AACIMP 2011 Summer School. Neuroscience stream. Lecture by John Rinzel.

2. References on Nonlinear Neuronal Dynamics References on Cellular Neuro, w/ modeling. Koch, C. Biophysics of Computation, Oxford Univ Press, 1998. Koch & Segev (eds): Methods in Neuronal Modeling, MIT Press, 1998. Johnston & Wu: Foundations of Cellular Neurophys., MIT Press, 1995. Tuckwell, HC. Intro’n to Theoretical Neurobiology, I&II, Cambridge UP, 1988. Rinzel & Ermentrout. Analysis of neural excitability and oscillations. In Koch & Segev (see above). Also “Live” on www.pitt.edu/~phase/ Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72. Izhikevich, EM: Dynamical Systems in Neuroscience. The Geometry of Excitability and Bursting. MIT Press, 2007. Strogatz, S. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994. Ermentrout & Terman. Mathematical Foundations of Neuroscience. Springer, 2010.

3. Software/Simulators for Cellular Neurophysiology/ HH and other modeling. HHsim: Graphical Hodgkin-Huxley Simulator By DS Touretzky , MV Albert , ND Daw, A Ladsariya & M Bonakdarpour http://www.cs.cmu.edu/~dst/HHsim/ NEURON: software simulation environment for computational neuroscience. NEURON calculates dynamic currents, conductances and voltages throughout nerve cells of all types. Developed by M Hines. http://www.neuron.yale.edu Carnevale NT, Hines ML (2005). The NEURON Book . Cambridge University Press. Neurons in Action: Tutorials and Simulations using NEURON. By JW Moore and AE Stuart (2009) 2 nd edition, Sinauer Associates. http://www.neuronsinaction.com/home/main XPP software: http://www.pitt.edu/~phase/ ModelDB: database of models. http://senselab.med.yale.edu/ModelDB/

4. Dynamics of Excitability and Repetitive Activity Auditory brain stem neurons fire phasically, not to slow inputs. w/ Svirskis et al, J Neurosci 2002

5. Take Home Messages Excitability/Oscillations : fast autocatalysis + slower negative feedback Value of reduced models Time scales and dynamics Phase space geometry Different dynamic states – “Bifurcations”; concepts and methods are general. XPP software:http://www.pitt.edu/~phase/ (Bard Ermentrout’s home page)

7. Excitability and Repetitive Firing

8. Electrically compact cell – the “point neuron” Current balance equation: A I app = A I m = A [C m dV m /dt + (V m -E rest )/R m ] = A (C m dV/dt + V/R m ) , where V=V m -E rest (dev’n from rest) or… divide by A and multiply by R m R m C m dV/dt = - V + I app R m (UNITS: 1/R m in mS/cm 2 , C m in μ F/cm 2 , I app in μ A/cm 2 , t in ms, V in mV) 1.1 Current balance – patch -- review Passive membrane: constant conductance. area A – response to current step. I app R m I app V I app t=0 t=t off

11. Nobel Prize, 1959

13. Development of the Hodgkin-Huxley model for the squid giant axon. Space clamp: developed by Cole/Marmont late ‘40s.

14. HH Recipe: V-clamp  I ion components Predict I-clamp behavior? I K (t) is monotonic; activation gate, n I Na (t) is transient; activation, m and inactivation, h e.g., g K (t) = I K (t) /(V-V K ) = G K n 4 (t) with V=V clamp gating kinetics: dn/dt = α (V) (1-n) – β (V) n = (n ∞ (V) – n)/  n (V) n ∞ (V) increases with V. I Na (t) = G Na m 3 (t) h(t) (V-V Na ) OFF ON P P * α (V) β (V) mass action for “subunits” or HH-”particles”

15. "The Squid and its Giant Nerve Fiber" was filmed in the 1970s at Plymouth Marine Laboratory in England. Dissection and anatomy (J.Z. Young) (7 MB) Voltage clamping (P.F. Baker & A.L. Hodgkin) (10 MB) http://www.science.smith.edu/departments/NeuroSci/courses/bio330/

16. HH Equations C m dV/dt + G Na m 3 h (V-V Na ) + G K n 4 (V-V K ) +G L (V-V L ) = I app [+d/(4R) ∂ 2 V/∂x 2 ] dm/dt = [m ∞ (V)-m]/  m (V) dh/dt = [h ∞ (V) - h]/  h (V) dn/dt = [n ∞ (V) – n]/  n (V) space-clamped φ φ φ φ , temperature correction factor = Q 10 **(temp-temp ref ) HH: Q 10 =3 V Reconstruct action potential Time course Velocity Threshold – strength duration Refractory period Ion fluxes Repetitive firing?

