Axon Models
Axon Models
Abstract and Keywords
The Axon model simulates the equations and parameters derived from experiments by Hodgkin and Huxley on the squid giant axon. Because of the exact correspondence between the equations incorporated into this model and the equations developed in the studies of Hodgkin and Huxley, this model generates graphs that mirror precisely the theoretical curves depicted in the HodgkinHuxley papers on the squid axon. Three similar models are included in this chapter: the single spaceclamped axon, simultaneous simulations of several spacedclamped axons to compare model output when parameters are altered, and a simulation of the spatially extended axon to illustrate impulse propagation.
Keywords: parameter, variable, axon, voltage clamp, ionic currents, activation, inactivation, parallel conductance model, action potential, conduction velocity
The Axon model simulates the equations and parameters derived from experiments on the squid giant axon by Hodgkin and Huxley. Because of the exact correspondence between the equations incorporated into this model and the equations developed in the modeling studies of Hodgkin and Huxley, this model generates graphs that mirror precisely the theoretical curves depicted in the Hodgkin–Huxley papers on the squid giant axon. Three similar models are included in this chapter: the single spaceclamped axon, simultaneous simulations of several spacedclamped axons to compare model output when parameters are altered, and a simulation of the spatially extended axon to illustrate impulse propagation.
II.4.1 SpaceClamped Axon Model
The spaceclamped axon impulses calculated by Hodgkin and Huxley are recreated in this model. The aim of this model, then, is to replicate Hodgkin–Huxley’s parallel conductance model in order to illustrate their experimental data and to examine the implications of their theoretical conclusions (Fig. II.41).
The model can perform simulations in either of two experimental modes. First, in voltageclamp mode, the model provides a means for recording total membrane currents generated by controlled, step changes in the membrane potential. Second, in currentclamp mode, the system provides a means of recording the membrane potential changes induced by current steps.
Specific properties illustrated with this simulation include (1) the time and voltage dependence of sodium and potassium currents, and conductances in voltage clamp; (2) the effect of temperature on rate constants and nerve impulses; (3) the relationship between conductance changes and (p.158)
Glossary of Variable and Parameter Names
Graphed Variables displayed in the Scope windows (units)

Vm (mV): membrane potential; set by the user in voltageclamp mode, computed by the model in currentclamp mode

INa (mA/cm^{2}): sodium current through the axon membrane

IK (mA/cm^{2}): potassium current through the axon membrane

INaK (mA/cm^{2}): sum of the sodium and potassium currents

ILeak (mA/cm^{2}): leakage current through the axon membrane

Iions (mA/cm^{2}): total ionic current; the sum of I _{Na}, I _{K}, and I _{l}

gNa (mS/cm^{2}): calculated conductance of the axon membrane for sodium ions

gK (mS/cm^{2}): calculated conductance of the axon membrane for potassium ions

ENa (mV): equilibrium potential for sodium current, set on the Parameters window

EK (mV): equilibrium potential for potassium current, set on the Parameters window

ELeak (mV): equilibrium potential for the leakage current, set on the Parameters window

inAct (0%–100%): inactivation of sodium conductance: (1  h) × 100

Istim (mA/cm^{2}): current applied by the Stimulator

gsum (mS/cm^{2}): sum of sodium, potassium, and leakage conductances
(p.159) Main Parameters (Units)
Toggles

Inactivation_{On}: “check” to include inactivation for the sodium conductance

VltClmp_{On}: “check” to activate voltageclamp mode rather than currentclamp mode
Column 1

E_{Na} (mV): sodium equilibrium potential; can be graphed as a Variable

g_{Na} (mS/cm^{2}): maximum value of the sodium conductance per unit area, this is a measure of the density of sodium channels in the squid axon

V_{hold} (mV): the steadystate membrane potential (holding potential) of the axon when in voltageclamp mode

Temp (°C): temperature at which the (model) experiments are conducted
Column 2

E_{K} (mV): potassium equilibrium potential; can be graphed as a Variable

g_{K} (mS/cm^{2}): maximum value of the potassium conductance per unit area, this is a measure of the density of potassium channels in the squid axon

I_{hold} (mA/cm^{2}): current injected into the axon in currentclamp mode

Noise (mV): amplitude of random noise added to V _{m}, the membrane potential
Tabs

Istim tab (generates a current for injection into the axon in currentclamp mode; mA/cm^{2})
The following tabs are used during voltageclamp simulations. These tabs can be “chained” to generate a series of voltage steps (prepotential, step (p.160) potential and postpotential) to simulate the experiments performed by Hodgkin and Huxley on the squid giant axon.

