The neural origins of shell structure and patternin aquatic mollusksAlistair Boettigera, Bard Ermentroutb, and George Ostera,1

aBiophysics Graduate Group and Department of Molecular and Cellular Biology, University of California, 216 Wellman Hall, Berkeley, CA 94720; andbDepartment of Mathematics, University of Pittsburgh, 512 Thackeray, Pittsburgh, PA 15260

Edited by Eve Marder, Brandeis University, Waltham, MA, and approved February 13, 2009 (received for review October 20, 2008)

We present a model to explain how the neurosecretory system ofaquatic mollusks generates their diversity of shell structures andpigmentation patterns. The anatomical and physiological basis of thismodel sets it apart from other models used to explain shape andpattern. The model reproduces most known shell shapes and patternsand accurately predicts how the pattern alters in response to envi-ronmental disruption and subsequent repair. Finally, we connect themodel to a larger class of neural models.

neurosecretory � mathematical model � bifurcation

Seashells display a remarkable variety of ornate pigmentationpatterns. Accumulating evidence now indicates clearly that shell

growth and patterning are under neural control and that shellgrowth and pigmentation is a neurosecretory phenomenon. Most ofthis evidence has been gained by detailed studies of the mantle, atongue-like protrusion of the mollusk that wraps around the edgeof the shell and deposits new shell material and pigment at theshell’s growing edge (see Fig. 1 A and B) (1, 2). The shell itself iscomposed of crystal structures of calcium carbonate interspersedwith associated proteins and other organic compounds, some ofwhich are pigmented and arrayed in intricate patterns (3, 4). Thishard shell is covered by a thin organic layer of proteinaceoussecretions, believed to function in regulating calcium crystallization(5). Early EM studies of the mantle recorded an extensive distri-bution of nerve fibers among the secretory cells (1, 6, 7). Thesefibers were later shown to have active synapses with secretory glandcells and synaptic inputs from other sensory organs in the mantle.From this evidence, it was proposed that neural-stimulated secre-tion controls shell growth (4, 7). The original evidence fromgastropods has been extended to other mollusk taxa, includingbivalves (8) and cephalopods, where improved experimental meth-ods confirm clearly the role of neural control (9, 10). Neuralrecordings and neural cell ablation experiments have further ver-ified the role of neural control in shell growth and repair (6, 11, 12).

This experimental data on mollusk shell construction and pig-mentation allow us to formulate a new neural network model anddevelop a unified explanation for the generation of both shellshapes and patterns. Unlike the purely geometric representationsproposed earlier to model shell shape (13–18), our model linksshape generation directly to the dynamics of the underlying neuralnetwork. Recent experimental work describes how differentialgrowth patterns can lead to shell-like structures (19–21) but doesnot explain the biological origin or mechanism of these growthpatterns. The neural model presented here closes this gap byexplaining how the mantle neural net can encode the appropriateinformation required for shell growth as well as pigment deposition.

Early attempts to reproduce shell patterns used cellular automatamodels, in which arbitrary rules determine the pigmentation of cellson a grid (22–24). Although they can reproduce some observedpatterns, these models have shed little light on how such patternsactually arise in the animal. Inspired by the chemistry of diffusingmorphogens, Meinhardt and coworkers (25–29) used a variety ofdifferent diffusion–reaction (DR) models to reproduce a widevariety of shell pigmentation patterns. Although no experimentalevidence has been found for diffusing morphogens in patterning,

the models can be viewed as an incomplete analogy for neuralactivity (chapter 12.4 in ref. 30).

Both the neural and DR models allow different ways of describ-ing the phenomenon of local excitation with lateral inhibition(LALI). Ernst Mach (31) first described this phenomenon toexplain the visual illusions now called ‘‘Mach Bands’’ and empha-sized LALI’s property of enhancing boundaries. Nearly a centurylater, Alan Turing (32) showed how LALI could be modeled bysystems of nonlinear DR equations. This property was exploited bylater workers, most notably Murray (30) and Meinhardt et al.(33–35) to model an extraordinary range of biological patterns.Indeed, DR models have become an all-purpose LALI metaphorin many domains wherein the underlying physics are clearly notdiffusing substances (30, 36). All of these models exhibit spatialinstabilities that lead to spatial patterns. The neural shell modelpresented here combines spatial with temporal instability becausethe mantle can sense previously laid patterns by, in a sense, lookingbackwards in time. Indeed, LALI in time is equivalent to arefractory period that leads to temporal oscillations. Meinhardt’sDR models have succeeded in reproducing almost all of thepatterns quite accurately, and we can hardly do better here. Ourgoal, however, is not to merely reproduce the patterns but to showhow a single neural network model, based directly on the mantleanatomy, can capture all of the pattern complexity, as well asconstructing the shell shape, and to relate the model to a broaderclass of experimentally observed neural network behavior.

Our exposition proceeds as follows. First we describe the neuralnet model as applied to the construction of the shell shape. We thenproceed to define the most important classes of shell patterns andshow how the model reproduces them. We also describe how theshell patterns respond to perturbations, such as injuries.

ResultsA Neurosecretory Model of Shell Growth and Pattern Formation.Because we are interested only in the origin of shape and patternand not the structural composition of the shell, we shall ignore thesubsequent biomineralization that strengthens the shell distallyfrom the leading edge. Secretions in the periostracal groove arecontrolled by the underlying neural network synapsing on thesecretory cells. The activity of this neural network is stimulated bythe existing pattern of shell deposition and pigment at the mantleedge. A schematic representation of this system is shown in Fig. 1B.

The shell is constructed by periodic—usually daily—bouts ofsecretion (40–42). These periodic increments are robust againstmany kinds of environmental variations (41). We model the secre-tions in daily steps, wherein the pattern of each day’s layer ofsecretions, Pt, is a function of the preexisting layers, P(t � �). Weadopt the hypothesis of Bauchau (41) that the pigment pattern

Author contributions: G.O. designed research; A.B. performed research; B.E. contributednew reagents/analytic tools; A.B., B.E., and G.O. analyzed data; and A.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

1To whom correspondence should be addressed: E-mail: goster@nature.berkeley.edu.

This article contains supporting information online at www.pnas.org/cgi/content/full/0810311106/DCSupplemental.

www.pnas.org�cgi�doi�10.1073�pnas.0810311106 PNAS Early Edition � 1 of 6

NEU

ROSC

IEN

CE

allows the mantle to position itself in register with the existingpattern. Fig. 1C shows a schematic diagram of how the model sensesthe pattern from previous days’ secretions and lays down the currentday’s pattern.

The basic property of the neural net control of secretion lies inthe phenomenon of LALI, common to many—if not all—neuralnetworks. As we shall demonstrate, this general feature of neuralnets will be sufficient to generate all of the observed shell patternsfrom a single mathematical model. A noteworthy aspect of this workis that LALI takes place both in space (along the mantle edge) andbackwards in time (perpendicular to the mantle edge). A precisemathematical derivation of the model is given in SI Appendix,section A. Here we present an intuitive description.

Sensory organs in the mantle detect the presence of pigment andstimulate pigment secretion in local secretory cells. Consider a celllocated at position x along the growing shell edge. The strength ofits lateral stimulation of surrounding cells is represented by thecurve, WE(x) (red trace in Fig. 1C). The same sensory input excitesa wider inhibitory field in the mantle, which we describe by thecurve, WI(x) (blue trace in Fig. 1C). Because the mantle wrapsaround the top edge of the shell, it senses, in addition to the exposededge, some previous history of pigmentation a distance � back fromthe leading edge. Pigmentation near the leading edge has anexcitatory effect on local secretion, whereas pigmentation sensedfurther from the edge has an inhibitory effect. We incorporate thiseffect into the excitatory and inhibitory kernels by making themfunctions of � and integrating back in time (i.e., distance from theshell edge). Only the qualitative features of the functions WE andWI, not their precise shape, are important for the pattern. Forcomputational analysis, we represent the excitation and inhibitionfunctions with Gaussian curves of different widths and differentrelative amplitudes (43).

Neurons have a nonlinear, saturating response to their netstimulation (44, 45). There is a characteristic threshold for theneuron, around which it is most sensitive to changes in the stimu-lation rate. Once the stimulation exceeds this threshold, the rate ofchange in activity in response to change in stimulation saturates. Ingeneral, the characteristics of this sigmoid input–output responsecurve will be different for inhibitory and excitatory synapses.Therefore we subject the excitation and inhibition to separatesaturating input–output functions SE (Fig. 1C, red) and SI (Fig. 1C,

blue). This step filters the sensory input that stimulates the secre-tory cells.

The secretion of pigment in the current layer, Pt, is determinedby the net stimulation of the secretory cells. Because of lateralinhibition in the � direction, the current secretion is affected bothdirectly and indirectly by the previous pattern. The pigmentedportion of these depositions provides ‘‘markers’’ and allows us to seehow this sensory network propagates the pattern from layer to layerover time.

The model can be cast in several nearly equivalent mathematicalforms; these forms are presented in SI Appendix, section A. Thespecific analytical forms of the kernels representing the lateralconnections and the saturation functions representing the nonlinearneural input–output responses do not affect the patterns generatedby the model.

Explaining Shell Structure. The location and shape of the initial shellsecretions are determined during embryonic morphogenesis. Afterthe initial shell secretion, the mollusk constructs the rest of the shellenclosure, regularly expanding the enclosure to accommodate itsgrowth. In gastropods (snails), the shell grows outward and spiralsdownward from the original region of shell secretion by successivelyadding small increments of additional shell material to the leadingedge. A variety of mathematical representations of the final geom-etry have been proposed, (see SI Appendix, section C) (13, 16, 29,

Fig. 2. Explaining structure. (A) The neural model explains how the aperture-growthvectorsarise fromtheneuralarchitecture in themantle.Thebottomplotsshow neural excitation at 2 different times (4 days apart). The effective growthvectors resulting from this pattern of excitation is shown at the top. (B) Aperture-growth vectors generated by the model to create spiral-shaped gastropods.

