# ch12: neural synchrony

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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) - PowerPoint PPT PresentationTRANSCRIPT

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CH12: Neural SynchronyJames Sulzer11.5.111BackgroundStability 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?

2Coupled Nonlinear OscillatorsCoupled nonlinear oscillators are a nightmareCohen et al. (1982) If the coupling is weak, only the phase is affected, not the amplitude or waveform12

312.1 Stability of Nonlinear Coupled Oscillators

Two neurons oscillating at frequency Now theyre coupled by some function H

Introducing , the phase difference (2-1)

How do we know if this nonlinear oscillator is stable?Iff

And... (Hint: Starts with Jacob, ends with ian)

Phase-locked

Synchronized

4

Stability of Nonlinear Coupled OscillatorsNow we substitute a sinusoidal function for H, is the conduction delay, and varies between 0 and /2Solving for ,

Must be between -1 and 1 for phase locking to occur

Final phase-locked frequency:5Demo 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 stabilityPhase locked frequency increases with connection strengthSpikes are 180 deg out of phase612.2 Coupled neurons

Synaptic time constant plays a large role Synaptic strength and conductance reduce potentialPresynaptic neuron dictates conductance thresholdModified Hogkin-Huxley:

77Demo 2Do qualitative predictions match with computational?What does it say about inhibition?

Qualitative and quantitative models agree

812.3 The Clione Inhibitory network

9Action potentials underlying movement

What phenomenon is this?10Post-inhibitory Rebound (PIR)VRdR/dt = 0dV/dt = 0High dV (e.g. synaptic strength) and low dt of stimulus facilitate limit cycle11Modeling PIR

12Demo 3: Generating a CPG with inihibitory couplingHow is PIR used to generate a CPG?

What are its limitations?PIR from Inhibitory stimulus on inhibitory neurons can generate limit cycleTime constant strongly influences dynamics of limit cycle1312.4: Inhibitory SynchronyWhy 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 synchronization14Demo 4: Effect of Time Constant

How does time constant differentially affect excitatory and inhibitory oscillators?1512.5: Thalamic Synchronization

16Demo 5: Synchrony of a networkHow can a mixed excitatory and inibitory circuit express synchrony?17Summary12.1Phase oscillator model shows that connection strength must be sufficiently high and frequency difference must be sufficiently low for phase-lockingConduction delays make instability more likely12.2Mathematical models of conductance and synchronization agree with qualitative models (to an extent)12.3PIR shows how reciprocal inhibition facilitates CPG12.4Time constant must be sufficiently high for inhibitory synchronous oscillations12.5Mixed Excitatory-Inhibitory networks can be daisy-chained for a traveling wavefront of oscillations

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