The neural origins of shell structure and pattern
in aquatic mollusks
Alistair Boettiger a, Bard Ermentrout b, and George Oster a,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 of
aquatic mollusks generates their diversity of shell structures and
pigmentation patterns. The anatomical and physiological basis of this
model sets it apart from other models used to explain shape and
pattern. The model reproduces most known shell shapes and patterns
and accurately predicts how the pattern alters in response to envi-
ronmental disruption and subsequent repair. Finally, we connect the
model to a larger class of neural models.
S eashells display a remarkable variety of ornate pigmentation
patterns. Accumulating evidence now indicates clearly that shell
growth and patterning are under neural control and that shell
growth and pigmentation is a neurosecretor y phenomenon. Most of
this evidence has been gained by detailed studies of the mantle, a
tongue-like protrusion of the mollusk that wraps around the edge
of the shell and deposits new shell material and pigment at the
shell’s growing edge (see Fig. 1AandB) (1, 2). The shell itself is
composed of cr ystal structures of calcium carbonate interspersed
with associated proteins and other organic compounds, some of
which are pigmented and arrayed in intricate patterns (3, 4). This
hard shell is covered by a thin organic layer of proteinaceous
secretions, believed to function in regulating calcium cr ystallization
(5). Early EM studies of the mantle recorded an extensive distri-
bution of ner ve fibers among the secretor y cells (1, 6, 7). These
fibers were later shown to have active synapses with secretor y gland
cells and synaptic inputs from other sensor y organs in the mantle.
From this evidence, it was proposed that neural-stimulated secre-
tion controls shell growth (4, 7). The original evidence from
gastropods has been extended to other mollusk taxa, including
bivalves (8) and cephalopods, where improved experimental meth-
ods confirm clearly the role of neural control (9, 10). Neural
recordings 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 and
develop a unified explanation for the generation of both shell
shapes and patterns. Unlike the purely geometric representations
proposed earlier to model shell shape (13–18), our model links
shape generation directly to the dynamics of the underlying neural
network. Recent experimental work describes how differential
growth patterns can lead to shell-like structures (19 –21) but does
not explain the biological origin or mechanism of these growth
patterns. The neural model presented here closes this gap by
explaining how the mantle neural net can encode the appropriate
information required for shell growth as well as pigment deposition.
Early attempts to reproduce shell patterns used cellular automata
models, in which arbitrar y rules determine the pigmentation of cells
on a grid (22–24). Although they can reproduce some obser ved
patterns, these models have shed little light on how such patterns
actually arise in the animal. Inspired by the chemistr y of diffusing
morphogens, Meinhardt and coworkers (25–29) used a variety of
different diffusion–reaction (DR) models to reproduce a wide
variety of shell pigmentation patterns. Although no experimental
evidence has been found for diffusing morphogens in patterning,the models can be viewed as an incomplete analogy for neural
activity (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
(LA LI). Ernst Mach (31) first described this phenomenon to
explain the visual illusions now called ‘‘Mach Bands’’ and empha-
sized LA LI’s property of enhancing boundaries. Nearly a centur y
later, Alan Turing (32) showed how LA LI could be modeled by
systems of nonlinear DR equations. This property was exploited by
later workers, most notably Murray (30) and Meinhardt et al.
(33–35) to model an extraordinar y range of biological patterns.
Indeed, DR models have become an all-purpose LA LI metaphor
in many domains wherein the underlying physics are clearly not
diffusing substances (30, 36). All of these models exhibit spatial
instabilities that lead to spatial patterns. The neural shell model
presented here combines spatial with temporal instability because
the mantle can sense previously laid patterns by, in a sense, looking
backwards in time. Indeed, LA LI in time is equivalent to a
refractor y period that leads to temporal oscillations. Meinhardt’s
DR models have succeeded in reproducing almost all of the
patterns quite accurately, and we can hardly do better here. Our
goal, however, is not to merely reproduce the patterns but to show
how a single neural network model, based directly on the mantle
anatomy, can capture all of the pattern complexity, as well as
constructing the shell shape, and to relate the model to a broader
class of experimentally obser ved neural network behavior.
