Transcript Slide 1

CH12: Neural Synchrony
James Sulzer
11.5.11
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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
f1  21t 
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f2  22t 
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12.1 Stability of Nonlinear Coupled
Oscillators
d1
d 2
 1 ,
 2
dt
dt
d1
 1  H ( 2  1 )
dt
Two neurons oscillating at frequency 
Now they‘re coupled by some function H
d 2
 2  H (1   2 )
dt
d
 1  2  H 2 ( )  H1 ( )
dt
Introducing , the phase difference (2-1)
How do we know if this nonlinear oscillator is stable?
Iff
1  2  H2 ( )  H1 ( )  0
1
2
Phase-locked
And... (Hint: Starts with Jacob, ends with ian)
d d d
  H 2 ( )  H1 ( )  0
d dt d
Synchronized
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Stability of Nonlinear Coupled
Oscillators
d1
Now we substitute a sinusoidal function for H,
 1  a1 sin(   )
 is the conduction delay, and varies between 0 and /2
dt
d
 1  2  a1 sin(   )  a2 sin(   )
dt
Solving for ,
Must be between -1 and 1 for phase locking to occur
 2  1 

 A 
  arcsin 
 a 2  a1  cos     a 2  a1  sin  
  a 2  a1 sin  
  arctan 

a
2

a
1
cos




A
2
2
2
2
Final phase-locked frequency:
d1
 1  a1 sin(   )
dt
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Demo 1: Effects of coupling strength and
frequency difference on phase locking
- What happens when frequencies differ?
Connection strength has to be high enough to maintain stability
-What happens when frequencies are equal and
connection strengths change?
Phase locked frequency increases with connection strength
- What about inhibitory connections?
Spikes are 180 deg out of phase
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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
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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)
dV/dt = 0
dR/dt = 0
R
V
High dV (e.g. synaptic strength) and low dt
of stimulus facilitate limit cycle
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Modeling PIR
dR
1

 -R+ 1.35V + 1.03
dt 5.6
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Demo 3: Generating a CPG with
inihibitory coupling
• How is PIR used to generate a CPG?
PIR from Inhibitory stimulus on inhibitory neurons can generate limit cycle
• What are its limitations?
Time constant strongly influences dynamics of limit cycle
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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
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large for synchronization
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 network
How can a mixed excitatory and inibitory circuit express synchrony?
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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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