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)