Difference-of-Gaussians connectivity kernel

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A narrow positive Gaussian models local excitation and a broader negative Gaussian models inhibition. The body places 2σ² in both exponential denominators, while the comment displays σ², so the code uses the usual Gaussian width convention. The sign and reach of the net kernel depend on both amplitudes and widths.

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Original context

// where f is the firing rate function and w is the connectivity kernel
template<size_t N>
struct NeuralField {
    float tau = 10.0f;
    float threshold = 0.0f;
    float steepness = 1.0f;

    // Mexican hat connectivity: w(x) = A*exp(-x²/σ_e²) - B*exp(-x²/σ_i²)
    float A = 1.5f;      // Excitation amplitude
    float B = 0.75f;     // Inhibition amplitude
    float sigma_e = 1.0f;// Excitation width
    float sigma_i = 2.0f;// Inhibition width

    std::array<float, N> u;      // Activity field
    std::array<float, N> input;  // External input

    void initialize() {
        u.fill(0.0f);

Mexican hat connectivity: w(x) = A*exp(-x²/σ_e²) - B*exp(-x²/σ_i²)

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