Our exposition proceeds as follows. First we describe the neural
net model as applied to the construction of the shell shape. We then
proceed to define the most important classes of shell patterns and
show how the model reproduces them. We also describe how the
shell patterns respond to perturbations, such as injuries.
A Neurosecretory Model of Shell Growth and Pattern Formation.
Because we are interested only in the origin of shape and pattern
and not the structural composition of the shell, we shall ignore the
subsequent biomineralization that strengthens the shell distally
from the leading edge. Secretions in the periostracal groove are
controlled by the underlying neural network synapsing on the
secretor y cells. The activity of this neural network is stimulated by
the existing pattern of shell deposition and pigment at the mantle
edge. A schematic representation of this system is shown in Fig. 1B.
The shell is constructed by periodic— usually daily— bouts of
secretion (40 – 42). These periodic increments are robust against
many kinds of environmental variations (41). We model the secre-
tions in daily steps, wherein the pattern of each day’s layer of
t, is a function of the preexisting layers,P(t ). We
adopt the hypothesis of Bauchau (41) that the pigment pattern
Author contributions: G.O. designed research; A.B. performed research; B.E. contributed
new reagents/analytic tools; A.B., B.E., and G.O. analyzed data; and A.B. wrote the paper.
The authors declare no con?ict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed: E-mail: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/cgi/content/full/
www.pnas.orgcgidoi10.1073pnas.0810311106 PNAS April 21, 2009 vol. 106 no. 16 6837– 6842
allows the mantle to position itself in register with the existing
pattern. Fig. 1Cshows a schematic diagram of how the model senses
the pattern from previous days’ secretions and lays down the current
The basic property of the neural net control of secretion lies in
the phenomenon of LA LI, common to many—if not all—neural
networks. As we shall demonstrate, this general feature of neural
nets will be sufficient to generate all of the obser ved shell patterns
from a single mathematical model. A noteworthy aspect of this work
is that LA LI takes place both in space (along the mantle edge) and
backwards in time (perpendicular to the mantle edge). A precise
mathematical derivation of the model is given inSI Appendix,
section A. Here we present an intuitive description.
Sensor y organs in the mantle detect the presence of pigment and
stimulate pigment secretion in local secretor y cells. Consider a cell
located at positionxalong the growing shell edge. The strength of
its lateral stimulation of surrounding cells is represented by the
E(x) (red trace in Fig. 1C). The same sensor y input excites
a wider inhibitor y field in the mantle, which we describe by the
I(x) (blue trace in Fig. 1C). Because the mantle wraps
around the top edge of the shell, it senses, in addition to the exposed
edge, some previous histor y of pigmentation a distance
the leading edge. Pigmentation near the leading edge has an
excitator y effect on local secretion, whereas pigmentation sensed
further from the edge has an inhibitor y effect. We incorporate this
effect into the excitator y and inhibitor y kernels by making them
and integrating back in time (i.e., distance from the
shell edge). Only the qualitative features of the functionsW
I, not their precise shape, are important for the pattern. For
computational analysis, we represent the excitation and inhibition
functions with Gaussian cur ves of different widths and different
relative amplitudes (43).
Neurons have a nonlinear, saturating response to their net
stimulation (44, 45). There is a characteristic threshold for the
neuron, around which it is most sensitive to changes in the stimu-
lation rate. Once the stimulation exceeds this threshold, the rate of
change in activity in response to change in stimulation saturates. In
general, the characteristics of this sigmoid input– output response
cur ve will be different for inhibitor y and excitator y synapses.
Therefore we subject the excitation and inhibition to separate
saturating input– output functionsS
E(Fig. 1C, red) andS I(Fig. 1C,blue). This step filters the sensor y input that stimulates the secre-
tor y cells.
