,

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. Both may modulate patterns, but in­ nized across an entire shell (cf Figures 2a, b), sometimes fluences that differentially affect discrete parts of the man­ into an entirely different sort of pattern (Figure 3a). A tle simultaneously seem more plausibly mediated by the variety of new patterns may arise in this manner, rather nervous system. than arising locally and propagating as a wave across the Entrainment of lines. Shells in which oblique lines shell. Such changes in the "state" of the pattern can also become entrained in the middle of a longitudinal band be initiated by a break in the shell (Figure 3b). It is hard also suggest coordination of pattern across sizable dis­ to see how such local perturbations could have such global tances, measured in cell diameters. A particular example effects by means other than nervous activity. of this is the shell in Figure 4a, on which a band appears It should be noted that physiological and (or) environ­ spaced equidistant from the neighboring bands. mental factors can influence the entire mantle simulta­ Termination of lines. On the shell in Figure 4b three neously. Indeed, it has been demonstrated that changes in oblique lines terminated anomalously at about the same diet can alter not only the color of the pattern, but the time. These events occurred in regions of shell separated pattern itself (D. Lindberg, personal communication). This by uninterrupted oblique bands. If these changes were due fact does not argue for or against the neural hypothesis, to a signal that propagated from one locale to another, for it is relatively common for dietary factors to affect that signal would have to have migrated cryptically past nervous activity, as well as other physiological systems. the unaffected domains in the mantle. The simpler inter- B. Ermentrout et aI., 1986 Page 373 I

b

Figure 4 a, appearance of a band spaced equidistantly between adjacent bands; b, simultaneous termination of several separated zebra stripes without noticeable concurrent alteration of the stripes in between.

pretation is that the three separate areas were acted upon THE NEURAL MODEL by a signal that could be conveyed to multiple local regions simultaneously. In this section we present a qualitative description of the Blotching. A polymorphism (not otherwise described shell pattern model. The mathematical discussion is given here) among Bankivia fasciata shells is blotching (Figure in the Appendices. The model we shall present here is the 5). On blotched shells areas of pigmentation abruptly dis­ simplest possible neural model, and we do not expect (0 appear or appear incrementally across large blocks of shell. reproduce every shell pattern, even those observed on the Alternatively, various segments of the mantle can be af­ two species we have selected for study. However, the mod­ fected simultaneously by blotching. Also, for some blotched el is capable of producing sufficiently diverse patterns that shells the zone of pigment deposition did gradually spread we consider it a reasonable first approximation; we shall along the mantle, indicating that blotching can be con­ suggest a number of improvements which will enlarge the trolled in a variety of ways. class of patterns, but at the expense of computational sim­ Global appearance of patterns. On some shells a gen­ plicity. eral type of pattern gradually develops across the entire mantle, but with no indication that the change sweeps across the mantle; the saw-tooth pattern in Figure 6 il­ BIOLOGICAL ASSUMPTIONS OF lustrates this phenomenon. THE MODEL Checkerboard patterns. (Figure 7) It is possible to The basic assumption of the model is that the secretory create a checkerboard pattern from two sets of colliding activity of the epithelial cells that generate shell patterns waves that propagate by strictly local interactions. How­ is regulated by nervous activity. Specifically, we assume ever, it is remarkable that the checkerboard as a whole that the secretory cells are enervated from the central gan­ can stay in register without drifting in alignment. This glion and secrete or not as they are activated and inhibited synchronicity implies that a substantial segment of the by the neural network that interconnects them with the mantle cycles back and forth between an active and in­ ganglion. Although arguments in favor of this hypothesis active state in precise coordination. Adjacent subzones were presented above, the issue can only be settled em­ switch states of activity simultaneously, but in opposite pirically, and experiments are under way to test the neu­ directions. ral hypothesis directly. Figure 8 shows a schematic of the ------...... £tQ

Page 374 The Veliger, Vol. 28, No.4

5

7

Explanation of Figures 5 to 7 Figure 5. Blotched patterns on Bankivia fasciata shells. Figure 7. Checkerboard patterns on Bankivia fasciata. Figure 6. Sawtooth patterns on Nerita tumta. mantle and the secreting cells (EMBERTON, 1963; KAPUR daily) bursts of activity. At the beginning of each & GIBSON, 1967; NEFF, 1972; KNIPRATH, 1977). session the mantle aligns with the previous pattern The specific assumptions that underlie the model are: and extends it by a small amount. This alignment process probably depends on the ability of the mantle (1.) Cells at the mantle edge secrete in intermittent (e.g., to sense (taste) the pigmented and (or) non-pigment- ..

B. Ermentrout et al., 1986 Page 375

extrapaZZial space

periostracU7n shell

0000 0 000

ventral epithelium

Figure 8 Diagram of the anatomy of the mantle region.

ed regions from the previous period of secretion. period t. This will depend on sensing the pigment Equivalently, a section of pigmented shell laid down secreted during the previous period, Pt-1(x). during the previous period will stimulate the mantle Then the equation governing the neural activity in the neurons locally to continue the pattern. mantle during period t 1 is related to the pigment se­ (2.) The secretion during a given period depends on two + factors: cretion during period t by the equation (a) the neural stimulation, S, from surrounding re­ A/+l(X) = S[P,(x)] - R, [ 1] gions of the mantle. (b) the buildup of an inhibitory substance, R, within Equation [1] says that the average neural activity, the secretory cell. A'+l(X), at location x on the mantle during day t + 1 (3.) The net neural stimulation of the secretory cells is depends on the net neural stimulation at that location, the difference between excitatory and inhibitory in­ which is stimulated by sensing the previous day's pigment puts from surrounding tissue. S[P,(x)]. In the absence of stimulation, this nervous activ­ ity decays as the inhibitory substance R,(x) builds up. The We incorporate these assumptions into the model as fol­ inhibitory substance, R, builds up as pigment, P, is man­ lows. ufactured, and is degraded at a constant rate (0 < 1):

Secretion of Pigment Depends on Current [2] Neural Activity Finally, we assume that secretion of pigment will only occur if the mantle activity is above a threshold value, A *: Consider a line of secretory cells whose position along the mantle edge is located by the coordinate x (Figure 9). [3] Let where H(A - A *) is a threshold function for pigment P,(x) = the amount of pigment secreted by a cell at x secretion: It IS zero for A < A *, and one for A > A *. during the time period t (e.g., one day). Equations [1] and [2] describe how the activity, A, and A,(x) = the average activity of the mantle neural net at refractory substance, R, evolve in time; having computed position x on the mantle edge during one secretion A, the actual pigment secretion is given by [3]. In the period, t. computer simulations we have simplified the model even R ,(x) = the amount of inhibitory substance produced by further by incorporating equation [3] into equation [1], cells at location x in day t. and writing equations for P directly (ef Appendix A). S[P] = the net neural stimulation at location x during This modification makes little difference in the computed Page 376 The Veliger, Vol. 28, No.4

Figure 9 Diagram of the model: MPG, mantle pigment cells; PGN, pigment cell neurons; MNN, mantle neural net; GG, central ganglion; r, receptor cells sensing pigment laid down in time period t; P, pigment cells secreting pigment in time period t + 1; E, excitatory neurons; I, inhibitory neurons.

patterns, but is somewhat simpler to simulate. Figure 9 two categories: (A) those controlling the shape of the neu­ shows a schematic of the model's structure. ral stimulation function, and (B) the production and deg­ radation rates of the inhibitory substance. Each parameter Neural Activity Depends on the Difference Between corresponds to a definite physiological quantity, and so is Excitatory and Inhibitory Stimulation measurable, at least in principle. Neural parameters. The neural stimulation function, N ext, we must model the process of neural stimulation S, in equation [1] contains the curves for excitation, in­ that regulates the secretion of the pigment. We regard the hibition, and firing threshold shown in Figure 10. Each net stimulation of a cell at x to be the difference between of these functions must be described by formulae that con­ excitatory and inhibitory stimulations from nearby cells. tain parameters to control their shapes. The functions we The situation is illustrated in Figure lOa: a cell located have employed in our simulations are described in Ap- at a position x on the mantle edge received excitatory inputs and inhibitory inputs. The inhibitory signals are generally more "long range" than the excitatory inputs; that is, the mantle edge exhibits the property of short­ range excitation and long-range inhibition characteristic of neural nets (BERNE & LEVY, 1983; ERMENTROUT & COWAN, 1979). Moreover, we assume that the response of a nerve cell is a saturating function of its inputs; that (a) is, both excitation and inhibition are sigmoidal functions of their arguments, as shown in Figure lOb. The mathe­ matical form of the neural stimulation term we have em­ ployed is given in the Appendix. When these assumptions are incorporated into the mod­ el equations there results a set of functional difference s equations that determines the pigment pattern, P,(x) (ef equations [Al, 2]). In Appendix A we perform a linear analysis on these equations. This gives some idea of the repertoire of patterns the model can generate, and pro­ vides a guide to the numerical simulations presented be­ low. (b) u The Model Parameters Figure 10 Any model contains adjustable parameters, and equa­ Diagram of the neural influence function and threshold tions [1]-[3] contain several. These parameters fall into function. r