Vpre tab (generates a voltage applied to the axon in voltageclamp mode; mV)

Vstep tab (generates a voltage applied to the axon in voltageclamp mode, chained to follow prepotential; mV)

Vpost tab (generates a voltage applied to the axon in voltageclamp mode, chained to follow Vstep; mV)
II.4.2 Axon Comparisons Model
This model is an expansion of the spaceclamped Axon model to allow simultaneous graphing of membrane currents or potentials for simulations that embody differing sets of parameters. The aim for this model is to explore the consequences of employing parameter sets that differ from those adopted by Hodgkin and Huxley. For this purpose, the entire parameter set required to replicate Hodgkin–Huxley’s parallel conductance model is available for each of the model equations. On activating lessons associated with this model, the user is asked to specify the number of simultaneous simulations.
The model can perform simulations in either of two experimental modes. First, in voltageclamp mode, the model provides a means for recording total membrane currents generated by controlled, step changes in the membrane potential. Second, in currentclamp mode, the system provides a means of recording the membrane potential changes induced by current steps. Parameters and variables are similar to those described for the Axon model; however, these are now indexed to designate particular axon models (Fig. II.42).
Glossary of Variable and Parameter names
Graphed Variables displayed in the Scope windows [n designates axon #; the “0” value is not used] (units)

Vm[n] (mV): membrane potential; set by the user in voltageclamp mode, computed by the model in currentclamp model

INa[n] (mA/cm^{2}): sodium current through the axon membrane

IK[n] (mA/cm^{2}): potassium current through the axon membrane

INaK[n] (mA/cm^{2}): sum of the sodium and potassium currents

ILeak[n] (mA/cm^{2}): leakage current through the axon membrane

Iions[n] (mA/cm^{2}): total ionic current; the sum of I _{Na}, I _{K}, and I _{l}

gNa[n] (mS/cm^{2}): calculated conductance of the axon membrane for sodium ions
(p.161)

gK[n] (mS/cm^{2}): calculated conductance of the axon membrane for potassium ions

ENa[n] (mV): equilibrium potential for sodium current, set on the Parameters window

EK[n] (mV): equilibrium potential for potassium current, set on the Parameters window

ELeak[n] (mV): equilibrium potential for the leakage current, set on the Parameters window

inAct[n] (0%–100%): inactivation of sodium conductance: (1  h) × 100

gsum[n] (mS/cm^{2}): sum of sodium, potassium, and leakage conductances

Istim (mA/cm^{2}): current applied by the Stimulator
Main Parameters (Units)
Tabs

CntrlParms tab (sets general parameters for each axon)

Vhold (mV): the steadystate membrane potential (holding potential) of the axon when in voltageclamp mode

Ihold (mA/cm^{2}): current injected into the axon in currentclamp mode

cap (μF/cm^{2}): specific capacitance of the axon membrane excluding gating capacitance

capNamax (μF/cm^{2}): specific capacitance of the axon membrane due to sodium channel gating

Temp (°C): temperature at which the (model) experiments are conducted

gEParms tab (sets sodium and potassium conductance and Nernst values for each axon)

gNamax (mS/cm^{2}): maximum value of the sodium conductance per unit area, this is a measure of the density of sodium channels

gKmax (mS/cm^{2}): maximum value of the potassium conductance per unit area, this is a measure of the density of potassium channels

setgLeak (mS/cm^{2}): value of the leakage conductance per unit area, this is a measure of the density of leakage channels in the squid axon

setENa (mV): sodium equilibrium potential; can be graphed as a variable

setEK (mV): potassium equilibrium potential; can be graphed as a variable

setELeak (mV): leak current reversal potential; can be graphed as a variable

nParms tab (sets potassium conductance activation parameters)

alphan1; alphan2; alphan3; betan1; betan2; betan3; powern (power to which n is raised)

mParms tab (sets sodium conductance activation parameters)

alpham1; alpham2; alpham3; betam1; betam2; betam3; powerm (power to which m is raised)
(p.163)

hParms tab (sets sodium conductance inactivation parameters)

alphah1; alphah2; alphah3; betah1; betah2; betah3

Istim tab (generates a current that is applied to the axon in currentclamp mode; mA/cm^{2})

