- Email: [email protected]
The effects of static magnetic field on action potential propagation and excitation recovery in nerve
ARTICLE IN PRESS
Progress in Biophysics and Molecular Biology 87 (2005) 321–328 www.elsevier.com/locate/pbiomolbio
The effects of static magnetic field on action potential propagation and excitation recovery in nerve$ R. Hincha,c, K.A. Lindsayb, D. Noblec,, J.R. Rosenbergd a Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, UK Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK c University Laboratory of Physiology, Parks Road, Oxford OX1 3PT, UK d Division of Neuroscience and Biomedical Systems, University of Glasgow, Glasgow G12 8QQ, UK b
Available online 2 November 2004
Abstract Calculations using the Hodgkin–Huxley and one-dimensional cable equations have been performed to determine the expected sensitivity of conduction and refractoriness to changes in the time constant of sodium channel deactivation at negative potentials, as reported experimentally by Rosen (Bioelectromagnetics 24 (2003) 517) when voltage-gated sodium channels are exposed to a 125 mT static magnetic field. The predicted changes in speed of conduction and refractory period are very small. r 2004 Elsevier Ltd. All rights reserved. Keywords: Action potential; Nerve conduction; Magnetic field; Refractory period; Sodium channel kinetics
1. Introduction The preceding article (Saunders, 2004) describes a paradox in relation to nerve conduction and excitation. This is that Rosen (2003) has reported an effect of static magnetic fields on voltagegated sodium channel kinetics, whereas work by Gaffey and Tenforde (1983) and others (see Saunders, 2004, for further references) show no effect on either conduction velocity or on $
The authors are listed in alphabetical order.
Corresponding author. Tel.: +44 1865 272528; fax: +44 1865 272554.
E-mail address: [email protected] (D. Noble). 0079-6107/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pbiomolbio.2004.08.013
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refractory period. In principle, changes in ion channel kinetics should be reflected in changes in excitation, conduction and recovery. A possible explanation for the paradox is that since the largest effect reported by Rosen is an increase in the sodium channel activation (m) time constant, tm, at negative potentials only, the results might be explained by a reduction of the deactivation rate function bm ; with little or no effect on the other rate functions of sodium channel activation and inactivation, i.e., am, ah and bh. Conduction velocity must be strongly dependent on am but may be only weakly dependent on bm. Most of the early depolarization, at potentials where bm would be significant, is a passive spread of current from already excited regions (Jack et al., 1975, Figure 10.2), and the later stages of depolarization take place at potentials at which bm is negligible. Moreover, even during the early stage, the back reaction, determined by bm, will be small because m is small. In this article we describe computations and analyses to test this explanation quantitatively. We encountered a technical difficulty, which is that in the Hodgkin–Huxley formulation of channel kinetics, any change in one or both of the rate functions automatically changes the voltagedependence of the steady-state activation curve. Thus, decreasing bm to achieve an increase in tm ¼ 1=ðam þ bm Þ automatically increases the steady-state value of m, namely m1 ðV Þ ¼ am =ðam þ bm Þ; at negative potentials. Since the sodium channel current is very large compared to the ionic currents at rest, even a relatively small increase at the foot of the activation curve (which may not be detectable experimentally) will disturb the resting potential. We have used two methods to avoid this. The first (in Section 3) involves keeping the activation curve constant and using curves fitted to the voltage dependence of tm to compute the rate at which m approaches its steady-state value. The second (in Sections 4 and 5) involves adjusting the leakage current, within experimental limits, to keep the resting potential constant.