Fig. 1. Shell-making machinery. (A) EM of the mollusk mantle. The EM of a nautilus mantle is shown, with secretory epithelial cells stained green and nerve axonsstained red. [Images reproduced with permission from ref. 10 (Copyright 2005, Wiley).] (B) Schematic representation of the mantle, showing the neurosensory cells,the circumpallial axons connecting these cells, and the neurons that control shell and pigment secretion. (C) Schematic illustration of the model. The existing edgepattern induces excitatory firing (red); the older, previous pattern is inhibitory (blue), indicated in the upper trace. For simulation purposes we use a Gaussian spatialkernel WE,I � �E,Iexp(�x2/�E,I

2). Excitatory stimulation leads to sharp stimulation of the local region and weak inhibition of an even wider surrounding region. Neuralstimuli are passed through saturating sigmoidal filters to determine the spatial pattern of activation or inhibition on the pigment secreting cells. We use SE,I � 1/[1 � exp(�E,I[�E,I � K])], where K is the input from the spatial kernel. The pigment is then Pt�1 � SE(WE�Pt) � SI(WE�Pt) � Rt, with Rt � �Pt�1 � �Rt�1 (see section A of SIAppendix for derivation, discussion, and alternate forms).

2 of 6 � www.pnas.org�cgi�doi�10.1073�pnas.0810311106 Boettiger et al.

46). The underlying phenomena that generate this spiral growthpattern are unspecified, as are the differences between species thataccount for the different spirals—or no spiral at all, as in thebivalves. Rice (19) provided some of the first insights into thesequestions. This work was extended to growth vector models whichdemonstrated that many different coiled shell forms could bereproduced by varying secretion rates appropriately around theaperture (as shown in Fig. 2B) (20, 21).

We propose that neural activity controls the amount and direc-tion in which shell material is secreted. The neural activity along themantle determines the local secretion rate and, thus, the angle andmagnitude of the growth vectors, as illustrated in Fig. 2A. Thecentral point here is that the same model that generates the patternscan generate the shell geometry as well. In the discussion to follow,we shall augment the pigment patterns generated by the model witha few examples of shell growth generated by the same model. Thedynamics of shell growth are best appreciated as movies computedfrom the neural secretion model, a frame of which is shown in Fig.2B. Examples are given in Movie S1 and Movie S2.

Understanding Pigmentation Patterns. The neural architecture oflocal excitation and long-range inhibition gives rise to a broad arrayof stable patterns. The type of pattern depends on the relativeranges and strengths of the interactions and the steepness andthresholds of the firing response curves. Remarkably, all of thepatterns initiate from 3 basic mathematical phenomena arisingfrom the LALI property: spatial instabilities (Turing bifurcations),

temporal instabilities (Hopf bifurcations), and traveling waves*.Analytical demonstrations of these instabilities are presented in SIAppendix, section B.

Bifurcations Create Periodic Patterns in Space and Time. LALI readilygives rise to patterns of parallel stripes orthogonal to the shell’sleading edge. This development takes place because the spatiallyuniform state is unstable to small, random perturbations or slightheterogeneities in the neural network (i.e., a Turing instability) (30,47). Briefly, the process works in the following manner. A slightlymore active local group of cells exerts a stronger inhibitory effecton its neighbors. This lateral inhibition weakens the activity in theneighbors and consequently weakens the inhibitory effect they exerton the original population; this weakening of the neighbor popu-lation causes the activity of the original group of cells to increase.At the same time, the activity reduction in the neighbor clusterallows the next cluster over to increase its activity because itsinhibition is lowered. The result is a standing wave of neural activitythat deposits stable pigmentation stripes normal to the aperture.The width of the stripes reflects the extent of the excitatory regionin the mantle and the width of the gaps reflects the extent of theinhibitory connections. Some examples are shown in Fig. 3D.

If we increase the strength of the stimulation arising from theprevious day’s pattern (i.e., the amplitude of the activation kernel

*The waves probably originate from an infinite dimensional saddle-node bifurcation, butwe have not proven this here.

A C E

B D

Fig. 3. Simple bifurcation patterns. In all images, the real shell is shown on the left, and the simulated shell on the right. (A) The gradual stabilization from randomnoise (Bottom) into periodic stripes (Top) shows how Turing instabilities give rise to patterns of stable bands perpendicular to the growing shell edge. Note thatactivation centers separate and shift as each one carves out a domain of influence. (B) Turing patterns. B. fasciata exhibits Turing bands of pigment, and Turritellaexhibits structural ridge bands. (C) Phase plot of model variables shows the periodic orbits of a limit cycle created by a Hopf bifurcation. (D) Patterns of periodic stripes.This periodic activity may influence secretion of structural elements instead of pigment, resulting in the periodic flanges seen on E. scalare. The zigzag stripes shownon N. communis were generated by a combination of Hopf bifurcation with wave generation as described in Fig. 4. (E) Hopf bifurcations and Turing instabilities canoccur simultaneously, leading to patterns like that of N. tigrina.

Boettiger et al. PNAS Early Edition � 3 of 6

NEU

ROSC

IEN

CE

in the � direction), the pattern can switch to alternating bandsparallel to the shell edge, as shown in Fig. 3E. To create theseperiodic patterns, secretion-stimulating neurons must cycle be-tween successive periods of stimulation and quiescence. A plot ofthe current pigmentation versus the past pigmentation (i.e., a phaseplane portrait) will trace out a single loop around which the systemcontinuously cycles (see Fig. 3C). Such periodic orbits on the phaseplane (limit cycles) arise from parameter changes triggering so-called Hopf bifurcations, which are well-known in models ofrepetitively firing and bursting neurons (45). These periodic insta-bilities can arise in several ways. For example, when there aredifferent thresholds for excitatory and inhibitory signals, the neuralnetwork may be excitable at low-pigment-induced stimulation butinhibitory under strong activity. A low basal level activity in theabsence of signaling triggers a slow, positive feedback, whichgradually amplifies the signal and eventually triggers the high-threshold inhibitory response, which shuts neuronal firing backdown to its basal level, from which the process begins again. Thisprocess leads to stable cycles of oscillation, which produce periodicbands of pigment parallel to the growing edge of the shell, whoseperiod can be many multiples of the secretory period (e.g., 1 day).Excitatory lateral connections tend to synchronize the populationinto parallel bands like those seen on shells of Amoria ellioti.Stronger lateral inhibition induces the patterns to lag each other,forming instead periodically distributed zigzag patterns, like thoseof Natica communis. For a mathematical derivation of synchroniz-ing phenomena among neural limit cycle oscillators, see ref. 45.Finally, Hopf bifurcations may coincide with a Turing instability,

leading to patterns periodic in both space and time, like thecheckerboard patterns of Natica tigrina shown in Fig. 3B.

In Fig. 3 D and E, we show how network-induced secretion ofshell material, instead of pigment, leads to the growth of the shellgeometry in Turritella and Epitonium scalare.

Asymmetric Activity Creates Traveling Waves. Traveling waves ofpigmentation arise when previous firing activity represses futurefiring activity while exciting lateral activity. An asymmetric regionof activation (as in the wedge shape in Fig. 4A) induces strongerstimulation toward its high side than toward its tail. This unevenlateral excitation induces the pigment in the next layer to spreadlaterally in front of the wedge. The repression from having firednarrows down the tail, and the whole wedge shifts sideways towardthe high side. Successive depositions thus create a traveling line ofpigment at an oblique angle to the shell edge. When two such wavesof pigmentation collide, they may mutually annihilate (as in Conusclerii), singularly annihilate (as Conus vicweei), or reflect (Tapeslitarus) (Figs. 4 and 5E). Close examination of reflecting waves showthat each wave is actually quenched but then reignites because theactivation width is broader than the inhibition region.

Depending on the size and shape of the kernels, when wavesapproach each other they can either slow down or speed up. Thus,regular reflecting waves can create patterns of spots as on Naticastercusmucarum (SI Appendix, section H) or teardrops like onConus marblus (Fig. 4A), depending on how the overlappingexcitation and inhibition kernels either accelerate or decelerateapproaching and separating waves. If the previous firing repression

Fig. 4. Wave patterns. (A) Patterns formed by traveling waves of excitation. Asymmetric regions of excitation travel toward the stronger side, as illustrated by C. cleriiin B. When these waves collide, they may reflect, as shown in this simulation and on the shells of C. clerii and C. marblus. The waves may annihilate, as shown on theshells of Conus viceweei. A wave may also emit ‘‘reverse’’ waves, creating the beautiful tent patterns of Olivia porphyria. (B) Graphs of neural activity across the leadingedge and close-up of resulting secretions for select shells. (C) Some traveling waves of excitation leave excited regions behind as they cross the mantle. At a critical width,the cumulative inhibition shuts down signaling, creating a region devoid of pigment. This region is slowly reclaimed by waves traveling back into it, as shown in thesimulation of C. innexa and C. marblus in B and C. (D) Some of the diverse patterns produced by the model by combinations of wave collisions and emissions.

4 of 6 � www.pnas.org�cgi�doi�10.1073�pnas.0810311106 Boettiger et al.

activity has a high threshold, waves traveling apart remain con-nected at the tails until the stimulation abruptly crosses the thresh-old and all pigmentation stimulation shuts off abruptly. The still-stimulated edges, however, travel back into the unpigmentedregion, leaving a triangular gap devoid of pigment. This gap is seenon many shells (e.g., Conus bullatus and Conus thailandis, seen inFig. 4C). Fig. 4B also shows the detailed steps that lead to thefractal-like triangles of Cymbiola innexa in Fig. 4C. Additionalpatterns generated by the model are shown in Fig. 4D and in SIAppendix, section H.

Effects of Shape and Environment on Patterns. The pigmentationpatterns on many shells change qualitatively as a result of shellgrowth or environmental disruption. Our simple neural modelprovides a mechanistic explanation for many of these patternchanges.

