hodgkin-huxley & the nonlinear dynamics of neuronal excitability
DESCRIPTION
AACIMP 2011 Summer School. Neuroscience stream. Lecture by John Rinzel.TRANSCRIPT
Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability.
John Rinzel, AACIMP, 2011
1. The Hodgkin-Huxley modelMembrane currents‘Dissection’ of the action potential
2. Excitability in the phase planeMorris-Lecar model
3. Onset of repetitive firing (Type I & II) and phasic firing (Type III).
4. Other currents and firing patterns
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 geometricaldissection 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.
Software/Simulators for Cellular Neurophysiology/ HH and other modeling.
HHsim: Graphical Hodgkin-Huxley SimulatorBy DS Touretzky, MV Albert, ND Daw, A Ladsariya & M Bonakdarpourhttp://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.eduCarnevale 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) 2nd 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/
Dynamics of Excitability and Repetitive Activity
Auditory brain stem neurons fire phasically, not to slow inputs.
w/ Svirskis et al, J Neurosci 2002
Take Home Messages
Excitability/Oscillations : fast autocatalysis + slowernegative feedback
Value of reduced models
Time scales and dynamics
Phase space geometry
Different dynamic states – “Bifurcations”; concepts andmethods are general.
XPP software:http://www.pitt.edu/~phase/ (Bard Ermentrout’s home page)
Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability.
John Rinzel, AACIMP, 2011
1. The Hodgkin-Huxley modelMembrane currents‘Dissection’ of the action potential
2. Excitability in the phase planeMorris-Lecar model
3. Onset of repetitive firing (Type I & II) and phasic firing (Type III).
4. Other currents and firing patterns
Excitability and Repetitive Firing
Electrically compact cell – the “point neuron”
Current balance equation: A Iapp = A Im = A [CmdVm/dt + (Vm-Erest)/Rm]
= A (Cm dV/dt + V/Rm) , where V=Vm-Erest (dev’n from rest)
or… divide by A and multiply by Rm
RmCm dV/dt = - V + Iapp Rm
(UNITS: 1/Rm in mS/cm2, Cm in μF/cm2, Iapp in μA/cm2, t in ms, V in mV)
Iapp
Rm Iapp
V
Iapp
t=0 t=toff
1.1 Current balance – patch -- review
Passive membrane: constant conductance. area A – response to current step.
Electrically compact cell – the “point neuron”
Iapp
Rm Iapp
V
Iapp
t=0 t=toff
1.1 Current balance – patch -- review
RmCm dV/dt = - V + Rm Iapp
Define τ = RmCm, , the membrane time constant (in ms) (τ or τm)
τ dV/dt = -V + Rm I app
Time course: V(t) = Rm Iapp [1-exp(-t/τ) ] for 0 ≤ t ≤ toff
= V(toff) exp[-(t-toff)/τ] for t ≥ toff
τ, “typical”: 10 ms if Cm=1μF/cm2, Rm=10,000 Ohm-cm2 (cortical cells, motoneurons ) 1 ms or less, … Rm=103 ohm-cm2 (auditory brain stem – RN ≈ 10s Meg-ohm).
Passive membrane: const conductance. area A – response to current step.
Electrical Activity of Cells• V = V(x,t) , distribution within cell
• uniform or not?, propagation?•Coupling to other cells•Nonlinearities•Time scales
∂V ∂ t
∂2 V ∂ x2Cm +Iion(V)= + Iapp + coupling
Current balance equation for membrane:
capacitive channels cable properties other cells
d4Ri
∑ gc,j(Vj–V)
∑ gsyn,j(Vj(t)) (Vsyn-V)
Coupling: “electrical” - gap junctions
j
j
chemical synapsesother cells
= ∑ gk(V,W) (V–Vk )
Iion = Iion(V,W)
kchannel types
∂W/∂ t = G(V,W) gating dynamics
generally nonlinear
Nobel Prize, 1959
Development of the Hodgkin-Huxley model for the squid giant axon.
Space clamp: developed by Cole/Marmont late ‘40s.
HH Recipe:
V-clamp Iion components
Predict I-clamp behavior?
IK(t) is monotonic; activation gate, nINa(t) is transient; activation, m and
inactivation, h
e.g., gK(t) = IK(t) /(V-VK) = GK n4(t)with V=Vclamp
gating kinetics: dn/dt = α(V) (1-n) – β(V) n = (n∞(V) – n)/n(V) n∞(V) increases with V.
