Computational Neural Modeling and Neuroengineering

The Hodgkin-Huxley Model for Action Potential Generation

Action Potential Propagation in Dendrites

Stochastic influences on dendritic computation

The Hodgkin-Huxley Model of Action Potential Generation

MotivationsAction Potentials

(A) Giant squid axon at 16C (B) Axonal spike from the node of Ranvier in a myelinated frog fiber at 22C (C) Cat visual cortex at 37C (D) Sheep heart Purkinje fiber at 10C (E) Patch-clamp recording from a rabbit retinal ganglion cell at 37C (F) Layer 5 pyramidal cell in the rate at room temperatures, simulataneuous recordings from the soma and apical trunk (G) A complex spike consisting of several large EPSPs superimposed on a slow dendritic calcium spike and several fast somatic spikes from a Purkinje cell body in the rat cerebellum at 36C (H) Layer 5 pyramidal cell in the rat at room temperature - three dendritic voltage traces in response to three current steps of different amplitudes reveal the all-or-none character of this slow event. Notice the fast superimposed spikes. (I) Cell body of a projection neuron in the antennal lobe of the locust at 23C

Historical BackgroundBernstein

The membrane “breakdown” hypothesis

Prior to 1940, the excitability of neurons was only known via extracellular electrodes A major mystery was the underlying mechanism By the turn of the 20th century it was known that

1) cell membranes separated solutions of different ionic concentrations 2) [K+]o << [K+]i

3) [Na+]o >> [Na+]i

In 1902, Bernstein, reasoning that the cell membrane was semi-permeable to K+ and

should have a Vm ~ -75mV, proposed that neuronal activity (measured extracellularly) represented a “breakdown” of the cell membrane resistance to ionic flow and the resulting redistribution of ions would lead from -75mV to 0mV transmembrane potential (Vm=0)

Historical BackgroundCole et al.

The space clamp

The voltage clamp

Marmont (1949) and Cole (1949) developed the space clamp technique to maintain a uniform spatial distribution of Vm over a region of the cell where one tried to record currents

This was accomplished by threading the squid axon with silver wires to provide a very low axial resistance and hence eliminating longitudinal voltage gradients

Cole and colleagues developed a method for maintaining Vm at any

desired voltage level

Required monitoring voltage changes, feeding it through an amplifier to then drive current into or out of the cell to dynamically maintain the voltage while recording the current required to do so

Schematic of the voltage clamp apparatus for the giant squid axon (reproduced from Hille, 1992)

Historical BackgroundHodgkin and Katz

The “sodium hypothesis”

Hodgkin and Katz (1949) had demonstrated that both sodium and potassium make significant contributions to the ionic current underlying the action potential

First to realize that, in contrast to Bernstein’s theory of increased permeability for all ions, the “overshoot” and “undershoot” of the AP could be explained by bounded changes in the permeabilites for a few different ions

Hodgkin and Katz postulated that during the upstroke of the AP, Na+ was the most permeable ion and so the voltage of Vm moved towards its Nernst potential of ~ 60mV.

iKiNa

oKoNarest KPNaP

KPNaP

F

RTE

ln

They predicted and then demonstrated that the AP amplitude would therefore depend critically on the external concentration of Na+.

They generalized the Nernst equation to predict the steady-state Vm for the case of

multiple permeable ions. Goldman-Hodgkin Katz Voltage Equation

Historical BackgroundHodgkin and Huxley

Following Hodgkin and Katz (1949), the big remaining question was how is the permeability of the membrane to specific ions linked to time and Vm?

This was not answered until the tour-de-force of physiology and modeling presented in four papers in 1952 by Hodgkin and Huxley. This work represents one of the highest-points in cellular biophysics and the quantitative model they developed forms the basis for understanding and modeling the excitable behavior of all neurons.

The mechanism of action potential generation

Hodgkin and Huxley realized that by manipulating the ionic concentrations, combined with the techniques of the space and voltage clamps, they could disentangle the temporal contributions of different ions assuming that they responded differently to changes in Vm.

• Removing Na+ from the bathing medium, INa becomes negligible so IK can be

measured directly. Subtracting this current from the total current yielded INa.

Disentangling the ionic currents (reproduced from Hodgkin and Huxley, 1952a)

Historical BackgroundNeher and Sakmann

Ion channels

Following Hodgkin & Huxley’s results in the 1950’s two classes of transport mechanisms competed to explain their results: carrier molecules and pores - and there was no direct evidence for either. It was not until the 1970’s that the nicotinic ACh receptor and the Na+ channel were chemically isolated, purified, and identified as proteins.

The technical breakthrough of the patch-clamp techniques developed by Neher and Sakmann (1976) allowed them to report the first direct measurement of electrical current flowing through a single channel for which they received the 1991 Nobel prize.

Patch-clamp recording from a single ACh-activated channel on a cultured muscle cell with the patch clamped to -80mV. Openings of the channel (downward events) caused a unitary 3 nA current to flow, often interrupted by a brief closing. Notice the random openings and closing, characteristic of all ion channels. Fluctuations in the baseline are due to thermal noise. Reproduced from Sigworth FJ (1983) An example of analysis in Single Channel Recording, eds. Sakmann B, Neher E. Pp 301-321. Plenum Press.

