1. Presented by Rhee, Je-Keun
Ch 9. Rhythms and Synchrony 9.9 ~ Adaptive Cooperative Systems, Martin Beckerman, 1997.
Presented by Rhee, Je-Keun

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
9.9 Spindle Waves
9.9.1 Ionic Mechanisms
9.9.2 Network Mechanisms
9.10 Calcium Oscillations, Excitable Media, and Cellular Automata
Calcium Oscillations and Spiral Waves
Excitable Media
Cellular Automata
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3. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Ionic Mechanisms
Three ionic currents cooperate to produce spindling oscillations.
The first of these is the mixed sodium/potassium current Ih, activated by hyperpolarization in the subthreshold range of membrane potentials. The second is the low-threshold calcium current IT, and the third is the voltage-dependent potassium current IK2.
A cellular model of spindle oscillations in thalamic neurons (Hodgkin-Huxley scheme)
(C) 2009, SNU Biointelligence Lab, 

4. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Ionic Mechanisms
A four variable model for the low-threshold, or T-type, calcium current IT has been developed by Wang, Rinzel, and Rogawski.
activation variable
the pair of inactivation
describes the slow recovery of IT from inactivation
maximum conductance of the calcium current
calcium reversal potential
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5. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
The voltage clamp data show that Ih is a noninactivating current.
The model adopted for the hyperpolarization activated current Ih
In the above system there are two activation gates, fast (F) and slow (S).
During activation F1 opens rapidly, while S1 opens more slowly. The dependence of the current on the product of S1 and F1 ensure that the activation kinetics is determined mainly by S1.
During inactivation the situation is reversed. Since F1 closes rapidly, the product form guarantees that the inactivation kinetics is determined primarily by F1.
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6. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Possible mechanisms for the waxing and waning oscillations were presented by Destexhe et al.
silent phase (SP)
oscillatory phase (OP)
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7. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Network Mechanisms
Consider network mechanisms that promote synchronization.
The full model describing the dynamics of a network of identical, mutually inhibiting neurons contains T-type calcium, leakage, and synaptic currents:
The elements of the connectivity matrix
postsynaptic conductance denoting the fraction of maximum arising from activity in neuron i
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8. Calcium Oscillations and Spiral Waves
The entry of ionized calcium through NMDA and AMPA receptors initiates a sequence of steps leading to synaptic modification.
Calcium is a common signal-transducing device in both non-excitable and excitable cells.
In nonexcitable and excitable cells, a series of steps involving IP3 or Ry cell surface receptors, respectively, leads to the release of calcium from the intracellular stores.
In many instances complex spatiotemporal patterns are generated.
These patterns include pulsate signaling, oscillations, plane waves, and spiral patterns.
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9. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Excitable Media
In simple excitable media, each element interacts with its nearest neighbors through diffusive coupling. The reaction-diffusion equations used to model excitable media of this type take the form
diffusion coefficients
nonlinear kinetics of the medium
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10. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Cellular Automata
The excitation variable u assume one of two values, 0 or 1.
The recovery variable v increases whenever u=1, and decrease when u=0.
Excitation occurs if v is sufficiently recovered, and deexcitation take place when v is adequately raised.
In its simplest form, this is a description of a system with two states, one in which both u and v are zero, one in which both variables are unity.
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Cellular Automata
The excitation variable can assume one of two values, 0 and 1, while the recovery variable may take integer values from 0 to vmax. The state u=0, v=0 is the resting state. If u=1, the unit is excited, and if u=0, u>0, the cell is in a recovering state.
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12. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Consider the interaction model.
The neighborhood system is defined as all cells lying within a square of linear dimension 2r+1 centered on the unit.
A resting or recovering cell undergoes a transition to an excited state if the number of excited units within its neighborhood exceeds a threshold value kex
An excited cell undergoes a transition to a recovering state if the number of resting or recovering units within its neighborhood exceeds a threshold value
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