Neural Adaptation and Bursting or: A dynamical taxonomy of neurons

1. Neural Adaptation and Burstingor: A dynamical taxonomy of neurons April 27th, 2011 Lars Kasper

2. Introduction and Link to last sessions

3. Symbols & Numbers • V membrane potential • R recovery variable (related to K+) • H conductance variable (related to slow K+ current, IAHP) • C very slow K+ (IAHP) conductance mediated by intracellular Ca2+ concentration • X Ca2+ conductance, rapid depolarizing current • IA rapid transient K+ current • IAHP slow afterhyperpolarizing K+ current • IADP slow afterdepolarizing current (fast R and slow X comb.) • +55, +48 mV Na+ equilibrium potential • +140 mV Ca2+ equilibrium potential • -95, -92 mV K+equilibrium potential • -70, -75.4 mV Resting membrane potential Chapter 10 – Neural Adaptation and Bursting

4. Overview of Introduced Neuron Models Chapter 10 – Neural Adaptation and Bursting

5. Take home message: More fun with currents • Essentially deepest insight of today’s session: Spike frequency and AP creation are dependent on external, stimulating current. • Today some intrinsic currents will partially counteract the effect of the external driving current. • This will be done in a dynamic manner via the introduction of 1 or 2 additional currents modelling • Afterhyperpolarizing effects (very slow K+) • Additional depolarizing effects (fast Ca2+) • This dynamic net current fluctuation will lead to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries Chapter 10 – Neural Adaptation and Bursting

6. Today: Completing the single neuron taxonomy Chapter 10 – Neural Adaptation and Bursting

7. Topics • Introduction and scope • There’s much more to neurons than spiking • Spike frequency adaptation • Neural bursting and hysteresis • Class II Neurons • Endogenous bursting • Class I neurons • Separating limit cycles using a neurotoxin • Constant current-driven bursting • Neocortical neurons • Summary: The neuron model zoo

8. Spike Frequency Adaptation • What is spike rate adaptation? • Threefold reduction of spike rates within 100 ms of constant stimulation typical for cortical neurons • Which current is introduced? • Very slow hyperpolarizing K+ current • Mediated by Ca2+ influx • What function does it enable? • Short-term memory • Neural competition

9. Spike Rate Adaptation 25 Hz 70 Hz

10. Recap: Rinzel-model with transient K+ current Voltage V Recovery variable R Transient K+ via quadratic voltage-dependence of recovery Chapter 10 – Neural Adaptation and Bursting

11. After-hyperpolarization via slow K+ current K+ No resting state effect H: Conductance of slow K+ (after-)hyperpolarizing current IAHP Chapter 10 – Neural Adaptation and Bursting

12. Explanation via reduction of effective driving current Simulation: RegularSpiking.mwith I=0.85, 1.8 • H has no effect on action potential (slow time constant) • H is driven by supra-threshold voltages • Then counteracts driving current in dV/dt Chapter 10 – Neural Adaptation and Bursting

13. Capability of the model • Predicts current-independent threefold reduction in spike rate from transient to steady state • Predicts linear dependence of spike rates on input current • But: fails to explain high-current saturation effects • Voltage dependent recovery time constant of R needed • Pharmacological intervention model: IAHPcan be blocked or reduced by neuromodulators (ACh, histamine, norepinephrine, serotonin) Chapter 10 – Neural Adaptation and Bursting

14. Wrap-up: Completing the single neuron taxonomy Chapter 10 – Neural Adaptation and Bursting

15. Neural Bursting and Hysteresis – Class II neurons • What is Bursting? • Short train of several spikes interleaved with phases of silence • Which current is introduced? • Might be the same as for spike rate adaptation • Very slow hyperpolarizing K+ current • What function does it enable? • Complex behavioral change of network • Synchronization • “Multiplexing”: driving freq-specific neurons

16. Slow hyperpolarization in a squid axon Standard Class II neuron: Class II neuron with slow hyperpolarization IAHP due to K+current:

17. Bursting Neurons Simulation: HHburster.mwith I=0.14, 0.18 Chapter 10 – Neural Adaptation and Bursting

18. Bursting Neurons Simulation: HHburster.mwith I=0.14, 0.18 V-R projection of phase space trajectories (red) Chapter 10 – Neural Adaptation and Bursting

19. Bursting analysis of bifurcation diagram Chapter 10 – Neural Adaptation and Bursting

20. Bursting analysis of bifurcation diagram Inet ↑ (a) V ↑ H ↓ (b) AP vanishes (d) Action potential H ↑ V ↓ Inet ↓ (c) Chapter 10 – Neural Adaptation and Bursting

21. Bursting Analysis of Bifurcation diagram (c) (d) (b) (a) Chapter 10 – Neural Adaptation and Bursting

22. Endogenous Bursting Californian Aplysia (Seehase) • Rinzel model for Class I – neurons • More realistic 4-current model

23. Endogenous Bursting • What is endogenous bursting? • Occurrence of bursting neuronal activity in the absence of external stimulation (via a current I) • Which currents are introduced? • Fast depolarizing Ca2+-influx conductance X • Slow hyperpolarizing K+ conductance C • What function does it enable? • Pacemaker neurons (heartbeat, breathing) • synchronization

