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NeuroDynamix IIConcepts of Neurophysiology Illustrated by Computer Simulations$

W. Otto Friesen and Jonathon Friesen

Print publication date: 2009

Print ISBN-13: 9780195371833

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780195371833.001.0001

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Axon Models

Axon Models

Chapter:
(p.157) II.4 Axon Models
Source:
NeuroDynamix II
Author(s):

W. Otto Friesen

Jonathon A. Friesen

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195371833.003.0011

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 Hodgkin-Huxley papers on the squid axon. Three similar models are included in this chapter: the single space-clamped axon, simultaneous simulations of several spaced-clamped 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 space-clamped axon, simultaneous simulations of several spaced-clamped axons to compare model output when parameters are altered, and a simulation of the spatially extended axon to illustrate impulse propagation.

II.4.1 Space-Clamped Axon Model

The space-clamped 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.4-1).

The model can perform simulations in either of two experimental modes. First, in voltage-clamp mode, the model provides a means for recording total membrane currents generated by controlled, step changes in the membrane potential. Second, in current-clamp 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)

                      Axon Models

Figure II.4-1 Equivalent circuit for the Axon model. There are three ionic conductance paths through the membrane: a constant leakage conductance and voltage-gated sodium and potassium conductances. Current pulses provided by the constant current generator (I stim) are used to elicit impulses or to voltage clamp the membrane.

shape of the axon impulse; and (4) the mechanisms that underlie the generation of nerve impulses, repetitive firing, threshold, refractory period, and membrane accommodation.

Glossary of Variable and Parameter Names

Graphed Variables displayed in the Scope windows (units)

  • Vm (mV): membrane potential; set by the user in voltage-clamp mode, computed by the model in current-clamp mode

  • INa (mA/cm2): sodium current through the axon membrane

  • IK (mA/cm2): potassium current through the axon membrane

  • INaK (mA/cm2): sum of the sodium and potassium currents

  • ILeak (mA/cm2): leakage current through the axon membrane

  • Iions (mA/cm2): total ionic current; the sum of I Na, I K, and I l

  • gNa (mS/cm2): calculated conductance of the axon membrane for sodium ions

  • gK (mS/cm2): 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/cm2): current applied by the Stimulator

  • gsum (mS/cm2): sum of sodium, potassium, and leakage conductances

(p.159) Main Parameters (Units)

Toggles

  • InactivationOn: “check” to include inactivation for the sodium conductance

  • VltClmpOn: “check” to activate voltage-clamp mode rather than current-clamp mode

Column 1

  • ENa (mV): sodium equilibrium potential; can be graphed as a Variable

  • gNa (mS/cm2): maximum value of the sodium conductance per unit area, this is a measure of the density of sodium channels in the squid axon

  • Vhold (mV): the steady-state membrane potential (holding potential) of the axon when in voltage-clamp mode

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

Column 2

  • EK (mV): potassium equilibrium potential; can be graphed as a Variable

  • gK (mS/cm2): maximum value of the potassium conductance per unit area, this is a measure of the density of potassium channels in the squid axon

  • Ihold (mA/cm2): current injected into the axon in current-clamp mode

  • Noise (mV): amplitude of random noise added to V m, the membrane potential

Column 3

  • ELeak (mV): leakage current reversal potential; can be graphed as a Variable

  • gLeak (mS/cm2): value of the leakage conductance per unit area, this is a measure of the density of leakage channels in the squid axon

  • Cmem (μF/cm2): specific capacitance of the axon membrane

Tabs

  • Istim tab (generates a current for injection into the axon in current-clamp mode; mA/cm2)

    The following tabs are used during voltage-clamp 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 voltage-clamp mode; mV)

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

  • Vpost tab (generates a voltage applied to the axon in voltage-clamp mode, chained to follow Vstep; mV)

II.4.2 Axon Comparisons Model

This model is an expansion of the space-clamped 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 voltage-clamp mode, the model provides a means for recording total membrane currents generated by controlled, step changes in the membrane potential. Second, in current-clamp 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.4-2).

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 voltage-clamp mode, computed by the model in current-clamp model

  • INa[n] (mA/cm2): sodium current through the axon membrane

  • IK[n] (mA/cm2): potassium current through the axon membrane

  • INaK[n] (mA/cm2): sum of the sodium and potassium currents

  • ILeak[n] (mA/cm2): leakage current through the axon membrane

  • Iions[n] (mA/cm2): total ionic current; the sum of I Na, I K, and I l

  • gNa[n] (mS/cm2): calculated conductance of the axon membrane for sodium ions

  • (p.161)
                          Axon Models

    Figure II.4-2 Equivalent circuits for the Axon Comparisons model. The model comprises several Hodgkin–Huxley circuits, each of which can have a unique set of parameter values.