The length scale of the pattern is determined by the distributionof axon and dendrite lengths. As the animal grows and moreneurons incorporate into the mantle, existing patterns may becomeunstable. Parallel lines perpendicular to the shell edge may widenor bifurcate as the overlap between lateral inhibition decreases.Both effects are apparent in the limpet Tectura testudinalis in Fig.5A. Another common observation is that patterns of pigmentationare homogeneous on small domains (relative to the average neu-ronal connection range). This observation explains why many shellsstart with either no pigment or uniform pigment and developintricate patterns only after the mantle edge grows to the appro-priate length. An example is the shell of Babylonia spirata in Fig. 5D.Note the lack of pigmentation in the small twists that form the topof the spiral. In contrast, regular patterns, such as oscillating bands,are stable only when the domain size is small. As the domainbecomes large, synchrony across the mantle is lost and the patterndegenerates into a uniform pigmentation as seen on the shells ofAmoria grayi (shown in Fig. 5B). Additionally, loss of patternsynchrony may cause the pattern to degenerate from alternatingbands into a mesh of dots, as the domain becomes too large tomaintain global synchrony in the presence of small background

noise. This effect is seen on a variety of shells, including the Mitramitra stictica shells shown in Fig. 5C. Other pattern bifurcations arealso captured by the model. The bivalve T. litarus exhibits regulartraveling waves across its shell. Near the top of the shell these wavescollide and terminate in V-shaped patterns. However, as thedomain size increases, the waves become reflecting, bifurcating inshaped intersections when they collide (Fig. 5E).

In addition to predicting how patterns are created ab initio, themodel predicts how the system responds to perturbations in thepattern. Ablation of a small portion of the ridge pattern on a Strigillashell allows for spontaneous activation of new waves (manifested bythe V’s formed) and acceleration of existing waves, as evidenced inthe lines becoming more closely parallel to the growing edge.Simulations of pattern ablation capture both of these effects, whichappear in a field of traveling waves and at an annihilation point, asshown in Fig. 5F. Bankivia fasciata synchronize in steps fromrandom initiation provided by variable background rates. First amesh-like pattern of dots emerges; these dots subsequently syn-chronize into uniform bands as explained above. If the pattern isdisrupted by an injury, the pattern restabilizes from dots to stripesagain, a property readily illustrated by the model in Fig. 5G.

Insights into Mollusk Evolution. An attractive feature of the neuronalmodel is its suggestive mechanism for the evolution of the observedpattern diversity. Because very similar species can exhibit signifi-cantly different patterns, the pattern difference cannot be the resultof dramatic anatomical differences. The neural model provides acommon mechanism whereby small changes in the parameters leadto large changes in the patterns. The model also makes testablepredictions about which sets of patterns one is likely to find withina genus and which sets of patterns are not likely to occur togetherin a genus. Our model predicts a greater evolutionary separationwould be required. Surprisingly, some of the dramatically differ-ently patterned species of cone shells can be reasonably well-reproduced by parameter sets that are quite close to each other, asshown in SI Appendix, section H and Fig. S5.

Fig. 5. The effects on patterns of shell growth and perturbations. As the shell grows, the width of the pattern domain increases leading to changes in the pattern.(A–E) These patterns include line bifurcations of T. testudinalis (A), collapse of oscillations in Amoria grayi (B), destabilization of waves into patchy dots on M. mitrastictica (C), emergence of a pattern from a uniform field in Babylonia spirita (D), and transition from annihilating to reflecting waves on T. litarus (E). Patterns changein response to scratches, which remove information about the previous pattern. (F) Traveling waves in Strigilla shells slow down and deflect away from the growingedge. [Photo adapted and reproduced with permission from ref. 27 (Copyright 1987, Elsevier.)] (G) B. fasciata goes through repeated stabilizations from dots to stripes.Shells in B–D are from ref. 52.

Boettiger et al. PNAS Early Edition � 5 of 6

NEU

ROSC

IEN

CE

DiscussionIn this work, we have shown that a single neurosecretory model canreplicate both the growth of mollusk shells and the enormousdiversity of pigment patterns they exhibit. The model is built aroundthe general property of local excitation coupled with lateral inhi-bition common to most neural networks. A noteworthy feature inthis model is that the same network architecture operates in boththe spatial and time directions because the pigment patternsdevelop sequentially as the mantle lays down periodic increments ofshell and pigment. Thus, the shell pattern records the complete timehistory of its neurosecretory activity. One might think of the patternas an electroencephalogram, or the history of the thoughts of amollusk! In general, waves propagating through a 3-dimensionalneural network (e.g., a cortical column) have this same property:Local excitation/lateral inhibition extends laterally, as well as back-wards in space from where the excitation came, which is essentiallybackwards in time.

By exploiting the permanent record of neural activity thatmollusks have incorporated into their shells, we have achieved amechanistic understanding of how these diverse, and seeminglyvery different, patterns arise. We have shown that all of the patternsemerge from combinations of 3 types of bifurcations: Turing andHopf bifurcations, wave propagation, and collisions, which probablyoriginate in saddle-node bifurcations. The intuitions we have de-veloped in the study of mollusk shells may provide a usefulfoundation for studying other types of neural patterns, such as thedynamic patterning of cuttlefish skin controlled by a neuromuscularnetwork that exposes and hides chromatophores. We have foundthat the mollusk model can reproduce many patterns observed inthe cuttlefish mantle (see SI Appendix, section E) that form

sequentially as a wave. The cuttlefish patterns change much fasterthan the mollusk patterns, but their origin is fundamentally thesame: Both are products of neural net activity. Insights from thestudy of complex visible neural patterns like these may prove usefulin understanding normally invisible patterns of neural activity, suchas the structured spatial organization of neural activity distributedover the mammalian cortex. Here we have seen that the molluskwaves slow down and reflect in a manner similar to that observedin cortical waves (48). Such wave collisions might allow for com-parison of cortical predictions with sensory input: Annihilationoccurs when waves are identical and an error wave propagates outwhen they are different.

Our mechanistic explanation of how a neural system determinesthe future pattern from the previous pattern suggests an interestingparallel with other cortical processes. It is not entirely unlike thechallenge the brain addresses of predicting the future from itsinternal neurological model of the world. Most theories of thisprocess make analogies to Bayesian inference (49–51). Our worksuggests how these models may be recast in terms of physiologicalneural parameters that may underlie such Bayesian predictionmodels (see SI Appendix, section F). We suspect many new insightsawait in the study of spatial patterns of neural activity.

Materials and Methods.The model was simulated and results plotted as graphics in MATLAB 2007b.

ACKNOWLEDGMENTS. We thank B. Westermann and H. Meinhardt for permis-siontoreproducethenotedimages inFigure1AandFigure5F.A.B.andG.O.weresupported by National Science Foundation Grant 0414039. B.E. was supported bythe National Science Foundation Division of Mathematical Sciences. A.B. wassupportedbyNational InstituteofBiomedical ImagingandBioengineeringTrain-ing Grant in Physical Biosciences T32 EB005586-01A2.

1. Saleuddin ASM, Dillaman RM (1976) Direct innervation of the mantle edge gland byneurosecretory axons in helisoma duryi (Mollusca: Pulmonata). Cell Tissue Res 171:397–401.

2. Saleuddin A, Kunigelis S, Khan H, Jones G (1980) Possible control mechanisms inmineralization in Helisoma duryi (mollusca: pulmonata). The Mechanisms of Biomin-eralization in Animals and Plants eds Omori M, Watabe N (Tokai Univ Press, Tokyo).

3. Checa AG, Okamoto T, Ramírez J (2006) Organization pattern of nacre in Pteriidae(Bivalvia: Mollusca) explained by crystal competition. Proc R Soc London Ser B273:1329–1337.

4. Saleuddin ASM (1979) Pathways in Malacology (Junk, The Hague), pp 47–81.5. Taylor JD, Kennedy WJ (1969) The influence of the periostracum on the shell structure

of bivalve molluscs. Calcif Tissue Int 3:274–283.6. Dillaman RM, Saleuddin ASM, Jones GM (1976) Neurosecretion and shell regeneration

in Helisoma duryi. Can J Zool 54:1771–1778.7. Saleuddin ASM, Kunigelis SC (1984) Neuroendocrine control mechanisms in shell

formation. Am Zool 24:911–916.8. Saleuddin A (1974) An electron microscopic study of the formation and structure of the

periostracum in Astarte (Bivalvia). Can J Zool 52:1463–1471.9. Westermann B, Beurelein K, Hempelmann G, Shipp R (2002) Localization of putative

neurotransmitters in the mantle and siphuncle of the mollusc Nautilus pompilius.Histochem J 34:435–440.

10. Westermann B, Schmidtberg H, Beurelein K (2005) Mantle organization of Nautiluspompilius. J Morphol 264:277–285.

11. Dogterom AA, van Loenhout H, van der Schors RC (1979) The effects of growthhormone of Lymnaea stagnalis on shell calcification. Gen Comp Endocrinol 39:63–68.

12. Wilbur K, Saleuddin ASM (1983) The Mollusca (Academic, New York).13. Thompson DW (1952) On Growth and Form (Cambridge Univ Press, Cambridge, UK).14. Raup D (1961) The geometry of coiling in gastropods. Proc Nat Acad Sci USA 47:602–629.15. Raup D (1962) Computer as aid in describing form in gastropod shells. Science 138:150–152.16. Raup D (1965) Theoretical morphology of the coiled shell. Science 147:1294–1295.17. Raup D (1966) Geometric analysis of shell coiling: General problems. J Paleontol

40:1178–1190.18. Raup DM (1969) Modeling and simulation of morphology by computer. Proc N Am

Paleontol Conv B 71–83.19. Rice S (1998) The bio-geometry of mollusc shells. Paleobiology 24:133–149.20. Hammer O, Bucher H (2005) Models for the morphogenesis of the molluscan shell.

