Donald Coxeter (1907 - 2003) Santiago Ramón y Cajal (1852 ... · stant membrane p otential V is...

Santiago Ramón y Cajal (1852-1934)Donald Coxeter (1907 - 2003)

Santiago Ramón y Cajal (1852-1934)

A mathematician is a blind man in a dark room looking for a black cat which isn't there

Donald Coxeter (1907 - 2003)

Santiago Ramón y Cajal (1852-1934)

Every new body of discovery is mathematical in form because there is no other guidance we can have

A mathematician is a blind man in a dark room looking for a black cat which isn't there

Donald Coxeter (1907 - 2003)

The action potential

Action potentials were characterised by Hodgkin & Huxley in squid. They are a rapid regenerative change in membrane potential that is initially triggered by depolarization. The depolarization opens voltage-gated sodium channels and triggers a transient sodium current. Then as the membrane potential becomes positive, slower voltage-gated potassium channels open to repolarize the membrane to the resting potential. There is a refractory period of insensitivity to depolarization before another action potential can be generated. The action potential threshold is -50 to -55 mV (approached from more negative values) in many cells.

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The Hodgkin-Huxley model

The conceptual idea behind current electrophysiological models is that cell membranes behave like electrical circuits. The basic circuit elements are 1) the phospholipid bilayer, which is analogous to a capacitor in that it accumulates ionic charge as the electrical potential across the membrane changes; 2) the ionic permeabilities of the membrane, which are analogous to resistors in an electronic circuit; and 3) the electrochemical driving forces, which are analogous to batteries driving the ionic currents. These ionic currents are arranged in a parallel circuit. Thus the electrical behaviour of cells is based upon the transfer and storage of ions such as K+ and Na+.

The equivalent electrical circuit for the Hodgkin-Huxley model of squid giant axon. Thecapacitance is due to the phospholipid bilayer separating the ions on the inside and the outsideof the cell. The three ionic currents, one for Na+, one for K+, and one for a non-specific leak,are indicated by resistances. The conductances of the Na+ and K+ currents are voltagedependent, as indicated by the variable resistances. The driving force for the ions is indicated bythe symbol for the electromotive force, which is given in the model by the difference betweenthe membrane potential, V = Vin − Vout and the reversal potential.

Batteries and the Nernst potential

Ion specific pores create voltage differences. Balancing the electrical and osmotic forces gives theNernst potential for each ionic species:

∆V ∝ log[ion]out

[ion]in.

This is the emf (or battery) that drives each ionic species in the electrical circuit shown above.Note that the resting potential is a weighted sum of individual ionic Nernst potentials.

The resting membrane potential is the point at which there is no net current across the membrane.The Goldman-Hodgkin-Katz (GHK) equation gives this value – see Appendix A.

A dynamical electrical circuit

The standard dynamical system for describing a neuron as a spatially isopotential cell with con-stant membrane potential V is based upon conservation of electric charge, so that

CdV

dt= Iion + Iapp.

where C is the cell capacitance, Iapp the applied current and Iion represents the sum of individualionic currents:

Iion = −gK(V − VK) − gNa(V − VNa) − gL(V − VL).

In the Hodgkin-Huxley model the membrane current arises mainly through the conduction ofsodium and potassium ions through voltage dependent channels in the membrane. The contri-bution from other ionic currents is assumed to obey Ohm’s law (and is called the leak current).The gK, gNa and gL are conductances (conductance=1/resistance).

Voltage gated channels

Channels are known to have gates that regulate the permeability of the pore to ions (so thatgK = gK(V) and gNa = gNa(V)). These gates can be controlled by membrane potential

3

Batteries and the Nernst potential: Ion specific pores create voltage differences. Balancing the electrical and osmotic forces gives the Nernst potential for each ionic species:

This is the emf (or battery) that drives each ionic species in the electrical circuit shown above. Note that the resting potential is a weighted sum of individual ionic Nernst potentials.


Balance currents (conserve charge)

Input current = capacitive current + ohmic currents

Q = C x V

ohmic current = V/R

= g x V

current = rate of change of Q= C x rate of change of V

C

R=1/g

g is conductanceR is resistanceC is capacitance

Symbol for rate of change of V





[ion]in.





