Fig. 10.

Models for a 2-variable decision-making from linear, perfect integrator to cubic, nonlinear attractor. a The parameter a, dictates the degree of nonlinearity, by adding a cubic term to the piece-wise linear portion of the firing-rate curve. b, c The steady state probability distribution of the pair of firing rates in the absence of input for the linear perfect integrator in b and the cubic nonlinear model in c. Spontaneous rates are constrained in the 2D system (unlike the perfect integrator in 1D) because of the threshold nonlinearity at zero rate (firing rates can not be negative, so noise-driven drift is constrained). b For the linear system (perfect integrator) for a given sum of the two rates, the probability density is independent of the difference in firing rates, whereas c) the nonlinear system constrains the difference in firing rates to produce elliptical contours of constant probability. Results are with D′ = 450Hz ² s ⁻¹, equivalent to D = 900Hz ² s ⁻¹ in the 1D system. Steady state distributions are shown, so no time-limit is included