Lethaia 38:111–122.21. Ubukata T (2005) Theoretical morphology of bivalve shell sculptures. Paleobiology

31:643–655.22. Waddington CH, Cowe J (1969) Computer simulations of a molluscan pigmentation

pattern. J Theor Biol 25:219–225.23. Kusch I, Markus M (1996) Mollusc shell pigmentation: Cellular automaton simulations

and evidence for undecidability. J Theor Biol 178:333–340.24. Wolfram S (1984) Cellular automata as models of complexity. Nature 311:419–424.25. Meinhardt H (1984) Models for positional signalling, the threefold subdivision of

segments and the pigmentation pattern of molluscs. J Embryol Exp Morphol83(Suppl):289–311.

26. Meinhardt H, Klingler M (1986) Pattern formation by coupled oscillations: The pig-mentation patterns on the shells of molluscs. Lecture Notes in Biomathematics(Springer, Berlin), Vol 7.

27. Meinhardt H, Klingler M (1987) A model for pattern formation on shells of molluscs. JTheor Biol 126:63–89.

28. Meinhardt H, Prusinkiewicz P, Fowler D (2003) The Algorithmic Beauty of Sea Shells(Springer, New York), 3rd Ed.

29. Fowler DR, Meinhardt H, Prusinkiewicz P (1992) Modeling seashells. Comput Graphics26:379–387.

30. Murray JD (2003) Mathematical Biology (Springer, New York), 3rd Ed, pp 638–655.31. Mach E (1865) On the effect of the spatial distribution of light stimulus on the retina,

I. Sitzungsber Math Nat Klasse Kaiserl Akad Wiss 52:303–332.32. Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc Ser B

237:37–72.33. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik

12:30–39.34. Meinhardt H (1982) Models of Biological Pattern Formation (Academic, New York).35. Meinhardt H, Gierer A (2000) Pattern formation by local self-activation and lateral

inhibition. BioEssays 22:753–760.36. Winfree A (2001) The Geometry of Biological Time (Springer, Berlin), 2nd Ed.37. Amari S-I (1977) Dynamics of pattern formation in lateral-inhibition type neural field.

Biol Cybern 27:77–87.38. Ellias S, Grossberg S (1975) Pattern formation, contrast control, and oscillations in the short

term memory of shunting on-center off-surround networks. Biol Cybern 20:69–98.39. Wilson H, Cowan J (1973) A mathematical theory of the functional dynamics of cortical

and thalamic nervous tissue. Kybernetik 13:55–80.40. Berard H, Bourget E, Frechette M (1992) Mollusk shell growth: External microgrowth

ridge formation is uncoupled to environmental factors in Mytilus edulis. Can J FishAquat Sci 49:1163–1170.

41. Bauchau V (2001) Developmental stability as the primary function of the pigmentationpatterns in bivalve shells? Belg J Zool 131(Suppl 2):23–28.

42. Lutz RA, Rhoads DC (1980) Skeletal growth of aquatic organisms; Biological records ofenvironmental changes (Plenum, New York), pp 203–254.

43. Dayan P, Abbott LF (2001) Theoretical neuroscience: Computational and mathematicalmodeling of neural systems (MIT Press, Cambridge, MA), pp 62.

44. Cowan JD (1968) Neural Networks (Springer, Berlin), pp 181–188.45. Ermentrout B (1998) Neural networks as spatio-temporal pattern-forming systems. Rep

Prog Phys 61:353–430.46. Checa A (2002) Fabricational morphology of oblique ribs in bivalves. J Morphol

254:195–209.47. Oster G (1988) Lateral inhibition models of developmental processes. Math Biosci

90:265–286.48. Xu W, Huang X, Takagaki K, Wu J-Y (2007) Compression and reflection of visually

evoked cortical waves. Neuron 55:119–129.49. Knill DC, Pouget A (2004) The Bayesian brain: The role of uncertainty in neural coding

and computation. Trends Neurosci 27:712–719.50. Doya K, Ishii S, Pouget A, Rajesh PNR, eds (2007) The Bayesian Brain: Probabilistic

Approaches to Neural Coding (MIT Press, Cambridge, MA).51. Huang G (2008) Is this a unified theory of the brain? New Sci 2658:30–33.52. Oliver APH, Nicholls J (1975) The Larousse Guide to Shells of the World (Larousse, New

York).

6 of 6 � www.pnas.org�cgi�doi�10.1073�pnas.0810311106 Boettiger et al.

The Neural Origins of Aquatic MolluskStructure and Pattern:

Supplementary Information

Alistair Boettiger, Bard Ermentrout, and George Oster

Contents

A Mathematical Formulation 2A.1 Deriving a Mollusk Model from a General Neural Model . . . 2A.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 6A.3 Model Parameters and Parameter Reduction . . . . . . . . . . 7A.4 Relation to Cell Automata Models . . . . . . . . . . . . . . . 8

B Mathematical Analysis 10B.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 10B.2 Bifurcation Conditions . . . . . . . . . . . . . . . . . . . . . . 12

C Shell Shape and Morphogenesis 14

D Quantitative Analysis of Patterns 18D.1 Mathematical Comparison of Simulated and Natural Patterns 18D.2 Phase Plane Analysis of Oscillations . . . . . . . . . . . . . . 18

E Extensions to Cuttlefish Patterns 20

F Extensions to Bayesian Models and Cognition 21

G Simulation Parameters 25

H Supplemental Figures 31

1

A Mathematical Formulation

In this Appendix, we first derive a neurosecretory model of shell formationand patterning by applying the Wilson-Cowan equation to the physiologicalcontext presented in the text. We then reduce this model into a simplier formwhich can be simulated numerically. We discuss next the parameterization ofthis model and the physical interpretation of these parameters. In AppendixB, we present a mathematical stability and bifurcation analysis of both thegeneral and reduced, simulated model.

A.1 Deriving a Mollusk Model from a General NeuralModel

Here we derive a specific description of the neural network in the shell modelby adapting the widely accepted formulation of the Wilson-Cowan equations[3, 22, 23]. We then further reduce these equations to a simplified form whichwe simulate to reproduce shell structure and pattern.

We start by defining some terminology. We shall assume that the mantlecovers some region of already created shell and uses this to determine theactivity for new pigmentation. Let x be the spatial coordinate along theaperture of the shell and y be the coordinate perpendicular to the aperture(i.e. backwards in time). For simplicity, we assume a rectangular shell andthat the y = 0 is the lower edge of the shell. This assumption we will relaxlater. Thus points on the shell have coordinates, 0 < x < L and 0 < y < T.Note that T is the total length of the shell from its beginning; this willincrease as shell material deposited.

The mantle consists of a neural network of distributed excitatory andinhibitory cells with activities, uE(ξ, η, t), uI(ξ, η, t). We will work in mantlecoordinates rather than in shell coordinates since this is invariant. (i.e. allpoints are measured from the growing edge). Let the mantle coordinatesbe defined as a rectangle 0 < ξ < L and 0 < η < D where ξ is parallelto the shell edge and η orthogonal. The pigmentation pattern, P (x, y), ofthe shell is independent of time, and only shell that is already pigmented isaccessible to the mantle sensory system. We use a firing rate formulation forthe mantle neural network. We first consider a general formulation and thensimplify this considerably to derive the ultimate network which we simulate.This general formulation allows us to better connect the model with existingtheories of neuronal networks.

2

We adopt the hypothesis that the neural networks have first order tem-poral kinetics. The activity of the excitatory cells, u(η, ξ, t), then takes theform of the Wilson-Cowan equations [23]. These equations describe activityof neurons in a general layered array (like the shell mantle or the corticalcolumns of the brain).

∂uE(η, ξ, t)

∂t=−1

τEuE(η, ξ, t) + SE(I) (1)

I =∫ L

0

∫ D

0[KEE(ξ − ξ′, η − η′)uE(ξ′, η′, t)

− KEI(ξ − ξ′, η − η′)uI(ξ′, η′, t)] +ME(ξ, η) (2)

where KEE(ξ, η) represents the recurrent connections between excitatorycells in the mantle, KEI(ξ, η) are recurrent connections between inhibitorycells and excitatory cells in the mantle, SE(I) is the firing rate as a function ofinputs and, most importantly, ME(ξ, η) is the sensory input felt by a mantlecell at location ξ, η. In a more general neural network, this will be a functionof time (at the neural time-scale level) but since this will be related to thepigmentation which is constant during a bout of secretion, we do not includethe time dependence. The sensory input to the mantle is dependent on thepigment/structure sensing cells and this, of course, depends on the pigmentwhich as already been laid down:

ME(ξ, η) =∫ L

0

∫ T

0WE(ξ − x, η − y)P (x, y) dx dy. (3)

There are equations analogous to (1) and (3) for the inhibition. To closethe system, we need to compute the activity of the secretory cells. Thesecells presumably lie only along the lower edge of the mantle. In priciple,these could integrate the activity of many mantle cells. Thus we assume thatthe secretion at y = 0 (the aperture at edge of the shell) is

P (x, 0) =

[∫ L

0

∫ D

0(VE(x− ξ,−η)uE(ξ, η)− VI(x− ξ,−η)uI(ξ, η)) dξ dη

]+

(4)Here VE,I is the spatial spread of the mantle neural network to the secretorycells. This is the most general formulation of the shell model. The output ofthe model is a leading edge of structural or pigmentation, P (x, 0). [A]+ indi-cates take the positive part of A or choose 0 if A < 0. This captures the fact

3

that when we subtract total inhibitory stimuli from total excitatory stimulionly positive results are meaningful. Inhibition can quell the excitation andreduce the perceived firing rate, but one cannot have negative firing rates.

