A Model for Shell Patterns Based on Neural Activity By

, THE VELIGER The Veliger 28(4):369-388 (April 1, 1986) © CMS, Inc., 1986 A Model for Shell Patterns Based on Neural Activity by BARD ERMENTROUT Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A. JOHN CAMPBELL Department of Anatomy, University of California, Los Angeles, California 90024, U.S.A. AND GEORGE OSTER Departments of Biophysics and Entomology, University of California, Berkeley, California 94720, U.S.A. Abstract. The patterns of pigment on the shells of mollusks provide one of the most beautiful and complex examples of animal decoration. Recent evidence suggests that these patterns may arise from the stimulation of secretory cells in the mantle by the activity of the animal's central nervous system. We present here a mathematical model based on this notion. A rather simple scheme of nervous activation and inhibition of secretory activity can reproduce a large number of the observed shell patterns. INTRODUCTION so as to give interesting patterns, rather than to correspond THE GEOMETRICAL patterns found on the shells of mol to known physiological processes (WADDINGTON & COWE, lusks comprise some of the most intricate and colorful 1969; LINDSAY, 1982a, b; WOLFRAM, 1984). In the most patterns found in the animal kingdom. Their variety is recent attempt, MEINHARDT (1984) modeled the growing such that it is difficult to imagine that any single mecha edge of the shell as a line of cells subject to activator nism can be found. Adding to their mystery is the dis inhibitor kinetics and a refractory period. He was able to turbing fact that, since many species hide their pattern in obtain a variety of shell-like patterns, suggesting that an the bottom mud, or beneath an opaque outer layer, it is activator-inhibitor mechanism is likely to be involved in doubtful they could serve any adaptive function. Perhaps the actual process. these wonderful patterns arise as an epiphenomenon of Recently, CAMPBELL (1982) proposed a novel expla the shell secretion process. This may account for the ex nation for the shell patterns. He reasoned that the pigment treme polymorphism exhibited by certain species-a phe cells of the mantle behaved much like secretory cells in nomenon characteristic of traits shielded from selection. other organisms; that is, they secreted when stimulated by Several authors have attempted to reproduce some of nervous impulses. Therefore, the shell patterns could be these patterns using models that depend on some assumed a recording of the nervous activity in the mantle. Because behavior of the pigment cells in the mantle that secrete the phylogeny of mollusks is well represented in the fossil the color patterns (WADDINGTON & COWE, 1969; COWE, record, the implications of this view for the study of the 1971; WANSHER, 1972; HERMAN & LIU, 1973; HERMAN, evolution of a nervous system are obvious. 1975; LINDSAY, 1982a, b; WOLFRAM, 1984; MEINHARDT, Building on Campbell's notion, and the suggestive sim 1984). These models have generally been of the "cellular ulations of Lindsay, Meinhardt, and Wolfram, we have automata" variety, and the postulated rules were chosen constructed a model for the shell patterns based on nerve- DR - --------------------------------------------------------------------------------------------------------------~.... Page 370 The Veliger, Vol. 28, No.4 a b c Figure 1 Three fundamental classes of shell pigment markings on Bankivia fasciata: a, longitudinal bands; b, incremental lines; c, oblique stripes. stimulated secretion of the mantle epithelial cells. This OBSERVATIONS ON SHELL PATTERNS model differs from previous models in at least one impor tant aspect: it depends on the "nonlocal" property of nerve The variety of shell patterns is so enormous that it appears nets. That is, because innervations may connect secreting that any attempt to classify them will inevitably leave out cells that are not nearest neighbors, the possibility of co many special cases. However, we do not hope to explain operative, long-range interactions is present. This greatly all of the patterns; rather, we seek to model the global enlarges the pattern-generating repertoire over nearest features shared by all patterns in a restricted class. In neighbor models, and has the virtue of relating directly to particular, we shall focus mostly on the patterns exhibited the anatomy of the mantle. Despite its simplicity, the mod by Nerita turrita and Bankivia Jasciata (Figures 1, 2, 3). el is remarkably successful in mimicking a wide variety These animals exhibit a representative variety of shell of shell patterns. patterns from which we can draw some inferences. The paper is organized as follows. First, we catalog a Many pigment patterns of gastropod shells are com number of regularities in the shell pattern that bear on posites of three basic types of patterns: (a) longitudinal the neural hypothesis. In particular, those phenomena that bands that run perpendicular to the lip of the shell, (b) implicate a global organizer and preclude strictly local incremental patterns arranged parallel to the growing