CdV

dt= Iion + Iapp.






3

Putting it all together





[ion]in.





CdV

dt= Iion + Iapp.






3





[ion]in.





CdV

dt= Iion + Iapp.






3


Channels are known to have gates that regulate the permeability of the pore to ions (so that gK=gK(V) and gNa=gNa(V)). These gates can be controlled by membrane potential (and are known as voltage gated channels). Gates are often modelled with a simple two state (open/closed) process, each described by a first order nonlinear ordinary differential equation. Gates can either be activating or inactivating.

The sigmoidal function is related to the fraction of channels in the open state.

(and are known as voltage gated channels). Gates are often modelled with a simple two state(open/closed) process, each described by a first order nonlinear ordinary differential equation.Gates can either be activating or inactivating.

Activation and inactivation. The sigmoidal function f∞ is related to the fraction of channels inthe open state.

The great insight of Hodgkin and Huxley was to realise that gK depend upon four activationgates:

gK = gKn4,

whereas gNa depends upon three activation gates and one inactivation gate:

gNa = gNam3h.

The stead-state value of the gate is denoted f∞(V) (where f stands for m,n, h).

The parameters in conductance-based models (i.e the detailed choice of f∞(V) etc.) are deter-mined from empirical fits to voltage-clamp experimental data, assuming that the different currentscan be adequately separated using pharmacological manipulations and voltage-clamp protocols.The details (fitted forms) of the Hodgkin-Huxley are listed for completeness in Appendix B.

For an animation of channel gating and a simultaneous simulation of the Hodgkin-Huxley modelseehttp://arrhythmia.hofstra.edu/java/neuron/apneuron.html

-80

-40

0

40

0 20 40 60 80 100t

V

An example of a periodic spike train that can be generated by the Hodgkin-Huxley model underconstant current injection.

In summary, the basic assumptions in all conductance-based models are: the different ion channelsin the cell membrane are independent from each other, activation and inactivation gating variables

4

The great insight of Hodgkin and Huxley

... was to realise that gK depend upon four activation gates:




gK = gKn4,


gNa = gNam3h.




-80

-40

0

40

0 20 40 60 80 100t

V



4





gK = gKn4,


gNa = gNam3h.




-80

-40

0

40

0 20 40 60 80 100t

V



4

The stead-state value of the gate is denoted (where f stands for m,n,h).




gK = gKn4,


gNa = gNam3h.




-80

-40

0

40

0 20 40 60 80 100t

V



4

-80

-40

0

40

0 20 40 60 80 100t

V

In summary, the basic assumptions in all conductance-based models are: the different ion channels in the cell membrane are independent from each other, activation and inactivation gating variables are voltage-dependent and independent of each other for a given ion channel type,each gating variable follows first-order kinetics, and the model cell compartment is isopotential.

I

0

0.04

0.08

0.12

0.16

0 20 40 60 80 100I

f

Mathematical details of the Hodgkin-Huxley model

Appendix A: The Goldman-Hodgkin-Katz

The Goldman-Hodgkin-Katz (GHK) voltage equation is used in cell membrane physiology todetermine the potential across a cell’s membrane taking into account all of the ions that arepermeant through that membrane.

The GHK equation is a variation on the Nernst equation. The Nernst equation can essentiallycalculate the membrane potential of a cell when only one ion is permeant, as long as the con-centrations of that ion both inside and outside the cell are known. The Nernst equation cannot,however, deal with cells having permeability to more than one ion. The GHK voltage equation isnot exact and makes assumptions with regards to the mechanism of diffusion, which influencesthe final result.

The GHK equation for N positive ionic species and M negative is

Em =RT

Fln

(∑Ni PM+

i[M+

i ]out +∑M

j PA−j[A−

j ]in∑Ni PM+

i[M+

i ]in +∑M

j PA−j[A−

j ]out

)It is similar to the Nernst equation but has a term for each permeant ion.

• Em is the membrane potential.

• Pion is the permeability for that ion.

