Part 2: Cellular Automata
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Examples of Excitable Media
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Slime mold amoebas
Cardiac tissue (& other muscle tissue)
Cortical tissue
Certain chemical systems (e.g., BZ
reaction)
• Hodgepodge machine
Excitable Media
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Characteristics of
Excitable Media
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Behavior of Excitable Media
• Local spread of excitation
– for signal propagation
• Refractory period
– for unidirectional propagation
• Decay of signal
– avoid saturation of medium
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Stimulation
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Relay (Spreading Excitation)
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Continued Spreading
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Recovery
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Typical Equations for
Excitable Medium
(ignoring diffusion)
Restimulation
• Excitation variable:
u˙ = f (u,v)
• Recovery variable:
v˙ = g(u,v)
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Nullclines
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Rest State
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Stability
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Super-threshold Excitation
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Phase 1: Increasing Excitation
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Phase 2: Start of Extinction
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Phase 3: End of Extinction
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Phase 4: Recovery
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Elevated Thresholds During
Recovery
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Modified Martiel & Goldbeter
Model for Dicty Signalling
Variables (functions of x, y, t):
= intracellular concentration of cAMP
= extracellular concentration of cAMP
= fraction of receptors in active state
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Equations
Positive Feedback Loop
• Extracellular cAMP increases
( increases)
• Rate of synthesis of intracellular cAMP
increases
( increases)
• Intracellular cAMP increases
( increases)
• Rate of secretion of cAMP increases
• ( Extracellular cAMP increases)
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Negative Feedback Loop
See Equations
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Dynamics of Model
• Extracellular cAMP increases
( increases)
• Unperturbed
cAMP concentration reaches steady state
• Small perturbation in extracellular cAMP
returns to steady state
• Perturbation > threshold
large transient in cAMP,
then return to steady state
• Or oscillation (depending on model
parameters)
• cAMP receptors desensitize
(f1 increases, f2 decreases, decreases)
• Rate of synthesis of intracellular cAMP
decreases
( decreases)
• Intracellular cAMP decreases
( decreases)
• Rate of secretion of cAMP decreases
• Extracellular cAMP decreases
( decreases)
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See Equations
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Circular & Spiral Waves
Observed in:
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Cause of
Concentric Circular Waves
Slime mold aggregation
Chemical systems (e.g., BZ reaction)
Neural tissue
Retina of the eye
Heart muscle
Intracellular calcium flows
Mitochondrial activity in oocytes
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• Excitability is not enough
• But at certain developmental stages, cells
can operate as pacemakers
• When stimulated by cAMP, they begin
emitting regular pulses of cAMP
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Some Explanations
of Spiral Formation
Spiral Waves
• Persistence & propagation of spiral waves
explained analytically (Tyson & Murray,
1989)
• Rotate around a small core of of nonexcitable cells
• Propagate at higher frequency than circular
• Therefore they dominate circular in
collisions
• But how do the spirals form initially?
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• “the origin of spiral waves remains
obscure” (1997)
• Traveling wave meets obstacle and is
broken
• Desynchronization of cells in their
developmental path
• Random pulse behind advancing wave front
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Step 0: Passing Wave Front
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Step 1: Random Excitation
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Step 2: Beginning of Spiral
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Step 3
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Step 4
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Step 5
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Step 6: Rejoining & Reinitiation
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Step 7: Beginning of New Spiral
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Formation of Double Spiral
Step 8
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StarLogo Simulation
Of Spiral Formation
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Response of Patch
• Amoebas are immobile at timescale of wave
movement
• A fraction of patches are inert (grey)
• A fraction of patches has initial
concentration of cAMP
• At each time step:
– chemical diffuses
– each patch responds to local concentration
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from Pálsson & Cox (1996)
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if patch is not refractory (brown) then
if local chemical > threshold then
set refractory period
produce pulse of chemical (red)
else
decrement refractory period
degrade chemical in local area
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Observations
• Excitable media can support circular and spiral
waves
• Spiral formation can be triggered in a variety of
ways
• All seem to involve inhomogeneities (broken
symmetries):
Demonstration of StarLogo
Simulation of Spiral Formation
– in space
– in time
– in activity
Run SlimeSpiral.slogo
• Amplification of random fluctuations
• Circles & spirals are to be expected
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