B. Ermentrout et al., 1986 Page 377

'." . "'-:::""'11'5' i.:S'~Ff'1"'U":~~1:;:.ijji.W:::!iljill~WiHi~;.till;*ii:l.ri-~I~1i:i~: :;:I:Z:i'i:IWi.:!i.mitJ~~~_

~':-~,=r=~~~]j;~!Ol_~~;'itP.fc:.i:w.A.~lSlf:::jiiP.~1:'.Mii.ri.~1il!~1~lk~).;:t~;1lm.Y_~~,i:i;ru~;;.;Lt;i!.!J~jjf!~;L;jjj~~

~...,~_~~:t__ :WJb.lt1L~=¥~'ti~~-=h.~~i¥:.£~~}'~e:i~~!llJi_~i~;~:'~~!~if~'!£7f~:t.';'~:-~~!i?::rg.;;;1~y'~~~. ~_ii!!!!!I~.. 1t.:¥.~~~~~lr~!'ri~7.il~i!;*'f!!~·~F.:i1\!':~!=t"1"ri=-"~I"~i'~*;i;~~';!!,~":!;1:~;~!i\:P;~~~~: .', ~~ .. , .... :~.~I~'l"I"I:;';;:'!~~'~"~!~'~~'of:~!;' ~!t;,.r~;r~~~·r~;r::-f':fi!~:~;~":h:;:~;l'!fi!r~~:\F:a'i~

:';'1.: '-.~:.:~:' ~-=-:-: ~·i";"~i· .• :·:.:·,·~~~!r~"'~r.:r~ln.!k~)!:'~:!l!: .. ·,,,:.\,.'::'::.r~~~~ ~::~:.:.:. :.;:'~:-' :.:.:.:.:.~: .. ~::~.:~:;~,::!~.:~~.:>:~~:I~:I:":: ~~:i ~':': l':'~. ~ :'~:'~':'~'::: I'~~'~':':jj~~~~~ ±E~:~:~~':·;:·:~~~J~~::~;'·:··::·I:~~==~;:

., ...... ~ ,' ... -···~~li~fi~Jiifif~~~;&,~ii!!iiitt~~~i;*i\i\lf~-:-··.:·::;:~.·:·· ... :

~iiHiiiiii~~Ii!Iii!iiiiiRii_Iiilii!iIiiiii~:a.~"'&ili*1tJl~::liiim··~~~.I'~I1-!·:~:ld~~:~: . 1::i:i~i!UtalMm·· ~"!fi!f:t!!:l±f,t!1.1!:t!I:~~'t:t:tl·t!t·:tft.l~ Figure 11 Simulations of: a, vertical stripes of constant width; b, vertical stripes of variable width; c, horizontal stripes.

pendix A; however, experience has shown that the qual­ stance, however, requires the two parameters: 'Y to regu­ itative predictions of the model depend only on the general late the growth rate of R, and 0 to control the decay rate shapes of the functions, not on their particular algebraic of R. form. Even though each of the model parameters has a direct Cellular parameters. Each secretory cell is character­ physiological interpretation, with enough parameters one ized by its production rate of pigment under neural stim­ might feel that any variety of patterns is possible. How­ ulation and its production and degradation of refractory ever, this is not true. For a fixed neural structure, there substance, R. The production rate of pigment is controlled are but two adjustable parameters: 'Y and o. Varying the entirely by the neural stimulation, S, and so no new pa­ neural interactions involves changes in their shape-con­ rameters are required to describe it. The refractory sub- trolling parameters, and analysis and simulation studies a

Page 378 The Veliger, Vol. 28, No.4

toriness is very low and the neural influence functions are 1------~ strong and thresholds small. There are two mechanisms for producing stripes: one is similar to the Turing mech­ 0:= anism in diffusion-reaction models. That is, short-range activation creates a laterally spreading zone of activity, ~ 0 which is eventually quenched by the longer range inhib­ itory activity. This produces stripes whose width is con­ t:I stant, as shown in Figure lla. The stripe width is a func­ 0 tion of the parameters (being roughly the width of the .--foJ , ttl activation-inhibition zone), and the locations of the stripes '"tj are determined by the width of the domain (i.e., the size ttl S-4 of the mantle). A different mechanism produces stripes of b.O unequal widths, as shown in Figure llb. It is also possible Q) to produce vertical stripes by simply activating certain 0 lUll regions of the mantle permanently, so that secretion is always turned on. Only experiments can distinguish be­ 0 tween these two possibilities. Horizontal stripes, or incremental lines (Figures 1 b, 0 1 llc), are produced when the refractory parameters are production of R small and thresholds are high. This results from a syn­ Figure 12 chronized, or homogeneous oscillation along the entire -y-o parameter plane showing domain of stripes and obliques. mantle (not to be confused with the incremental pattern associated with the episodic nature of shell deposition). Diagonal stripes, or zebra bands (Figures 1c, 2, 3, 4), show that the resulting patterns can be classified into a are characterized by very low thresholds and gradual cut­ relatively small number of types. Within each distinct type, offs. These arise as waves of activity propagate along the variations of the parameters merely alter the relative di­ mantle. If the neural structure is constant, the presence of mensions of the pattern, and not its qualitative appear­ oblique stripes or vertical bands depends on the values of ance. However, parameter variations that exceed certain the two parameters controlling the refractoriness, -y and thresholds, cause the pattern to shift not just its scale, but o. Figure 12 shows the parameter domain that character­ its qualitative type as well. This "bifurcation" behavior izes each pattern type. will be discussed further below. The direction of the stripes produced by the model de­ pends on the parameter values. However, downward ori­ PATTERNS GENERATED BY ented stripes (i.e., away from the apex of the shell) are THE MODEL more common in Bankivia fasciata and Nerita turrita and In this section we describe the patterns generated by the exhibit far fewer irregularities. Moreover, upward-di­ neural model. We shall present numerical simulations of rected stripes appear to be more unstable, reverting to the neural model which mimic certain patterns observed downward stripes after a short progression. This points on the shells of Bankivia fasciata and Nerita turrita. to a consistent inhomogeneity in the mantle. Indeed, su­ perimposing a parameter gradient (e.g., in 0 and [or] -y) Basic Patterns on the model equations strongly biases the direction of striping in one direction. Interestingly, the direction of Equations [1]-[3] constitute the simplest possible model stable striping is in the same direction as the spiral of the for a neural net; consequently, we cannot hope to repro­ shell. Because shell patterns are associated with shell con­ duce all of the known shell patterns. However, we can struction, this could indicate a physiological (anatomical) reproduce aU of the basic patterns; moreover, it is easy to correlation between the direction of shell growth and the see how the model can be elaborated to incorporate a pattern direction, such as an asymmetry in the muscle wider variety of patterns. We shall briefly discuss these mass of the mantle. The direction of the zebra stripes can modifications here, and present a more detailed study in switch at certain times, especially-but not exclusively­ a subsequent paper. at a varix. The three fundamental patterns exhibited by Nerita Divaricate patterns. Zebra patterns may reverse di­ turrita and Bankiviafasciata are longitudinal bands, incre­ rections giving a herring-bone pattern. We have used the mental lines, and oblique stripes (Figure 1). The param­ observation that synchronous switching of the direction of eter values that realize these patterns are given in Table stripes indicates a global coordinating mechanism for the 2 in Appendix A. Qualitatively, the conditions that yield pattern. In terms of the model, switching of the direction these patterns are as follows. of obliques involves a jump in a parameter value. The Vertical stripes (Figures la, 11) occur when refrac- model does not address what the underlying signal for B. Ermentrout et at., 1986 Page 379