The secretion of pigment in the current layer,P
t, is determined
by the net stimulation of the secretor y cells. Because of lateral
inhibition in the
direction, the current secretion is affected both
directly and indirectly by the previous pattern. The pigmented
portion of these depositions provides ‘‘markers’’ and allows us to see
how this sensor y network propagates the pattern from layer to layer
The model can be cast in several nearly equivalent mathematical
forms; these forms are presented inSI Appendix, section A. The
specific analytical forms of the kernels representing the lateral
connections and the saturation functions representing the nonlinear
neural input– output responses do not affect the patterns generated
by the model.
Explaining Shell Structure. The location and shape of the initial shell
secretions are determined during embr yonic morphogenesis. After
the initial shell secretion, the mollusk constructs the rest of the shell
enclosure, regularly expanding the enclosure to accommodate its
growth. In gastropods (snails), the shell grows outward and spirals
downward from the original region of shell secretion by successively
adding small increments of additional shell material to the leading
edge. A variety of mathematical representations of the final geom-
etr y have been proposed, (seeSI Appendix, section C) (13, 16, 29,
Fig. 2.Explaining structure. (A) The neural model explains how the aperture-
growth vectors arise from the neural architecture in the mantle. The bottom plots
show neural excitation at 2 different times (4 days apart). The effective growth
vectors 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 axons
stained 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 edge
pattern induces excitatory ?ring (red); the older, previous pattern is inhibitory (blue), indicated in the upper trace. For simulation purposes we use a Gaussian spatial
E,IE,Iexp(x 2/E,I2). Excitatory stimulation leads to sharp stimulation of the local region and weak inhibition of an even wider surrounding region. Neural
stimuli are passed through saturating sigmoidal ?lters to determine the spatial pattern of activation or inhibition on the pigment secreting cells.We useS E,I1/
[1exp( E,I[E,IK])], whereKis the input from the spatial kernel. The pigment is thenP t1 S E(WEPt)S I(WEPt)R t, withR tPt1 Rt1 (see section A ofSI
Appendixfor derivation, discussion, and alternate forms).
6838 www.pnas.orgcgidoi10.1073pnas.0810311106Boettiger et al.
46). The underlying phenomena that generate this spiral growth
pattern are unspecified, as are the differences between species that
account for the different spirals— or no spiral at all, as in the
bivalves. Rice (19) provided some of the first insights into these
questions. This work was extended to growth vector models which
demonstrated that many different coiled shell forms could be
reproduced by var ying secretion rates appropriately around the
aperture (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 the
mantle determines the local secretion rate and, thus, the angle and
magnitude of the growth vectors, as illustrated in Fig. 2A. The
central point here is that the same model that generates the patterns
can generate the shell geometr y as well. In the discussion to follow,
we shall augment the pigment patterns generated by the model with
a few examples of shell growth generated by the same model. The
dynamics of shell growth are best appreciated as movies computed
from the neural secretion model, a frame of which is shown in Fig.
2B. Examples are given in Movie S1and Movie S2.
Understanding Pigmentation Patterns. The neural architecture of
local excitation and long-range inhibition gives rise to a broad array
of stable patterns. The type of pattern depends on the relative
ranges and strengths of the interactions and the steepness and
thresholds of the firing response cur ves. Remarkably, all of the
patterns initiate from 3 basic mathematical phenomena arising
from the LA LI property: spatial instabilities (Turing bifurcations),temporal instabilities (Hopf bifurcations), and traveling waves*.
Analytical demonstrations of these instabilities are presented inSI
Appendix, section B.
Bifurcations Create Periodic Patterns in Space and Time. LA LI readily
gives rise to patterns of parallel stripes orthogonal to the shell’s
leading edge. This development takes place because the spatially
uniform state is unstable to small, random perturbations or slight
heterogeneities in the neural network (i.e., a Turing instability) (30,
47). Brief ly, the process works in the following manner. A slightly
more active local group of cells exerts a stronger inhibitor y effect
on its neighbors. This lateral inhibition weakens the activity in the
neighbors and consequently weakens the inhibitor y effect they exert
on 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 cluster
allows the next cluster over to increase its activity because its
inhibition is lowered. The result is a standing wave of neural activity
that deposits stable pigmentation stripes normal to the aperture.