18. Moore & Stuart: Neurons in Action I app Strength-Duration curve time, ms Voltage, mV I app Threshold for spike generation Membrane is refractory after a spike.

19. 1 μ m 2 has about 100 Na + and K + channels.

20. Dissection of the HH Action Potential Fast/Slow Analysis - based on time scale differences V t Idealize the Action Potential (AP) to 4 phases Mathematically, this is construction of a solution by the methods of (geometric) singular perturbation theory (Terman, Carpenter, Keener…)

21. I-V relations: I SS (V) I inst (V) steady state “instantaneous” HH: I SS (V) = G Na m ∞ 3 (V) h ∞ (V) (V-V Na ) + G K n ∞ 4 (V) (V-V K ) +G L (V-V L ) h, n are slow relative to V,m I inst (V) = G Na m ∞ 3 (V) h (V-V Na ) + G K n (V-V K ) +G L (V-V L ) fast slow, fixed at holding values e.g., rest

22. Dissection of HH Action Potential Fast/Slow Analysis - based on time scale differences V t h, n are slow relative to V,m Idealize AP to 4 phases h,n – constant during upstroke and downstroke V,m – “slaved” during plateau and recovery

23. Dissecting the HH Action Potential The upstroke: m, fast and h, n slow – fixed at rest. C m dV/dt = -I inst (V; h R , n R ) +I app V depolarizes to E Then, recovery phase: h increases, n decreases … . the return to rest. Then, plateau phase: h decreases, n increases When E & T coalesce: downstroke

24. Upstroke… R and E – stable T - unstable C dV/dt = - I inst (V, m ∞ (V), h R , n R ) + I app neglect thus, dv/dt = - λ v where λ =C -1 dI inst /dV, at V=V R solutions are exptl: v(t) = v 0 exp(- λ t) V R is stable if λ >0 and unstable if λ <0 (negative resistance Linear stability analysis: Do small perturbations grow or decay with time? V(t) = V R + v(t) Substitute into ode: C dV/dt = C dv/dt = - I inst (V R +v) + I app = - [I inst (V R ) + (dI inst /dV) v + …v 2 +…] +I app cancel

25. HH, dissection of single action potential I inst vs V changes as h & n evolve during AP V equilibrates to I inst (V; h,n) =0. V I inst

26. HH, dissection of repetitive firing I inst vs V changes as h & n evolve during AP V equilibrates to I inst (V; h,n) =0. I app = 40 V I inst

27. Repetitive Firing, eg, HH model Response to current step I app frequency subthreshold nerve block I app

28. Repetitive firing in HH and squid axon -- bistability near onset Rinzel & Miller, ‘80 Interval of bistability Linear stability: eigenvalues of 4x4 matrix. For reduced model w/ m=m ∞ (V): stability if ∂ I inst /∂V + C m /  n > 0. HH eqns Squid axon Guttman, Lewis & Rinzel, ‘80

29. Exercises: 1. Consider HH without I K (ie, g k =0). Show that with adjustment in g Na (and maybe g leak ) the HH model is still excitable and generates an action potential. (Do it with m=m ∞ (V).) Study this 2 variable (V-h) model in The phase plane: nullclines, stability of rest state, trajectories, etc. Then consider a range of I app to see if get repetitive firing. Compute the freq vs I app relation; study in the phase plane. Do analysis to see that rest point must be on middle branch to get limit cycle. 2. Convert the HH model into “phasic model”. By “phasic” I mean that the neuron does not fire repetitively for any I app values – only 1 to a few spikes and then it returns to rest. Do this by, say, sliding some channel gating dynamics along the V-axis (probably just for I K ) . [If you slide x ∞ (V), you must also slide  x (V).] If it can be done using h=1-n and m=m ∞ (V) then do the phase plane analysis.

31. Two-variable Morris-Lecar Model  Phase Plane Analysis V V K V L V Ca I Ca – fast, non-inactivating I K -- “delayed” rectifier, like HH’s I K Morris & Lecar, ’81 – barnacle musclel V rest ML model has the features of excitability: Threshold, refractoriness, SD, repetitive firing

33. Get the Nullclines dV/dt = - I inst (V,w) + I app dw/dt = φ [ w ∞ (V) – w] /  w (V) dV/dt = 0 I inst (V,w) = I app w= w ∞ (V) dw/dt = 0 w = w rest rest state w= w rest w > w rest

34. Case of small φ traj hugs V-nullcline - except for up/down jumps. ML model - excitable regime

35. FitzHugh-Nagumo Model (1961) See. http://www.scholarpedia.org/ dv/dt = - f(v) – w +I dw/dt = ε (v- γ w) Where, f(v) = v ( v-a) (v-1) and γ ≥ 0 and 0 < ε << 1.