The Hodgkin-Huxley FormalismBasic Assumptions

Vm

Cm Rm

Em ENa EK

gNa gK

Im Iionic

dt

dVCtItI m

mionicm

leakKNaionic IIII

The Hodgkin-Huxley FormalismOhmic Currents

V m

C m R m

E m E N a E K

g N a g K

I N a

Currents are linearly related to the driving potential Vm

NaNaNa EtVttVgtI ,

The Nernst potential, here for Na+, gives the reversal potential ENa or the ionic battery – it is a function of the

intra- and extracellular concentrations of the ion

i

oNa Na

Na

zF

RTE lnThe Nernst Equation

Ohm’s law

The Hodgkin-Huxley FormalismVoltage-Dependence of Conductances

Experimentally recorded (circles) and theoretically calculated (smooth curves) changes in gNa and gK in the squid giant axon at 6.3C C during depolarizing voltage steps away from the resting potential (here set to 0). Inactivation is demonstrated by the decay of gNa following its initial rise. Reproduced from Hodgkin AL (1958) Ionic movements and electrical activity in giant nerve fibres, Proc R Soc Lond B 148:1-37

The Hodgkin-Huxley FormalismGating Particles

V m

C m R m

E m E N a E K

g N a g K

I N a

4

3

ngg

hmgg

KK

NaNa

Gating particles (m,h,n, etc.) were introduced to describe the dynamics of the conductances (time- and voltage-dependent) and scale a maximal conductance. They can be activating or inactivating.

The values range from 0 to 1 and (knowing what we know today with respect to ion channels) can be thought of as the percentage of channels in the activated or inactivated state.

The Hodgkin-Huxley FormalismGating particles obey first order kinetics

pi = probability (or fraction of) gate(s) i being in permissive state(1-pi) = probability (or fraction of) gate(s) i being in non-permissive state

iiiii pVpV

dt

dp)()1)((

Steady state solution

)()(

)()(, VV

VVp

ii

iti

)()(

1)(

VVV

iii

Time constant

Activation and Inactivation Kinetics Potassium Current IK

n

nn

dt

dn

n

t

ennntn

0

KKK EVngI 4Non-inactivating current

Activation particle n i.e.

Time-dependent solution

2mS/cm 36KgHodgkin and Huxley’s Parameterization

1100

1010/)10(

Vn e

VV

80/125.0 VeV

nVnVdt

dnnn )()1)((

Activation and Inactivation KineticsSodium Current INa

Activating and inactivating current

activation inactivation

NaNaNa EVhmgI 3

Gating particles m and h

2mS/cm 120NagHodgkin and Huxley’s Parameterization

110

2510/)25(

Vm e

VV

18/4 Vm eV

20/07.0 Vh eV

1

110/)30(

Vh eV

m

mm

dt

dm

h

hh

dt

dh

Activation and Inactivation KineticsGraphical Representation

n

h

m

n

h

m

Time constants (upper plot) and steady-state activation and inactivation (lower plot) as a function of the relative membrane potential V for sodium activation m (solid line) and inactivation h (long dashed line) and potassium activation n (short dashed line).

Reproduced from Koch C (1999) Biophysics of Computation, Oxford University Press.

m

Generation of Action PotentialsThe Complete Hodgkin-Huxley Model

Computed action potential in response to a 0.5 ms current pulse of 0.4 nA amplitude (solid lines) compared to a subthreshold response following a 0.35 nA current pulse (dashed lines).

(A) Time course of the two ionic currents – note their large size relative to the stimulating current

(B) Membrane potential in response to threshold and subthreshold stimuli

(C) Dynamics of the gating particles – note that the Na+ activation m changes much faster than h or n

Reproduced from Koch C (1999) Biophysics of Computation, Oxford University Press.

Generation of Action PotentialsThe Complete Hodgkin-Huxley Model

Results of the complete model:

1) Action potential generation

2) Threshold for spike initiation

3) Refractory period

For an overview on the Loligo’s axon (Giant squid acon) see

http://www.mbl.edu/publications/Loligo/squid/science.html

Activation and Inactivation Kinetics Temperature Dependence

Q10

Kinetics of channels/currents (i.e. and are strongly dependent on temperature while the peak conductance remains unchanged – be very careful when reading the methods section of a neurophysiology paper!!!

Hodgkin and Huxley recorded from the Loligo axon at 6.3C and so the rate constants shown above are for that temperature

To adjust for different temperature, and must be scaled by

10/)(10

measuredTTQ

Where the Q10 measures the increase in the rate constant for every 10C change from the temperature at which the kinetics were measured – this is typically between 2 and 4

The Hodgkin-Huxley Formalism Summary

1) The Hodgkin-Huxley 1952 model of action potential generation and propagation is the single most successful quantitative model in neuroscience

2) The model represents the cornerstone of quantitative models of neuronal excitability

3) The heart of the model is a description of the time- and voltage-dependent conductances for Na+ and K+ in terms of their gating particles (m, h, and n)

4) Gating particles can be of the activation or inactivation variety – activation implies its amplitude (from 0 to 1) increases with depolarization while the converse is true of inactivation

5) Kinetics of gating are represented either by the rate constants and or the steady-state activation/inactivation and time constant (e.g. n and n)

6) Without any a priori assumptions about action potentials, this model generates APs of appropriate shape, threshold and refractory periods (both absolute and relative)

7) Temperature can have a dramatic effect on the kinetics of gating and, ideally, should be accounted for in a model by incorporation of the Q10 scaling factor – this is an experimentally-determined

quantity