Convergence and stability in networks with spiking neurons

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

2. 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

3. The neural code Firing rate Recruitment Neuronal assembly Synchronous firing Neuronal assemblies are flexible

4. Why flexible synchronization ? Stimulus driven; bottom-up process From Fries et al. Nat Rev Neurosci.

5. Synchronization of firing related to attention Evidence for Top-Down processes on coherent firing Riehle et al. Science, 1999

6. Coherence between sensori-motor cortex MEG and muscle EMG Before and after a visual warning signal for the “go” signal to start a movement Schoffelen, Oosterveld & Fries, Science in press

7. Functional role of synchronization Schoffelen, Oosterveld & Fries, Science in press

8. Questions regarding initiation/disappearence of temporal coding • Bottom-up and/or top-down mechanisms for initiation of neuronal synchronization ? • Stability of oscillations of neuronal activity • functional role of synchronized neuronal oscillations

9. 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

10. Leaky-Integrate and Fire neuron For constant input I Small input Large input

11. Conductance-based Leaky-Integrate and Fire neuron Membrane conductance is a function of total input, and so is the time-constant. With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector.

12. Synaptic processes

13. Conductance-based Leaky-Integrate and Fire neuron Membrane conductance is a function of total input, and so is the time-constant. With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector.

14. Conductance-based Leaky-Integrate and Fire neuron With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector. τ = 2 ms τ = 40 ms

15. Hodgkin-Huxley neuron

16. 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)

17. 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:

18. Actionpotential

19. 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

20. Phase diagram for the Morris-Lecar model

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

22. Phase diagram

23. Phase diagram of the Morris-Lecar model

24. 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

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

26. 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

27. 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

28. 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)

29. 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

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

31. 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

32. Excitatory/inhibitory interactions excitation-excitation inhibition-inhibition excitation-inhibition Behavior depends on synaptic strength ε and size of delay Δt

33. Excitatory interactions excitation-excitation Mirollo and Strogatz (1990) proved in a rigorous way that excitatory coupling without delays always leads to in-phase synchronization.

34. Stability for two excitatory neurons with delayed coupling Return map if tk is time when oscillator A fires For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995

35. Summary for excitatory coupling between two neurons • In-phase behavior for excitatory coupling without time delays • tight coupling with a phase-delay for time delays with excitatory coupling.

36. Inhibitory interactions excitation-excitation

37. Inhibitory couplingfor two identical leaky-integrate-and-fire neurons Out-of-phase stable In-phase stable Lewis&Rinzel, J. Comp. Neurosci, 2003

38. Phase-shift function for 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

39. I=1.2 I=1.4 I=1.6 Phase-shift functionfor inhibitory coupling for stable attractor Increasing constant input to the LIF-neurons

40. Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

41. Time constant for inhibitory synaps Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

42. Summary for inhibitory coupling Stable pattern corresponds to • out-of-phase synchrony when the time constant of the inhibitory post synaptic potential is short relative to spike interval • in-phase when the time constant of the inhibitory post synaptic potential is long relative to spike interval

43. Inhibitory coupling with time delays

44. Stability for two inhibitory neurons with delayed coupling Return map For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995

45. Stability for two excitatory neurons with delayed coupling Return map if tk is time when oscillator A fires For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2 Ernst et al. PRL 74, 1995

46. Summary about two-neuron coupling with delays • Excitation leads to out-of-phase behavior • Inhibition leads to in-phase behavior

47. 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

48. A network of oscillators with excitatory coupling

49. Winfree model of coupled oscillators N-oscillators with natural frequency ωi P(Θj) is effect of j-th oscillator on oscillator I (e.g. P(Θj) =1+cos(Θj) R(Θi) is sensitivity function corresponding to contribution of oscillator to mean field. Ariaratnam & Strogatz, PRL 86, 2001

50. Averaged frequency ρi as a function of ωi locking partial locking partial death slowest oscillators stop incoherence Frequency range of oscillators Ariaratnam & Strogatz, PRL 86, 2001