Since the pigmentation pattern during a secretory bout is fixed in time,the neural network reaches a steady state, so that uE(x, y, t) loses its fasttime-dependence. However, it is almost impossible in general to find steadystates of general recurrent neural networks of the form (1) (along with thecorresponding equations for the inhibition). Thus, let us suppose that thereare no recurrent connections. In cortical networks and presumably the cut-tlefish (see Appendix E), the recurrent connections may be the dominantfactors in determining the output of the network. However, for mollusk pig-mentation the input from the previous pattern alone determines secretionand there is no need for recurrent connections. Thus, in steady state:

uE(ξ, η) = SE(ME(ξ, η)) = SE

(∫ L

0

∫ T

0WE(ξ − x, η − y)P (x, y) dx dy

).

Similarly

uI(ξ, η) = SI(MI(ξ, η)) = SI

(∫ L

0

∫ T

0WI(ξ − x, η − y)P (x, y) dx dy

).

As a final simplification, we assume that secretion occurs only at the leadingedge. This means the functions VE,I are localized to the edge are deltafunctions and we obtain the model

P (x, 0) = [AE(x, 0)− AI(x, 0)] (5)

AE(x, 0) := SE

(∫ L

0

∫ T

0WE(x− x′,−y′)P (x′, y′) dx′ dy′

)(6)

AI(x, 0) := SI

(∫ L

0

∫ T

0WI(x− x′,−y′)P (x′, y′) dx′ dy′

). (7)

Note the input to these secretory cells at the edge still depends on the pat-tern some distance T away from the edge. One can regard this equation asan evolution equation in y since the pigmentation at y = 0 depends onlyon the pigment laid down earlier. In our simulations, we transfer space iny−direction to pigmentation bouts (e.g. days) which are time-like variables.That is, a point y in the shell represents pigmentation that was laid down

4

n days earlier: n∆ = y, where ∆ is the spatial thickness of a single bout ofshell construction.

To put this in “bout” coordinates, let p(x, t) be the pigmentation atposition x at time t (in bout time and not in terms of real time).

p(x, t) = [aE(x, t)− aI(x, t)]+

aE(x, t) := SE

(∫ L

0

∫ t

0WE(x− x′,−∆s)p(x′, t− s) dx′ ds

)

aI(x, 0) := SI

(∫ L

0

∫ t

0WI(x− x′,−∆s)P (x′, t− s) dx′ ds

).

We simplify this notation by considering the discrete time interval withstep-size ∆, which we denote by subscripts indexed by n. The most generalform of the shell model consists of a spatial and temporal summation ofthe previous patterns put through a nonlinear filter and then differenced toproduce the pigmentation:

Pn+1(x) = [SE(En(x))− SI(In(x))]+ (8)

where the excitation and inhibition activities are given by the convolutions

En(x) =M∑j=0

∫ L

0WE(x− y, j)Pn−j(y) dy (9)

In(x) =M∑j=0

∫ L

0WI(x− y, j)Pn−j(y) dy. (10)

The weighting functions, W (y, j) take into account the spatial and temporalintegration of the sensory input, along the edge (y) or back in time j. Thesecould be measured directly by recording the firing rate in response to stimuliat different points throughout the perceptive field of the neuron, or estimatedby analogy to known characteristics. M is the temporal distance (number ofdays back in time) that the mantle senses the pattern.

In equation (8), there are two non-linearities. These firing rate functionsare sigmoids, with little response to low stimulation and saturating responseto high stimulation. Two simple forms for the spatio-temporal filters are:

W (y, j) = w(y)v(j) (11)

5

which we call a product or space-time separable kernel. Alternatively,

W (y, j) = w(y2 + d2j2) (12)

where d is the spatial width of a single row of pigment. Most of our analysiswill be devoted to the product case, equation (11), where w(y) is symmetricand v(j) decreases with larger j. We first show that this general model ismathematically equivalent to the form simulated in the text, and closelyrelated the the earlier neural model of Ermentrout et al [5]. We shall alsointroduce an alternative form of the model that can more easily analyzed forpattern formation.

A.2 Simulation Model

First we derive the equations which are simulated in the text. For notationalconvenience, let w(x) ∗ p(x) :=

∫ L0 w(x − y)p(y) dy denote the convolution.

we let v(0) = 1,v(j) = −βcj with 0 < c < 1 and M =∞. The latter assump-tion means that the mantle has infinite memory. This is not physiologicallypossible, but since for c < 1, cj will be small for j large so that as long ascM is very small, we can approximate the finite memory case with an infinitememory model which is mathematically reducible to a simpler model. Let

Rn(x) = β∞∑j=1

cjPn−j(x).

Then equation (8) becomes

Pn+1(x) = [SE(wE(x) ∗ [Pn(x)−Rn(x)])− SI(wI(x) ∗ [Pn(x)−Rn(x)])]+Rn+1(x) = βcPn(x) + cRn(x)

Let us finally introduce the pigment-like variable, Yn(x) = Pn(x) − Rn(x)obtaining at last

Yn+1(x) = [SE(wE(x) ∗ Yn(x))− SI(wI(x) ∗ Yn(x))]+ −Rn+1(x) (13)

Rn+1(x) = βcYn(x) + c(1 + β)Rn(x) := γYn(x) + δRn(x). (14)

These are the equations which are simulated in the text with the nonlinear-ities as above and with

w(x) = α exp[−(x/σ)2]/√πσ (15)

6

with different σE,I for the excitatory and inhibitory connections respectively.For simulation, we choose sigmoids of the form,

S(x) =1

1 + exp[ν(θ − x)](16)

with different νE,I and θE,I for each population of neurons.This model is slightly different from the earlier Ermentrout, et al. model

since the term Rn+1 appears instead of Rn in equation (13). We have alsochosen the more standard difference of Gaussian spatial kernel which is sep-arable rather than powers of cosines.

These equations define the model which we solve by numerical simulationand graph the solutions to create all of the images presented in the text.Random initial pattern (or neural) states are chosen to initialize the simu-lation. For all patterns and shapes shown in the text the initial conditionshave no effect on the emergent pattern far from the initialization point (savefor the metastable solutions resulting from perfectly uniform patterns). Asthere is always some background firing rate activity in most known neuralsystems, the perfectly uniform state is not likely biological plausible. Thechoice of parameters used to generate the images in each figure are givenin appendix G. In the next section, we discuss the interpretation of theseparameters and the minimum parameter representation of this model.

A.3 Model Parameters and Parameter Reduction

For clarity we rewrite the simulation model with all of the parameters. Eachof the parameters is designated by a different Greek letter. All of the letterswith double subscripts are actually two parameters – one for the inhibitoryconnections and one for the excitatory connections.

Yn+1(x) = [SE(wE(x) ∗ Yn(x))− SI(wI(x) ∗ Yn(x))]+ −Rn+1(x) (17)

Rn+1(x) = γYn(x) + δRn(x) (18)

SE,I(x) =βE,I

1 + exp(νE,I(θE,I − x))(19)

wE,I(x) =αE,I√πσE,I

e−(x/σE,I)2 (20)

We stress that each of these parameters has a direct physiological inter-pretation, and in theory could be measured independently by experiment.

7

The lateral connections are controlled by four parameters. Two parametersare needed to specify the strength, α and lateral range, σ of these connec-tions. The net stimulation is filtered through a sigmoidal response characterof the target neuron in order to regulate its behavior. In general, three pa-rameters are required to specify the sigmoid, ν, the sharpness of the responsebehavior, and θ, the threshold of the response. The third parameter, βE,Idetermines the saturated firing rate of the neuron at maximal stimulation.As ν →∞ the sigmoid becomes a step function located at x = θ, so the neu-ron exhibits no response for stimulation less than θ and fires at rate β whenstimulated at by excitatory rate greater than θ. Two parameters, γ and δcontrol the strength and range respectively of inhibitory inputs from the thepigment away from the edge. The length of the mantle edge L, finishes theparameter set.

We can eliminate 4 of these parameters by creating the appropriate di-mensionless ratios of parameters. All the spatial length scales can be writtenas fractions of the length of the mantle, σ∗E,I = σE,I/L. Only the relativecontributions of excitation and inhibition are important in determining thefinal firing rate, and so we normalize α = αE/αI and αI = αI/αI . Forsimplicity we choose both βE,I = 1 and control the values of SE,I only bythe appropriate choice of νE,I and θE,I . This leaves the model with but 9parameters,

α, σE,I , νE,I , θE,I , δ, γ,

each of which still relates directly to an mathematically independent physi-ologically distinct property of the described neurons or neurosecretory cells(with the exception of δ and γ in which several different neural propertieshave been compressed by the simplifying assumptions described above). Nofurther reduction is readily accessible with sacrificing the basic properties of aindependent spatial response kernels and input-output firing response curves.The contribution of each of these parameters to the types of stable patternsand bifurcations between patterns which arise in the model are explored inAppendix B.

A.4 Relation to Cell Automata Models

In the simplest case where the only the most recent pattern affects secretion,the discrete shell model (13) can be written as:

Pn+1(x) = [SE[WE(x) ∗ Pn(x)]− SI [WI(x) ∗ Pn(x)]]+

8

Suppose, now, that we take the limit as the sharpness of the nonlinearitiesgoes to infinity, resulting in a step function:

Pn+1(x) = [H(WE(x) ∗ Pn(x)− θE)−H(WI(x) ∗ Pn(x)− θI)]+ . (21)

This means that at each point x, Pn(x) is either 0 or 1. If the weight func-tions have a finite interaction distance then the convolutions, W (x) ∗ Pn(x)approximate adding up the total number of 1’s in this neigborhood. Thus,the right-hand sides of the equation are roughly functions of the total numberof 1s in a neighborhood. This makes the right-hand side look exactly likea totalistic cellular automata (CA) [24]. Recall in a CA, there is a discretearray of cells that take values of 0 or 1. In a totalistic rule, the new valueat site j is just a function of the sum of the current value of site j plusits m nearest neighbors. This sum, S can be any of the 2m + 2 numbers,0, 1, . . . , 2m + 1. The totalistic rule simply assigns a value of 0 or 1 to eachof these numbers.