The width of the stripes ref lects the extent of the excitator y region
in the mantle and the width of the gaps ref lects the extent of the
inhibitor y connections. Some examples are shown in Fig. 3D.
If we increase the strength of the stimulation arising from the
previous day’s pattern (i.e., the amplitude of the activation kernel
*The waves probably originate from an in?nite dimensional saddle-node bifurcation, but
we have not proven this here.
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 random
noise (Bottom) into periodic stripes (Top) shows how Turing instabilities give rise to patterns of stable bands perpendicular to the growing shell edge. Note that
activation centers separate and shift as each one carves out a domain of in?uence. (B) Turing patterns.B. fasciataexhibits Turing bands of pigment, andTurritella
exhibits 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 in?uence secretion of structural elements instead of pigment, resulting in the periodic ?anges seen onE. scalare. The zigzag stripes shown
onN. communiswere generated by a combination of Hopf bifurcation with wave generation as described in Fig. 4. (E) Hopf bifurcations and Turing instabilities can
occur simultaneously, leading to patterns like that ofN. tigrina.
Boettiger et al.PNAS April 21, 2009 vol. 106 no. 16 6839
in the direction), the pattern can switch to alternating bands
parallel to the shell edge, as shown in Fig. 3E. To create these
periodic patterns, secretion-stimulating neurons must cycle be-
tween successive periods of stimulation and quiescence. A plot of
the current pigmentation versus the past pigmentation (i.e., a phase
plane portrait) will trace out a single loop around which the system
continuously cycles (see Fig. 3C). Such periodic orbits on the phase
plane (limit cycles) arise from parameter changes triggering so-
called Hopf bifurcations, which are well-known in models of
repetitively firing and bursting neurons (45). These periodic insta-
bilities can arise in several ways. For example, when there are
different thresholds for excitator y and inhibitor y signals, the neural
network may be excitable at low-pigment-induced stimulation but
inhibitor y under strong activity. A low basal level activity in the
absence of signaling triggers a slow, positive feedback, which
gradually amplifies the signal and eventually triggers the high-
threshold inhibitor y response, which shuts neuronal firing back
down to its basal level, from which the process begins again. This
process leads to stable cycles of oscillation, which produce periodic
bands of pigment parallel to the growing edge of the shell, whose
period can be many multiples of the secretor y period (e.g., 1 day).
Excitator y lateral connections tend to synchronize the population
into parallel bands like those seen on shells ofAmoria ellioti.
Stronger lateral inhibition induces the patterns to lag each other,
forming instead periodically distributed zigzag patterns, like those
ofNatica 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 the
checkerboard patterns ofNatica tigrinashown in Fig. 3B.
In Fig. 3DandE, we show how network-induced secretion of
shell material, instead of pigment, leads to the growth of the shell
geometr y inTurritellaandEpitonium scalare.
Asymmetric Activity Creates Traveling Waves. Traveling waves of
pigmentation arise when previous firing activity represses future
firing activity while exciting lateral activity. An asymmetric region
of activation (as in the wedge shape in Fig. 4A) induces stronger
stimulation toward its high side than toward its tail. This uneven
lateral excitation induces the pigment in the next layer to spread
laterally in front of the wedge. The repression from having fired
narrows down the tail, and the whole wedge shifts sideways toward
the high side. Successive depositions thus create a traveling line of
pigment at an oblique angle to the shell edge. When two such waves
of pigmentation collide, they may mutually annihilate (as inConus
clerii), singularly annihilate (asConus vicweei), or ref lect (Tapes
litarus) (Figs. 4 and 5E). Close examination of ref lecting waves show
that each wave is actually quenched but then reignites because the
activation width is broader than the inhibition region.
