Neural-field dynamics and the implemented discrete average

Read the equation

The field loses activity through a decay term and gains input through a spatially weighted sum of firing responses plus external drive. The implementation samples integer positions, sums all pairwise kernel contributions, and divides by N. This is a particular normalized discrete network model; reproducing a continuum integral requires a spatial grid and quadrature convention.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

        E = std::clamp(E, 0.0f, 1.0f);
        I = std::clamp(I, 0.0f, 1.0f);
    }
};

// Neural Field Theory (Amari Equation)
// τ * ∂u(x,t)/∂t = -u(x,t) + ∫w(x-x')*f(u(x',t))dx' + s(x,t)
// 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

τ * ∂u(x,t)/∂t = -u(x,t) + ∫w(x-x')*f(u(x',t))dx' + s(x,t)

Related builds