LNCS 4527, Pp

Physiological Laws of Sensory Visual System in Relation to Scaling Power Laws in Biological Neural Networks Isabel Gonzalo-Fonrodona1 andMiguelA.Porras2 1 Departamento de Optica.´ Facultad de Ciencias F´ısicas. Universidad Complutense de Madrid. Ciudad Universitaria s/n. 28040-Madrid. Spain [email protected] 2 Departamento de F´ısica Aplicada. ETSIM. Universidad Politécnica de Madrid. Rios Rosas 21. 28003-Madrid. Spain Abstract. Measurements of some visual functions (visual fields, acuity and visual inversion) versus intensity of stimulus, including facilitation, carried out by Justo Gonzalo in patients with central syndrome, are seen to follow Stevens’ power law of perception. The characteristics of this syndrome, which reveals aspects of the cerebral dynamics, allow us to conjecture that Stevens’ law is in these cases a manifestation of the universal allometric scaling power law associated with biological neural networks. An extension of this result is pointed out. 1 Introduction Half a century ago, Stevens [1] formulated his well-known relation between sen- sation or perception P and the physical intensity of a stimulus S, expressed mathematically as a power law of the type P = pSm , (1) where p is a constant and m depends on the nature of the stimulus. This law is regarded as more accurate than the logarithmic Fechner’s law, but is not exempt from criticism. In a different, and somewhat more general context, it was argued that in biological organisms, mass is the determinant factor for the scaling of the physiological behavior. If M is the mass of the organism, many observable biological quantities, for instance Y , are statistically seen to scale with M according to a power law of the form Y = kMn , (2) where k is a constant and, in formal similitude with Stevens’ law, the exponent n changes from one observable to another, leading to different (allometric) behavior of observables with respect to mass M. Most of exponents in this law are surpris- ingly found to be multiples of the power 1/4. Biological variables that follow these J. Mira and J.R. Alvarez´ (Eds.): IWINAC 2007, Part I, LNCS 4527, pp. 96–102, 2007. c Springer-Verlag Berlin Heidelberg 2007 Physiological Laws of Sensory Visual System 97 quarter-power allometric laws are, for instance, the metabolic rate (n 3/4), lifes- pan (1/4), growth rate (−1/4), height of trees (1/4), cerebral gray matter (5/4), among many others (see [2,3] and references therein, and also [4,5]). The allometric scaling laws are supposed to arise from universal mechanisms in all biological systems, as the optimization to regulate the activity of its subunits, as cells. According to West and Brown [6], optimization would be achieved through natural selection by evolving hierarchical fractal-like branching networks, which exchange energy and matter between the macroscopic reservoir and the microscopic subunits. Some examples of these networks are the animal circulatory, respiratory, renal and neural systems, the plant vascular system, etc. The quarter-power allometric laws can be theoretically derived from simple di- mensionality reasonings that derive from the geometrical constraints inherent to these networks. As the same authors remark, powers proportional to a multiple of 1/4 would be strictly verified only in ideal biological organisms, while in real organisms the power may slightly depart from these values, since they are af- fected by stocastic factors, environmental conditions and evolutionary histories. For other authors [7], however, the scaling power laws are valid independently of the network type, and hence also for those without hierarchical or fractal structure. Though Stevens’ law for perception-stimulus relation and biological scaling law with mass relate to different phenomena, their formal similitude indicates a possible connection between them. A more fundamental connection is pointed out here on the basis of the measurements of some visual functions versus stimulation intensity, carried out by Gonzalo [8,9,10] in patients with the central syndrome that he described. After recalling the characteristics of this syndrome, we first verify that the measured data of the visual functions versus stimulus intensity fit well to Stevens’ power laws. Second, under reasonable assumptions on the relation between physical stimulus and activated neural mass, we conclude that Stevens’ law of perception is, in the cases studies at least, a manifestation of the universal scaling power laws. 2 Characteristics of the Central Syndrome The central syndrome (or symmetric poly-sensory syndrome) seems to be partic- ularly suitable for the observation of the unfolding of the sensory functions. This syndrome originates from a unilateral parieto-occipital lesion equidistant from the visual, tactile and auditory projection areas, and is featured by [8,10] (a) poly-sensory affection with symmetric