• [ion]out is the extracellular concentration of that ion.

• [ion]in is the intracellular concentration of that ion.

• R is the ideal gas constant.

• T is the temperature in Kelvin.

• F is the Faraday constant.

Appendix B: Mathematical details of the Hodgkin-Huxleymodel

The gating variables satisfy the nonlinear ordinary differential equations

m =m∞(V) − m

τm(V), n =

n∞(V) − n

τn(V), h =

h∞(V) − h

τh(V).

The six functions τX(V) and X∞(V), X ∈ {m,n, h}, are obtained from fits with experimentaldata. It is common practice to write

τX(V) =1

αX(V) + βX(V), X∞(V) = αX(V)τX(V)

The details of the final Hodgkin-Huxley description of nerve tissue are completed with:

7

αm(V) =0.1(V + 40)

1 − exp[−0.1(V + 40)]αh(V) = 0.07 exp[−0.05(V + 65)]

αn(V) =0.01(V + 55)

1 − exp[−0.1(V + 55)]βm(V) = 4.0 exp[−0.0556(V + 65)]

βh(V) =1

1 + exp[−0.1(V + 35)]βn(V) = 0.125 exp[−0.0125(V + 65)]

C = 1µF cm−2, gL = 0.3mmho cm−2, gK = 36mmho cm−2, gNa = 120mmho cm−2, VL =−54.402mV, VK = −77mV and VNa = 50mV. (All potentials are measured in mV, all times inms and all currents in µA per cm2).

XPP code

par J=10par VK=-77par VNa=50

C=1gL=0.3gK=36gNa=120VL=-54.402

v(0)=-75m(0)=0.08n(0)=0.7h(0)=0.1

alpham(x)=0.1*(x+40)/(1-exp(-0.1*(x+40)))alphah(x)=0.07*exp(-0.05*(x+65))alphan(x)=0.01*(x+55)/(1-exp(-0.1*(x+55)))betam(x)=4.0*exp(-0.0556*(x+65))betah(x)=1.0/(1+exp(-0.1*(x+35)))betan(x)=0.125*exp(-0.0125*(x+65))taum(x)=1.0/(alpham(x)+betam(x))taun(x)=1.0/(alphan(x)+betan(x))tauh(x)=1.0/(alphah(x)+betah(x))minfty(x)=alpham(x)*taum(x)ninfty(x)=alphan(x)*taun(x)hinfty(x)=alphah(x)*tauh(x)

v’=1.0/C*(-gL*(v-VL)-gK*n*n*n*n*(v-VK)-gNa*h*m*m*m*(v-VNa)+J)m’=1.0/(taum(v))*(minfty(v)-m)n’=1.0/(taun(v))*(ninfty(v)-n)h’=1.0/(tauh(v))*(hinfty(v)-h)

aux INa=-gNa*h*m*m*m*(v-VNa)aux IK=-gK*n*n*n*n*(v-VK)

8

m =dm

dt

A modern application: dynamic clamp

Dynamic-clamp conductances act in parallel with thenormal membrane conductances of the neuron, and theinteraction between the added and existing conductancesis what makes such manipulations interesting. In theexample shown in Figure 1b, current pulses of constantamplitude were introduced to show that the dynamicclamp was modifying the conductance of the neuron(Figure 1b, bottom) in exactly the same way as a bathapplication of GABA (Figure 1b, top). The dynamic clampmimics both the GABA-induced hyperpolarization and thereduction in the voltage response to constant-amplitudecurrent pulses caused by the GABA conductance. In arelated approach, the dynamic clamp has been used to addartificial GABA conductances in thalamocortical relaycells to elucidate the role of GABA-mediated inhibitorypostsynaptic potentials (IPSPs) in rebound burst firingand burst inhibition [15,16].

