Thresholded population response sigmoid

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The function maps net input into (0,1), crosses one half at input θ, and has central slope a/4 when a>0. It converts weighted network input into an activation fraction in the population model. The parameter θ is a threshold in input units rather than a phase angle here.

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Original context

// ═══════════════════════════════════════════════════════════════════════════════
// LEVEL 2: NEURAL NETWORK / POPULATION MATHEMATICS
// ═══════════════════════════════════════════════════════════════════════════════

// Wilson-Cowan Equations (population dynamics)
// τ_E * dE/dt = -E + (1 - r_E*E) * S(w_EE*E - w_EI*I + I_E)
// τ_I * dI/dt = -I + (1 - r_I*I) * S(w_IE*E - w_II*I + I_I)
// where S(x) = 1/(1 + exp(-a*(x-θ))) is the sigmoid
struct WilsonCowan {
    // Time constants (ms)
    float tau_E = 10.0f;
    float tau_I = 20.0f;

    // Connection weights
    float w_EE = 12.0f;   // E → E
    float w_EI = 4.0f;    // I → E (inhibition of excitatory)
    float w_IE = 13.0f;   // E → I
    float w_II = 11.0f;   // I → I

where S(x) = 1/(1 + exp(-a*(x-θ))) is the sigmoid

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