Sensory and motor mechanisms interact to control amplitude of pupil noise

Pupil noise was the example for the pioneering’ application of communication engineering to a stochastic biological process (Stark et al., 1958) almost twenty years ago and considerable experimental, theoretical and modelling studies have since been accomplished (Stark, 1959; Bouma, 1965; Hess, 1965; Loewenfeld, 1966; Stanten and Stark, 1966; Hansmann et al., 1974; Semmlow et al., 1975) in extending the earlier pupillary noise findings and in relating them to other approaches and pupil phenomena. We now present new experimental results (plotted as open circles in Fig. 1) that show pupillary noise to be multiplicative in a particular fashion-with greatest variance in mid-range and smaller variance at high and low ranges. This confirms the finding of multiplicative noise put forward by Stanten and Stark (1966), but modifies and extends their relationship between standard deviation and mean pupil diameter. A variety of experimental methods have been used to acquire the noise data presented in Fig. 1 ranging from photography by de Launay (1949), infra-red photocell recordings by Stanten and Stark (1966) and Miller and Stark (1964) to infra-red TV pupillometry by us. Left inset of Fig. 1 collects several comparative sets of experimental data using deterministic inputs, such as light, accommodation and vergence stimuli, ail of which drive the pupil as a member of synkinetic near-response triad (Stark, 1964; O’Neill and Stark, 1968; Loenwenfeld and Newsome, 1971; Krishnan et al., 1973; Semmlow and Stark, 1973). The earliest of these studies on the A-multiplier by Stark (1964) focused attention only on the positively sloped portion of the expansive range nonlinearity. These results demonstrate the nonlinear characteristics of deterministic pupillary gain as a function of pupil diameter to be similar to pupillary noise results. An imporant expansive range nonlinearity having its locus in the ‘plant’ or neuromuscular mechanism of the pupil system appears to be operative in controlling the multiplicative nature of pupil noise and as well as the nonlinear pupil gain. In order to establish this we simulate a biomechanical pupil model and demonstrate quantitative agreement of model and experimental results. A biomechanical model of the iris sphincter and dilator operating as a reciprocally-innervated pair of muscles with similar static and dynamic properties has been developed (Terdiman et al., 1969; Semmlow, 1970; Terdiman et al., 1971; Hansmann, 1972; Hansmann et al., 1974; Semmlow et al., 1975). The lengthtension, L-T, characteristics of smooth muscle are not well known (Meiss, 1971) but appear similar to skeletal muscle (Gordon et al., 1966) where it has been related directly to degree of overlap in the sliding filament model (Huxley, 1959). Some recent findings modifying straightforward dependence upon the sliding filament model have been put forward, but considerable debate has not yet clarified the underlying physiological and molecular mechanisms (Huxley and Hanson, 1954). We have used the Carlson length-tension equation with quadratic terms (Carlson, 1957; Hansmann, 1972; Hansmann et al., 1974) and passive parallel elasticities to calculate the model relationship between gain and operating level, or pupil diameter, shown in Fig. 1, left inset; this is the ‘expansive range nonlinearity’. By using a measure-preserving transformation of probability theory (Thomasian, 1969) we have also extended the model so that it can treat the noise problem. Essentially, the second moment or variance of the stochastic equation representing the static model is proportional to the slope of the static inputoutput curve, or equivalently to the nonlinear gain curve. This provides the basic model curve for Fig. 1. The third moment or skewness of the stochastic equation similarly is proportional to the second derivative of the input-output curve, or equivalently to the slope of the nonlinear gain curve. Skewed curves from de Launay (1949) and Stanten and Stark (1966) appear to experimentally confirm this prediction of the model. The nonlinear stochastic model characteristic is also shown in Fig. 1, main part, and represents the action of the expansive range nonlinearity on the noise process. The classical S-shaped curve of pupil diameter as a function of the logarithmic of light intensity (Branchard, 1918;,Reeves, 1920; Crawford, 1936; Wagman and Nathanson, 1942; Moon and Spencer, 1944; Spring and Stiles, 1948; de Groot and Gebhard, 1952; ten Doesschate and Alpern, 1967; Usui, 1974) is additionally explained by our model results. A pure logarithmic operation would yield a straight line relationship on the linear-log plot of Fig. 1, right inset, (Stevens, 1951); a power law might show curvature