36. Anode Break Excitation or Post-Inhibtory Rebound (PIR) I K - deactivated

38. Onset is via Hopf bifurcation Repetitive Activity in ML (& HH) “ Type II” onset Hodgkin ‘48

39. V max V min Frequency vs I app Amplitude vs I app Bistability near onset - subcritical Hopf

41. Adjust param’s  changes nullclines: case of 3 “rest” states Stable or Unstable? 3 states – not necessarily: stable – unstable – stable. 3 states  I ss is N-shaped Φ small enough, then both upper/middle unstable if on middle branch.

42. ML: φ large  2 stable steady states Neuron is bistable: plateau behavior. V t I app switching pulses e.g., HH with V K = 24 mV e.g., Hausser lab: Bistability of cerebellar Purkinje cells… Nature Neurosci, 2005 Saddle point, with stable and unstable manifolds

43. ML: φ small  both upper states are unstable Neuron is excitable with strict threshold . threshold separatrix long Latency I ss must be N-shaped. I K-A can give long latency but not necessary. V rest saddle

44. Onset of Repetitive Firing – 3 rest states SNIC- saddle-node on invariant circle V w I app excitable saddle-node limit cycle homoclinic orbit; infinite period emerge w/ large amplitude – zero frequency

45. ML: φ small Response/Bifurcation diagram low freq but no conductances very slow I K-A ? (Connor et al ’77) Firing frequency starts at 0. freq ~√ I–I 1 “ Type I” onset Hodgkin ‘48

46. Transition from Excitable to Oscillatory Type II, min freq ≠ 0 I ss monotonic subthreshold oscill’ns excitable w/o distinct threshold excitable w/ finite latency Type I, min freq = 0 I SS N-shaped – 3 steady states w/o subthreshold oscillations excitable w/ “all or none” (saddle) threshold excitable w/ infinite latency Hodgkin ’48 – 3 classes of repetiitive firing; Also - Class I less regular ISI near threshold

48. Type II Type I I app frequency Noise smooths the f-I relation

49. FS cell near threshold RS cell, w/ noise FS cell, w/ noise

51. Bullfrog sympathetic Ganglion “B” cell Cell is “compact”, electrically … but not for diffusion Ca 2+ MODEL: “ HH” circuit + [Ca 2+ ] int + [K + ] ext g c & g AHP depend on [Ca 2+ ] int Yamada, Koch, Adams ‘89

52. Bursting mediated by I K-Ca C V = - I Ca - I K – I leak – I K-Ca + I app .... gating variables… I K-Ca = g K-Ca [Ca/(Ca+Ca o )] (V-V K )

53. Bursting mediated by I K-Ca Ca C V = - I Ca - I K – I leak – I K-Ca + I app .... gating variables… I K-Ca = g K-Ca [Ca/(Ca+Ca o )] (V-V K ) Spike generating, V-w, phase plane Bistability: “lower-V” steady state “ upper-V” oscillation Ca, fixed

54. The “definitive” Type 3 neuron. Coincidence detection for sound localization in mammals. Blocking I KLT may convert to tonic firing. Auditory brain stem (MSO) neurons fire phasically, not repetitively to slow inputs. Steady state is stable for any I app .

55. I KLT msec mV I KLT I Na /4 I KHT I KLT-frzn Rothman & Manis, 2003 Golding & Rinzel labs, 2009

56. Auditory brain stem, DCN pyramidal neuron. Transient K + current, I KIF : fast activating and slow inactivating I KIF de-inactivates… I KIF inactivates… h f h s

57. Noise gating: detecting a slow signal.

58. Noise-gated response to low frequency input. Gai, Doiron, Rinzel PLoS Computl Biol 2010 Noise-free With noise

59. Noise-gating: experimental, gerbil Gai, Doiron, Rinzel PLoS Computl Biol 2010

60. Threshold for phasic model: ramp slope.

61. Take Home Message Excitability/Oscillations : fast autocatalysis + slower negative feedback Value of reduced models Time scales and dynamics Phase space geometry Different dynamic states – “Bifurcations” Excitability: Types I, II, III XPP software:http://www.pitt.edu/~phase/ (Bard Ermentrout’s home page)