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Neurobiological Models of Two-Choice Decision Making Can Be Reduced to a One-Dimensional Nonlinear Diffusion Equation

Figure 1

The bifurcation structure (A,C), typical dynamics (B,D) and behavioral measures (E) for the system of three coupled rate equations, Equations 11–13.

For all panels shown, τ = g = c = 1 and I_I = 0.2. The nonlinear transfer function is taken as with α = 1.5, β = 2.5. (A) Below a critical value of the recurrent excitation the system exhibits a supercritical bifurcation. Shown are the fixed points of Equations 11–13 without noise using a Newton-Raphson solver, in black. Also shown are approximations of the fixed points given by Equations 8 with coefficients Equations 14–17, in red. Here s = 1.5 and I₁ = I₂ = 0. (B) Typical dynamics for a single trial given a supercritical bifurcation. Shown are the time-dependent variables r₁ and r₂ 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 s = 1.5, I = I_cr = 0.6502, I₁ = 0.0025, I₂ = −0.0025, σ_E = σ_I = 0.01. Initial conditions for rate equations, r₁(0) = r₂(0) = r_I(0) = 0. Initial condition for nonlinear diffusion equation, X(0) = 0. (C) Above a critical value of recurrent excitation s the system exhibits a subcritical bifurcation. Lines are as in (A). Symbols show the value of the common inputs used in the four cases shown in (E). Here s = 1.9 and I₁ = I₂ = 0. (D) Typical dynamics for a single trial given a subcritical bifurcation. Lines and insets are as in (C). Here s = 1.9, I = I_cr = 0.3679, I₁ = 0.001, I₂ = −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 I and correspond to symbols in panel C. Specifically, I−I_cr = −0.001 (triangles), 0 (squares), 0.0012 (circles), and 0.0321 (diamonds). Initial conditions for rate equations: I = 0.2 for t = −100 to t = 0 with appropriate steady state solutions. Initial condition for nonlinear diffusion equation X(0) = 0. Thresholds were 0.7 for the rate equation.

doi: https://doi.org/10.1371/journal.pcbi.1000046.g001