Depending on the size and shape of the kernels, when waves
approach each other they can either slow down or speed up. Thus,
regular ref lecting waves can create patterns of spots as onNatica
stercusmucarum(SI Appendix, section H) or teardrops like on
Conus marblus(Fig. 4A), depending on how the overlapping
excitation and inhibition kernels either accelerate or decelerate
approaching 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. clerii
in B. When these waves collide, they may re?ect, as shown in this simulation and on the shells ofC. cleriiandC. marblus. The waves may annihilate, as shown on the
shells ofConus viceweei. A wave may also emit ??reverse?? waves, creating the beautiful tent patterns ofOlivia porphyria.(B) Graphs of neural activity across the leading
edge 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 the
simulation ofC. innexaandC. marblusinBandC.(D) Some of the diverse patterns produced by the model by combinations of wave collisions and emissions.
6840 www.pnas.orgcgidoi10.1073pnas.0810311106Boettiger 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 unpigmented
region, leaving a triangular gap devoid of pigment. This gap is seen
on many shells (e.g.,Conus bullatusandConus thailandis, seen in
Fig. 4C). Fig. 4Balso shows the detailed steps that lead to the
fractal-like triangles ofCymbiola innexain Fig. 4C. Additional
patterns generated by the model are shown in Fig. 4Dand inSI
Appendix, section H.
Effects of Shape and Environment on Patterns. The pigmentation
patterns on many shells change qualitatively as a result of shell
growth or environmental disruption. Our simple neural model
provides a mechanistic explanation for many of these pattern
The length scale of the pattern is determined by the distribution
of axon and dendrite lengths. As the animal grows and more
neurons incorporate into the mantle, existing patterns may become
unstable. Parallel lines perpendicular to the shell edge may widen
or bifurcate as the overlap between lateral inhibition decreases.
Both effects are apparent in the limpetTectura testudinalisin Fig.
5A. Another common obser vation is that patterns of pigmentation
are homogeneous on small domains (relative to the average neu-
ronal connection range). This obser vation explains why many shells
start with either no pigment or uniform pigment and develop
intricate patterns only after the mantle edge grows to the appro-
priate length. An example is the shell ofBabylonia spiratain Fig. 5D.
Note the lack of pigmentation in the small twists that form the top
of the spiral. In contrast, regular patterns, such as oscillating bands,
are stable only when the domain size is small. As the domain
becomes large, synchrony across the mantle is lost and the pattern
degenerates into a uniform pigmentation as seen on the shells of
Amoria grayi(shown in Fig. 5B). Additionally, loss of pattern
synchrony may cause the pattern to degenerate from alternating
bands into a mesh of dots, as the domain becomes too large to
maintain global synchrony in the presence of small backgroundnoise. This effect is seen on a variety of shells, including theMitra
mitra sticticashells shown in Fig. 5C. Other pattern bifurcations are
also captured by the model. The bivalveT. litarusexhibits regular
traveling waves across its shell. Near the top of the shell these waves
collide and terminate in V-shaped patterns. However, as the
domain size increases, the waves become ref lecting, bifurcating in
shaped intersections when they collide (Fig. 5E).
In addition to predicting how patterns are created ab initio, the
model predicts how the system responds to perturbations in the
pattern. Ablation of a small portion of the ridge pattern on aStrigilla
shell allows for spontaneous activation of new waves (manifested by
the V’s formed) and acceleration of existing waves, as evidenced in
the lines becoming more closely parallel to the growing edge.
Simulations of pattern ablation capture both of these effects, which
appear in a field of traveling waves and at an annihilation point, as
shown in Fig. 5F.Bankivia fasciatasynchronize in steps from
random initiation provided by variable background rates. First a
mesh-like pattern of dots emerges; these dots subsequently syn-
chronize into uniform bands as explained above. If the pattern is
disrupted by an injur y, the pattern restabilizes from dots to stripes
again, a property readily illustrated by the model in Fig. 5G.