2. Summary of changes due to magnetic field In this section we review the experimental changes in the sodium channel kinetics due to the application of a magnetic field (Rosen, 2003). The (TTX-sensitive) sodium channel current was recorded in GH3 cells using the whole cell patch clamp method. The cells were exposed to a 125 mT magnetic field for 150 s and the sodium currents were recorded before, during and after the field was applied. Experiments were carried out at a range of temperatures from 25 to 37 1C, although significant (and large) changes were only observed at 35 and 37 1C. This strong temperature dependence of the effect of the magnetic field is similar to the highly non-linear dependence of tm on temperature (Rosen, 2001). The following results are from recordings in six cells at 35 and 37 1C. 1. When the field was applied, there was a ‘slight shift’ in the current–voltage relationship and less than a 5% reduction in the peak current. These changes were transient and reversed within 100 s of the field being turned off. Since these changes were small, we shall exclude them from our model. 2. The activation time constant tm changed by a significant amount at low potentials when the field was applied, and these changes persisted for at least l00 s after the field was removed. Rosen’s results are re-plotted in Fig. 1. 3. There was no significant change in the inactivation time constant th when the field was applied.
ARTICLE IN PRESS R. Hinch et al. / Progress in Biophysics and Molecular Biology 87 (2005) 321–328
323
250
% change in τm
200
150
100
50
0 -30
-25
-20
-15 -10 -5 membrane potential (mV)
0
5
10
Fig. 1. The change in the activation time constant tm as a function of membrane potential when the cell was exposed to a 125 mT magnetic field. The crosses are the experimental measurements re-plotted from Rosen (2003), and the line is a Gaussian curve fitted by least-squares regression.
It has been suggested that the change in the properties of a fast sodium channel could be due to the slow realignment of diamagnetic anisotropic molecules in the membrane (Rosen, 2003). The sodium channel activation is due to a sequence of voltage-induced changes in the S4 segment of the a-subunit (Catterall, 1988) which lies entirely in the membrane. This segment would therefore be particularly vulnerable to changes in orientation of diamagnetic anisotropic molecules in the membrane.
3. Model of change in sM due to magnetic field The effect of the magnetic field on the sodium channel kinetics can be calculated by solving the Hodgkin–Huxley equations (Hodgkin and Huxley, 1952) for a propagating nerve impulse with a modified activation time constant t^ m ðV Þ: The change induced in tm ðV Þ by the magnetic field is modelled by fitting a Gaussian curve to the experimental data to obtain a B ðV V 0 Þ2 ; (1) r exp t^ m ðVÞ ¼ tm ðVÞ 1 þ 125 2D2 where tm ðVÞ is the Hodgkin–Huxley activation time constant. The parameters V0, D and r were estimated by least-squares regression (see Fig. 1), and B and a are parameters which control the magnitude of the change in tm : When B ¼ 125; the least-squares fitting procedure gives V 0 ¼ 11:71 mV; D ¼ 6:04 mV and r ¼ 2:077: Note that if the change in tm is linear with respect to the applied field, then a ¼ 1 and B is the applied field in mT. However, Rosen did experiments for one field strength, so the change in tm ðVÞ as a function of field is unknown.
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The propagation velocity of a nerve impulse can be calculated by solving the Hodgkin–Huxley equations for the sodium, potassium and leakage currents with the cable equation. The equations are (Keener and Sneyd, 1998) CM
@V agA @2 V gNa m3 hðV V Na Þ gK n4 ðV V K Þ gL ðV V L Þ; ¼ @t 2 @x2
@m m1 ðV Þ m ¼ ; @t tm ðV Þ @h h1 ðV Þ h ¼ ; @t th ðV Þ @n n1 ðV Þ n ¼ ; @t tn ðV Þ
ð2Þ
where CM is the specific capacitance of the axonal membrane, a is the radius of the axon and gA is the conductance of its axoplasm. The remaining parameters in Eqs. (2) have their usual meanings within the Hodgkin–Huxley membrane model. Fig. 2 shows the membrane potential at a fixed point on the axon for three different values of B. Note that when the change in tm is identical to that observed by Rosen ðB ¼ 125Þ; the profile is practically unchanged compared with that when no field is applied ðB ¼ 0Þ: The relative change in the velocity as a function of the control parameter B is shown in Fig. 3A and the maximum depolarised potential is shown in Fig. 3B. Note that when the change in tm is the value reported in Rosen’s experiment ðB ¼ 125Þ; then the wave speed is reduced by 6% suggesting that the change in sodium channel kinetics has only a minimal effect on the propagation velocity. This result agrees with the experimental observations of Gaffey and Tenforde (1983) and is due to the fact that the increase in tm occurs only over a 40 B=0
membrane potential (mV)
20 B = 125 0 B = 1000 -20 -40 -60 -80 -100
0
5
10
15 20 time (ms)
25
30
35
Fig. 2. The membrane potential at a fixed position for three different values of B. Note that when B ¼ 125 (the Rosen case) the profile of the action potential hardly changes. However, when B ¼ 1000 (so that the field is eight times greater than that in the Rosen experiments) the profile is changed dramatically.