INa(t) = GNam3(t) h(t) (V-VNa)
OFF ON P P*
α(V)
β(V)
mass action for “subunits” or HH-”particles”
"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/
HH Equations
Cm dV/dt + GNa m3 h (V-VNa) + GK n4 (V-VK) +GL (V-VL) = Iapp [+d/(4R) ∂2V/∂x2]
dm/dt = [m∞(V)-m]/m(V)dh/dt = [h∞(V) - h]/h(V)dn/dt = [n∞(V) – n]/n(V)
space-clamped
φφφ
φ, temperaturecorrection factor= Q10**(temp-tempref) HH: Q10=3
V
Reconstruct action potential
Time courseVelocityThreshold – strength durationRefractory periodIon fluxesRepetitive firing?
Iapp
Strength-Duration curve
time, ms
Vol
tage
, m
V
Iapp
Threshold for spike generation
Membrane is refractory after a spike.
Moore & Stuart: Neurons in Action
1 μm2 has about100 Na+ and K+ channels.
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 methodsof (geometric) singular perturbation theory (Terman, Carpenter, Keener…)
I-V relations: ISS(V) Iinst(V) steady state “instantaneous”
HH: ISS(V) = GNa m∞3(V) h∞(V) (V-VNa) + GK n∞
4(V) (V-VK) +GL (V-VL)
Iinst(V) = GNa m∞3(V) h (V-VNa) + GK n (V-VK) +GL (V-VL)
fast slow, fixed at holding values e.g., rest
h, n are slow relative to V,m
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
Dissecting the HH Action Potential
The upstroke: m, fast and h, n slow – fixed at rest.
CmdV/dt = -Iinst(V; hR, nR) +Iapp
V depolarizes to E
Then, plateau phase: h decreases, n increases
When E & T coalesce: downstroke
Then, recovery phase: h increases, n decreases…. the return to rest.
Upstroke…
R and E – stable
T - unstable
C dV/dt = - Iinst(V, m∞(V), hR, nR) + Iapp
Linear stability analysis: Do small perturbations grow or decay with time?
V(t) = VR + v(t)Substitute into ode: C dV/dt = C dv/dt = - Iinst(VR+v) + Iapp
= - [Iinst(VR) + (dIinst/dV) v + …v2 +…] +Iappcancel
neglect
thus, dv/dt = -λ v where λ=C-1 dIinst/dV, at V=VR
solutions are exptl: v(t) = v0 exp(-λt)
VR is stable if λ>0 and unstable if λ<0 (negative resistance
HH, dissection of single action potential
V
Iinst
Iinst vs V changes as h & n evolve during AP
V equilibrates to Iinst (V; h,n) =0.
HH, dissection of repetitive firing
V
Iinst
Iinst vs V changes as h & n evolve during AP
V equilibrates to Iinst (V; h,n) =0.
Iapp = 40
Repetitive Firing, eg, HH model
Response to current step
Iapp
Iapp
frequency
subthreshold nerve block
Repetitive firing in HH and squid axon -- bistability near onset
Rinzel & Miller, ‘80
HH eqns Squid axon
Guttman, Lewis & Rinzel, ‘80
Interval of bistability
Linear stability: eigenvalues of4x4 matrix. For reduced model w/ m=m∞(V): stability if∂Iinst/∂V + Cm/n > 0.
Exercises:
1. Consider HH without IK (ie, gk =0). Show that with adjustment in gNa (and maybe gleak) 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 Iapp 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 branchto get limit cycle.
2. Convert the HH model into “phasic model”. By “phasic” I meanthat the neuron does not fire repetitively for any Iapp 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 IK) .[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.
Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability.
John Rinzel, AACIMP, 2011
1. The Hodgkin-Huxley modelMembrane currents‘Dissection’ of the action potential
2. Excitability in the phase planeMorris-Lecar model
3. Onset of repetitive firing (Type I & II) and phasic firing (Type III).
4. Other currents and firing patterns
Two-variable Morris-Lecar Model Phase Plane Analysis
VVK VL VCa
ICa – fast, non-inactivatingIK -- “delayed” rectifier, like HH’s IK
Morris & Lecar, ’81 – barnacle musclel
Vrest
ML model has the features of excitability:Threshold, refractoriness, SD, repetitive firing
Get the Nullclines
dV/dt = - Iinst (V,w) + Iapp
dw/dt = φ [ w∞(V) – w] / w(V)
dV/dt = 0
Iinst (V,w) = Iapp
w= w∞(V)
dw/dt = 0
w = w rest
rest state
w= w rest
w > wrest
Case of small φ
traj hugs V-nullcline - except for up/downjumps.
