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Convergence and stability in networks with spiking neurons

Convergence and stability in networks with spiking neurons. Stan Gielen Dept. of Biophysics Magteld Zeitler Daniele Marinazzo. Hodgkin-Huxley neuron. Membrane voltage equation. 0 mV. 0 mV. I Na. K. I C. V mV. V mV.

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Convergence and stability in networks with spiking neurons

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  1. Convergence and stability in networks with spiking neurons Stan Gielen Dept. of Biophysics Magteld Zeitler Daniele Marinazzo

  2. Hodgkin-Huxley neuron

  3. Membrane voltage equation 0 mV 0 mV INa K IC V mV V mV -Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)

  4. m m V (mV) Gating kinetics m Open m m Closed State: m m (1-m) Probability: m.m.m.h=m3h Channel Open Probability:

  5. Actionpotential

  6. Fast variables membrane potential V activation rate for Na+ m Slow variables activation rate for K+n inactivation rate for Na+h Simplification of Hodgkin-Huxley -C dV/dt = gNam3h(V-Ena)+gKn4(V-EK)+gL(V-EL) + I dm/dt = αm(1-m)-βmm dh/dt = αh(1-h)-βhh dn/dt = αn(1-n)-βnn Morris-Lecar model

  7. Phase diagram for the Morris-Lecar model

  8. Phase diagram for the Morris-Lecar model Linearisation around singular point :

  9. Phase diagram

  10. Phase diagram of the Morris-Lecar model

  11. Overview • What’s the fun about synchronization ? • Neuron models • Phase resetting by external input • Synchronization of two neural oscillators • What happens when multiple oscillators are coupled ? • Feedback between clusters of neurons • Stable propagation of synchronized spiking in neural networks • Current problems

  12. Neuronal synchronization due to external input T Δ(θ)= ΔT/T ΔT Synaptic input

  13. Neuronal synchronization T Δ(θ)= ΔT/T ΔT Phase shift as a function of the relative phase of the external input. Phase advance Depolarizing stimulus Hyperpolarizing stimulus

  14. Neuronal synchronization T Δ(θ)= ΔT/T ΔT • Suppose: • T = 95 ms • external trigger: every 76 ms • Synchronization when ΔT/T=(95-76)/95=0.2 • external trigger at time 0.7x95 ms = 66.5 ms

  15. T=95 ms Example For strong excitatory coupling, 1:1 synchronization is not unusual. For weaker coupling we may find other rhythms, like 1:2, 2:3, etc. P=76 ms = T(95 ms) - Δ(θ)

  16. Neuronal synchronization T Δ(θ)= ΔT/T ΔT • Suppose: • T = 95 ms • external trigger: every 76 ms • Synchronization when ΔT/T=(95-76)/95=0.2 • external trigger at time 0.7x95 ms = 66.5 ms Unstable Stable

  17. P T • Convergence to a fixed-point Θ* requires • Substitution of and expansion near gives • Convergence requires • and constraint gives

  18. Overview • What’s the fun about synchronization ? • Neuron models • Phase resetting by external input • Synchronization of two neural oscillators • What happens when multiple oscillators are coupled ? • Feedback between clusters of neurons • Stable propagation of synchronized spiking in neural networks • Current problems

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