ARTICLE IN PRESS R. Hinch et al. / Progress in Biophysics and Molecular Biology 87 (2005) 321–328 1.2 maximum depolarised potential (mV)
40
1 relative velocity
325
0.8 0.6 0.4 0.2
20 0 -20 -40 -60
0 0
200
(A)
400
600
800
B
1000
0
(B)
200
400
600
800
1000
B
Fig. 3. The relative velocity (A) of the action potential when the magnetic field is applied compared to the velocity when no field is present, and the maximum depolarised potential (B). If the change in tm is linear with respect to the applied field, then the control parameter B is the applied field. In Rosen’s experiments B ¼ 125; and the wave speed is reduced by only 6%.
narrow range of potentials. However, when the increase in tm is six times greater than that observed by Rosen ðB 770Þ; there is a dramatic reduction in the maximum depolarised potential (Fig. 3B) and shape of the action potential (Fig. 2). Note that while there is a large jump in the maximum depolarised potential, the jump in relative velocity is small (E1%).
4. Model of effect of magnetic field on conduction speed Suppose now that it is specifically the m deactivation rate function bm that is influenced by applied magnetic fields. To gauge the sensitivity of the conduction speed of a propagated action potential to changes in the value of bm, a simulation exercise involving 2000 trials was conducted. Each simulation of the conduction speed used the parameter values and rate functions proposed originally by Hodgkin and Huxley with the exception of bm which was chosen so that its ratio with the functional form proposed by Hodgkin and Huxley was an N(1,s2) deviate, that is, bm ðVÞ ¼ bHH m ðV Þx;
x Nð1; s2 Þ:
(3)
Note that a change in the functional form of bm produces a change in the value of mN which in turn requires the leakage equilibrium potential to be recomputed to maintain a membrane equilibrium potential of 60 mV. In effect, the randomness in bm generates a distribution of leakage potentials. Fig. 4A illustrates the distribution of leakage potentials and Fig. 4B shows the distribution of conduction speeds for a simulation exercise based on 2000 trials in which the deactivation rate function used in each trial is chosen by expression (3) with s ¼ 0:1: The choice s ¼ 0:1 will generate widely varying functional forms for bm. For example, approximately 5% of all functional forms will be a least 20% different from the mean. By contrast, the conduction speed exhibits a much smaller variability. Its coefficient of variation (standard deviation/mean) is less than 2% and the most extreme variation achieved in the simulation exercise was approximately 6% of the mean. The conclusion is clear—the conduction
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Fig. 4. The distributions of leakage potential (A) and conduction speed (B) based on 2000 simulations of the speed of the propagated action potential are shown for an axon with membrane obeying Hodgkin–Huxley kinetics. The combined influence of magnetic field and biological variability in the axon are modelled by expression (3) with s ¼ 0:1 (i.e. 10% variability).
speed is not sensitive to large variations in the deactivation rate function for m. Similarly, the distribution of leakage potentials lies well within the range of acceptable values reported by Hodgkin and Huxley, and is also tightly controlled despite the large variability in bm. The observations in this section independently confirm those of Section 3—large changes in tm or bm are required to generate significant changes in the shape of the action potential or its conduction speed.