ML model- excitableregime
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.
Anode Break Excitation or Post-Inhibtory Rebound (PIR)
IK - deactivated
Onset is via Hopf bifurcation
Repetitive Activity in ML (& HH)
“Type II” onset
Hodgkin ‘48
Vmax
Vmin
Frequency vs Iapp
Amplitude vs Iapp Bistability near onset - subcritical Hopf
Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability.
John Rinzel, AACIMP, 2011
1. The Hodgkin-Huxley modelMembrane currents‘Dissection’ of the action potential
2. Excitability in the phase planeMorris-Lecar model
3. Onset of repetitive firing (Type I & II) and phasic firing (Type III).
4. Other currents and firing patterns
Adjust param’s changes nullclines: case of 3 “rest” states
Stable or Unstable?
3 states – not necessarily:stable – unstable – stable.
3 states Iss
is N-shaped
Φ small enough,then both upper/middle unstable if on middle branch.
ML: φ large 2 stable steady states Neuron is bistable: plateau behavior.
Saddle point, with stable and unstable manifolds
V
t
Iapp switching pulses
e.g., HH with VK = 24 mV
e.g., Hausser lab: Bistability of cerebellar Purkinje cells… Nature Neurosci, 2005
ML: φ small both upper states are unstable Neuron is excitable with strict threshold.
thresholdseparatrix long
Latency
Vrest
saddle
Iss must be N-shaped.
IK-A can give long latency but not necessary.
Onset of Repetitive Firing – 3 rest states
SNIC- saddle-node on invariant circle
V
wIapp
excitable
saddle-node
limit cycle
homoclinic orbit;infinite period
emerge w/ large amplitude – zero frequency
ML: φ smallResponse/Bifurcation diagram
Firing frequency starts at 0.
freq ~√ I–I1
low freq but no conductancesvery slow
IK-A ? (Connor et al ’77)
“Type I” onset
Hodgkin ‘48
Transition from Excitable to Oscillatory
Type II, min freq ≠ 0Iss monotonicsubthreshold oscill’nsexcitable w/o distinct thresholdexcitable w/ finite latency
Type I, min freq = 0 ISS N-shaped – 3 steady statesw/o subthreshold oscillationsexcitable w/ “all or none” (saddle) thresholdexcitable w/ infinite latency
Hodgkin ’48 – 3 classes of repetiitive firing; Also - Class I less regular ISI near threshold
Type II
Type I
I app
frequency
Noise smooths the f-I relation
FS cellnear threshold
RS cell, w/ noise FS cell, w/ noise
Hodgkin-Huxley & the nonlinear dynamics of neuronal excitability.
John Rinzel, AACIMP, 2011
1. The Hodgkin-Huxley modelMembrane currents‘Dissection’ of the action potential
2. Excitability in the phase planeMorris-Lecar model
3. Onset of repetitive firing (Type I & II) and phasic firing (Type III).
4. Other currents and firing patterns
Bullfrog sympathetic Ganglion “B” cell
Cell is “compact”, electrically … but notfor diffusion Ca 2+
MODEL:
“HH” circuit+ [Ca2+] int
+ [K+] ext
gc & gAHP depend on [Ca2+] int
Yamada, Koch, Adams ‘89
Bursting mediated by IK-Ca
C V = - ICa - IK – Ileak – IK-Ca + Iapp
.... gating variables…
IK-Ca = gK-Ca [Ca/(Ca+Cao)] (V-VK)
Bursting mediated by IK-Ca
Ca
C V = - ICa - IK – Ileak – IK-Ca + Iapp
.... gating variables…
IK-Ca = gK-Ca [Ca/(Ca+Cao)] (V-VK)
Spike generating, V-w, phase planeBistability: “lower-V” steady state
“upper-V” oscillation
Ca, fixed
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 Iapp.
IKLT
msec
mV
IKLT
INa/4
IKHTIKLT-frzn
Rothman & Manis, 2003Golding & Rinzel labs, 2009
Auditory brain stem, DCN pyramidal neuron.
Transient K+ current, IKIF:fast activating and slow inactivating
IKIF de-inactivates… IKIF inactivates…
hf
hs
Noise gating: detecting a slow signal.
Noise-gated response to low frequency input.
Noise-free
With noiseGai, Doiron, Rinzel PLoS Computl Biol 2010
Noise-gating: experimental, gerbil
Gai, Doiron, Rinzel PLoS Computl Biol 2010
Threshold for phasic model: ramp slope.
Take Home Message
Excitability/Oscillations : fast autocatalysis + slowernegative 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)