Insights into Mollusk Evolution. An attractive feature of the neuronal
model is its suggestive mechanism for the evolution of the obser ved
pattern diversity. Because ver y similar species can exhibit signifi-
cantly different patterns, the pattern difference cannot be the result
of dramatic anatomical differences. The neural model provides a
common mechanism whereby small changes in the parameters lead
to large changes in the patterns. The model also makes testable
predictions about which sets of patterns one is likely to find within
a genus and which sets of patterns are not likely to occur together
in a genus. Our model predicts a greater evolutionar y separation
would 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, as
shown inSI 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 ofT. testudinalis(A), collapse of oscillations inAmoria grayi(B), destabilization of waves into patchy dots onM. mitra
stictica(C), emergence of a pattern from a uniform ?eld inBabylonia spirita(D), and transition from annihilating to re?ecting waves onT. litarus(E). Patterns change
in response to scratches, which remove information about the previous pattern. (F) Traveling waves inStrigillashells slow down and de?ect away from the growing
edge. [Photo adapted and reproduced with permission from ref. 27 (Copyright 1987, Elsevier.)] (G)B. fasciatagoes through repeated stabilizations from dots to stripes.
Shells inB–Dare from ref. 52.
Boettiger et al.PNAS April 21, 2009 vol. 106 no. 16 6841
In this work, we have shown that a single neurosecretor y model can
replicate both the growth of mollusk shells and the enormous
diversity of pigment patterns they exhibit. The model is built around
the general property of local excitation coupled with lateral inhi-
bition common to most neural networks. A noteworthy feature in
this model is that the same network architecture operates in both
the spatial and time directions because the pigment patterns
develop sequentially as the mantle lays down periodic increments of
shell and pigment. Thus, the shell pattern records the complete time
histor y of its neurosecretor y activity. One might think of the pattern
as an electroencephalogram, or the histor y of the thoughts of a
mollusk! In general, waves propagating through a 3-dimensional
neural 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 essentially
backwards in time.
By exploiting the permanent record of neural activity that
mollusks have incorporated into their shells, we have achieved a
mechanistic understanding of how these diverse, and seemingly
ver y different, patterns arise. We have shown that all of the patterns
emerge from combinations of 3 types of bifurcations: Turing and
Hopf bifurcations, wave propagation, and collisions, which probably
originate in saddle-node bifurcations. The intuitions we have de-
veloped in the study of mollusk shells may provide a useful
foundation for studying other types of neural patterns, such as the
dynamic patterning of cuttlefish skin controlled by a neuromuscular
network that exposes and hides chromatophores. We have found
that the mollusk model can reproduce many patterns obser ved in
the cuttlefish mantle (seeSI Appendix, section E) that formsequentially as a wave. The cuttlefish patterns change much faster
than the mollusk patterns, but their origin is fundamentally the
same: Both are products of neural net activity. Insights from the
study of complex visible neural patterns like these may prove useful
in understanding normally invisible patterns of neural activity, such
as the structured spatial organization of neural activity distributed
over the mammalian cortex. Here we have seen that the mollusk
waves slow down and ref lect in a manner similar to that obser ved
in cortical waves (48). Such wave collisions might allow for com-
parison of cortical predictions with sensor y input: Annihilation
occurs when waves are identical and an error wave propagates out
when they are different.
Our mechanistic explanation of how a neural system determines
the future pattern from the previous pattern suggests an interesting
parallel with other cortical processes. It is not entirely unlike the
challenge the brain addresses of predicting the future from its
internal neurological model of the world. Most theories of this
process make analogies to Bayesian inference (49 –51). Our work
suggests how these models may be recast in terms of physiological
neural parameters that may underlie such Bayesian prediction
models (seeSI Appendix, section F). We suspect many new insights
await 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-
sion to reproduce the noted images in Figure 1Aand Figure 5F. A.B. and G.O. were
supported by National Science Foundation Grant 0414039. B.E. was supported by
the National Science Foundation Division of Mathematical Sciences. A.B. was
supported by National Institute of Biomedical Imaging and Bioengineering Train-
ing Grant in Physical Biosciences T32 EB005586-01A2.
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