In addition to being added, conductances can, with somerestrictions, be subtracted using the dynamic clamp.Figure 1c shows an example. The leakage conductanceintroduced by the electrode penetration required forintracellular recordings made with sharp electrodes is apotential source of distortion of the natural activity of therecorded neuron. Figure 1c shows an example in which thedynamic clamp was used to simulate a negative conduc-tance designed to cancel out the impact of the leakageintroduced by electrode penetration [17]. Addition of anartificial leak conductance had been shown previously toswitch leech heartbeat interneurons from an active statewith high-frequency bursting to an inactive state [18].Because of this sensitivity of bursting to additional leakconductance, the electrode leak was removed by thedynamic clamp. The bursting activity that is the naturalmode of operation for this neuron was revealed only afterthe leakage conductance introduced by electrode pene-tration was subtracted using the dynamic clamp.

Taken together, dynamic-clamp studies that simulatevoltage-independent conductances in different prepar-ations demonstrate important roles for seemingly simpleleak and ligand-gated currents in shaping neural activity.The importance of voltage-independent conductances isfurther supported by reports that dynamic-clamp simu-lated leak current can increase motoneuron spiking in themammalian spinal cord [19] and that adding a Ca2þ

window current or subtracting leak current can renderthalamocortical neurons bistable [20].

Figure 1. Using the dynamic clamp to simulate voltage-independent conductances.(a) Schematic of the experimental configuration. The dynamic clamp computesthe current, I, flowing through a voltage-independent conductance, g, as g multi-plied by the instantaneous driving force, V–E, where E is the reversal potential andV is the membrane potential. In every cycle of dynamic-clamp operation, V ismeasured and fed into the computer, I is computed based on the momentaryvalue of V, and I is injected into the cell. Voltage measurement and current injec-tion can be made through the same electrode with discontinuous clamp tech-niques, or through two separate electrodes. (b) Voltage traces recorded from acultured crab stomatogastric neuron during 30 s bath application of 0.1 mM GABA(top) and during dynamic-clamp injection of an exponentially rising (t ¼ 5 s) andfalling (t ¼ 15 s) GABA conductance with a reversal potential of 275 mV (bottom).The starts of bath and dynamic-clamp application are indicated by asterisks.During both runs, current pulses of 20.5 nA were applied every 3 s to illustrate thechange in input conductance. Adapted, with permission, from Ref. [2]. (c) Voltagetrace from a leech heart interneuron before and during injection of a negative leakconductance of 26 nS with a reversal potential at 0 mV. The leak subtraction com-pensates for the effect of sharp microelectrode penetration, which suppressesbursting. Adapted, with permission, from Ref. [17] q (2002) by the Society forNeuroscience.

I = g*(V–E)

V

I

GABA

Dynamic clamp

*V

V

*

–46 mV

–46 mV

(a)

(b)

(c)

10 mV

4 s

Simulatingvoltage-independent

conductances

–50 mV

V

10 mV2 s

Dynamic clamp leak subtraction

V

TRENDS in Neurosciences

Figure 2. Adding or subtracting voltage-dependent conductances. (a) Schematic ofthe experimental configuration. The dynamic-clamp current is computed as inFigure 1, but in this case the conductance, g, varies with time and depends on themembrane potential, V. (b) Each panel shows 65 superimposed spikes from anAplysia R20 neuron in response to 7 Hz current pulse injection. In control con-ditions, the action potential is initially narrow and broadens during the spike train(top-left). Spike broadening is abolished in 50 mM tetraethylammonium (TEA) and10 mM 4-aminopyridine (4-AP; top-middle) and rescued when an A-type and adelayed-rectifier Kþ current are added with the dynamic clamp (top-right). In adifferent cell (bottom), the action of the blockers was approximated by subtractingthese two conductances with the dynamic clamp. Adapted, with permission, fromRef. [21]q (1996) by the Society for Neuroscience.

TRENDS in Neurosciences

I

VI = g(V)*(V–E)

V

Cell 1

Cell 2

(a)

(b)

Adding or subtractingvoltage-gatedconductances

Dynamic-clampsubtraction

Control

Control

Pharmacologicalblock

Dynamic-clamprescue

40 mV

20 ms

Review TRENDS in Neurosciences Vol.27 No.4 April 2004220

www.sciencedirect.com

Donald Coxeter (1907 - 2003) Santiago Ramón y Cajal (1852 ... · stant membrane p otential V is...

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