  • gK[n] (mS/cm2): 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/cm2): sum of sodium, potassium, and leakage conductances

  • Istim (mA/cm2): current applied by the Stimulator

Main Parameters (Units)

Toggles

  • Inactivation: “check” to include inactivation for the sodium conductance

  • VltClmp: “check” to activate voltage mode rather than current-clamp mode

(p.162) Column 3

  • Setnoise (mV): amplitude of random noise added to V m, the membrane potential

    The following tabs are used during simulations to set specific parameters for individual axons.

Tabs

  • CntrlParms tab (sets general parameters for each axon)

  • Vhold (mV): the steady-state membrane potential (holding potential) of the axon when in voltage-clamp mode

  • Ihold (mA/cm2): current injected into the axon in current-clamp mode

  • cap (μF/cm2): specific capacitance of the axon membrane excluding gating capacitance

  • capNamax (μF/cm2): 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/cm2): maximum value of the sodium conductance per unit area, this is a measure of the density of sodium channels

  • gKmax (mS/cm2): maximum value of the potassium conductance per unit area, this is a measure of the density of potassium channels

  • setgLeak (mS/cm2): 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 current-clamp mode; mA/cm2)

  • Vpre tab (generates a voltage applied to the axon in voltage-clamp mode; mV)

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

  • Vpost tab (generates a voltage applied to the axon in voltage-clamp mode, chained to V step; mV)

II.4.3 Axon Propagation Model

This model divides an axon into a user-selectable 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.4-3). 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

  •                       Axon Models

    Figure II.4-3 Equivalent circuit for the Axon Propagation model. Four compartments are shown, but the number of compartments can be as large as 200. All model parameters are available for manipulation and are then applied identically to all compartments.

    (p.164)
  • VmPlot (mV): values for plotting the membrane potential at all points along the axon

  • gNaPlot (mS/cm2): Variable for plotting sodium conductance for each compartment

  • gKPlot (mS/cm2): Variable for plotting potassium conductance for each compartment

  • gsumPlot (mS/cm2): 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/cm2): current applied by the Stimulator

Main Parameters (Units)

Column 1

  • gNa max (mS/cm2): maximum value of the sodium conductance per unit area

  • ENa (mV): sodium equilibrium potential

  • AxonLngth (cm): length of the simulated axon

  • AxonDiameter (cm): diameter of the simulated axon

  • nalpha1: potassium activation parameter alphan1

  • nalpha2: potassium activation parameter alphan2

  • nalpha3: potassium activation parameter alphan3

  • nbeta1: potassium activation parameter betan1

  • nbeta2: potassium activation parameter betan2

  • nbeta3: potassium activation parameter betan3

  • npower: power to which n is raised

  • stim#1compart: compartment number to be stimulated with Stimulator

Column 2

  • gK max (mS/cm2): maximum value of the potassium conductance per unit area

  • EK (mV): potassium equilibrium potential

  • Cm (μF/cm2): specific capacitance of the axon membrane; without gating capacitance

  • ResistAxplsm (Ohm * cm): specific resistivity of squid axoplasm (Ri)

  • malpha1: sodium activation parameter alpham1

  • malpha2: sodium activation parameter alpham2

  • malpha3: sodium activation parameter alpham3

  • (p.165)
  • mbeta1: sodium activation parameter betam1

  • mbeta2: sodium activation parameter betam2

  • mbeta3: sodium activation parameter betam3

  • mpower [#]: power to which m is raised

  • stim#2compart: compartment number to be stimulated (may, or may not, be the same as #1)

Column 3

  • gLeak (mS/cm2): leakage conductance per unit area

  • ELeak (mV): leakage current reversal potential

  • CgateNamax (μF/cm2): capacitance per unit area due to sodium channel gating

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

  • halpha1: sodium inactivation parameter alphah1

  • halpha2: sodium inactivation parameter alphah2

  • halpha3: sodium inactivation parameter alphah3

  • hbeta1: sodium inactivation parameter betah1

  • hbeta2: sodium inactivation parameter betah2

  • hbeta3: sodium inactivation parameter betah3

  • Vmnoise (mV): amplitude of random noise added to V m

Tabs

  • stimulator tab (generates a current for injection into the designated axon compartments; mA/cm2)