An intrinsic property of human motor behaviour is a trade-off between speed and accuracy. This is classically described by Fitts' law, a model derived by assuming the human body has a limited capacity to transmit information in organizing motor behaviour. Here, we propose an alternative foundation, based on the neurodynamics of the motor circuit, wherein Fitts' law is an approximation to a more general relationship. In this formulation, widely observed inconsistencies with experimental data are a consequence of psychomotor delay. The methodology developed additionally provides a method to estimate the delay within the motor circuit from the speed-accuracy trade-off alone.

We discuss natural limitations on motor performance caused by the time delay required for feedback signals to propagate within the human body or mechanical control systems. By considering a very simple delayed linear servomechanism model, we show there exists a best possible speed-accuracy trade-off similar to Fitts' law that cannot be exceeded when delay is present. This is strictly a delay effect and does not occur for the ideal case of instantaneous feedback. We then examine the performance of the vector integration to endpoint (VITE) circuit as a model of human movement and show that when this circuit is generalized to include delayed feedback the performance may not exceed that of the servomechanism with an equal delay. We suggest the existence of such a limitation may be a ubiquitous consequence of delay in motor control with the implication that the index of performance in Fitts' law cannot arbitrarily large.

Quasi-Spherical harmonics, Y m ℓ (θ, φ) are derived and presented for half-odd-integer values of ℓ and m. The form of the φ factor is identical to that in the case of integer ℓ and m: exp (imφ). However, the domain of these functions in the half-odd-integer case is 0≤φ<4π rather than the domain 0≤φ<2π in the case of integer ℓ and m (the true spherical harmonics). The form of the θ factor, P |m| ℓ (θ) (an associated Legendre function) is (as in the integer case) the factor (sin θ) |m| multiplied by a polynomial in cos θ of degree (ℓ−|m|) (an associated Legendre polynomial). A substantial difference between the spherical (integer ℓ and m) and quasi-spherical (half-odd-integer ℓ and m) Legendre functions is that the latter have an irrational factor of √ sin θ whereas the factor of the truly spherical functions is an integer power of sin θ. The domain of both the true and quasi spherical associated Legendre functions is the same: 0≤θ<π. A table of the Associated Legendre Functions is presented for both integer and half-odd-integer values of ℓ and m, for |m| = 0, 1 2 , 1 . . . 11 2 , and for (ℓ−|m|) = 0, 1, 2, 3, 4, 5. The table displays the similarity between the functions for integer ℓ and m (which are well known) and those for half-odd-integer ℓ and m (which have not been recognized previously).

An intrinsic property of human motor behavior is a trade-off between speed and accuracy. This is classically described by Fitts' law, a model derived by assuming that the human body has a limited capacity to transmit information in organizing motor behavior. However, Fitts' law can also be realized as an emergent property of movements generated by delayed feedback. In this article, we describe the relationship between the Fitts' law coefficients and the physiological parameters of the underlying delayed feedback circuit: the relaxation rate or time constant, and the psychomotor delay of the feedback process. This relationship is then used to estimate the motor circuit delay of several tasks for which Fitts' law data are available in the literature. We consistently estimate the delay to be between 0 and 112 ms. A further consequence of this model is that not all combinations of slope and Y-intercept in Fitts' law are possible when movements are generated by delayed feedback. In fact, it is only possible for an observed speed-accuracy trade-off to be generated by delayed feedback if the Fitts' law coefficients satisfy -0.482 < or = a/b < or = 3.343 [bits] where b represents the slope in bits per second and a represents the Y-intercept in seconds. If we assume human movement is generated by delayed feedback, then the Fitts' law coefficients should always be restricted to this range of values.