Vpre tab (generates a voltage applied to the axon in voltageclamp mode; mV)

Vstep tab (generates a voltage applied to the axon in voltageclamp mode, chained to prepotential; mV)

Vpost tab (generates a voltage applied to the axon in voltageclamp mode, chained to V _{step}; mV)
II.4.3 Axon Propagation Model
This model divides an axon into a userselectable number of compartments, from 1 to about 200. The length specified for the axon then determines the length of the individual compartments. The membrane potential of each compartment is computed from the Hodgkin–Huxley equations, with electrical coupling between compartments to allow for longitudinal current flow (Fig. II.43). All of the parameters of the Hodgkin–Huxley equations are available for manipulation in the main Parameters window. The axon can be stimulated by current injection at any two compartments. The Vm Variable is indexed, and can be graphed, for the individual compartments of the axon. Variable names ending in “Plot” are meant to be graphed against distance along the axon in the parametric plot window [Scope(ParametricPlot)].
Glossary of Variable and Parameter Names
Graphed Variables displayed in the Scope windows [n] (units)

Vm[n] (mV): membrane potential of any of the n compartments (n not equal to 0)

Xdstnce (cm): distance along the axon for plotting the membrane potential at points along the axon

VmPlot (mV): values for plotting the membrane potential at all points along the axon

gNaPlot (mS/cm^{2}): Variable for plotting sodium conductance for each compartment

gKPlot (mS/cm^{2}): Variable for plotting potassium conductance for each compartment

gsumPlot (mS/cm^{2}): Variable for plotting total conductance for each compartment

CondVel (m/s): velocity of impulse propagation along the axon

inactPlot (%): percentage of sodium channels that are inactivated

Istim (mA/cm^{2}): current applied by the Stimulator
Main Parameters (Units)
Column 1

gNa _{max} (mS/cm^{2}): maximum value of the sodium conductance per unit area

E_{Na} (mV): sodium equilibrium potential

Axon_{Lngth} (cm): length of the simulated axon

Axon_{Diameter} (cm): diameter of the simulated axon

n_{alpha1}: potassium activation parameter alphan1

n_{alpha2}: potassium activation parameter alphan2

n_{alpha3}: potassium activation parameter alphan3

n_{beta1}: potassium activation parameter betan1

n_{beta2}: potassium activation parameter betan2

n_{beta3}: potassium activation parameter betan3

n_{power}: power to which n is raised

stim#1_{compart}: compartment number to be stimulated with Stimulator
Column 2

gK _{max} (mS/cm^{2}): maximum value of the potassium conductance per unit area

E_{K} (mV): potassium equilibrium potential

C_{m} (μF/cm^{2}): specific capacitance of the axon membrane; without gating capacitance

Resist_{Axplsm} (Ohm * cm): specific resistivity of squid axoplasm (Ri)

m_{alpha1}: sodium activation parameter alpham1

m_{alpha2}: sodium activation parameter alpham2

m_{alpha3}: sodium activation parameter alpham3
(p.165)

m_{beta1}: sodium activation parameter betam1

m_{beta2}: sodium activation parameter betam2

m_{beta3}: sodium activation parameter betam3

m_{power} [#]: power to which m is raised

stim#2_{compart}: compartment number to be stimulated (may, or may not, be the same as #1)
Column 3

g_{Leak} (mS/cm^{2}): leakage conductance per unit area

E_{Leak} (mV): leakage current reversal potential

Cgate_{Namax} (μF/cm^{2}): capacitance per unit area due to sodium channel gating

Temperature _{°C} (°C): temperature at which the (model) experiments are conducted

h_{alpha1}: sodium inactivation parameter alphah1

h_{alpha2}: sodium inactivation parameter alphah2

h_{alpha3}: sodium inactivation parameter alphah3

h_{beta1}: sodium inactivation parameter betah1

h_{beta2}: sodium inactivation parameter betah2

h_{beta3}: sodium inactivation parameter betah3

Vm_{noise} (mV): amplitude of random noise added to V _{m}