Shells

The neural origins of sea shell patterns
George Oster
Bard Ermentrout
Professor, University of Pittsburgh
Alistair Boettiger
Graduate student, UCB Biophysics
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Introduction
Among the most striking and diverse patterns in nature are
those seen on gastropod shells.
Underlying them are common dynamical concepts:
Local Activation with Lateral Inhibition (LALI) &
Slow negative feedback (“LALI in time”)
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Shapes
ANGULATE WENTELTRAP PINK MOUTHED MUREX PAPER NAUTILUS
VENUS COMB ROSE BRANCHED MUREX TOOTH SHELL
http://xahlee.org/xamsi_calku/xamsi_calku.html
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Colors & Patterns
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What purpose is the pigment
Most species found in sediment or covered with
the opaque periostracum
Camouflage possible in only a few species
Many species (bivalves) with colorful shells have
no eyes
Comfort (1950) suggests pigment is a waste
product! (But why so complex)
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Developmental stability
“…pigmentation and growth are functionally related. Both co-occur
temporally and spatially, as the pigmentation is deposited
when and where the shell grows. The same (richly innervated)
organ, the mantle, would be responsible for growing the shell,
depositing the pigment, and sensing the actual shell shape and the
current pigmentation.”
Bauchau (Belg. J. Zool., 131:23, 2001)
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Model for neurosecretion of shell and pigment
Time
-L
0
X
L
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Mathematical model
Sensory
Integration
Memory
E( x,t)
Excitation
L
∫
− L
M
∫
0
( )
= dx' dτ W E
x − x',τ
Spatial-Temporal
weighting
P( x',t − τ )
Pigment or Structure
elements
E( x,t)
Inhibition
L
∫
− L
M
∫
0
( )
= dx' dτ W I
x − x',τ
( )
P x',t − τ
P( x,t)
New pigment
secretion
( )
( ) − S I
I ( x,t)
= S E
E(x,t
Neurally stimulated secretion
SE
SI
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Reduction to simpler models
Discretize
time
Z t+1
= S E
R t +1
= γ Z t
+ δ R t
( W E
∗ Z ) t
− S ( I
W I
∗ Z ) t
− R t +1
where: Z t
≡ P( x) t
− R ( t
x)
‘Refractory substance’
Cellular
Automata (CA)
P t +1
= H ( W E
∗ P t
− θ ) E
− H ( W I
∗ P t
− θ ) I
S E,I
( u) = H ( u)
SE
SI
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Other models
Waddington & Crowe (1969) Cellular automata model
Wolfram (in ANKS, 2002) CA models
Meinhardt (2003)
The Algorithmic Beauty of Seashells
(many different reaction-diffusion models)
Kusch & Markus (1996), complex CA model
Ermentrout, Campbell, Oster (1986), mechanistic model
based on neural activity 10
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Experimental foundations
Richly innervated sensory region (circumpallial nerve).
Secretion is neurally driven.
Increased mantle neural activity associated with shell repair
Ablation of mantle neurons impairs shell growth
Mantle contains many neurotransmitters (ACh, dopamine,
peptides, Westerman et al, 2002)
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Summary so far…
It is clear that there is a ʻmemoryʼ: the new state
is determined by previous states.
The simplest way to read this memory is via the
shell itself, either pigment or shape.
Pigment and shape serve to mark the shell
margin, without which growth of the shell could
not be regulated (Bauchau).
Has the advantage of being clocked to the act of
shell deposition rather than any internal or
external synchronizer that is easily interruptible
(tides, weather, etc.)
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Shell anatomy
Inner Outer
Epitheli
Middle
Polyaxial
Periostracal groove
C ircum pallial
axon
Pigm ent sensing cells
shell
Extrapallial space
Secretory gland cells
D orsal epithelium
Pallial nerves
Ventral epithelium
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Westerman et al. (2005)J. Morphology 264
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Shells can be described geometrically
r(t,ω) = r 0
(t)e − k r ω '
θ(t,ω) = ω + θ 0
(t)
z(t,ω) = z 0
(t)e − k z ω 14
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The model can ‘grow’ shells dynamically
Existing structural
features
N eurosensory
stim ulation
R esulting neurosecretory
activity
C orres ponding aperture growth
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The model can ‘grow’ shells dynamically
Existing structural
features
N eurosensory
stim ulation
R esulting neurosecretory
activity
C orres ponding aperture growth
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All shell patterns are generated from 2 basic patterns
1. Bifurcation patterns
• Oscillations
• Stripes
2. Traveling waves
• Triangle patterns arise from wave collisions
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Bifurcations from the homogeneous state
Start with homogeneous steady state: A = I = P = SE(P) − SI:
M =
⎛
⎝
⎜
( ) ˆ
a + (1− a )S '
P W
E E E E
(k)
(1− a E
)S ' I
(P) Wˆ
E
(k)
−(1− a E
)S I ' (P) ˆ W I
(k)
a I
+ (1− a I
)S I
'
( P) ˆ
⎞
W I
(k) ⎠
⎟
Fourier transform
of kernel
If a E,I
= 1− c E,I
,
Turing-Hopf
0 < 1+ S I ' (P) ˆ W I
(k) − S E
'
(P) ˆ W E
(k)
Turing
0 > c E
c I
(
'
S E
(P) Wˆ
E
(k) − 1) − 1+ S ' I
(P) ˆ
( W I
(k)) Hopf
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Turing instablity → Stripes
Pigment secretion
Bankivia fasciata
Model simulations
Turitella sp.
Structual secretion
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Hopf instability → Oscillations
Pacemakers/hot spots
Model
Amoria ellioti
Strigella sp
Conus zebroides
Narita communis
The stripes are actually propagating waves, as we will see…
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Excitable → Waves
C. vicweei
C. clerii
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Waves can annihilate or reflect
Colliding waves annihilate then reignite, appearing to cross
Waves with broad inhibitory tails can reignite
Tapes literus
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Triangle patterns arise from wave collisions
Model
Model
Model
conus_bullatus
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Complex patterns
Clithon oualaniensis
Lioconcha castrensis
Model
Model
Model
Conus gloriamaris
Model
Conus vicweei
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The role of shell geometry
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The effect of shell defects on patterns
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Bifurcations separate different patterns
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Quantifying patterns
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Cowries require a 2-D model
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Connection to cortical waves
Neuron
Compression and Reflection of
Visually Evoked Cortical Waves
Weifeng Xu, Xiaoying Huang, Kentaroh Takagaki, and Jian-young Wu
Department of Physiology and Biophysics, Georgetown University Medical Center, Washington, DC
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What’s next: the amazing cuttlefish!
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What’s next: the amazing cuttlefish!
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