AAAU; +UAAAAA: AAAAAAA .AAAAAAt: :IIAAAAA AllAM.: ,IAAAAAA. IAAAAAA AAAAAAAt ·tAUAA1; •• AAAAN;, IAAAAAA+ ;AAA&. ,AAAAAU. ,AAMA%: •• • ;3AAAM, ,HAAAAAW:, ;%AAAAAAA1, :%lAAAAA%:. , ; &MAAM+. ,lAAAAAAA; : AAAAAAAt. %AAAAA%: • • : &AAAAA, :3AAAAAA AAAAAAA. ,%AAAAAAAl: ,IAAAAAAAI; :AAAAAA; AMAAAA :AAAAA'+: , ,:+&AAAAI ,+%lAMI :AAAAAAI • :IMAAAAI ;%8M AUl, +AAAAAAAI; ,%AAAAAAAAA;: .IAAMA+, ,+IAA AMI ,%AAAAAA : AAAAAAAI tMAAAAAAA, : UIAAA AAAH I; +AAAAAAAI; AAMAAA AAAAAA%, : AMAAA AAAAAI • :UAAAA ;IMMM AAAAAM+ • ;%AAAAM+ +AAAAAAU+;. tAMAIIAt:, +I; ;AAAAAIt. .AAAA!t,. • : IAAAAA. ,IAAAAAAAt I&AAAAAAAL ;IAAAAAAI;. ,IAAMAN+ IAAAAAAA+ AAAAAAI, %AAAAAI:. :SAAAAAA: :$AAAAAA AAAAAAAA tlAAAAAAAI .IAAAAAAiI+, AAAAAA: ,AAAAA&+ MMAAU;, , +. .: %AAAAA, •• alAAA+,. :%AAAAAI ,UAAAAAAA+ :%&U AAll, IAAAAAA%, ; lAAAAAAA.; ; AMAAS: ,lAM AAAI, ; MAAAAAA. ;AAAAAAAI +AAAAAAAI ,tlAAAA MAn: IAAAAAAA1, AAAAAAA AAAAM.: ,AlIMA' AAAAA&. ,;$AAAAM SAAAAAI :AAAAAAA; .;XAAAAU+ ;IAAAAAU+, &AAAAA$:.. ;&U: SAAAA1+: ,AMA';, .. • ;SAAAAl :$AAAAAAMt. +lAAAAAAAS .+IAAAAAAI; : IAAAAAI+ &AAAAAAAt ,AAAAAAA; :AAAAAI:. :HAAAAA: ,IAAAAAA AAAAAAAA ,%AAAAAAAf .&AAAAAAUt, .AAAAAA, .AAAAAA$ ;AAAAAlt; , + ., ;IAAAAA. ,XSUMt. ,1AAAAAA: ;UAAAAAA& .;131 AU+, ,MAAAAAA%, ; MAAAAAAAM;, • AAAAA&;. .+lAA AAAI. ;WAAAAAA ;AAAAAAA& tMAAAAAAA, ,l!lAAA AAA&% , IAAAAAAAL AAAAAAA AAAAAAS: ,AAAAAA AAAAH. ,+&AAAA& lAAAAA& :AAAAAAA+ :+IAAAAA1 +IAAAAAUt, !AAAAAl:. +UI: %AAAAl+:. MAAAH, •• .; IAAAAS ,SA AAAAA&+. HAAAAAAAI • +&AMAAA1+, :IAAAAAM+ MAAAAAAA+ .AAAAAAA; SAAAAA%: • • :%AAAMA, : &AAAAAA AAAAAAAA • UAAAAAAI ,!AAAAAAUi· ,AAAAAA, • AAAAAAI ,AAAAAA$+ • , • ,'SMAAa : IUAAA: +AAAAAA; ,UAAAAAAM • ;%$1 AU+. : AAAAAAA+, • ;&AAAAAAAMt: .IAAAA&;. ;IAA AAM. .;MAAAAAM +AAAAAAA& ; MAAAAAAA, ,ZUMA AAAU: SAAAAAAA% • AAAAAAA AAAAAA&; ,MAAAAA AAAAAI. :+AAAAA+ SAAAAA& • MAAAAAA! • ;&AAAAAI tAAAAAAU; • ·.AAAAAA+, • %AAI+ jAAAAM%;, lAAAU i, . , ; lAAAA+ ; SAAAAAAS;. ;nAAAAAAI. .+&AAAAAAMt : ; IAAAAA&; • MAAAAAAA+ • AAAAAAA+ • %AAAAAl: • • ,%AAAAAA. ; MAAAAAA &AAAAAAA +IAAAAAAI : HAAAAAU: +AAAAAA. AAAAAAl. AAAAAAU, , ,: IlAAAAI ;SIIAAl, :AAAAAA% : +lAAAAAAA; :+SI AU; +AAAAAAIt • : tlAAAAAAAA+; • lAMA!; • :%MA AAM+ • HAAAAAl ;AAAAAAA& : &AAAAAAA: :I&lAAA AAAf%: &AAAAAAA+ AAAAAAA, IMAAAN+ • ,IAAAAA AAAAAt. .: IAAAAA, &AAAAAM , &AAAAAAA : IAAAAAI IAAAAAU+ : ;AAAAAA; • &AAA$ :AAAAIl; , ;AAAAZ;: , ,+MAAAA, .+1AAAAA.+;. : tSAAAAAAM, • +&AAAAAAU: a • +AAAAAAI; : MAAAAAAA+ &AAAAAAZ, IAAAAAZ: • ' :%AAAAAA ; MAAAAAA ZAAAAAAA: ; !AAAAAAL b ;!AAAAAAH, MAAAAAA AAAAAAX, AAAAAA*l:

Figure 13 a, divaricate patterns on Bankivia fasciata showing open and closed V's; b, simulation of V's. such an event is, but does provide a mechanism for gen­ Wavy stripes (Figure 14). These are characterized by erating a coordinated reversal of the pattern orientation very sharp cutoffs of the excitatory and inhibitory thresh­ (Figure 13). Lines that converge as the shell grows will olds, small thresholds, and large turnover of refractory be called "closed V's"; those that diverge as the shell grows substance ('Y, a "" 1). Note the "shocklike" discontinuities are "open V's." Pattern reversals that produce a "closed in the stripes that the simulation reproduces. V" frequently extend beyond the intersection a small Streams (Figure 15) are irregular striped patterns that amount, forming a "snout" on the V. This is also a feature occur when the sharpness of the cutoff is quite large and of the simulations, because a collision of two obliques refractoriness is persistent (0 "" 0.8). admits a small overlap of the activation region extending' beyond the collision apex. Note also that the upward stripes Interaction Patterns are shorter than the downward stripes, suggesting a man­ tle inhomogeneity. This has been suggested previously by In addition to the basic patterns, additional designs WRIGLEY (1948). emerge from the interaction of the basic patterns. Typi- Q

Page 380 The Veliger, Vol. 28, No.4

1+, ,+UA'+i;aU& %AAAAAUU, XAAA+ii •• ·ij:j;:;;i:UAAI , •• AU;, +UA :AAl" ;'AAAtj~ +IX, •• , .SUAA ,+.AAAAi ::A%%:&lA;, ,:AAAAJ, :i; i:IAIl ,,+IAAAA%: lAAAAA. :AAAA., :;Ulij;jij;ij ,. UAUAt: iUS: .AAAAA, jj% :+AAAI,.,+UAAAUAUAUll AAAA%: :IAUA';: +U;: :jAAAl+ .jAAAj;+IU%%IIAIAAAAAI:; AU: :AAAAAAAAA AAAAAAA&; :AAAAA%% +%%%U%lAAIIAI;: .. :UAAA;:&AAA+ +UAAAAAAI :IAUS, "UI".IAAAAA :fAAAA' :AAA,. ::AAa •• jAAAAI; UI1.::: ••• i. :.AAA :SAAAAA. "AAAUAA, • jAAAAI: ,j :Ul:: :lIA • ,.AAAAA+. "AAAAAAA jAAAAA: %AAAU, ,++XI. .+ +AIIAAA+. ,&AA%; .AAAAI. ++1 .+AAAAS AAAAAAlj.AAAA&; AAAA+. +AAAAAAj .:. ifAAU+ ;AAAA; I:;:UUIIAAAAA;. AA&. .+AAAAAAAAI AAAAAAU: .AUAAI ;. US+,. jAAAAl, t, .+AAAA, %AAA+ +AAAUUAAI. .+AUA lAI. jllAAAA • +AAMA. • &AAX j: +IAA+ lUAA:, ,SAA, j :; ,: I,. :$AAA

