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Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability

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

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Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability

  1. 1. <ul><li>Hodgkin-Huxley & the nonlinear </li></ul><ul><li>dynamics of neuronal excitability . </li></ul><ul><li>John Rinzel, AACIMP, 2011 </li></ul><ul><li>The Hodgkin-Huxley model </li></ul><ul><ul><ul><li>Membrane currents </li></ul></ul></ul><ul><ul><ul><li>‘ Dissection’ of the action potential </li></ul></ul></ul><ul><li>Excitability in the phase plane </li></ul><ul><ul><li>Morris-Lecar model </li></ul></ul><ul><li>Onset of repetitive firing (Type I & II) and phasic firing (Type III). </li></ul><ul><li>Other currents and firing patterns </li></ul>
  2. 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. 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. 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. 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)
  6. 6. <ul><li>Hodgkin-Huxley & the nonlinear </li></ul><ul><li>dynamics of neuronal excitability . </li></ul><ul><li>John Rinzel, AACIMP, 2011 </li></ul><ul><li>The Hodgkin-Huxley model </li></ul><ul><ul><ul><li>Membrane currents </li></ul></ul></ul><ul><ul><ul><li>‘ Dissection’ of the action potential </li></ul></ul></ul><ul><li>Excitability in the phase plane </li></ul><ul><ul><li>Morris-Lecar model </li></ul></ul><ul><li>Onset of repetitive firing (Type I & II) and phasic firing (Type III). </li></ul><ul><li>Other currents and firing patterns </li></ul>
  7. 7. Excitability and Repetitive Firing
  8. 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
  9. 9. Electrically compact cell – the “point neuron” 1.1 Current balance – patch -- review <ul><li>R m C m dV/dt = - V + R m I app </li></ul><ul><li>Define τ = R m C m, , the membrane time constant (in ms) ( τ or τ m ) </li></ul><ul><li>τ dV/dt = -V + R m I app </li></ul><ul><li>Time course: V(t) = R m I app [1-exp(-t/ τ ) ] for 0 ≤ t ≤ t off </li></ul><ul><li> = V(t off ) exp[-(t-t off )/ τ ] for t ≥ t off </li></ul><ul><li>τ , “typical”: 10 ms if C m =1 μ F/cm 2 , R m =10,000 Ohm-cm 2 (cortical cells, motoneurons ) </li></ul><ul><li> 1 ms or less, … R m =10 3 ohm-cm 2 (auditory brain stem – R N ≈ 10s Meg-ohm). </li></ul>Passive membrane: const conductance. area A – response to current step. I app R m I app V I app t=0 t=t off
  10. 10. Electrical Activity of Cells <ul><li>V = V(x,t) , distribution within cell </li></ul><ul><ul><li>uniform or not?, propagation? </li></ul></ul><ul><li>Coupling to other cells </li></ul><ul><li>Nonlinearities </li></ul><ul><li>Time scales </li></ul>∂ V ∂ t ∂ 2 V ∂ x 2 C m +I ion (V)= + I app + coupling Current balance equation for membrane: capacitive channels cable properties other cells d 4R i ∑ g c,j (V j –V) ∑ g syn,j (V j (t)) (V syn -V) Coupling: “ electrical” - gap junctions j j chemical synapses other cells = ∑ g k (V, W ) (V–V k ) I ion = I ion (V, W ) k channel types ∂ W /∂ t = G (V, W ) gating dynamics generally nonlinear
  11. 11. Nobel Prize, 1959
  12. 13. Development of the Hodgkin-Huxley model for the squid giant axon. Space clamp: developed by Cole/Marmont late ‘40s.
  13. 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”
  14. 15. &quot;The Squid and its Giant Nerve Fiber&quot; 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/
  15. 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?
  16. 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.
  17. 19. 1 μ m 2 has about 100 Na + and K + channels.
  18. 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…)
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. 27. Repetitive Firing, eg, HH model Response to current step I app frequency subthreshold nerve block I app
  26. 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
  27. 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.
  28. 30. <ul><li>Hodgkin-Huxley & the nonlinear </li></ul><ul><li>dynamics of neuronal excitability . </li></ul><ul><li>John Rinzel, AACIMP, 2011 </li></ul><ul><li>The Hodgkin-Huxley model </li></ul><ul><ul><ul><li>Membrane currents </li></ul></ul></ul><ul><ul><ul><li>‘ Dissection’ of the action potential </li></ul></ul></ul><ul><li>Excitability in the phase plane </li></ul><ul><ul><li>Morris-Lecar model </li></ul></ul><ul><li>Onset of repetitive firing (Type I & II) and phasic firing (Type III). </li></ul><ul><li>Other currents and firing patterns </li></ul>
  29. 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
  30. 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
  31. 34. Case of small φ traj hugs V-nullcline - except for up/down jumps. ML model - excitable regime
  32. 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.
  33. 36. Anode Break Excitation or Post-Inhibtory Rebound (PIR) I K - deactivated
  34. 38. Onset is via Hopf bifurcation Repetitive Activity in ML (& HH) “ Type II” onset Hodgkin ‘48
  35. 39. V max V min Frequency vs I app Amplitude vs I app Bistability near onset - subcritical Hopf
  36. 40. <ul><li>Hodgkin-Huxley & the nonlinear </li></ul><ul><li>dynamics of neuronal excitability . </li></ul><ul><li>John Rinzel, AACIMP, 2011 </li></ul><ul><li>The Hodgkin-Huxley model </li></ul><ul><ul><ul><li>Membrane currents </li></ul></ul></ul><ul><ul><ul><li>‘ Dissection’ of the action potential </li></ul></ul></ul><ul><li>Excitability in the phase plane </li></ul><ul><ul><li>Morris-Lecar model </li></ul></ul><ul><li>Onset of repetitive firing (Type I & II) and phasic firing (Type III). </li></ul><ul><li>Other currents and firing patterns </li></ul>
  37. 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.
  38. 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
  39. 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
  40. 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
  41. 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
  42. 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
  43. 48. Type II Type I I app frequency Noise smooths the f-I relation
  44. 49. FS cell near threshold RS cell, w/ noise FS cell, w/ noise
  45. 50. <ul><li>Hodgkin-Huxley & the nonlinear </li></ul><ul><li>dynamics of neuronal excitability . </li></ul><ul><li>John Rinzel, AACIMP, 2011 </li></ul><ul><li>The Hodgkin-Huxley model </li></ul><ul><ul><ul><li>Membrane currents </li></ul></ul></ul><ul><ul><ul><li>‘ Dissection’ of the action potential </li></ul></ul></ul><ul><li>Excitability in the phase plane </li></ul><ul><ul><li>Morris-Lecar model </li></ul></ul><ul><li>Onset of repetitive firing (Type I & II) and phasic firing (Type III). </li></ul><ul><li>Other currents and firing patterns </li></ul>
  46. 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
  47. 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 )
  48. 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
  49. 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 .
  50. 55. I KLT msec mV I KLT I Na /4 I KHT I KLT-frzn Rothman & Manis, 2003 Golding & Rinzel labs, 2009
  51. 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
  52. 57. Noise gating: detecting a slow signal.
  53. 58. Noise-gated response to low frequency input. Gai, Doiron, Rinzel PLoS Computl Biol 2010 Noise-free With noise
  54. 59. Noise-gating: experimental, gerbil Gai, Doiron, Rinzel PLoS Computl Biol 2010
  55. 60. Threshold for phasic model: ramp slope.
  56. 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)

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