bilaterality, (b) functional disgregation of perception, in the sense that sensory qualities are gradually lost in a well-defined order as the stimulus intensity diminishes, and (c) capability to improve the perception by iterative temporal summation and by facilitation through other stimuli, as for instance, strong muscular stress. The syndrome was interpreted [10] as a deficit of cerebral integration due to a deficit of cerebral nervous excitation caused by the loss of a rather unspecific (multisensory) neural mass. This interpretation arises from a model in which functional sensory densities 98 I. Gonzalo-Fonrodona and M.A. Porras for each sensory system are distributed in gradation through the whole cortex [10,11]. There are other works dealing with this research [12,13,14,15,16,17], or related to it (e.g., [18,19,20]). A close connection can also be established with models based on a distributed character of cerebral processing, its adaptive and long-distance integrative aspects (e.g., [21]). The remarkable point here is that the central syndrome was explained in terms of a scale reduction of the normal cerebral system caused by the lesion [10,12]. From the concept of dynamic similitude, scaling laws were applied to the sensory decrease —or functional depression— observed in the patients. From the comparison between twelve cases with central syndrome in different degrees, their visual luminosity, acuity and other qualities were found to obey approximate power laws versus their respective visual field amplitudes, with different exponent for each quality, i.e., allometrically [10,12]. This is the formal description of the functional disgregation, or decomposition of the normal perception into its different qualities by their gradual loss, from the most to the less complex ones, as the nervous excitation diminishes, or equivalently, as the magnitude of the lesion grows. The organization of the sensorium can be then visualized, up to a certain extent, as displayed in patients with central syndrome. For a given individual with central syndrome, the sensory level grows by intensifying the stimulus, or by adding other different stimuli, which are able to compensate for the neural mass lost. This dynamic capability is greater as the neural mass lost is greater, and is null, or extremely low for some functions, in a normal man. [8,10,13]. 3 Stevens’ Law in Central Syndrome All experimental data presented are taken from Ref. [8], correspond to two different cases with central syndrome, called M and T (less intense), and refer to the change of visual functions or qualities with intensity of stimulation in a stationary regime. Stevens’ law [Eq. (1)] is used to fit the data. Figure 1(a) shows the experimental data for the visual field amplitude of right eye in cases M, M facilitated by strong muscular stress (40 kg held in his hands), and case T, as functions of the illumination of the test object. The reduction of the visual field in the central syndrome is concentric. In the log-log graphic, Stevens’ law [Eq. (1)] yields a straight line of slope equal to the exponent m. As seen, the data fit rather well to Stevens’ straight lines not very close to saturation. The slope m of the fitting straight lines is remarkably close to 1/4 for M and M facilitated, and 1/8 for T. In Fig. 1(b), similar representation is shown for the visual acuity in central vision, including a normal man. Straight lines with slope 1/4 fit well to the central part of the data for the two states of case M and for case T, and with slope 1/8 for normal man. In another series of experiments, the intensity of light on the test object was kept constant and low, whereas the variable stimulus was the facilitation sup- plied by muscular stress [Fig. 2(a) and (b)], or by light on the other eye (Fig. 3). Fig. 2(a) shows the measured visual field amplitude of right eye in case M Physiological Laws of Sensory Visual System 99 Fig. 1. (a) Visual field of right eye versus relative illumination (test object: 1 cm- diameter, white disk). Squares: M (fitting straight line with slope 1/4). Circles: M facilitated (straight line with slope 1/4). Triangles: T (straight line with slope 1/8). (b) Acuity of right eye versus illumination. Squares: M (fitting with slope 1/4). Circles: M facilitated (fitting with 1/4). Triangles: T (fitting with 1/4). Inverted triangles: Normal man (fitting with 1/8). Fig. 2. (a) Visual field amplitude in right eye versus facilitation by muscular stress. Squares: M, 0.5 cm test size (fitting straight line with slope 1/2).Circles:M,1.0cmtest size (fitting slope 1/3). (b) Visual direction (reinversion) in right eye versus facilitation by muscular stress. Squares: M (fitting slope 1/4). Fig. 3. Visual direction (reinversion) in right eye versus facilitation by illumination on left eye. Squares: M (fitting straight line with slope 1/8).

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