...... IA,. I:. I: :AAAUUUUAUA"" ,AAAUUU'AJ::;:.: :AU...... fAA.,: ;,:::SAA.. UUUIIAAAAAAIXAAAI&UUAA.:::::: :AAn .., ...... ,: I::: :AA .. 'UUU%%UUUSU%%%%UIIA •• ii i::: :AAn ...... AAX ••• : •• +A ...... AAAAAAAAAAAAA.UAAA ••• ,;;;;; ;$11 ...... A+::::: :.AUUXXXXX+++; ;+++++X+++UUA 0,:;;;;; ;AIMUU ...... UA :::::: IAAAAAAA •••••• ,'''''' , •• AA" •• " II;;;; ;AAI&UU'" ....Aua. I. I ::sAAIIAAA", •• I:'::::':::::'"" II; i ;;;UAAUSUII.A ...WAAAJ. ::.:: AAAIIAAA., : .::: ::;;;; ;;;;:,:::.;;;:: AAAAAUUS".U .... AAA:.", ,SAA ..... A.:.::;;;i;; i;; ii i'::::: :::AAAAA.UXXUII.AA IAAAAAI ,':: :AAA ...... ,::: iii i; i: i;;; i;::;'::: :AAAAAUSlXSSUII.1 ..AAIIAS '" :+AAII""',::;;; i i; i;;; iii;;:;::: :AAAA .. UUUU .... . • AAAAA .,::: :AA .....I&.::;; i i;;;;;; i;;; i::': ,AAA .. USU.U1U .... AAAAAS.,:::: .AII ...... : i; iii;;;;; i;;;;;::: :AAAA.U1UUU&"... . AAAAAA.::. ::%11 ...... 11 •• :; iii;; i;;;;; i;: II ;AAAUUuuUuuum 'AAAAII,,:: ::AII ...... A::;;;;;; i;;;;;;;:: :SAAAAU1UU ...... UA AAAAAA,,:: ::AII"1&"'A:: i;;;; i;;;;; i;:: ,AAAA.U%X%%SSlSUnUuU IAAAAA",:, :II"UU"A::;;;;; i;;;;;;:: SAAAAAlIlUUAAAAAAt ....AAAA AAAAAA,::: ::II ...." .. A::;;;; i;;;;i:: ,AAAUU%++++X%%lX+++++UUU UAMA::::: :AU.I&... A::;;;;;;;;;:: ,IIAUAA"" ...... UAAS. AAAAAA::::: : ...... A::;;;;; i;:: ,AAANlUUAA."::::::::::",, ,At KAAUA,:::: :UU ...... ::;;;;;;:: ;AAAAUUUA. ,::::::::::::::"", 'AAAAA:::: ::;nll..... ;:;;;;: ::AAAAA1U1UA •. ,::;;;;;;;;;;: ::::,: IAAAAA:::::, A..... IIA;::::: :&AAAA&&HUA'·.,:;;;;;;;;;;i;;;"::: UAAAA+:::,: •• 11 ...... :::" ,AAAAAUUUA+. 0' :i;;;;;;;;;; i; i ;;;::; ""AAA::,,: ...... ,'" ,AAAAUUUAA···,';i;;;;;;;;;;; ;i;;;;'; •••••• A."" .ItIUIA •• '" ,AAAAlali"A+ •• ~ ~::!!~ ~!:! !!:!:: :!! :!!~! ••••• IAA", , ... AAttAIIA , t , I AAAIS&&'AA .... I •• ,"" r " , "", "" '" " •••• ,.AA:::: , AAAA" ... ::: :AAA'.'AAA •• ,,:: ;;;; ,j;;;;;; ii;; i;;;;;;; .tI ..... AA::,,, .AAAI+ •• ,::: ;AII".IAII:.,:: j;;;;;;;;;;;;; i;;;;;; i; i, i MI .. " •• AA::: I:, ..... , ,:;;; ,""AAAIIAS., I::;;; i;; ;;,;;;; it;; it;;;; ii; •••••• AA i ; +++; r '" , :; i +; jAAAtlAAAA%. ,: : ; ;;; ;;; ;;;;;; ;;; j ; ;;;;; ;;;; •••• I&II.A.; i+i",,:;;; jAAAJ&.&J.IfIl&,;:;;;;;;;;;;; i;;;;;;;;;;;;;;;; I'&&S"IAAA;;;:::::: :IItAAASSSSSItA,:::;;;;;;;; i ;;;;;;; Ii;;;;;;;;;; AI&' •• SlAAAIIX:"", :AAAAAISSISIU .:;;;;;;;;; i;;;;;;;;;;;;; ii;;;;; AtI.' .. SISIAAAA+ •••• ,AI\AI\.'I$'I,. .• r;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;; AAtll' ••• I.AAA% ••••• AAAA •••• A••• ,,:; i;;;;;;;;;;;;;;;; i;;; i;;; ;i;;; ."'I ••• "AAA",,, ,IAAAAAIIII ••• t::;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; A.MtI&I&'.AAAI,::" ,AltAAAAAA.,,::;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; W'.IISSI'IAAA,:", ,AAAAAAAA., t:;;;;;;;;;;;;;;;;; i;;;;;;;;;;;; j;;; A••• &.1II •• AA&:::: =:AAAAAAAA,:::;;;;;;;;; i;;; ;;i;;;;;;;j;;;;;;;i;; •• 11&1&" ••"&::::: :A ••••••• :: ::;;;;;;;;;;;;; i;;;;;;;;;; i; ij;; ii;; A""""'AA,::::: :AAAAAItAA:::;;;;;;; j; i;;; i;;;;;;;;;;;;;;;;; j;;; ,.""IS •• A&.::::: :A"""I;::;;;;;; i; j;;; i; i;; i;;;; ;i;;;; i;;; i;; .'MII""AA, .. ::::: :AI.I ••• Ml:: i;;;;;; i;;;;;;;;;;;;;; i;;;; i;;;;;;; .I&&II ••• AS J; i;;; ;AII.I •• NA:: i;;; ;i;; i;,;;;;;;;;;;;;;; i; i;;;;;;; IUU' •••• AA .:;;;; ;'.ltl'I&IIAA: ::;;; i;;;;; i;;;;;; i;;;;;;; if;; ;;i;; •• ' •••• AII,.,:;;;; ;AI.IS •• IIAAA:::; ;i;;;;;;;;; ,i;;;; j;;;;;;;;; i;;; a ••••• AIIA, , =:;; ;;AAllt&S'$S&AAIIAI: =;; i; ii;;;;;;;;; i;; i;;; j;;;;;;;; , •• 'AAiu\+ .,::;; ,AltAI'.XXlX"IAAAA:::; i i;i;;;; j;;;;;;; i;; i;;; i;;;; •• AAAAA .,::: :$AAAA'&'$'."".AAIIAA:::;;;;;;;;;;; i;; ii;;;;; i,;;;; ""AAA"."" :AAAA'I"'SS$SI%$t"'AAAA:::;;;;;; i;;;;;;;;;;;;;;;;;; IIAAAA; ••• , ,AAAAUUU ..... UUUAAAAA$::;;i;;;;;;;;;; i;;;;;;;; i i Figure 15 AAAAA* •• ,' :AAAIUUX%s ..SUUnXU.AAAA,:::i::;;; i;;;;;;;;;; i i; i IIAAAA ,,:: :AAAAAU1UAAAAAAAUUUAAAAAA:::;;;;;;; i;; i;;;;;; i;;; a, wandering stripes on Bankiviajasciata; b, simulation. AAAA. ,::: :AAA,S'L%+++++++++++++++++XS'IAAAA:::;;;i; i;;;;;;;;;;;;; A••••• t : : :; AAAAAAA ...... , ..... IttiltlAAAAAA&: :;;;;;;:;; ;; i;;;; ;; •••• ,::;; ,AAAAAAA; t ,::::::::::::". AMI'SI"AAA:::;;;;;;;;;;;;;;;; "J'::;; ;AAA •••• A., I::::::::::::::, .AA ••••• AAAA:::;;;;;; i;;; i;;;; J , : ::: : ; AAA ...... :: : :';; ; ; ;; :;;; ; :,. IA., •• sa.AAAA:;:;; ;;;; ;; ;;; ;; could be produced in a deterministic fashion. In a subse­ :: :::: ;AAIcA.&I •• ,::; ;;;;;;; ;;i;;;:, .1A.IISla.AMIA.::;;; ;i; if;; if :: :::: AAAA...... :;;;;;;;;;;;;;;;;:: •• AAIJSSXS'.AAAAZ: :;; ; i;;;;; ; quent study we shall demonstrate the role of stochastic ::::,I\AI\A..... , ; ;; ; ; ; ;;; ; ; ;;; ; ; ;;: , •• 'AItJ'$SSIIAAAAA::; ;;; ;;;; ; influences on the structure of the patterns. :: :, AAAA""'A .. ,:;;;;;;;;;;;; i; ; ; ; ;;:,., AAAIU%%IS&&IAAAA: :: ;;;;;; "~A'II"A f::; ,;;;;;;;; ;t;;: ;;;;:: ,,,.AAAA"SSS"AAAAAtt::;;;;; b \I •• IA .. ,11;;;;;;;;;;;;;;;;;;;;;: 1, AAAAISttXIX"IAAAAu:; i; AI •• A .,::;;;;;;;;;;;;;;;;;;;;; i:: 1 .AAAA"I'&'I"IAIIAAAA:::;; DISCUSSION AA •••••••• ,::;; j;;;;;;;;;;;;;;;;;;:;;:, ~ IU,&.tX'%%%%ZSS'tlAAAA t:: AAAAAAAA .,";;;;;;;;;;;;;; i ;; ;; ;;; ; ;;: ::. AIIAAAAAAA'U.U.AAAAAA. We have constructed a model for shell patterns based on AAAAAAA+.,::;;;;;;;;;;;;; j;;;;;; i;;;;;;': ,"&&SS%%%%%%++++%"'AAA the hypothesis that the secretion of pigment is stimulated by neural activity. Our model postulates the simplest pos­ sible neural interactions: local activation and lateral in­ hibition, such as is found in the retina. Despite its sim­ and threshold for activation. Very short-range interactions plicity the model is able to reproduce a variety of observed and strong nonlinearities produce tentlike patterns char­ shell patterns, such as bands, diagonal stripes, and various acteristic of the cone shells, and which resemble the pat­ divaricate interference patterns that arise from the inter­ terns generated by the automata models of Lindsay and action of propagating bands. of Wolfram and his coworkers. Longer range interactions The type of pattern generated by the model depends on produce interference patterns, such as checks and wan­ the nature of the neural interaction, its range, persistence, dering streams seen on Bankivia fasciata and other shells. Page 382 The Veliger, Vol. 28, No.4

