Leakyintegraattoria

Leakyintegraattoria, or leaky integrator, is a dynamic system that accumulates input signals over time while allowing a fraction of the accumulated value to decay. In continuous-time form, the state x(t) evolves according to dx/dt = - (1/τ) x(t) + k u(t), where τ > 0 is a time constant and u(t) is the input. The parameter governs the leakage rate; larger leakage (smaller τ) causes faster decay.

In discrete time, y[n] = y[n-1] + α (u[n] - y[n-1]), or equivalently y[n] = (1 - α) y[n-1] + α u[n], with 0 < α < 1. The system behaves as a first-order low-pass filter with an exponential impulse response.

Implementation commonly uses an RC circuit: a resistor in series with a capacitor where input voltage charges the capacitor and leakage through the resistor causes discharging, producing a naturally leaky integration of the input. Digital realizations use a first-order infinite impulse response filter.

Applications: smoothing and integration of slowly varying signals, estimating smoothed states in control systems, and as a basic neuron model in computational neuroscience (leaky integrate-and-fire models describe how neurons accumulate input with leak and fire when a threshold is exceeded). In psychology and economics, leaky accumulators model evidence accumulation with forgetting.

Variants: leaky integrator with adaptive leakage, or multi-stage leaky integrators forming higher-order filters.

See also: Integrator, Exponential moving average, IIR filter, Leaky integrate-and-fire neuron.