shell interactions. Second, we sketch the model equations and edge, and (c) oblique patterns that run at an angle to the discuss their behavior. Third, we present patterns gener lines of the shell. Some mollusk taxa have more specialized ated by simulations of the model and compare them to types of patterns, such as the circular eye-spots on some actual shell patterns. Fourth, we discuss some experi cowry shells, or the intricate tent-like patterns on cone ments the model suggests and some generalizations of the shells. Species differ in the categories of patterns that they model. The Appendices contain the mathematical details display. Nerita turrita shells are always dominated by of the model and a discussion of how it relates to other oblique patterns without longitudinal bands, whereas oth models of shell patterns. er members of the genus have shells with bands as well , B. Ermentrout et at., 1986 Page 371 b a Figure 2 a, an incremental alternation in zebra stripes across the entire whorl of Bankiviafasciata; b, simultaneous termination of str! pes in B. fasciata. as modified oblique lines. Shells of Bankivia fasciata are building (producing about one-third to one-half whorl of highly polymorphic, with various combinations of these shell in the case of Bankivia fasciata), followed by "rest" pattern types, as illustrated in Figure 1. periods during which no shell is secreted. Shell patterns Longitudinal bands require only simple developmental often are reorganized at these major interruptions in shell controls. They could result from a mosaic mantle in which synthesis, and many sculptured shells produce flamboyant regions continuously deposit pigment, along with shell, ridges or spines along varices. separated by mantle zones that do not synthesize pigment. Oblique patterns are the most intricate, and have the In general, the number and position of bands appears to most implications for our theoretical model. They imply be a genetic characteristic of the species, or of the individ that the activities involved in pigment secretion are coor ual in a polymorphic species. A second possibility-which dinated laterally and proceed dynamically across the man we shall illustrate with the model-is that the band width tle. For example, the oblique lines shown in most of the and spacing are characteristic of the neural activity in the shell illustrations in this paper represent a patch or do mantle. The two mechanisms are not mutually exclusive, main of secretory activity that sweeps across the mantle, as we shall discuss. Banding indicates that variation can eventually migrating to its edge. These mobile domains of be a permanent (e.g.) programmed) feature of the mantle activity in the mantle behave in a variety of ways to pro edge. duce the diverse appearance of the patterns. Incremental markings have several sources. Some ap pear to result from haphazard physiological stresses or EVIDENCE IN FAVOR OF LONG-RANGE environmental factors that temporarily affect the activities COORDINATION OF PATTERNS of the mantle as a whole. In addition, some species of snails (other than the ones we shall consider here) show The neural network model we propose here allows for regular periodic incremental shell patterns, indicating that interactions and coordination beyond nearest neighbors. they are programmed in a deterministic and cyclic man As we shall demonstrate in the next section, this gener ner. One of the most important incremental features seen alization enormously enlarges the possible types of pat on shells of the two species we have chosen for analysis terns over previous models, which employ short range, or are varices: time periods during which shell synthesis was "nearest neighbor" interactions. What evidence do we have halted (Figures 2, 3). In general, mollusks do not produce that pigment secretion is indeed a neurally controlled pro shell continuously, but go through cyclic periods of shell cess? We can offer no direct experimental support, for we ..... Page 372 The Veliger, Vol. 28, No.4 b Figure 3 Abrupt reorganization of patterns on shells of Nerita turrita (a), and after a break in a shell (b). have not been able to find any anatomical studies of man However, because diet and other environmental factors tle innervation patterns nor of secretory cell physiology. affect the pattern formed on a shell, there must be some Therefore, aside from the general observation that secre physiological mechanism that relates the two. That is, tory cells in most organisms are influenced by neural ac there must be some mechanism whereby a systemic effect tivity, we can offer only the following indirect evidence in allows two separated regions of mantle tissue to manifest support of the neural activation hypothesis of shell pat coincidental patterning. The two main avenues for trans terns. mitting stimuli from the environment to the mantle cells Global reorganizations. At a varix, a shell pattern are soluble chemical factors (especially hormones) and may become systematically and simultaneously reorga nervous connections.

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