AAI, ,AAAAAAU : +AAAAA;. ,'AAAl: ,+UAAAA+ lates, a phenomenon commonly observed, especially in ,, • &AAAAAAA+ :+AAAAAAA: SAAAAAU: :AAAAAAAt: Nerita turrita. Such sudden shifts in behavior triggered by $AAAAAU+ UAAAAAI+ ,AAAAAAA, +&AAAAAA .%AAAAAAAI ,HUAAAAAAA, AAAAAAAA: AAAAAA smoothly varying a parameter are typical of models with • AAAAA$; • ;AMAl: ,.'. :lAM .IAAAA strong nonlinear terms (MAY & OSTER, 1976; , +AAAAAAA :+IAAAAAAAA: :lAAAAAA; • ZAAAAAAf: fAA :&AAAAII; : SAAAAAI&+. AAAAAAA tAAAAAAl. GUCKENHEIMER et ai., 1976). AAAAAAA% , %AAAAAAA+ SAAAAAAA+ ; AAAAAAAI The cowries have a mantle that imprints a pattern over AAAAA1: AAAAAUI; &AAUUAAI +AAAAAAAA AAAAS , :+UAAAAAAA+ : XAAAAAAAUX+;; lAAAt&+ ;UAAAAA+. a large expanse of shell, rather than just at the growing U+, :AAAAAl; • .MAAAAAI SAAAAA&: .+lAAMA!, edge. To model this, one must employ a two-dimensional ,IAAAAAAA, ;AAAAAAA: AAAAAAA% ;AAAAAAAA .AAAAAAAM+ :AAAAAAAA IAAAAAAA; .MAAAAAA version of the neural model. Two-dimensional automata • +AAAAAAAA .: +AAAAAAAA .::, •• AAAAAAAA%: :. ; AAAAA models with very local interactions can produce patterns :AAAAH: , lAAAMt ,:; +It • $AAAA, .lIAA : lAAAAAA' • +lAAAAAAA+ lAAAAAAA • &AAAAAA+, +A that bear a striking resemblance to those found on the ; AAAAAAA&; IAAAAAAA+ +AAAAAA& ;AAAAAAAI map cowrie (N. Packard & S. Wolfram, personal com­ AAAAAAAI • +AAAAAAA+ AAAAAAAA ,AAAAAAAA, AAAUI: ,AAAAU%;: AAASI+%1AI. ;+I1AAAAA munication), and preliminary analysis of the neural model AAAA; .;UAAAAAAA& .1AMAAAI%+;; •• ZAAAAAA&, :AAAAAAA+. indicates that the eye-spot pattern found on many cowries Alt :AAAAAIX. lAAAAAA, • AAAAAA& ; SMAAAA •• • +AAAAAAA, +AAAAAAA: • AAAAAAAt lAAAAAAA can be easily obtained . ,AAAAAA&%: 'AAAAAAAA &AAAAAAAA +&AAM4 The neural model also touches on the problem of shell ., ;&AAAAAAA, ,:+I1AAAAAAAS. ;+ .. , 1UAAAAAA%+;. XAAAA &AAAA&%: +AAAAS, ,1%%%n, .$AAAAI ,fAA construction, for as WRIGLEY (1948) and others have .+AAAAAAA+ :IAAAAAAA. &AAAAAAI IAAAAAA%, % pointed out, there is a correlation between the color pat­ SAAAAAAII: AAAAAAAA; SAAAAAAI : lAAAAAAA, AAAAAAA: , ; &AAAAAAA MAMMA: &AAAAAAAX. terns and the geometrical features of the shell (e.g., pig­ AAAII+ AAAAI .. ,. ,AAS;::;$AA%'. ,;+UAAAA+ ments may concentrate in the grooves between ridges, and AAA& • :XlAAAAAAA. ,$AAAAAA% ..... : &AAAAAA&, lAAAAAAl, AS, SAAAAA&+. MAAAAAA. .IAAAAAA ,UAAAAAA+ spines tend to be colorless). This is hardly surprising, • ;&AAAAAAA XAAAAAAA. AAAAAAA&, %AAAAAAA because the same mantle that deposits the color is busy MAAAAU%: AAAAAAAAA ;AAAAAAAA: .1AAAAA building the shell. However, this correlation between pat­ ,;AAAAAAAS • :UMAAAAAAI&; ;S&&1: IUAAAAAU;, ,AAAA tern and form suggests that the neural model might be Figure 16 extended to investigate the diversity of shell shapes and Checkerboard patterns. their mode of construction. If the neural hypothesis is correct, the shell is a hard­ copy record of the neural activity in the mantle. The fossil We have mapped out many, but not all, of the possible record for these creatures is as complete as for any known patterns that arise from the neural hypothesis. The model lineage. What can such an electroencephalogram tell us can be elaborated in several directions. For example, what about the evolution and ecology of mollusks? We shall not is the effect of postulating a more complex neural struc­ speculate here, but the model suggests an explanation for ture (such as long-range activation)? Many shells secrete the diversity of patterns found on the same species in several kinds of pigments; including more than one pig­ different environments, and the similarity of different ment into the neural model would increase enormously species in the same environment. Moreover, the enormous the possible patterns it could generate, including the char­ diversity of pattern within certain species may reflect the acteristic of stripes passing through one another-a com­ fact that the patterns in those species are not visible during mon phenomenon that the simple neural model presented the animal's life. Being invisible to selection generally leads here cannot reproduce. It is clear from many studies (e.g., to increased genotypic variance, and so we should expect WRIGLEY, 1948) that the mantle is not a homogeneous the color patterns in such species to be highly polymor­ tissue as we have assumed here. By adding to the model phic. spatial gradients and periodic variations in the parameters The usefulness of any model stems not only from its (e.g., refractoriness or density of innervation) a far greater specific predictions and its ability to unify disparate ex­ variety of patterns can be produced than from the homo­ perimental observations, but also from its fertility in sug­ geneous mantle we have assumed here. We shall present gesting further experiments. If the neural hypothesis is simulations of more complex mantle structures elsewhere. correct experiments that intervene with mantle neural ac­ In addition to spatial variations, a variety of transition tivity, without disrupting shell construction, need to be patterns can be produced if parameter values evolve slow­ devised. Perhaps the topical application of neuroactive ly as the simulation proceeds. These are distinct from the substances such as xylocaine, lysergic acid, or various kinds discontinuities and V -patterns that may involve a sudden, of neurotransmitters can provide information. Probably global perturbation of a system parameter. In particular, electrophysiological measurements will interrupt mantle shell size is an important determinant of pattern. Small, activity, but perhaps the neural connections between pig­ or young animals will typically exhibit less complex de­ ment cells can be explicated in sufficient detail to deter­ signs, because fewer stripes will "fit" into a smaller do­ mine the range of neural interactions characteristic of each main. Moreover, as shell size (i.e., domain size in the pattern type. It is a rich field for neurobiology and anat­ model) increases with growth, stripes widen until a omy which will have a direct impact on larger issues of threshold is reached, whereupon another stripe interca- evolution and adaptation. p.

