Examples of Excitable Media. Excitable Media. Characteristics of Excitable Media. Behavior of Excitable Media. Part 2: Cellular Automata 9/7/04
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1 Examples of Excitable Media Excitable Media Slime mold amoebas Cardiac tissue (& other muscle tissue) Cortical tissue Certain chemical systems (e.g., BZ reaction) Hodgepodge machine 9/7/04 1 9/7/04 2 Characteristics of Excitable Media Behavior of Excitable Media Local spread of excitation for signal propagation Refractory period for unidirectional propagation Decay of signal avoid saturation of medium 9/7/04 3 9/7/04 4 1
2 Stimulation Relay (Spreading Excitation) 9/7/04 5 9/7/04 6 Continued Spreading Recovery 9/7/04 7 9/7/04 8 2
3 Restimulation Typical Equations for Excitable Medium (ignoring diffusion) Excitation variable: u = f (u,v) Recovery variable: v = g(u,v) 9/7/04 9 9/7/04 10 Nullclines Rest State 9/7/ /7/
4 Stability Super-threshold Excitation 9/7/ /7/04 14 Phase 1: Increasing Excitation Phase 2: Start of Extinction 9/7/ /7/
5 Phase 3: End of Extinction Phase 4: Recovery 9/7/ /7/04 18 Elevated Thresholds During Recovery 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 9/7/ /7/
6 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) 9/7/ /7/04 See Equations 22 Negative Feedback Loop Extracellular camp increases ( increases) camp receptors desensitize (f 1 increases, f 2 decreases, decreases) Rate of synthesis of intracellular camp decreases ( decreases) Intracellular camp decreases ( decreases) Rate of secretion of camp decreases Extracellular camp decreases ( decreases) Dynamics of Model 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) 9/7/04 See Equations 23 9/7/
7 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 9/7/04 25 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 9/7/04 26 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? 9/7/04 27 Some Explanations of Spiral Formation 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 9/7/
8 Step 0: Passing Wave Front Step 1: Random Excitation 9/7/ /7/04 30 Step 2: Beginning of Spiral Step 3 9/7/ /7/
9 Step 4 Step 5 9/7/ /7/04 34 Step 6: Rejoining & Reinitiation Step 7: Beginning of New Spiral 9/7/ /7/
10 Step 8 Formation of Double Spiral 9/7/ /7/04 from Pálsson & Cox (1996) 38 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 9/7/ /7/
11 Observations Demonstration of StarLogo Simulation of Spiral Formation Run SlimeSpiral.slogo 9/7/04 41 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): in space in time in activity Amplification of random fluctuations Circles & spirals are to be expected 9/7/
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