5. Model of effect of magnetic field on refractory period The refractory property of an axonal membrane, like the conduction speed of an action potential, is another important biophysical property of an axon that is well explained by the Hodgkin–Huxley membrane model. Our investigation of the effect of magnetic fields on refractory period will again be based on a simulation exercise involving 2000 trials in which the outcome of each trial is the refractory period of a Hodgkin–Huxley membrane with parameters and rate functions chosen by the procedure described in Section 4. The definition of refractory period used in the simulation exercise follows exactly the protocol provided by Hodgkin and Huxley for their membrane. Consider a membrane at rest in its equilibrium state. At t ¼ 0; 15 nC/ cm2 is injected into the axon, instantaneously raising its potential by 15 mV (CM=1 mF/cm2) and causing an action potential to be initiated. The membrane potential first rises to a peak value and then falls back to its equilibrium value via a hyperpolarizing phase. Charge of 90 nC/cm2, which instantaneously raises the membrane potential by 90 mV, is now injected into the axon during the re-polarizing or hyperpolarizing phases of the membrane action potential. The membrane is said to be within its refractory period if this second injection of charge fails to elicit a second action potential shortly after its application. The refractory period is therefore defined as the minimum interval of time to elapse between the initiation of a membrane action potential in the resting membrane by the injection of 15 nC/cm2 and the initiation of a second membrane action potential
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Fig. 5. The distribution of refractory period based on 2000 simulations of a Hodgkin–Huxley membrane at 6.3 1C in which the influence of magnetic fields and biological variability are modelled by the random selection of the deactivation rate function bm ðV Þ using expression (3) with s ¼ 0:1 (10% variability). The first membrane action potential is initiated in the resting membrane by the injection of 15 nC/cm2 and the second action potential is initiated by an injection of 90 nC/cm2.
by the injection of 90 nC/cm2. In this analysis, the second injection of charge will be assumed to have initiated a second membrane action potential provided the membrane potential exhibits a minimum turning value within half a millisecond of its application. Based on this criterion, Fig. 5 shows the distribution of refractory period for a Hodgkin–Huxley membrane at 6.3 1C. As in the investigation of the dependence of conduction speed on the deactivation rate function bm ðV Þ; it is clear from Fig. 5 that the distribution of refractory period is also insensitive to large variations in bm ðV Þ: Indeed the coefficient of variation is approximately half of one percent, and the maximum deviation of the refractory period from its mean value is nowhere more than 3% of the mean value. Again the conclusion is clear—the refractory period of the Hodgkin–Huxley membrane is insensitive to large variations in the deactivation rate function for m. The associated distribution of leakage potentials is illustrated in Fig. 4A.
6. Discussion The hypothesis that conduction velocity and refractoriness should not be very sensitive to the sodium channel activation time constant at negative potentials is fully borne out by the computations presented here. The conclusion is not dependent on the precise method used to
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ensure that the resting potential is not disturbed. At the field strengths used by Rosen (2003) therefore, we would not expect major changes in electrical functioning of neurons. We cannot, however, exclude an effect at large field strengths. Fig. 2 shows that action potential changes might be expected at much larger field strengths.
Acknowledgements Work in the Oxford departments is supported by the Wellcome Trust, The British Heart Foundation and the Medical Research Council.