B. Ermentrout et al., 1986 Page 383

AAAAAAAAAAAAAAAAAAAA. SAAAAAAAAA lAAAAAAAAAAAAA AAA AA AA AAlA AAAA AIlA AAAA AAA AA AAA AAAAAAAAAAAAAAAA AA AAAAAAAAAAAAAAAAAAAAA AAAAAAA AAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAA AAAAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAA AAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAAA AAAAAAA AAAAAAAAAAA% AAAAAAAAAAAAAA ;AAAAAAAAAAAAAA AAAAAAAAA AAA AAA AA A AA AAAA AAAAAAA AAA AAMA AAAAAAAA AA AAA AAAAAAA AAAA AAAM AA AAAAAAA AAAAAAAAAAAAAAAAAAAAAA AAAAAAA AAAAAAA: AAAAAAA AAAAAAA AAAAAAA AAAAAAA AAAAAAA AAAAAA MAMA AAAAAAAAA AAAAAAAA MAAAAA AAAAAAAAAAA AAAAAAAAAAAA AAAAAAAA AAAAAAAAAAAAA AAAAAA AAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAA AAAAAAAA AAAAAA AAAAAAAAAAAAAAAAA AAAAAAAAAA AAAAAA AAAAAAAAAAAAAAAAAAA AAAAAAAAAAA AAAAA AAAAAAAAA AAAAAAAAAA AAAAAAAAAAAAAA AMAAA AAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAA AAAAAAA AAAAAAA AAAAAAAAAA AAAAAAAAAAAAAAAAAAAA: AAAAAAAAA AAAAAAAA A AAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAAAAAAA AAAAAAAAA. AAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAA AAAAAAAAA AAAAAAA ,MAAAA AAAAAAAAAAAAAAMAAAA AAAAAAAA MAMA AAAAAAAAAASAAAMAAAAAA AAAAAAAAA AAAAA AAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAA AAAAAA AAAAAAAAAAA SAAAAAMAAA AAAAAAAAAAAAAAAA ; AAAAAAA AA AAAAA AA AA MAMA AAAA AA AAA AAAAAAAA AAMA AM AAAAA AAAA AAA AA AAAAAAAAA AAAAAAAAAA AAAAAAAAAAA AAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAA AAAAAA +AAAAAA Figure 17 AAAAAAAAAA. AAAAAAA AAAAAA AAAAAAAAAAAA AAAAAAAA AAAM a, tent patterns characteristic of olive snails (e.g., tent olive or AAAAAAAAAAAAAA AAAAAAAAAAA AAAAAA AAAAAAAAAAAAAAAA AAAAAAAAAAAAAAA AAAAAAAA :o~al purple olive (Oliva porphyria); b, tent patterns character­ AAAAAAA AAAAAAAAAAA AAAAAAAAAAAAAAAAAAA AAAAAAAAA A [SHe of the textile or courtly cones-these patterns differ from AAAAAAAAAAAAAAAAAAAA AAAAAAAAA AAAAAAAA AAAAAAAAAAA (a) by slightly longer range neural interactions. AAA AAAA AAt AAA AAAAAA AA AAAA AAAAAAAAAAAAA AAAA AAAAAAAAA~AAAAAAA AA AAAAAAA AAAA%AAAAAAAAAAAAAAAAAAAA AAAAAAA AAAAAAAAAAAA. *AAAAAAAAAAA AAAAAA AAAA AA A A AA AAAA AAAAAAAAAAAAAA AAA AAA AAA Finally, we should mention the issue of the uniqueness AAAAAAAA% AAAAAAAAA AAAAAA of the model. It would be gratifying if we could claim that AAAAAAAAA AAAAAAAAAAA AAAAAAAA AAAAAAAAA AAAAAAAAAAAA AAAAAAAAAA our model can reproduce the observed patterns better than AAMAAAAA AAAAAAAAAAAAAAA AAAAAAMAAAA all competing models; however, this is not the case. Using I b AAAAAAA AAAAAAAAAAAAAAAAAAA :AAAAAAAAAAAAA ". AAAAMAAA AAAAMAAAA AAAAAAAAA AAAAAAAAAAAAAAA a model based on diffusion and reaction of chemical mor­ AA AA AAAA AA A A AAAAAAA AA AA AA AA AAAA AAAA AA AAA AAA AAA AAA AA AA AA AA AA A phogens, H. Meinhardt has produced simulations that are AMAAAA &A AA AA AA AAAA AA AAAAA AAAAAAAAAA A equally as convincing in reproducing the shell patterns as AMAAA. AAAAAA AAAAAAAAA the neural model. The reason is clear: one can model the phenomenon of local activation and lateral inhibition characteristic of neural nets in a variety of ways. Any number of diffusion-reaction mechanisms can produce this and GO by NSF Grant # MCS 8110557. Conversations effect by a slowly diffusing autocatalytic reaction that is with James Keener and Arthur Winfree were crucial to quenched by a fast diffusing inhibitor molecule (MEIN­ the completion of this work. Thanks also to E. R. Lewis HARDT, 1982). Even the mechanical models that OSTER and D. Lindberg for valuable comments and criticisms. et al. (1985) have employed to model the regular patterns The hospitality of James Murray and the Institute for of microvilli on cells can be viewed as a mechanical im­ Mathematical Biology, and Gerald Edelman and the plementation of this neural-like property. Therefore, we Neurosciences Institute are gratefully acknowledged. The are left with the disappointing conclusion that it may be computer graphics were performed by Greg Kovacs. quite difficult to infer mechanism from pattern alone, be­ cause several quite distinct cellular mechanisms can pro­ LITERATURE CITED duce identical patterns. Thus the issue of whether the BERNE, R. & M. LEVY. 1983. Physiology. Pp. 116-118. Mos­ patterns on mollusk shells arise from neural activity as by: St. Louis. we have suggested here will be settled only by experi­ CAMPBELL, J. 1982. Proposal submitted to the National Sci­ ments. Theory can provide only a shopping list of possible ence Foundation (Grant No. PCM-20923). mechanisms. COMFORT, A. 1951. Pigmentation of molluscan shells. Bio!. Rev. 26:285. ACKNOWLEDGMENTS COWE, R. J. 1971. Simulation of seashell pigment patterns using an interactive graphics system. Computer Bull. 15: GBE was supported by NSF Grant # MCS 8300888 and 290. the Sloan Foundation; JC by NSF Grant # PCM-20923; EMBERTON, L. 1963. Relationships between pigmentation of Page 384 The Veliger, Vol. 28, No.4

