CH12: Neural Synchrony James Sulzer

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? 2

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 12 3

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 ) 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 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: 5

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 6

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: 7

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

12.3 The Clione Inhibitory network 9

Action potentials underlying movement What phenomenon is this? 10

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 11

Modeling PIR 12

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 13

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 functionSolution 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 14

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

12.5: Thalamic Synchronization 16

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

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 18

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