24. A more complex model of 4 intrinsic currents“Plant-model” • X is voltage-dependent (voltage-gated Ca2+ channels) • C is Ca2+-concentration dependent (Ca2+-activated K+ channels) • No external currents occur

25. Comparison to 3-current model of spike rate adaptation Now termed C, IAHP Chapter 10 – Neural Adaptation and Bursting

26. Endogenous Bursting Neuron: in-vivo • Difference to former model: • No stimulating current • Modulation back- and forth a saddle-node bifurcation Chapter 10 – Neural Adaptation and Bursting

27. Endogenous Bursting Neuron: in silico Simulation: PlantBurster.m X-C-projection of Phase space Time course of voltage V • Burst phases again occur due to a crossingof a bifurcationpointenabling a limitcycle • Due to Rinzel model: saddle node bifurcation • Additional currents X&C follow a limit cycle themselves with slower time scale than V-R (visible as ripples in projection) Chapter 10 – Neural Adaptation and Bursting

28. Wrap-up: Completing the single neuron taxonomy Chapter 10 – Neural Adaptation and Bursting

29. Separating limit cycles via intoxication VS Californian Aplysia (Seehase) Puffer Fish (Kugelfisch)

30. Tetrodotoxin and Sushi • Tetrodotoxin (TTX) acts as nerve poison via blockingofthedepolarizing Na+channels • Neurons cannot create action potentials any longer Removed voltage dependency of Na+ conductance Chapter 10 – Neural Adaptation and Bursting

31. Silencing all Na+-channels – in vivo Chapter 10 – Neural Adaptation and Bursting

32. Silencing all Na+-channels: in silico Without TTX With TTX • Still fluctuation due to X-C dynamics • No action potentials created Chapter 10 – Neural Adaptation and Bursting

33. Remaining limit cycle without Na+ current Simulation: PlantBursterTTX.m • X-C-projection of Phase space exhibits same limit cycle behavior • Modulation of X due to voltage changes vanish Without TTX With TTX Chapter 10 – Neural Adaptation and Bursting

34. Current-driven Bursting in Neocortical Neurons • What is endogenous bursting? • Occurrence of bursting neuronal activity in response to a constant external stimulation (via a current I) • Which currents are introduced? • External, stimulating current I • Fast depolarizing Ca2+-influx conductance X • Slow hyperpolarizing K+ conductance C • What function does it enable? • Chattering sensory neurons

35. Sensory cell bursting Mouse somatosensory cortex neuron Cat visual cortex neuron Chapter 10 – Neural Adaptation and Bursting

36. Driving Current Chapter 10 – Neural Adaptation and Bursting

37. Driving Current: differences to endogenous bursting model Chapter 10 – Neural Adaptation and Bursting

38. Driven bursting in a neocortical neuron I=0.19 Simulation: Chattering.m Time course of voltage V X-C-projection of phase space I=0.2 • Hopf bifurcation of X-C at I=0.197 • Qualitatively similar behavior of X-C limit cycleabovethisthresholdtoendogenousspiking • X-C limit-cycle drives V-R subspace through saddle-node bifurcation • One limit cycle driving the other to create bursts • But not autonomous due to V-dependence of X I=0.2 Chapter 10 – Neural Adaptation and Bursting

39. Wrap-up: Completing the single neuron taxonomy Chapter 10 – Neural Adaptation and Bursting

40. Dynamical Taxonomy of Class I neurons Fast-Spiking Inhibitory interneurons • Only 2 ion channel currents (Rinzel-model) • fast Na+ depolarization • slow K+ recovery • Constant spike rate: 1-400 Hz Regular Spiking Excitatory Neurons • Additional 3rd current • very slow after-hyperpolarizing K+ current • Enables spike rate adaptation Neocortical Bursting Cells • Additional 3rd & 4th current • very slow after-hyperpolarizing K+ current, mediated by Ca2+ concentration • fast depolarizing Ca2+ current • Enables bursting, either intrinsic () as pacemaker or driven by an external current Chapter 10 – Neural Adaptation and Bursting

41. Dynamical Taxonomy of Class I neurons Fast-Spiking Inhibitory interneurons Regular Spiking Excitatory Neurons Neocortical Bursting Cells Chapter 10 – Neural Adaptation and Bursting

42. Take home message: More fun with currents • Spike frequency and AP creation are dependent on external, stimulating current. • Intrinsic currents partially counteract the effect of the external driving current. • This happens in a dynamic manner via the introduction of 1 or 2 additional currents modelling • Afterhyperpolarizing effects (very slow K+) • Additional depolarizing effects (fast Ca2+) • This dynamic net current fluctuation leads to complex behavior due to recurring back- and forth-crossings of bifurcation boundaries Chapter 10 – Neural Adaptation and Bursting

43. Picture Sources • http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Tetrodotoxin.svg/1000px-Tetrodotoxin.svg.png • http://upload.wikimedia.org/wikipedia/commons/7/77/Puffer_Fish_DSC01257.JPG • http://upload.wikimedia.org/wikipedia/commons/e/ef/Aplysia_californica.jpg • http://www.cvr.yorku.ca/webpages/spikes.pdf => Chapter 9 and 10 Chapter 10 – Neural Adaptation and Bursting