shell and of mantle in the snails Cepaea nemoralis (L.) and Rt+I(x) = 'YP,(x) + oR,(x) [2] Cepaea hortensis (Mull). Proc. Zoo1. Soc. Lond. 140:273. ERMENTROUT, B. &]. COWAN. 1979. A mathematical theory P,(x) = H(A - A*) [3] of visual hallucination patterns. Bio!. Cybernetics 34:137- 150. where 0 < 'Y < 1 is the rate R increases and 0 < 0 < 1 is GUCKENHEIMER,]., G. OSTER & A. IPAKTCHI. 1976. Density its degradation rate. dependent population models. J. Math. Bio!. 4:101-147. We can further simplify the model by assuming that HERMAN, G. T. & W. H. LIU. 1973. The daughter of Celia, the French flag, and the firing squad. Simulation 21:33. the pigment secretion, P, is simply proportional to the HERMAN, G. T. 1975. Developmental systems and language. activity, A, and let the function S take care of the threshold North-Holland: Amsterdam. for secretion. This does not affect the patterns signifi­ KApUR, S. & M. GIBSON. 1967. A histological study of the cantly, and is somewhat easier to treat numerically and development of the mantle-edged gland and shell in the theoretically. Thus the equations we shall deal with are freshwater gastropod Helisoma durvi en discus. Can. J. Zoo!. 45:1169. P'+l(X) = S[P,(x)] - R, [4] KNIPRATH, E. 1977. Ontogeny of the shell field in Lymnaea stagnalis. Roux. Arch. Dev. Bio!. 181:11. R,+I(x) = 'YP,(x) + oR,(x) [5] LINDSAY, D. 1982a. Simulating molluscan shell pigment lines and states: implications for pattern diversity. Veliger 24: The neural stimulation function, S[P,(x)] in equation 297-299. [4] is composed of excitatory and inhibitory effects. Note LINDSAY, D. 1982b. A new programmatic basis for shell pig­ that the pigment secretion on day t + 1 can depend only ment patterns in the bivalve mollusc Lionconcha castrensis on the excitation during day t + 1; however, according to (L.). Differentiation 21 :32-36. the assumptions of the model, each day's pattern of exci­ MAY, R. & G. OSTER. 1976. Bifurcation and dynamic com­ plexity in simple ecological models. Amer. Natur. 110:573- tation is stimulated by "tasting" the previous day's pig­ 599. ment pattern. We can safely assume that the time con­ MEINHARDT, H. 1984. A model for positional signalling, the stants for neural interactions are much shorter than those threefold subdivision of segments and the pigmentation pat­ of shell growth, so that we need deal only with the daily terns of molluscs. J. Embryo!. Exp. Morpho!. 83(Suppl.): average, or steady state firing rate of the neurons in the 289-311. mantle. Therefore, we define the following functionals: NEFF, J. 1972. Ultrastructure of the outer epithelium of the mantle of the clam Mercenaria mercenaria in relation to cal­ Excitation: cification of the shell. Tissue Cell 4:591. OSTER, G., J. MURRAY & G. ODELL. 1985. The formation of microvilli. In: G. Edelman (ed.), Molecular Determinants E,+I(x) = 1WE(x' - x)P,(x') dx' [6] of Animal Form. UCLA Sympos. Molec. CeiL BioL (in press). Inhibition: TIMMERMANS, L. 1969. Studies on shell formation in molluscs. Neth. J. ZooL 19:417. WADDINGTON, C. & R. COWE. 1969. Computer simulation of I,+I(x) = 1W/x' - x)P,(x') dx' [7J a molluscan pigmentation pattern. J. Theo. BioL 25:219- 225. W AUSHER, J. 1972. Considerations on phase-change and dec­ Here the kernels WE(x' - x) and W/x' - x) weight orations in snail shells. Hereditas 71:75-94. the effect of neural contacts between cells located at po­ WILBER, K. 1972. Shell formation in mollusks. In: M. Florkin sition x' and a cell at x; they effectively define the con­ & B. Scheer (eds.), Chemical zoology. VoL 7. Academic Press: New York, p. 113. nectivity of the mantle neuron population. In general, the WOLFRAM, S. 1984. Cellular automata as models of complex­ inhibitory kernel, W/(x' - x) is broader than the exci­ ity. Nature 311:419-424. tatory kernel, WE(x' - x); i.e., activation has a shorter WRIGLEY, A. 1948. The color patterns and sculpture of mol­ range than inhibition. n is the domain of the mantle; for luscan shells. Proc. Malaco!. Soc. Lond. 27:206. most shells this is a finite interval, but may be circular in the case of mollusks such as limpets and planar in cowries. APPENDICES The particular connectivity functions we have em­ A. The Model Equations ployed in our simulations are:

In this Appendix we give the complete mathematical Wj = 0 for I x I > Uj' j = E, I expression for the model equations given in the text, as well as the functional forms employed in the numerical ~. = qA2P - (1 - cos(1I'x/u)P] [8] simulations. for I x I :S Uj' j = E, I The model consists of the three difference-integral where qj is chosen so that equations

At+I(x) = S[P,(x)] - R, [1] 1W;(x) dx = 0/), j = E, I [9] B. Ermentrout et at., 1986 Page 385

Table 1 1m a

Neural influence function parameters 0:'.£ = amplitude of the excitatory influence function. Ct.1 = amplitude of the inhibitory influence function. ---\.--+--i-- b Uli = range of the excitatory influence function. -1 (jr = range of the inhibitory influence function. p£ = sharpness of the excitatory influence cutoff, or the "flat­ (a) ness" of the influence function. p, = sharpness of the inhibitory influence cutoff. Firing threshold functions PE = steepness of the excitatory cutoff (nonlinearity). PJ = steepness of the inhibitory cutoff. 1:>.1 0E = location of the excitatory threshold; i.e., the midpoint of the sigmoidal curve (threshold). (b) o f--::~--- Or = location of the inhibitory threshold. Refractory parameters -1 'Y = production rate of refractory substance. o = decay rate of refractory substance. c c The shape of the connectivity functions is controlled by p: for p very small the ~ are sharply peaked, for p large, B ~~---l-_=:;::) A ---I'---+--+---b the YVj become nearly rectangular. In our simulations p is in the range of 4-8. The range for lateral inhibition is made greater than the excitation by choosing (JI > (JE, and (c) c since the local excitation strength is generally greater than the inhibition, we choose OlE > Oil' Figure Al The responses of the secretory cells to neural stimula­ a, the unit circle and the stability triangle on the coefficient (a, tion are assumed to be sigmoidal functions of their inputs: b) plane; b, dispersion relation A(k) for spatial instability; c, trajectories for each type of bifurcation. S[P'(x)J = SE[E,+Jx)J - SAI'+l(X)J [10] For simulation purposes, we have employed the follow­ ing function for both S E and S / where we have suppressed the dependence on x for no­ tational simplicity. S(u)- 1 )=E,/ [11] Let Po be a homogeneous equilibrium, i.e., J 1 + e",Iu-Oj) , Po = S[PJ + oPo - (-yPo + oS[PoJ) [13J The parameter v) controls the sharpness of the nonlin­ or earity, and OJ the location of the threshold. Thus the raw parameter list consists of the 12 quan­ 1 - 0 tities: Po = S[PJ 1 - 0 + 'Y [ 14J If we shift the sigmoid S so that SIlO) = SlO) = 0, then This list can be reduced to nine because some parameters we can linearize about Po = 0 to obtain the linear differ­ enter only as products, and some may be rescaled. Table ence equation: 1 summarizes the model parameters. Pt+2 + L,p,+! + oP, - [yP, + aLo(p,)J = 0 [15]

B. Analysis and Simulation of the Model where L,(.) is the linear (convolution) operator A linear stability analysis of the model equations gives L,luJ(x) = S'E(PJ 1WE(x' - x)u(x') dx' some idea of the patterns the model will generate. There­ fore, we proceed as follows. The pair of equations [4, 5] are equivalent to the single - S'/(P,,) I W/(x ' - x)u(x ') dx ', [16J second order equation [12] where S'iP,) are derivatives of SJ. I Q