We now point out a particularly simple rule which is the one-dimensionalanalogue to the famous Conway “Game of Life” CA. Suppose that we assigna new state of 1 if 0 < k1 ≤ S ≤ k2 < 2m + 1 and 0 otherwise. (This saysthat if you are “lonely” or “crowded”, you die.) This type of automata leadsto what Wolfram called “Class III” authomata which produce fractal-liketriangles and were the motivation for Wolfram’s analogy to seashell patterns.There are two simple ways to get “Class III” behavior from equation (21).Suppose, first, that WE(x) = WI(x) and that θE < θI . Then if there are veryfew 1s in a neighborhood, the convolution will not exceed either thresholdand if there are many 1s in a neighborhood, the convolution will exceed boththresholds, so that the result will be a 0 in the next step. Contrarily, foran intermediate number of 1s, only the excitatory threshold will be exceededand the result will be a 1 in the next step. The second way to get the class IIIrule is to assume that θE = θI = θ but that WI has a greater spread than WE,that is, lateral inhibition. For example, suppose that WE,I(x) = 1/2bE,I for|x| < bE,I and 0 otherwise with bE < bI . The update rule for this shell modelis now easily read off since WE,I(x) ∗Pn(x) simply adds up the total numberof points in a neighborhood of size 2bE,I and divides by the normalizationfactor. Thus, if there are too few then neither convolution exceeds thresholdand if there are too many, both convolutions exceed threshold and we havea class III automata.

9

B Mathematical Analysis

In this Appendix we present a linearized analysis of the model which allowsan eigendecomposition of the model. This allows us to explore stability ofthe model and gain insight into the sort of bifurcations which arise in themodel and which parameters they depend on. By exploiting the Gaussianchoice of a kernel we then present an additional derivation of the Turing anHopf bifurcation properties.

B.1 Linear Stability Analysis

We analyze the sketch linear stability of this model around a homogeneousequilibrum point. The stability analysis is similar to that in the appendix ofErmentrout, et al. but there are slight differences due to minor changes inthe equations. Let (Y , R) be steady states

Y = SE(Y )− SI(Y )− R, R + γY

1− δ.

We work on an infinite domain so we can write explicit eigenfunctions as iscommon in pattern formation analysis. The linearized equations are

Yn+1 = [ηEwE(x)− ηIwI(x)] ∗ Yn(x)− γYn − δRn

Rn+1 = γYn + δRn.

Here ηE,I := S ′E,I(Y ). Let F (k) be the Fourier transform of the effectivekernel, ηEwE(x) − ηIwI(x). Note that if σE < σI , then this effective linearkernel is a “Mexican hat”, the hallmark of pattern formation. The eigenvaluesof this second-order system satisfy

λ2 − a(k)λ+ b(k) = 0

where b(k) = δF (k) and a(k) = F (k)− γ + δ. Hopf bifurcations occur whenb = 1 and −2 < a < 2. Bifurcations to patterned states occur when λ = 1,or b = a− 1. For a lateral-inhibition kernel, F (k) has a maximum at a valuek = ±k∗. Thus, if we choose δ such that δF (k∗) is slightly larger than 1 andγ large enough so that a(k) < 2 then patterns with spatial frequency closeto k∗ will grow in an oscillatory pattern leading to a so-called Turing-Hopfbifurcation. If the effective kernel is not of lateral inhibition type, the peak of

10

F (k) occurs at k = 0 and homogeneous oscillations will bifurcate. Stationarystripes bifurcate when λ = 1 or

F (k) = 1 +γ

1− δand |δF (k)| < 1. With no refractory period (γ = 0) is is readily achieved byvarying the magnitude of ηE,I through sharpening the nonlinearities. In sum,strong refractory feedback will give rise to Hopf bifurcations and if the inter-actions are effectively Mexican hat, this will lead to spatially and temporallyperiodic behavior. For weak refractory feedback and lateral inhibition, thenspatial patterns bifurcate.

Let us return to the full model equations (8) and assume, again, a productkernel, WE,I(x, j) = wE,I(x)vE,I(j). Let vE,I(j) = (1− cE,I)cjE,I and let M =∞. (Note, the multiplication by (1 − c) is to normalize so that the sum ofthe temporal weights is 1.) Define

An(x) := (1− cE)∞∑j=0

cjEPn−j(x)

Bn(x) := (1− cI)∞∑j=0

cjIPn−i(x).

Then equation (8) can be wriiten as the system of two equations

An+1(x) = (1− cE)G(An(x), Bn(x)) + cEAn(x)

Bn+1(x) = (1− cI)G(An(x), Bn(x)) + cIBn(x)

G(A(x), B(x)) := [SE(wE(x) ∗ A(x))− SI(wI(x) ∗B(x)]+ .

This equation is quite interesting as both the excitation, An and the inhi-bition, Bn receive the same input, G(An, Bn) but pass it through differenttemporal filters, cE,I , respectively. They have exactly the same steady states,but because of the temporal differences, if they can evolve to non-synchronousbehaviors. The analysis of the stability of these equations can be done forgeneral cE,I , but at the expense of a good deal of intuitively unappealingalgebra. Instead, suppose that cE,I = 1− εqE,I where ε is small and positiveand qE,I are positive numbers. This says that the decay is slow and that thememory is long. For small ε, the discrete model becomes the much simplercontinuous time system

∂A

∂t= dE[−A+G(A,B)]

11

∂B

∂t= dI [−B +G(A,B)].

The steady states satisfy A = B = C with

C = [SE(C)− SI(C)]+ .

The linearization is

∂A

∂t= dE[−A+ ηEwE(x) ∗ A(x)− ηIwI(x) ∗B(x)]

∂B

∂t= dI [−B + ηEwE(x) ∗ A(x)− ηIwI(x) ∗B(x)]

where ηE,I = S ′E,I(C). Solutions to this linear equation are of the form(A(x, t), B(x, t)) = (A0, B0) exp(ikx+λt). If λ has a positive real part for anyk, then the constant state is unstable. Let fE,I(k) be the Fourier transformof ηEwE,I(x). Conditions for stability are

0 < 1− [fE(k)− fI(k)] (22)

0 < (1 + fI(k))− dEdI

(fE(k)− 1). (23)

This is a very intuitively appealing equation. When condition (22) is vio-lated, there is a Turing bifurcation to pattern formation. Notice that it isindependent of the temporal filtering parameters, dE,I as it is a bifurcation toa new stationary state and thus independent of time. The term fE(k)−fI(k)rears its head and is our effective Mexican hat. If this has a large positivemaximum value at k ± k∗, then condition (22) will be violated and therewill be a real positive eigenvalue. when condition (23) is violated, there willbe a pair of complex eigenvalues with positive real parts leading to a Hopfbifurcation. The key component appearing in this is the ratio of dE/dI . Ifthe inhibition recovers slowly (lateral inhibition in time), then dE/dI can bequite large. This contributes to the destabilization as long as fE(k) > 1.With lateral inhibition in space, fI(k) will decay faster (as k increases) thanwill fE(k) so that it is possible to get a Turing-Hopf bifurcation for thismodel just as in the previous one.

B.2 Bifurcation Conditions

In the final part of this appendix, we examine the mask model (12) in thesimplest scenario of a difference of Gaussian’s mask. To do this, we will go

12

to the continuous time since this allows one to at least get some handle onthe stability analysis. The Gaussian has a distinct advantage as it is actuallya separable kernel since:

N exp(−(x2 + d2j2)/σ2) = N exp(−x2/σ2) exp(−d2j2/σ2).

The Gaussian is unique in this regard, but we do not expect other typesof kernels to differ substantially in terms of their behavior. Let w(x, t) =2 exp(−x2 − t2)/π and wE,I(x, t) = w(x/σE,I , t/σE,I)/σ

2. Note these kernelsintegrate to 1 over −∞ < x <∞ and 0 < t <∞. By rescaling time, we keepthe kernels isotropic in space and time. Let us define

w(x, t) ∗ ∗P (x, t) :=∫ ∞−∞

∫ ∞0

w(x− y, s)P (y, t− s) dy ds

to be the convolution over space time. The continuous time analogue ofequation (8) with the mask kernel is

P (x, t) = [SE(wE(x, t) ∗ ∗P (x, t))− SI(wI(x, t) ∗ ∗P (x, t))] .

The steady state satisfies, P (x, t) = C where

C = SE(C)− SI(C).

We assume this is a positive number. The linearized equations are

P (x, t) = ηEwE(x, t) ∗ ∗P (x, t))− ηIwI(x, t) ∗ ∗P (x, t).

Solutions to this equation have the form P (x, t) = exp(ikx + λt) where k isarbitrary and λ satisfies the following characteristic equation:

1 = ηEf(kσE)g(λσE)− ηIf(kσI)g(λσI) (24)

where f(k) = exp(−k2/4) and g(k) = exp(k2/4)(1 + erf(k/2)). Here

erf(x) =2√π

∫ x

0e−t

2

dt.

The roots of this should have negative real parts for stability. However,finding these roots is very difficult, so instead, we will look for bifurcationswhich will occur when λ = 0 or when λ = ±iω. The case of λ = 0 correspondsto spatially periodic patterns and leads to the simple equation:

1 = ηEf(kσE)− ηIf(kσI).

13

The right-hand side is the familiar Mexican hat, so that we can expect patternformation as, e.g. the sensitivities (ηE,I) increase. Conditions for the Hopfbifurcation are more complicated. If we write λ = iω, then two equationsmust be solved corresponding to the real and imaginary parts of (24):

1 = ηE exp(−σ2E/4[ω2 + k2])− ηI exp(−σ2

I/4[ω2 + k2])

0 = ηE exp(−σ2E/4[ω2 + k2])E(ωσ)− ηI exp(−σ2

I/4[ω2 + k2])E(ωσI)

where E(x) = −Ierf(Ix/2). The parameters, ηE,I are related to the slope ofthe sigmoid nonlinearity. If they are both scaled by some common factor,then the second equation can be solved for, say, ω and this factor will haveno effect. The value of ω can then be substituted into the right-hand sideof the first equation and the this common factor can be scaled to satisfy theequation. This provides a set of Hopf bifurcation curves.

