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CH12: Neural Synchrony

CH12: Neural Synchrony. James Sulzer 11.5.11. Background. Stability and steady states of neural firing, phase plane analysis (CH6) Firing Dynamics (CH7) Limit Cycles of Oscillators (CH8) Hodgkin-Huxley model of Oscillator (CH9) Neural Bursting (CH10)

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CH12: Neural Synchrony

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  1. CH12: Neural Synchrony James Sulzer 11.5.11

  2. Background • Stability and steady states of neural firing, phase plane analysis (CH6) • Firing Dynamics (CH7) • Limit Cycles of Oscillators (CH8) • Hodgkin-Huxley model of Oscillator (CH9) • Neural Bursting (CH10) • Goal: How do coupled neurons synchronize with little input? Can this be the basis for a CPG?

  3. Coupled Nonlinear Oscillators • Coupled nonlinear oscillators are a nightmare • Cohen et al. (1982) – If the coupling is weak, only the phase is affected, not the amplitude or waveform 1 2

  4. 12.1 Stability of Nonlinear Coupled Oscillators Two neurons oscillating at frequency  Now they‘re coupled by some function H Introducing , the phase difference (2-1) Phase-locked How do we know if this nonlinear oscillator is stable? Iff And... (Hint: Starts with Jacob, ends with ian) Synchronized

  5. Stability of Nonlinear Coupled Oscillators Now we substitute a sinusoidal function for H,  is the conduction delay, and varies between 0 and /2 Solving for , Must be between -1 and 1 for phase locking to occur Final phase-locked frequency:

  6. Demo 1: Effects of coupling strength and frequency difference on phase locking • What happens when frequencies differ? • What happens when frequencies are equal and connection strengths change? • What about inhibitory connections? Connection strength has to be high enough to maintain stability Phase locked frequency increases with connection strength Spikes are 180 deg out of phase

  7. 12.2 Coupled neurons Modified Hogkin-Huxley: Synaptic strength and conductance reduce potential Presynaptic neuron dictates conductance threshold Synaptic time constant plays a large role

  8. Demo 2 • Do qualitative predictions match with computational? • What does it say about inhibition? • Qualitative and quantitative models agree

  9. 12.3 The Clione Inhibitory network

  10. Action potentials underlying movement What phenomenon is this?

  11. Post-inhibitory Rebound (PIR) dV/dt = 0 dR/dt = 0 R V High dV (e.g. synaptic strength) and low dt of stimulus facilitate limit cycle

  12. Modeling PIR

  13. Demo 3: Generating a CPG with inihibitory coupling • How is PIR used to generate a CPG? • What are its limitations? PIR from Inhibitory stimulus on inhibitory neurons can generate limit cycle Time constant strongly influences dynamics of limit cycle

  14. 12.4: Inhibitory Synchrony Why does the time constant have such an effect on synchrony of inhibitory neurons? Predetermined model for conductance Convolving P with sinusoidal H function Solution for H H(-)-H() for neuronal coupling (equivalent ) Stable states at = 0, Calculate Jacobian, differentiating wrt  at = 0, Time constant must be sufficiently large for synchronization

  15. Demo 4: Effect of Time Constant • How does time constant differentially affect excitatory and inhibitory oscillators?

  16. 12.5: Thalamic Synchronization

  17. Demo 5: Synchrony of a network How can a mixed excitatory and inibitory circuit express synchrony?

  18. Summary 12.1 • Phase oscillator model shows that connection strength must be sufficiently high and frequency difference must be sufficiently low for phase-locking • Conduction delays make instability more likely 12.2 • Mathematical models of conductance and synchronization agree with qualitative models (to an extent) 12.3 • PIR shows how reciprocal inhibition facilitates CPG 12.4 • Time constant must be sufficiently high for inhibitory synchronous oscillations 12.5 • Mixed Excitatory-Inhibitory networks can be daisy-chained for a traveling wavefront of oscillations

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