Page 386 The Veliger, Vol. 28, No.4

finite range of unstable modes: I).. I (k) > 1 for a < k1 < k < kz < 00. That is, the dispersion relation )"(k) should look qualitatively as shown in Figure Alb. Such an instability can arise in three qualitatively dif­ ferent ways: as one of the model parameters is varied the (a) unstable eigenvalue can pass out of the unit circle through ,..--+--+---'r--, X +1, -1 or at a complex value (Figure Alc). Which one of these instabilities occurs depends on which parameter is varied and on the shape of the connectivity kernels, W)" The neural connectivity function, W(x) we have em­ ployed is the usual "short-range excitation/long-range in­ hibition" type shown in Figures 9 and 10. The linear operator L*(k) is essentially the Fourier transform of W(x). By the properties of the Fourier transform, L *(k) has the shape shown in Figure A2a. Because only positive values of k are physically relevant, the dispersion relation looks qualitatively as sketched in Figure A2b. The three paths to spatial instability shown in Figure Alc correspond to violating the following three inequali­ 1 (b) ties: (a) Bifurcation through +1 will occur if L*(k) > (1 - 'Y)/5 (path a in Figure Alc). This is a so-called Ot------k "equilibrium" bifurcation because in the spatially homo­ geneous case (k = 0) such a bifurcation creates a new equilibrium point (cf MAY & OSTER, 1976; GUCKENHEI­ Figure A2 MER et at., 1976). When k > a this creates a stationary spatial pattern of regularly spaced stripes as shown in a, the shapes of S(x) and L(k); b, the dispersion relation. Figure A3a. (b) Bifurcation through -1 will occur if L *(k) < On a periodic domain of length L (e.g., the limpet), the -(1 + 'Y/(1 + 5» (path b in Figure A1c). From Figure eigenfunctions for La are exp(27f"inx/L), n = 1,2, ... ; on A2b we see that this can occur only at k = 0, so that a finite linear domain these are approximate eigenfunc­ homogeneous instability results. (This can only happen in tions, since the domain size, L, is much greater than the this model for the kernel shown providing 8 Jo; ~ 8I> since range of the connectivity functions W. we have assumed that at: > a/.) The pattern resulting The characteristic equation for the spatially homoge­ from this bifurcation consists of fine horizontal stripes, as neous system is obtained by substituting P,(x) :::::: shown in Figure A2b. >"'exp(27f"ikx/L) into the linearized equation: (c) Bifurcation through A = e iO (8 .,. 0, 7f") occurs if L*(k) > 1 + 'Y/(1 - /J) (path c in Figure Alc). This >.,2 + (L *(k) + 0»).. - ['Y + 5L *(k)] generates periodic spatia-temporal patterns, as shown in == )..2 + a(k»)" + b(k) = 0 [17] Figure A3c (e.g., stripes and checks). Note that (1 - 'Y)//J < 1 + 'Y /(1 - {j) if and only if Here {j > 1 - Vi Thus + 1 bifurcations occur first when /J < L*(k) == S'E(Po)We(k) - S'/(Po)W/(k) [18] 1 - 'v0; otherwise the bifurcation is via a complex ei­ genvalue. where the ~ are (close to) the Fourier cosine transforms When 'Y = 0, so that the refractory substance cannot of the ~: build up, the model can take a particularly simple form.

If we make p large, and (J E, (J / equal, and v large, then ~(k) :::::: 1cos(27f"kx/ L) ~(x) dx [19] the model is approximated by the rule: The spatially homogeneous solution is stable if and only P'+I(X) = 1 if 8E < P,(x + x') dx' < 81 if the roots of the characteristic equation lie within the 1-' unit circle on the complex plane: I).. I < 1 for every k = 0, = a otherwise [20] 1, 2, .... This condition can be plotted on the coefficient plane (a, b), as shown in Figure Ala, where stability This is essentially a continuous space analog of Wol­ requires that a and b lie within the shaded triangle. fram's Class-3 cellular automata rule (WOLFRAM, 1984). Spatial instability requires that (i) the homogeneous This type of rule leads to "chaos" and the "tent" patterns. solution be stable: I).. I (k = 0) < 1, and (ii) there exists a The linear analysis was employed to guide the numer-

t~'A /Al__ ~ ______B. Ermentrout et ai., 1986 Page 387

~.~. ~. m

(0) (b) (c) Figure A3 a, spatial pattern arising from + 1 bifurcation; b, spatial pattern arising from -1 bifurcation; c, spatial pattern arising from complex bifurcation.

icaI simulations. The model equations were converted to C. Alternative Formulations a single second order difference equation and the integrals approximated by The behavior of the model equations can be illuminated by examining their continuous time limit. If we subtract P and R from both sides of [4] and [5], respectively, we f W(x' - x)P(x') dx' ::::: ~ ±W('_j)/NP j [21] obtain j-O P'+l - P, = S[P,] - R, - P, [22] Generally, N was taken to be 64, although when un­ Rt+l - R, = (0 - l)R, + ,,(P, [23] usual patterns were encountered N was set to 128 or 256 to check that they were not numerical artifacts. Initial By an appropriate choice of time scale, t, we can divide conditions were random, or small regions of the domain were excited. Typically, long transients generated com­ plicated patterns which gradually simplified as the tran­ . sients damped out. R R=O

Table 2 P=Q Fig. 8E 8/ 01", 01/ (1" (1/ 'Y 0 v lla 0.0 0 6 8 0.1 0.2 0.0 0.0 1 lIb 5.5 0.22 15 0.32 0.1 0.15 0.0 0.0 8 p lIe 4.5 0 15 0.5 0.1 0.12 0.05 0.6 8 14b 1 100 5.0 4.0 0.05 0.2 0.8 0.4 2 15b 4.5 0.32 15 0.5 0.1 0.15 0.1 0.8 8 16 a 0 8.8 6.6 0.1 0.2 0.4 0.6 1 17a 3 4 8.0 4.0 0.1 0.2 0 0 8 Figure A4 17b 5.5 5.5 10 4 0.1 0.2 0.3 0.2 8 Phase plane for the differential equations [A24j and [A25j. q

Page 388 The Veliger, Vol. 28, No.4 both sides by t and replace the differences by derivatives where PCP, j) is an S-shaped curve whose intercept is to obtain: regulated by j. By expanding the convolution to fourth ap order, this model can also be reduced to a biharmonic at = S[P] - R - P [24] diffusion-reaction model: aT 82j a2 a2j aR [32] at = (05 - l)R + 'YP [25] at = -D1 axz - D2axz ax2 + (aP - bf) 8P Now let us examine the phase plane of this system at at = F(P,f) [33] a fixed x = Xo. The operator S is sigmoidal in P, and so the right-hand side is a cubic-shaped curve (a sigmoid minus a linear term). The (P, R) phase plane is shown Somewhat different approaches were employed by WADDINGTON & COWE (1969), MEINHARDT (1984), and in Figure A4; it is qualitatively similar to the FitzHugh­ Nagumo model for excitable media. That is, each volume WOLFRAM (1984). They modeled the shell patterns by an element is excitable, and the volume elements are spatially automata wherein the activation-inhibition effect was rep­ coupled by the activation-inhibition operator W. resented by nearest neighbor interactions via diffusion. If only nearest neighbor cells interact inhibitorily, then Meinhardt's model employed two substances with differ­ W can be expanded in a Taylor series about x, and only ent diffusion constants (D I > D A). He obtained some of lowest order terms retained. Then a familiar diffusion­ the same patterns we obtain here by assuming that each reaction model emerges: cell of the automata could periodically fire and become refractory for a while. In a more recent simulation, Mein­ ap a2p hardt and Klingler (to appear) included longer range in­ [26] at = D ax2+ PCP, R) teractions by allowing morphogens to diffuse beyond near­ aR est neighbors. These simulations resemble ours and it at = 'YP + (05 - l)R [27] appears that most patterns can be created by either mech­ anism. However, it is not clear how the diffusion-reaction model handles the problem of pattern alignment between where D is a diffusion coefficient that can be expressed episodes of shell secretion, whereas this is intrinsic to the in terms of the expansion coefficients of the integrand. neural model. Wolfram's simulations mimic to a remark­ If activation-inhibition is to be retained in the model able extent the "tent" patterns observed on many cone then fourth order terms must be retained (odd order term~ shells. However, his rules were rather arbitrary, and have dropping out by symmetry), and we obtain the biharmonic diffusion-reaction system: no obvious physiological interpretation. The neural net model, in the limit of short-range interactions and sharp ap a2p a2[a 2P] threshold functions, reduces to the automata model, and at = - DI ax2 - D2 ax2 ax2 + PCP, R) [28] can also reproduce the tent patterns. All of these models have a similar structure: a locally aR excitable activator-inhibitor system that is coupled spa­ = 'YP + (05 - l)R [29] at tially to nearby points. In order to obtain spatial patterns, the activation-inhibition is essential. Moreover, it appears ~er~ the negative sign in DI corresponds to short-range that many of the patterns depend on the long-range (i.e., aCtivatIOn, and the negative sign in Dz corresponds to long­ beyond nearest neighbor) interactions characteristic of range inhibition. neural nets. Also, the episodic nature of the secretion pro­ A model quite similar to this was arrived at by J. Keen­ cess dictated our choice of a discretized model in time; this ~r (personal communication) by defining a net neural fir­ feature also appears essential to the formation of certain mg rate, f(x, t) according to the equation pattern types. In a subsequent publication we shall inves­ af tigate a broader class of neural models, including kernels = (aP - bf) r W(x/ - x)j(x/) dx' [30] at + Jo with long-range activation and two-dimensional mantles, such as are found in cowries. where W(x - x') is the activation-inhibition kernel shown in .Figure 9. Co~pling to the secretion is obtained by de­ finmg the secretIon rate to be a bistable function: ap at = P(P,!) [31]