C Shell Shape and Morphogenesis

The location and shape of the initial shell secretions are determined duringembryonic morphogenesis. After the initial shell secretion, the mollusk con-structs the rest of the shell enclosure, regularly expanding the enclosure toaccommodate its growth. In gastropods (snails) the shell grows outward andspirals downwards from the original region of shell secretion by successivelyadding small increments of additional shell material to the leading edge.

The shell geometry can be sufficiently described by propagating a generat-ing curve, C(t), around a helico-spiral, as first observed by d’Arcy Thompsonand first modeled by Raup in the 1960s [2, 15, 16, 17, 18]. In our models,the generating curves are defined by cubic spline interpolation of a givennumber of user defined points, selected to mimic the geometry of the shell.For simplicity, the geometric scaling of the curve is dependent on the scalingof the helico-spiral. This simplest geometric model is defined by one initialcurve and three parameters which define the coiling rate down kz and out krof the helicospiral (and the number of turns). In cylindrical coordinates onehas,

z(t, ω) = z0(t)e−kzω (25)

r(t, ω) = r0(t)e−krω (26)

θ(t, ω) = ω + θ0(t)) (27)

14

in cartesian coordinates of the spiral frame this becomes,

x(t, ω) = r0(t)e−krω cos(ω + θ0(t)) (28)

y(t, ω) = r0(t)e−krω sin(ω + θ0(t)) (29)

z(t, ω) = z0(t)e−kzω (30)

Figure 1: The Geometry of a Shell: The surface of a shell can be describedby wrapping a generating curve with dimensions of the shell opening around ahelico-spiral, as illustrated. The generating curve need not lie in the normal tohelico-spiral, and is specified on an independent set of axis u,v,w.

A more faithful reconstruction of shell-like geometry can be achieved byallowing the generating curve to lie in a coordinate plane that is indepen-dent of spiral axis coordinates [7]. The generating curve can be orientedindependently of the helico-spiral, rotated by some angles defined below.

Let the generating curve lies in the v,w plane of the u,v,w coordinatesystem. This origin of this frame is set at some position (xc, yc, zc) in thex,y,z spiral coordinate frame. The w axis is rotated away from the thez-axis by an angle ψ. The v axis is rotated away from the y-axis by anangle φ. If both these angles are zero, the Frenet frame is equivalent tothe frame of the generating spiral. Each point on the generating curve hasa corresponding point the spiral frame, denoted in cylindrical coordinates,r0, θ0, z0. Each point on the generating curve C(t), parameterized by the

15

variable t, is propigated along a spiral H(ω), parameterized by ω, as inequations (25) and (28). This geometry is illustrated in figure 1.

The helix is still specified by three parameters, the rate of increase inheight, kz, the rate of change of the radius kr, and the number of turnsthe spiral makes. The location of the generating curve relative to the spiralaxis, (xc, yc, zc), and the orientation of the Frenet frame relative to the Spiralframe, determined by the angles ψ and φ, only refines the basic shape. Theedge of the pattern of the generating curve, C(t) determines the patternalong the surface. For example, a wavy edge creates a ribbed shell.

Figure 2: A The neural model explains how the aperture-growth vectors arisefrom the neural architecture in the mantle. The bottom plots show neuralexcitation at 2 different times (4 days apart). The effective growth vectorsresulting from this pattern of excitation is shown at the top. B aperture-growth vectors generated by the model to create spiral-shaped gastropods.

The underlying phenomena that generate this spiral growth pattern areunspecified, as are the differences between species that account for the dif-ferent spiralsor no spiral at all, as in the bivalves. Rice (1998) provided someof the first insights into these questions [19]. This work was extended togrowth vector models, which demonstrated that many different coiled shell

16

forms could be reproduced by varying secretion rates appropriately aroundthe aperture [11, 21]. This worked removed any reference to spiral or helicalaxes, shifting the question to the origin of the growth vectors correspondingto the secretions. However no mechanism has been proposed to define whatdetermines these growth vectors or how they propagate and change over thelifetime of the animal.

We propose that neural activity controls the amount and direction inwhich shell material is secreted. The neural activity along the mantle de-termines the local secretion rate, and thus the angle and magnitude of thegrowth vectors, as illustrated in Figure A. Using precisely the same neurose-cretory model as described in the main text to produce patterns, we canproduce shell-shapes. The changing pattern of secretory stimulation leads todifferential growth along the leading edge of the shell, which may cause it toflare out, spiral down, or form ribs. We find in particular the analysis of pat-terns of propagating waves to provide insight into the relationship betweenthe parameters of the neural model, the stimulation strength, and the localgrowth. The dynamics of shell growth are best appreciated as movies com-puted from the neural secretion model, a frame of which is shown in FigureB. Example movies which integrate both structural growth and patterningare given in the online Supporting Materials.

Starting from the same initial generating curve, our neural model canbuild a shell through successive projection of growth vectors using only nineparameters. (Not counting the parametrization of the generating curve thisbasic geometric model as described by Fowler and colleagues requires eightparameters). However since these parameters define only local behaviors andthe resultant structure is emergent, it is much harder to guess the appropri-ate choice of parameters than to select them directly from the geometricmodel to match the observed geometry. For this reason, the majority ofshell patterns presented in the text are constructed from a direct geometricparameterization and the patterns from neural simulation are subsequentlywrapped around this geometry.

17

D Quantitative Analysis of Patterns

D.1 Mathematical Comparison of Simulated and Nat-ural Patterns

A variety of different models are capable of capturing the qualitative effectsseen on many shells. A more rigorous validation of how well a particularmodel manages to reproduce an observed shell pattern is through comparisonof quantitative pattern statistics. In this section, we present some spatial-temporal analysis of pattern properties for some of the shells that we havestudied, and confirm that our models indeed capture these traits. We alsoillustrate limit cycles exhibited by several of the periodic patterns.

One interesting feature of patterns is the overall periodicity. Some funda-mentally periodic patterns do not appear periodic to the eye, because they arecomposed of two many different frequencies to distinguish. Other patternsthat appear to have some alternating structure are not in fact characterizedby a single frequency related to a regular period. A simple Fourier analysis ofthe shell patterns allows us to investigate these subtle differences in a robustmanner [12].

In figure 3, we import several shell patterns and extract spatial or tem-poral statistics in the pattern (i.e. measurements from lines either perpen-dicular or parallel to the growing edge). We Fourier transform the resultingintensities profiles, and examine the power-spectra to identify the dominatefrequency modes. Oscillating bands, of the type discussed above, show twoclear peaks in the power-spectra. The peak near zero represents the gradedtransition from pigmented to unpigmented and back, as predicted in ourmodel. The solitary later peak occurs at the oscillation frequency for thisshell.

D.2 Phase Plane Analysis of Oscillations

A clearer understanding of the oscillatory behavior exhibited in many shellpatterns can be developed from an analysis of the phase-portraits of theunderlying dynamical system. Here we examine traces of P vs R and Pt+1

vs Pt. The stable points of the system for the chosen parameter values areplotted as small blue circles. Starting from random initial conditions, thesystem spirals around the stable point until it settles gradually into a stablelimit cycle, illustrated in green. The phase portraits are given for a particular,

18

Figure 3: Samples of Spatio-temporal Pattern Analysis Top and mid-dle, periodic structure of patterns arising from oscillatory networks. The intensityprofile is shown to the upper left of each image, the power spectra is shown onthe lower left. Bottom, Spatial and temporal spectral analysis of a more compli-cated shell pattern. The graphed spectra come form the indicated lines on theshell. These measured statistics are consistent with those exhibited by the modelspresented in the text.

randomly selected node, however aside from the transient behavior, the sameportrait will exist for all nodes in the system.

19

Figure 4: Phase Analysis of Oscillations: The stable points of the systemfor the chosen parameter values are plotted as small blue circles. Starting fromrandom initial conditions, the system spirals around the stable point until it settlesgradually into a stable limit cycle, illustrated in green. The phase portraits aregiven for a particular, randomly selected node, however aside from the transientbehavior, the same portrait will exist for all nodes in the system.

E Extensions to Cuttlefish Patterns

Understanding the neural origins of shell patterns gives us insight into avariety of other neuronally controlled forms of pattern formation. A strikingexample comes from the study of another peculiar marine organism, thecuttlefish.

Cuttlefish are cephalopods, and they are hence also members of the phylaMollusca. They build no shells, nor are they particularly fast swimmers orgenerally poisonous. Instead they avoid predators through excellent adap-tive camouflage, which they can adjust to match their environment. Othermore elaborate and sometimes dynamic patterns are exhibited during hunt-ing, aggression, and courtship. The pigmentation and texture pattern oftheir skin are controlled by a large network of nerves. These nerves synapseonto small muscles, whose contractions reveal or conceal different pigmentedcells (chromatophores). A large variety of overlapping pigments provides thecuttlefish with an almost complete color palette. Both the rapid speed ofpattern change and the anatomical evidence of its origin make it clear thesepatterns reflect spatial distribution of neural activity [14, 20, 10, 1].

Our simple model of neural interactions is able to explain many of thetypes of patterns also commonly exhibited on cuttlefish. Careful observation

20

of pattern changes reveals that many – and perhaps all – patterns growlaterally from an initiation zone, either from the center towards the perimeteror vice versa. This suggests a natural way to implement the mechanismdescribed for shell-patterning on the skin of the cuttlefish. Now each layerof the pattern will be represented by a different line of neurons. Insteadof the input to the neural network being previously deposited pigment, theinput comes directly from the activity state of the adjacent line of neurons.The activity state of these neurons is translated into a pattern through thecontraction of the muscles surrounding the appropriate pigment cells.