References Catterall, W.A., 1988. Structure and function of voltage-sensitive ion channels. Science 242, 50–61. Gaffey, C.T., Tenforde, T.S., 1983. Bioelectric properties of frog sciatic nerves during exposure to stationary magnetic fields. Radiat. Environ. Biophys. 22, 61–73. Hodgkin, A.L., Huxley, A.F., 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544. Jack, J.J.B., Noble, D., Tsien, R.W., 1975. Electric Current Flow in Excitable Cells. Oxford University Press, Oxford. Keener, J.P., Sneyd, J., 1998. Mathematical Physiology. Springer, Berlin. Rosen, A.D., 2001. Nonlinear temperature modulation of sodium channel kinetics in GH3 cells. Biochim Biophysica Acta: Biomembranes 1511, 391–396. Rosen, A.D., 2003. Effect of 125 mT static magnetic field on the kinetics of voltage activated Na+ channels in GH3 cells. Bioelectromagnetics 24, 517–523. Saunders, R., 2004. Static magnetic fields: animal studies. Progress in Biophysics and Molecular Biology, this volume. doi:10.1016/j.pbiomolbio.2004.09.001
Progress in Biophysics and Molecular Biology 87 (2005) 321–328 www.elsevier.com/locate/pbiomolbio
The effects of static magnetic field on action potential propagation and excitation recovery in nerve$ R. Hincha,c, K.A. Lindsayb, D. Noblec,, J.R. Rosenbergd a Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, UK Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK c University Laboratory of Physiology, Parks Road, Oxford OX1 3PT, UK d Division of Neuroscience and Biomedical Systems, University of Glasgow, Glasgow G12 8QQ, UK b
Available online 2 November 2004
Abstract Calculations using the Hodgkin–Huxley and one-dimensional cable equations have been performed to determine the expected sensitivity of conduction and refractoriness to changes in the time constant of sodium channel deactivation at negative potentials, as reported experimentally by Rosen (Bioelectromagnetics 24 (2003) 517) when voltage-gated sodium channels are exposed to a 125 mT static magnetic field. The predicted changes in speed of conduction and refractory period are very small. r 2004 Elsevier Ltd. All rights reserved. Keywords: Action potential; Nerve conduction; Magnetic field; Refractory period; Sodium channel kinetics
1. Introduction The preceding article (Saunders, 2004) describes a paradox in relation to nerve conduction and excitation. This is that Rosen (2003) has reported an effect of static magnetic fields on voltagegated sodium channel kinetics, whereas work by Gaffey and Tenforde (1983) and others (see Saunders, 2004, for further references) show no effect on either conduction velocity or on $
The authors are listed in alphabetical order.
Corresponding author. Tel.: +44 1865 272528; fax: +44 1865 272554.
E-mail address: [email protected] (D. Noble). 0079-6107/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pbiomolbio.2004.08.013
ARTICLE IN PRESS 322
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refractory period. In principle, changes in ion channel kinetics should be reflected in changes in excitation, conduction and recovery. A possible explanation for the paradox is that since the largest effect reported by Rosen is an increase in the sodium channel activation (m) time constant, tm, at negative potentials only, the results might be explained by a reduction of the deactivation rate function bm ; with little or no effect on the other rate functions of sodium channel activation and inactivation, i.e., am, ah and bh. Conduction velocity must be strongly dependent on am but may be only weakly dependent on bm. Most of the early depolarization, at potentials where bm would be significant, is a passive spread of current from already excited regions (Jack et al., 1975, Figure 10.2), and the later stages of depolarization take place at potentials at which bm is negligible. Moreover, even during the early stage, the back reaction, determined by bm, will be small because m is small. In this article we describe computations and analyses to test this explanation quantitatively. We encountered a technical difficulty, which is that in the Hodgkin–Huxley formulation of channel kinetics, any change in one or both of the rate functions automatically changes the voltagedependence of the steady-state activation curve. Thus, decreasing bm to achieve an increase in tm ¼ 1=ðam þ bm Þ automatically increases the steady-state value of m, namely m1 ðV Þ ¼ am =ðam þ bm Þ; at negative potentials. Since the sodium channel current is very large compared to the ionic currents at rest, even a relatively small increase at the foot of the activation curve (which may not be detectable experimentally) will disturb the resting potential. We have used two methods to avoid this. The first (in Section 3) involves keeping the activation curve constant and using curves fitted to the voltage dependence of tm to compute the rate at which m approaches its steady-state value. The second (in Sections 4 and 5) involves adjusting the leakage current, within experimental limits, to keep the resting potential constant.
