Reduction and Analysis of 2-D Neuronal Dynamics

1. Overview • Reduction of the HH model to a 2-D model • Principles of phase plane analysis; geometric aspects of neuronal dynamics; emergence of oscillations • A mathematical description of bursting in excitable cells

2. HH model Threshold behaviour V(0)=5 mV V(0)=4 mV

4. The “slow” variables: h, n

5. Two qualitative observations 1) the time scale of the dynamics of the gating variable m is much faster than that of the variables n, h. 2) the time constants and are roughly the same, whatever the voltage V. Moreover, the asymptotic functions , are similar.

6. The idea use quasi steady-state approximation 1) 2) lump together the slow variables

7. The V-equation can be rewritten as or which must be coupled to the equation for the “recovery” variable w,

8. Two dimensional models of excitable media fast variable slow variable Fitzhugh – Nagumo :

9. 2D models • Phase plane, trajectories • Vector fields, nullclines • Fixed points, stability, linearization • Periodic orbits, limit cycles • Bifurcations

10. Phase-plane integration of dynamical equations the infinitesimal displacement is in the direction of the “flow”

11. w- nullcline V- nullcline fixed point both the components vanish

12. Stability of fixed points Introduce the displacement from the F.P. Linearize the system at the F.P. Find the eigenvalues of the matrix and eigenvector

13. An example of bifurcation: saddle-node bifurcation

14. Hopf bifurcation

15. Homoclinic bifurcation

16. The Morris-Lecar model

17. A “spike” in the ML (type II) model graded threshold

18. Phase-portrait of the ML (type II) model

19. ML (type II) bifurcation diagram amplitude of oscillations The rest state turns unstable here, via a (subcritical) Hopf bifurcation

20. ML (type II) response function: narrow frequency range onset of oscillation with non-zero frequency

21. Bistability in ML (type II) model R UPO = Unstable periodic orbit SPO = Stable periodic orbit R = Stable rest state

22. A bistable cell model can switchfrom rest to oscilation and then back to rest

23. ML (type I) U T R unstable manifold stable manifold R = stable rest state ; T = saddle point threshold ; U = unstable node

24. ML (type I): threshold behaviour -20 -22.1 The model shows a distinct threshold -22.2 -25

25. When the stimulus increases, the stable and the threshold points coalesce, and a zero-frequency limit cycle is formed “narrow channel” = slow dynamics

26. Low frequency repetitive firing

27. ML (type I) bifurcation diagram high-voltage stable state unstable oscillations unstable states stable oscilllations a stable rest state exists here

28. ML (type I): response function Oscillations arise with zero frequency

29. Bursting cells • Widespread among neurons and non-neuronal cells • Involved in generation of rythmic pattern (e.g. breathing, pacemaking, hormone secretion .. ) • The membrane is switching periodically between an active state (rapid oscillations) and a rest state (lack of oscillations) Active phase Silent phase

30. Bursting with bistability: the “hystheresis” loop Busting requires more than two dynamical variables

31. Ca2+ is the fundamental “second messenger”: release • of hormones and neurotransmitters, cell motility or • contractions, gene expression and development, …. • Ca2+ ions flow into the cell via V-dependent channels • Ca2+ is highly buffered inside the cytoplasm, • low intracellular concentration phenomenological model buffering influx

32. Ca2+ controls the activation of ionic channels adaptation

33. Calcium dynamics provide a slow, negative feedback on membrane excitability

34. A dynamical interpretation

35. Bursting without bistability • Without the hystheresis loop, a negative feedback would only produce “adaptation” • In order to push the system into the oscillatory regime, we need a positive feedback, e.g. a slowly activating Ca2+ current: slow time scale

36. Bursting without bistability