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CH12: Neural Synchrony

James Sulzer11.5.11

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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?

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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

2 22f t

1 12f t

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12.1 Stability of Nonlinear Coupled Oscillators

1 21 2,d d

dt dt

1 2 2 1( ) ( )d H Hdt

Two neurons oscillating at frequency

Now they‘re coupled by some function H1

1 2 1( )d Hdt 2

2 1 2( )d Hdt

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)1 2 2 1( ) ( ) 0H H Phase-locked

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2

Synchronized

2 1( ) ( ) 0d d d H Hd dt d

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11 1 sin( )d a

dt

1 2 1 2sin( ) sin( )d a adt

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 ,

2 1

2 2 2 2

arcsin

2 1 cos 2 1 sin

2 1 sinarctan

2 1 cos

A

A a a a a

a aa a

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

11 1 sin( )d a

dt

Final phase-locked frequency:

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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

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12.2 Coupled neurons

Synaptic time constant plays a large role

Synaptic strength and conductance reduce potential

Presynaptic neuron dictates conductance threshold

Modified Hogkin-Huxley:

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Demo 2

• Do qualitative predictions match with computational?

• What does it say about inhibition?

• Qualitative and quantitative models agree

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12.3 The Clione Inhibitory network

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Action potentials underlying movement

What phenomenon is this?

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Post-inhibitory Rebound (PIR)

V

R

dR/dt = 0

dV/dt = 0

High dV (e.g. synaptic strength) and low dt of stimulus facilitate limit cycle

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Modeling PIR

1 -R+ 1.35V + 1.035.6

dRdt

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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

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12.4: 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 synchronization

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Demo 4: Effect of Time Constant• How does time constant differentially affect

excitatory and inhibitory oscillators?

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12.5: Thalamic Synchronization

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Demo 5: Synchrony of a networkHow can a mixed excitatory and inibitory circuit express synchrony?

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Summary12.1• 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 likely12.2• Mathematical models of conductance and synchronization agree with qualitative

models (to an extent)12.3• PIR shows how reciprocal inhibition facilitates CPG12.4• Time constant must be sufficiently high for inhibitory synchronous oscillations12.5• Mixed Excitatory-Inhibitory networks can be daisy-chained for a traveling

wavefront of oscillations

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