1. Biointelligence Laboratory, Seoul National University
Ch 9. Rhythms and Synchrony 9.1 ~ 9.2 Adaptive Cooperative Systems, Martin Beckerman, 1997.
Summarized by
B.-K. Min
Biointelligence Laboratory, Seoul National University

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
9.1 Biological Rhythms and Synchrony
9.1.1 Nonlinear Dynamics
9.1.2 Excitable Membranes
9.1.3 Population Oscillations
9.1.4 Neural Rhythms and Synchrony
9.2 Outline of the Chapter
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9.1.1 Nonlinear Dynamics
The nonlinear threshold for synaptic modification permits the system to dynamically adapt to the mean activity level.
Steady oscillatory states of a dynamic system are represented in phase space by limit cycles.
The limit cycle oscillators are self-sustaining oscillators, describing closed trajectories in phase space that are stable against small perturbations.
Nonlinear systems exhibit different response characteristics as their physical parameters are varied.
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9.1.2 Excitable Membranes
Van der Pol’s model of heartbeat: a pair of coupled relaxation oscillators: the Hodgkin-Huxley equation (FitzHugh and Nagumo; Morris and Lecar)
Excitable membrane: a dynamic cooperative system capable of propagating pulses and waves of activity over long distances: e.g. action potentials in nerve fibers (passive membrane properties: Hodgkin & Rushton)  cable theory and compartmental models of neurons. (
Dendritic spines are the main locus of excitatory synaptic input (important in synaptic plasticity).
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5. 9.1.3 Population Oscillations
Synchronization in nonlinear dynamic systems when coupled together.
Winfree (through the mean-field phase): the oscillatory units in the resulting phase-coupled system synchronize their motions through influences of the collective rhythms of all other units.
The synchrony depends on the width of the frequency distribution (cf. temperature) relative to the strength of the coupling; a phase transition from an incoherent to a coherent phase (cf. spin of an Ising systems) at some critical coupling strength (frequency width).
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6. 9.1.3 Population Oscillations (cont’)
Local clustering in place of global clustering and Traveling waves in chains of oscillators .
The properties of the coupling (nearest-neighbor) interaction (e.g. symmetry) have a role in determining the nature of the steady states generated by the system (Cohen et al.)  the notion of universality.
Oscillatory patterns: pacemakers (intrinsic electrophysiological properties), cells supporting rhythmicity, cells with networks properties.
Limit cycle phenomena can arise in populations of excitatory and inhibitory neurons reciprocally coupled to one another (Wilson & Cowan).
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7. 9.1.4 Neural Rhythms and Synchrony
Spectral analyses of EEG can often yield significant insight into the functional cognitive correlations of the signals.
Frequency
Name
Representative function
0.1-4 Hz
Delta (δ)
Deep sleep
4-8 Hz
Theta (θ)
Working memory (Hippocampus)
8-13 Hz
Alpha (α)
Relaxed wakefulness
13-30 Hz
Beta (β)
Movement, memory rehearsal
Above 30 Hz
Gamma (γ)
Attention, Perceiving meaningful objects
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8. 9.1.4 Neural Rhythms and Synchrony (cont’)
Simultaneous multielectrode recording devices: 40 Hz synchronized repetitive firing patterns (Gray et al. and Eckhorn et al.)  induced but not stimulus-locked.
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9. 9.1.4 Neural Rhythms and Synchrony (cont’)
•Phase-locking index: the homogeneity of the instantaneous phase across single trials (ranging from 0 to 1)
• Phase coherence:
• Temporally correlated activity:
Separate cortical columns,
across hemispheres etc.
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10. 9.1.4 Neural Rhythms and Synchrony (cont’)
The cells with high-frequency synchronous activity are capable of rapidly organizing themselves on time scales (~ a few tens of ms; rat dorsal cochlear nucleus by Gochin et al.).
The dynamic couplings may extend across multiple cortical regions at different frequencies and operate across several time scales (Ahissar et al., Bressler et al.)
Traveling waves of synchronous active clusters of cells in hippocampal tissue slices (Miles et al.)
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11. 9.1.4 Neural Rhythms and Synchrony (cont’)
Multiple steady states, bifurcations, clustering, and propagating waves of synchrony: nonlinear dynamic properties
Rhythmic discharges to propagate across the retina (Meister et al.): the refinement of the topographic maps, selective gain control.
Temporally correlated signaling: the reciprocal pathway from layer IV of the visual cortex to the LGN (Sillito et al.)
Correlated firings of an assembly of neurons convergent on a common postsynaptic target are effective mechanisms for activating NMDA receptors and selectively enhancing the synaptic efficiency of common target cells.
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12. 9.1.4 Neural Rhythms and Synchrony (cont’)
Abeles’s notion of Synfire chains
Malsburg’s assembly coding hypothesis: signaled by temporally correlated firing (e.g. Treisman’s feature integration), cooperative processes mediated by the network connectivity.
Receptive fields (RF) and topographic maps are dynamic entities able to encode sensory information spatially and temporally (e.g. using facial whiskers to make repetitive contacts).
The stimulus response of single neurons in the ventral posterior medial thalamus (VPM) of the rat: ~35ms from caudal-most to rostral-most whiskers: alpha synchronization begins as traveling waves in the cortex and spreads to the thalamus and then to the brainstem before the onset of rhythmic whisker twitching.
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9.2 Outline of the Chapter
The entrainment condition for a pair of coupled rotators: a phase-pulling mechanism synchronizes their motions.
A random pinning model, Nearest-neighbor couplings, Macroscopic clustering
Rhythmic behavior and Synchrony in populations of model neurons
Phase plane methods: Linear stability and nullcline analysis, the Poincaré-Bendixson theorem, the Hopf bifurcation theorem.
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14. 9.2 Outline of the Chapter (cont’)
The coarse-grained equations of motion for models of the Wilson-Cowan form (Two populations of cells, one excitatory and the other inhibitory, reciprocally coupled to one another).
Temporally correlated firing patterns arise naturally in networks characterized by recurrent excitation and global feedback inhibition, given the appropriate input and physiological parameters.
The Hodgkin-Huxley equations: how the ionic currents self-organize in a membrane to produce a variety of stable firing states.
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15. 9.2 Outline of the Chapter (cont’)
The Morris-Lecar model of membrane excitability and oscillations
The Somers and Kopell model of relaxation oscillators coupled to one another through nearest-neighbor interactions  Global synchronization
Spindle waves in the thalamocortical system
Intracelluar calcium oscillations
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