2. Summary of changes due to magnetic field In this section we review the experimental changes in the sodium channel kinetics due to the application of a magnetic field (Rosen, 2003). The (TTX-sensitive) sodium channel current was recorded in GH3 cells using the whole cell patch clamp method. The cells were exposed to a 125 mT magnetic field for 150 s and the sodium currents were recorded before, during and after the field was applied. Experiments were carried out at a range of temperatures from 25 to 37 1C, although significant (and large) changes were only observed at 35 and 37 1C. This strong temperature dependence of the effect of the magnetic field is similar to the highly non-linear dependence of tm on temperature (Rosen, 2001). The following results are from recordings in six cells at 35 and 37 1C. 1. When the field was applied, there was a ‘slight shift’ in the current–voltage relationship and less than a 5% reduction in the peak current. These changes were transient and reversed within 100 s of the field being turned off. Since these changes were small, we shall exclude them from our model. 2. The activation time constant tm changed by a significant amount at low potentials when the field was applied, and these changes persisted for at least l00 s after the field was removed. Rosen’s results are re-plotted in Fig. 1. 3. There was no significant change in the inactivation time constant th when the field was applied.
ARTICLE IN PRESS R. Hinch et al. / Progress in Biophysics and Molecular Biology 87 (2005) 321–328
323
250
% change in τm
200
150
100
50
0 -30
-25
-20
-15 -10 -5 membrane potential (mV)
0
5
10
Fig. 1. The change in the activation time constant tm as a function of membrane potential when the cell was exposed to a 125 mT magnetic field. The crosses are the experimental measurements re-plotted from Rosen (2003), and the line is a Gaussian curve fitted by least-squares regression.
It has been suggested that the change in the properties of a fast sodium channel could be due to the slow realignment of diamagnetic anisotropic molecules in the membrane (Rosen, 2003). The sodium channel activation is due to a sequence of voltage-induced changes in the S4 segment of the a-subunit (Catterall, 1988) which lies entirely in the membrane. This segment would therefore be particularly vulnerable to changes in orientation of diamagnetic anisotropic molecules in the membrane.
3. Model of change in sM due to magnetic field The effect of the magnetic field on the sodium channel kinetics can be calculated by solving the Hodgkin–Huxley equations (Hodgkin and Huxley, 1952) for a propagating nerve impulse with a modified activation time constant t^ m ðV Þ: The change induced in tm ðV Þ by the magnetic field is modelled by fitting a Gaussian curve to the experimental data to obtain a B ðV V 0 Þ2 ; (1) r exp t^ m ðVÞ ¼ tm ðVÞ 1 þ 125 2D2 where tm ðVÞ is the Hodgkin–Huxley activation time constant. The parameters V0, D and r were estimated by least-squares regression (see Fig. 1), and B and a are parameters which control the magnitude of the change in tm : When B ¼ 125; the least-squares fitting procedure gives V 0 ¼ 11:71 mV; D ¼ 6:04 mV and r ¼ 2:077: Note that if the change in tm is linear with respect to the applied field, then a ¼ 1 and B is the applied field in mT. However, Rosen did experiments for one field strength, so the change in tm ðVÞ as a function of field is unknown.
