The lognormal handwriter: learning, performing, and declining

Abstract

The generation of handwriting is a complex neuromotor skill requiring the interaction of many cognitive processes. It aims at producing a message to be imprinted as an ink trace left on a writing medium. The generated trajectory of the pen tip is made up of strokes superimposed over time. The Kinematic Theory of rapid human movements and its family of lognormal models provide analytical representations of these strokes, often considered as the basic unit of handwriting. This paradigm has not only been experimentally confirmed in numerous predictive and physiologically significant tests but it has also been shown to be the ideal mathematical description for the impulse response of a neuromuscular system. This latter demonstration suggests that the lognormality of the velocity patterns can be interpreted as reflecting the behavior of subjects who are in perfect control of their movements. To illustrate this interpretation, we present a short overview of the main concepts behind the Kinematic Theory and briefly describe how its models can be exploited, using various software tools, to investigate these ideal lognormal behaviors. We emphasize that the parameters extracted during various tasks can be used to analyze some underlying processes associated with their realization. To investigate the operational convergence hypothesis, we report on two original studies. First, we focus on the early steps of the motor learning process as seen as a converging behavior toward the production of more precise lognormal patterns as young children practicing handwriting start to become more fluent writers. Second, we illustrate how aging affects handwriting by pointing out the increasing departure from the ideal lognormal behavior as the control of the fine motricity begins to decline. Overall, the paper highlights this developmental process of merging toward a lognormal behavior with learning, mastering this behavior to succeed in performing a given task, and then gradually deviating from it with aging.

Keywords: aging; handwriting analysis and generation; kinematic theory; learning; lognormal models; lognormality; neuromuscular systems.

Figure 1

(A) Example of the speed profile of a delta-lognormal modeling for a typical fast reaching movement. (B,C) Example of the trace (B) and the speed profile (C) of the sigma-lognormal modeling of a typical triangular movement. (D,E) Example of the trace (D) and the speed profile (E) of the sigma-lognormal modeling of a typical handwritten letter “a.”

Figure 2

(A) Distribution of age and gender for the NRF subsample. (B) Idem, but for the WRF subsample. (C) Distribution of the risk factors in the WRF subsample. The height of the bars shows the overall number of subjects with each risk factor. As subjects may have more than one risk factor simultaneously, the bars are separated into colored sections indicating how many risk factors the subjects have. For example, 40 subjects have HT (height of the HT bar); among these, about 8 subjects have only the HT (the light gray portion of the HT bar) whereas one subject who has HT also has 4 other risk factors, for a total of 5 (the red portion of the HT bar).

Figure 3

Guiding sheets used for the simple reaction time experiments (A), the choice reaction time protocol (B), and the triangular drawings (C–E). The starting position is shown as a dark circle and the target zones as gray areas. For triangular drawing sheets, the targets are 15 mm in diameter and are positioned at the apexes of equilateral triangles with vertexes of 135 mm (C), 90 mm (D), and 45 mm (E) long.

Figure 4

SNR (A), nbLog (B), and SNR/nbLog ratio (C) as functions of the writer class for all movements.

Figure 5

Scatter plots of the proportion of rejected movements when considering the rejection criterion SNR < λ, for different values of λ (in dB). A logistic regression curves has been added to the plots to show the average tendency. The plots (A,C,E,G) refer to the population with No Risk Factor (NRF) having a SNR < 30,25, 20 and 15 dB, respectively. The plots (B,D,F,H) refer to the population With Risk Factor (WRF) under the same SNR conditions.

Figure 6

Scatter plot linking the average SNR to the subjects' age. The lines show the best linear models associated with these data. The plot (A) is for the population with No Risk Factor (NRF) and the plot (B) for the population With Risk Factor (WRF).