A variety of different cuttlefish patterns are shown in Figure 5. Camou-flage patterns are shown in their natural background. The striped patternsexpressed by the Giant Cuttlefish and the Sepia officinalis in Figure 5A,Bare common mating patterns. 5C-E exhibit stable camouflage patterns thatmatch the characteristic spatial frequency of the coral or gravel background.The Giant Cuttlefish in 5F puts on a display of traveling dark V’s that prop-agate laterally from the periphery. The simple neural model we have usedfor the mollusk patterns can capture the salient features of each of thesepatterns. Of course, further study will be required to obtain a deeper under-standing of the origins of these neural patterns. Meanwhile, the cuttlefishprovides another stunning example of the importance of neural systems inunderstanding many of natures most intricate patterns.

F Extensions to Bayesian Models and Cogni-

tion

In this section we discuss the relation between neural control of shell pat-terning and neural mechanisms of prediction. A variety of mathematicalformulations of prediction exist. We first review the common Bayesian pre-diction formulation, and then show how such a formulation can be derivedin terms of our neural model for shells. This derivation allows one to seea possible mechanistic way to realize the abstract Bayesian model and toanswer such question like how that model is influenced by cellular details ofthe system (like basal firing rates or excitation thresholds).

Let us denote all the observations we would like to predict by the vectory(t). The dimensionality of y is equal to the dimensionality of the observa-tions we need to predict, all of which change in time, which may be continuous

21

Figure 5: Cuttlefish Patterns

or discrete.We denote by the vector x(t) all our observations of the past and present

with which we’ll use to predict y. The Bayesian estimate for y, we denotey. In general one allows for an arbitrary cost function, c(y, y) to denote thecost of predicting y when y is correct. The best estimate is the choice of ythat minimizes the expected cost,

22

y = argminyE[c(y, y)|x] = argminy

∫c(y, y)p(y|x)dy (31)

Intuitively, this equation tells us to consider all possible guesses for thefuture. Some possibilities predictions are very expensive if we get the wrong,and therefore we only chose those if they are very likely. A common choicewhich illustrates some of the mechanics of this prediction is the traditionalmean square error cost function, equation (31) becomes,

y = argminy

∫(y − y)2p(y|x)dy (32)

differentiating the integral on the left with respect to y and setting the resultequal to zero ,

∂

∂y

(∫(y − y)2p(y|x)dy

)= −2

∫(y − y)p(y|x)dy

Setting this equal to zero and rearranging gives the minimum solution

y =∫

yp(y|x)dy

which is just the mean of the conditional prior probability distribution ofy given x. The predictive model In p(y|x) is a model of how the past historyand additional observations determine the future. This probability modelcan be learned through repeated sampling of the data (Empirical Bayes orExpectation Maximization (EM) algorithm) [4, 8]. But it remains phenomi-logical, telling us nothing of the physical mechanisms that extrapolate y fromx. Given a mechanistic prediction framework like the one we have proposedfor ‘predicting’ the future shell pattern, we can derive just such a prior condi-tional probability distribution in terms of the cellular parameters. From sucha derivation we can now imagine other mechanistic, neural ways in which torealize the probability models employed to describe the cortical predictionprocesses such as those of Knill, Pouget or Friston [13, 6, 8].

Given a physical model, such as we have proposed, can we gain someinsight into what factors determine the corresponding Bayesian conditionalpriors (or priors and cost functions)? The general shell model given by equa-tions (8)-(10) provides a prediction of how the future pigment Pn+1 is deter-mined from the past bout(s) of secreted pigment Pn−M : Pn. Yet in realitythis is not a deterministic function, but a stochastic equation, due to the

23

stochastic nature of the underlying biology. In the simplest formulation,we can consider additive noise, where we add a small random variable η tothe right hand side of equation (8) drawn from some underlying probabilitydistribution pη(η).

Pn+1 = g[Pn−M , Pn−M+1, . . . Pn] + η

In this case,

p(Pn+1|Pn−M , Pn−M+1, . . . Pn) = pη(Pn+1 − g[Pn−M , Pn−M+1, . . . Pn])

In this trivial case the conditional probability distribution depends only onthe structure of the noise(shifted by the model function. However, a morerealistic treatment of the intrinsic noise in this system is to recognize theparameters of our model are not fixed values but samples from a distributionwith some unique mean. The firing rate associated with a given stimuli, thethreshold for when a nerve starts firing, and other cellular parameters willfluctuate stochastically around this mean value with probability distribu-tions determined by the underlying cellular physiology. Then the conditionalprobability distribution depends not only on the distribution of η but alsothe functional form with which the randomness enters the model, g(x, η).Simplifying notation, we let y = Pn+1, and x = Pn−M , Pn−M+1, . . . Pn

y = g(x, η) (33)

p(y|x) =∫p(y, η|x)dη

using Bayes rule we can rewrite this as

p(y|x) =∫p(y|η,x)pη(η|x)dη (34)

Now we observe since η is independent of x we can drop the secondconditional probability. We also observe that y is completely determinedonce η and x are known, so we can replace this probability function with adelta function, centered at the solution to η from equation (33).

p(y|x) =∫δ(η − h(y,x))pη(η)dη (35)

24

where h(y,x) is the solution to equation (33) for η. In the case of addi-tive noise, this reduces precisely to the solution we had before. Otherwisethe centering of the broadening distribution from the noise depends on thespecifics of where the noise enters. The shape of the distribution however stilldepends exclusively on the distributions of the noise as far as the conditionalprobability distribution for y is concerned.

G Simulation Parameters

Here we present the model parameters used to produce each of the shellimages presented in the paper.

25

26

27

28

References

[1] Cloney RA and Brocco SL. (1983) Chromatophore Organs, ReflectorCells, Iridocytes and Leucophores in Cephalopods. Amer. Zool. 23(3):581-592.

[2] Thompson d’Arcy (1952) On Growth and Form (University Press, Cam-bridge).

[3] Dayan, P and Abbott LF (2001) Theoretical Neuroscience; Computa-tional and Mathematical Modeling of Neural Systems. MIT Press, Cam-bridge MA.

[4] Dempster A. Laird N. and Rubin D (1977) Maximum likelihood fromincomplete data via the EM algorithm J. Royal Stat. Soc. Series B, 39(1):1-38.

[5] Ermentrout B, Campbell J, and Oster G (1986) A Model for Shell Pat-terns Based on Neural Activity. The Veliger 28:369-388

[6] Deneve S, Latham PE, and Pouget A. Efficient computation and cueintegration with noisy population codes. Nature Neuroscience 4: 826-831.

29

[7] Fowler DR, Meinhardt H, and Prusinkiewicz P. (1992) Modeling seashells.Computer Graphics 26:379-387.

[8] Friston, K (2005) A theory of cortical responses. Phil. Trans. R. Soc. B360: 815-836.

[9] Friston, K Trujillo-Barreto, N and Dauniezeau, J. (2008) DEM: A varia-tional treatment of dynamical systems. NeuroImage 41: 849-885.

[10] Gaston MR and Tublitz NJ. (2004) Peripheral innervation patterns andcentral distribution of chormatphore motorneurons in the cuttlefish Sepiaofficinalis. J. Exp. Bio 207(17): 3089-3098.

[11] Hammer, O. and Bucher, H. (2005) Models for the morphogensis of themolluscan shell. Lethaia 38(2):111-112.

[12] Hayes, MH. (1996) Statistical Digital Signal Processing and Modeling.John Wiley & Sons Inc., Hoboken, NJ.

[13] Knill DC and Pouget A (2004) The Bayesian brain: the role of uncer-tainty in neural coding and computation. TRENDS in Neuroscience 27:712-719.

[14] Messenger, J. (2001) Cephalopod chromatophores: neurobiology andnatural history. Biol. Rev. 76: 473-528.

[15] Raup D (1961) The geometry of coiling in gastropods. PNAS 47:602-629.

[16] Raup D (1962) Computer as aid in describing form in gastropod shells.Science 138:150-152.

[17] Raup D (1965) Theoretical morphology of the coiled shell. Science147:1294-1295.

[18] Raup D (1966) Geometric analysis of shell coiling: general problems. J.of Paleontology 40:1178-1190.

[19] Rice S. (1998) The bio-geometry of mollusc shells. Paleobiology 24: 133-149.

30

[20] Tublitz, NJ, Gaston MR and Loi PK. (2006) Neural regulation of a com-plex behavior: body patterning in cephalopod molluscs. Integr. Comp.Biol 46(6): 880-889.

[21] Ubukata T. (2005) Theoretical morphology of bivalve shell sculptures.Paleobiology 31(4):643-655.

[22] Wilson, HR and Cowan, JD (1972) Excitatory and inhibitory interac-tions in localized populations of model neurons. Biophysical Journal 12:1-24.

[23] Wilson HR, and Cowan, JD (1973) A mathematical theory of the func-tional dynamics of cortical and thalamic nervous tissue. Kybernetik 13:55-88.

[24] Wolfram, S (1984) Cellular automata as models of complexity. Nautre311: 419-24.

H Supplemental Figures

31

Figure 6: Neural similarities of different Cone shells: Parameter changesneeded to make transitions between five rather different cone shells patterns. Onlytransitions that can be made by moderate changes of three or fewer parametersare shown. Note, some patterns live in moderately large parameter spaces, sotransitions between patterns are not necessarily transitive.

32

Figure 7: Additional Shell Images. A As seen in the inset, pigment bound-aries which appear sharp (due to LALI processing by the neural field in our oureyes) they are in fact graded, as postulated by the model. B Patterns of dotson Natica stercusmucarum (left) and simulated model (right). C Rib patterns ona clam shell (right) and simulation model (left). D Simulation of Nautilus shellpatterns (left) recapitulate the natural patterns (right).

33