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The propagation velocity of a nerve impulse can be calculated by solving the Hodgkin–Huxley equations for the sodium, potassium and leakage currents with the cable equation. The equations are (Keener and Sneyd, 1998) CM
@V agA @2 V gNa m3 hðV V Na Þ gK n4 ðV V K Þ gL ðV V L Þ; ¼ @t 2 @x2
@m m1 ðV Þ m ¼ ; @t tm ðV Þ @h h1 ðV Þ h ¼ ; @t th ðV Þ @n n1 ðV Þ n ¼ ; @t tn ðV Þ
ð2Þ
where CM is the specific capacitance of the axonal membrane, a is the radius of the axon and gA is the conductance of its axoplasm. The remaining parameters in Eqs. (2) have their usual meanings within the Hodgkin–Huxley membrane model. Fig. 2 shows the membrane potential at a fixed point on the axon for three different values of B. Note that when the change in tm is identical to that observed by Rosen ðB ¼ 125Þ; the profile is practically unchanged compared with that when no field is applied ðB ¼ 0Þ: The relative change in the velocity as a function of the control parameter B is shown in Fig. 3A and the maximum depolarised potential is shown in Fig. 3B. Note that when the change in tm is the value reported in Rosen’s experiment ðB ¼ 125Þ; then the wave speed is reduced by 6% suggesting that the change in sodium channel kinetics has only a minimal effect on the propagation velocity. This result agrees with the experimental observations of Gaffey and Tenforde (1983) and is due to the fact that the increase in tm occurs only over a 40 B=0
membrane potential (mV)
20 B = 125 0 B = 1000 -20 -40 -60 -80 -100
0
5
10
15 20 time (ms)
25
30
35
Fig. 2. The membrane potential at a fixed position for three different values of B. Note that when B ¼ 125 (the Rosen case) the profile of the action potential hardly changes. However, when B ¼ 1000 (so that the field is eight times greater than that in the Rosen experiments) the profile is changed dramatically.
ARTICLE IN PRESS R. Hinch et al. / Progress in Biophysics and Molecular Biology 87 (2005) 321–328 1.2 maximum depolarised potential (mV)
40
1 relative velocity
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0.8 0.6 0.4 0.2
20 0 -20 -40 -60
0 0
200
(A)
400
600
800
B
1000
0
(B)
200
400
600
800
1000
B
Fig. 3. The relative velocity (A) of the action potential when the magnetic field is applied compared to the velocity when no field is present, and the maximum depolarised potential (B). If the change in tm is linear with respect to the applied field, then the control parameter B is the applied field. In Rosen’s experiments B ¼ 125; and the wave speed is reduced by only 6%.
narrow range of potentials. However, when the increase in tm is six times greater than that observed by Rosen ðB 770Þ; there is a dramatic reduction in the maximum depolarised potential (Fig. 3B) and shape of the action potential (Fig. 2). Note that while there is a large jump in the maximum depolarised potential, the jump in relative velocity is small (E1%).
4. Model of effect of magnetic field on conduction speed Suppose now that it is specifically the m deactivation rate function bm that is influenced by applied magnetic fields. To gauge the sensitivity of the conduction speed of a propagated action potential to changes in the value of bm, a simulation exercise involving 2000 trials was conducted. Each simulation of the conduction speed used the parameter values and rate functions proposed originally by Hodgkin and Huxley with the exception of bm which was chosen so that its ratio with the functional form proposed by Hodgkin and Huxley was an N(1,s2) deviate, that is, bm ðVÞ ¼ bHH m ðV Þx;
x Nð1; s2 Þ:
(3)
Note that a change in the functional form of bm produces a change in the value of mN which in turn requires the leakage equilibrium potential to be recomputed to maintain a membrane equilibrium potential of 60 mV. In effect, the randomness in bm generates a distribution of leakage potentials. Fig. 4A illustrates the distribution of leakage potentials and Fig. 4B shows the distribution of conduction speeds for a simulation exercise based on 2000 trials in which the deactivation rate function used in each trial is chosen by expression (3) with s ¼ 0:1: The choice s ¼ 0:1 will generate widely varying functional forms for bm. For example, approximately 5% of all functional forms will be a least 20% different from the mean. By contrast, the conduction speed exhibits a much smaller variability. Its coefficient of variation (standard deviation/mean) is less than 2% and the most extreme variation achieved in the simulation exercise was approximately 6% of the mean. The conclusion is clear—the conduction
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Fig. 4. The distributions of leakage potential (A) and conduction speed (B) based on 2000 simulations of the speed of the propagated action potential are shown for an axon with membrane obeying Hodgkin–Huxley kinetics. The combined influence of magnetic field and biological variability in the axon are modelled by expression (3) with s ¼ 0:1 (i.e. 10% variability).
