| Reading |
| Everyone: Flake, ch. 16 | |
| CS594: Bar-Yam, Sections 7.1, 7.2.1-7.2.2 (pp. 621-48) |
| Universal Properties |
| What leads to these expanding rings and spirals in very different systems? | |
| Under what conditions do these structures form? | |
| What causes the rotation? | |
| These are all examples of excitable media |
| Excitable Media |
| Examples of Excitable Media |
| Slime mold amoebas | |
| Cardiac tissue (& other muscle tissue) | |
| Cortical tissue | |
| Certain chemical systems (e.g., BZ reaction) | |
| Hodgepodge machine |
| Characteristics of Excitable Media |
| Local spread of excitation | ||
| for signal propagation | ||
| Refractory period | ||
| for unidirectional propagation | ||
| Decay of signal | ||
| avoid saturation of medium | ||
| Behavior of Excitable Media |
| Stimulation |
| Relay (Spreading Excitation) |
| Continued Spreading |
| Recovery |
| Restimulation |
| Typical Equations for Excitable Medium (ignoring diffusion) |
| Excitation variable: | |
| Recovery variable: |
| Nullclines |
| Rest State |
| Stability |
| Super-threshold Excitation |
| Phase 1: Increasing Excitation |
| Phase 2: Start of Extinction |
| Phase 3: End of Extinction |
| Phase 4: Recovery |
| Elevated Thresholds During Recovery |
| Modified Martiel & Goldbeter Model for Dicty Signalling |
| Variables (functions of x, y, t): | |
| b = intracellular concentration of cAMP | |
| g = extracellular concentration of cAMP | |
| = fraction of receptors in active state |
| Equations |
| Positive Feedback Loop |
| Extracellular cAMP increases | ||
| (g increases) | ||
| Þ Rate of synthesis of intracellular cAMP increases | ||
| (F increases) | ||
| Þ Intracellular cAMP increases | ||
| (b increases) | ||
| Þ Rate of secretion of cAMP increases | ||
| (Þ Extracellular cAMP increases) | ||
| Negative Feedback Loop |
| Extracellular cAMP increases | ||
| (g increases) | ||
| Þ cAMP receptors desensitize | ||
| (f1 increases, f2 decreases, r decreases) | ||
| Þ Rate of synthesis of intracellular cAMP decreases | ||
| (F decreases) | ||
| Þ Intracellular cAMP decreases | ||
| (b decreases) | ||
| Þ Rate of secretion of cAMP decreases | ||
| Þ Extracellular cAMP decreases | ||
| (g decreases) | ||
| Dynamics of Model |
| Unperturbed Þ cAMP concentration reaches steady state |
|
| Small perturbation in extracellular
cAMP Þ returns to steady state |
|
| Perturbation > threshold Þ large transient in cAMP, then return to steady state |
|
| Or oscillation (depending on model parameters) |
| Circular & Spiral Waves Observed in: |
| Slime mold aggregation | |
| Chemical systems (e.g., BZ reaction) | |
| Neural tissue | |
| Retina of the eye | |
| Heart muscle | |
| Intracellular calcium flows | |
| Mitochondrial activity in oocytes |
| Cause of Concentric Circular Waves |
| 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 |
| Spiral Waves |
| Persistence & propagation of spiral waves explained analytically (Tyson & al., 1989) | |
| Rotate around a small core of of non-excitable cells | |
| Propagate at higher frequency than circular | |
| Therefore they dominate circular in collisions | |
| But how do the spirals form initially? |
| Some Explanations of Spiral Formation |
| Òthe origin of spiral waves remains obscureÓ (1997) | |
| Traveling wave meets obstacle and is broken | |
| Desynchronization of cells in their developmental path | |
| Random pulse behind advancing wave front |
| Step 0: Passing Wave Front |
| Step 1: Random Excitation |
| Step 2: Beginning of Spiral |
| Step 3 |
| Step 4 |
| Step 5 |
| Step 6: Rejoining & Reinitiation |
| Step 7: Beginning of New Spiral |
| Step 8 |
| Formation of Double Spiral |
| StarLogo Simulation Of Spiral Formation |
| 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 | ||
| Response of Patch |
| 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 |
| Demonstration of StarLogo Simulation of Spiral Formation |
| Run SlimeSpiral.slogo |