing decision is made” [17] . Further support for this interpretation comes from a study showing a direct correspondence between the time-evolution of trial-averaged firing rates in the superior colliculus and the dynamics of the decision variable in the Diffusion model [21] . ”

“ These two facts, coupled with the reflection symmetry, lead to the form of the equation describing the time evolution of Y
(7)
where I = Icr only when α = β identically, i.e. at point of instability, and ∂T is a time derivative with respect to the slow time T. Note that for I1−I2 the equation is invariant under Y−Y as it should be (indeed, Y3 is the lowest order nonlinearity which obeys reflection symmetry) . The coefficients , μ and γ can be calculated analytically from Equations 2–5. ”

“ 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 xcr = (1,−1,0,...,0) which corresponds to a mode for which either x1 or x2 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 n−1 eigenvalues is negative to avoid unwanted instabilities unrelated to the dynamics of interest and sufficiently distant from zero. ”

“ Mean reaction-times for error trials can, in fact, be slower or faster than those for correct trials depending on the value of the common input . Performance increases with increasing Δν, owing to a more pronounced asymmetry in the energy. Indeed, it can be shown analytically that , see supporting material (Text S1) . An analogous effect is obtained in the linear diffusion equation, Equation 1, by changing the constant drift term. ”

“ Strong recurrent excitation therefore naturally leads to a subcritical bifurcation and hence hysteresis. Network models describing the activity of neuronal populations in two-choice decision-making tasks have made use of this working-memory property which allows for a decision to be made and then ‘held in mind’, e.g. ”

“ The data shown in Figure 4 are from three sets of experiments in which human subjects are told to respond within 0.5, 1 and 2 seconds, and are shown in blue, red and black symbols respectively [9] . These data clearly exhibit precisely the speed-accuracy trade-off. ”

“ A computational model.”, J Neurosci 26: 9761–9770, pmid:16988047 ↑ Haken, H, Synergetics ↑ Strogatz, SH, Nonlinear dynamics and chaos, Addison Wesley ↑ Nayfeh, AH & Balachandran, B, Applied nonlinear dynamics, Wiley ↑ Holmes, MH, Introduction to perturbation methods, Springer ↑ Brunel, N & Hakim, V, “Fast global oscillations in networks of integrate-and-fire neurons with low firing rates.”, Neural Comp 11: 1621–1671 ↑ Brunel, N ”