speed is not sensitive to large variations in the deactivation rate function for m. Similarly, the distribution of leakage potentials lies well within the range of acceptable values reported by Hodgkin and Huxley, and is also tightly controlled despite the large variability in bm. The observations in this section independently confirm those of Section 3—large changes in tm or bm are required to generate significant changes in the shape of the action potential or its conduction speed.
5. Model of effect of magnetic field on refractory period The refractory property of an axonal membrane, like the conduction speed of an action potential, is another important biophysical property of an axon that is well explained by the Hodgkin–Huxley membrane model. Our investigation of the effect of magnetic fields on refractory period will again be based on a simulation exercise involving 2000 trials in which the outcome of each trial is the refractory period of a Hodgkin–Huxley membrane with parameters and rate functions chosen by the procedure described in Section 4. The definition of refractory period used in the simulation exercise follows exactly the protocol provided by Hodgkin and Huxley for their membrane. Consider a membrane at rest in its equilibrium state. At t ¼ 0; 15 nC/ cm2 is injected into the axon, instantaneously raising its potential by 15 mV (CM=1 mF/cm2) and causing an action potential to be initiated. The membrane potential first rises to a peak value and then falls back to its equilibrium value via a hyperpolarizing phase. Charge of 90 nC/cm2, which instantaneously raises the membrane potential by 90 mV, is now injected into the axon during the re-polarizing or hyperpolarizing phases of the membrane action potential. The membrane is said to be within its refractory period if this second injection of charge fails to elicit a second action potential shortly after its application. The refractory period is therefore defined as the minimum interval of time to elapse between the initiation of a membrane action potential in the resting membrane by the injection of 15 nC/cm2 and the initiation of a second membrane action potential
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Fig. 5. The distribution of refractory period based on 2000 simulations of a Hodgkin–Huxley membrane at 6.3 1C in which the influence of magnetic fields and biological variability are modelled by the random selection of the deactivation rate function bm ðV Þ using expression (3) with s ¼ 0:1 (10% variability). The first membrane action potential is initiated in the resting membrane by the injection of 15 nC/cm2 and the second action potential is initiated by an injection of 90 nC/cm2.
by the injection of 90 nC/cm2. In this analysis, the second injection of charge will be assumed to have initiated a second membrane action potential provided the membrane potential exhibits a minimum turning value within half a millisecond of its application. Based on this criterion, Fig. 5 shows the distribution of refractory period for a Hodgkin–Huxley membrane at 6.3 1C. As in the investigation of the dependence of conduction speed on the deactivation rate function bm ðV Þ; it is clear from Fig. 5 that the distribution of refractory period is also insensitive to large variations in bm ðV Þ: Indeed the coefficient of variation is approximately half of one percent, and the maximum deviation of the refractory period from its mean value is nowhere more than 3% of the mean value. Again the conclusion is clear—the refractory period of the Hodgkin–Huxley membrane is insensitive to large variations in the deactivation rate function for m. The associated distribution of leakage potentials is illustrated in Fig. 4A.
6. Discussion The hypothesis that conduction velocity and refractoriness should not be very sensitive to the sodium channel activation time constant at negative potentials is fully borne out by the computations presented here. The conclusion is not dependent on the precise method used to
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ensure that the resting potential is not disturbed. At the field strengths used by Rosen (2003) therefore, we would not expect major changes in electrical functioning of neurons. We cannot, however, exclude an effect at large field strengths. Fig. 2 shows that action potential changes might be expected at much larger field strengths.
Acknowledgements Work in the Oxford departments is supported by the Wellcome Trust, The British Heart Foundation and the Medical Research Council.
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