Adaptation

In adaptation, a vigorous early response gives way to little or no chronic response. The output is determined by changes in the input rather than the magnitude of the input.

In the figure above, the input is a step function of amplitude 1 that occurs at time equal to 4 (Red). The output is initially equal to the input, but then it decays toward 0 as time passes (Blue).

Since many biological processes adapt, this can be a useful tool to have in your toolbox.

The equations for this model are

    dA/dt = K * Output
    Output = Input - A

A is the adaptation term, shown in the figure above (Magenta).

K is the adaptation rate constant. The adaptation time constant (Tau) is the reciprocal of the rate constant.

    Tau = 1 / K

For a step change in input, when time = Tau, Output = 0.63 * Input.

The adaptation half-time is the time required for the output to fall to one-half of its initial value.

    t _1/2 = 0.69 * Tau

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