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Modelling of the excitation and the propagation of nerve impulses by natural and artificial stimulations
.,,¢~ M.ATHEMATICS AND ~ ) COMPUTERS IN SIMULATION ELSEVIER
Mathematics and Computers in Simulation 39 (1995) 589-595
Modelling of the excitation and the propagation of nerve impulses by natural and artificial stimulations Frank Rattay Technical University Vienna, A-1040 l,Tenna, Austria
Abstract
The functional properties of a neuron can be simulated by electrical circuits. This technique is of use for natural as well as for electrical stimulation. In particular, a new model is proposed which shows the influence of the myelinated parts of the neuron.
Introduction In contrast to other cells neurons are remarkable because of their shapes showing a very complicated fine structure and because of the electrical properties of the membrane which separates the ionic components of the intracellular and the extracellular fluids. In the resting state the neuron's inside potential is constant with a value of about - 7 5 mV compared to that of the outside. In general, however, this membrane voltage essentially depends on location. As an example, we assume that a neuron generates a spike train consisting of 250 spikes/s. These action potentials are the answer to synaptic activities in the dendritic region and at the cell body. An action potential with a duration of 1 ms will propagate with a velocity of 50 m / s ( = 50 m m / m s ) along the nerve fibre (axon) which may have a diameter of 10 ~ m and a length of 50 cm. Thus, 5 cm intervals of excited membrane regions which are separated by 15 cm regions with resting state voltages are traveling along the nerve fibre from the cell body towards the branching part of the nerve fibre (output region). Of course, membrane voltage is also a function of space within the dendritic area and in the different branches in the terminal region. The main elements of a neuron are the soma (cell body), the dendrites and the axon. Dendrites and soma are usually covered with synapses from other neurons (input zone), whereas the task of the axon is the transport of neural information via action potentials into distant regions. The mammalian axons with diameters from 1-20 ~ m are usually myelinated, i.e. only in the nodes of 'Ranvier', which have a length of about 1 ~ m the membrane is active. The internode is covered with many layers of membranes of Schwann cells which are tightly wrapped around the axon. The internode has a length of about 100 times the fibre diameter. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4754(95)00122-7
590
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
The insulating properties of the internode allow a 20 ~ m mammalian fibre to obtain a higher velocity of propagation as it is reached in the unmyelinated giant axons of a squid with diameters up to 1 mm. The purpose of this paper is to discuss several models which finally can be used to predict the input-output relations of a single neuron.
Membrane models A patch of the membrane of a neuron can be described by an electrical circuit consisting of a voltage source, which accounts for the resting voltage, a capacitance (1-3 ~ F / c m 2) and a resistance. The conductance of the active membrane changes within a large range depending on the open/closed status of the ionic channels. The membrane models which are mostly used are the Hodgkin-Huxley model for unmyelinated (parts of) neurons (Box 1, see Fig. 1) and the CRRSS model for the nodal membrane of myelinated mammalian nerve fibres (Box 2, see Fig. 1). Further discussions about membrane models can be found, e.g. in [9-11].
Boxl I
HODGKIN-HUXLEY MODEL
= [-9~omsh(V
- VNo) - g K ~ ( V
- VK)
(HH-1) (HH-2) (HH-3) (HH-4)
--gL(V -- VL) + i,t]/c
rh = [ - ( a m + tim)" m + am]- k h = [ - ( a n + ft,). n + an]" k = [--(ah + fib)" h + ah]" k
Box 2 I
CRRSS MODEL
SWEENEY el al. (1987) transformed the original data ofOHIU, RITCHIE, ROGEKT & STAGG (1979) from experimental temperature of T = 14°C to T = 37°C. We name the following equations after the investigators the CRRSS model.
with the coefficient k for temperature T (in °C) k = 3 °1T-°63 (HH-5
= [-gNom2h(Y
- YNo) - gL(V -
CI:LKSS-1) CRRSS-2)
,'h = - ( a . ~ + Z . , ) - m + a m /~ =
and am
~
2.5-0.1V exp(2.5-O.1V)-I 1-0.1V
'
a~ = lo.(~p0_o.tv)_o , = 0.07.
V~e,t = -70[mV],
Vg =
-12[mY],
INs = 120[kf2-lcm-2], gL = 0 . 3 [ k ~ - I c m - 2 ] , m(0) = 0.05, h(0) = 0.6
e = = 4.,
p
z . = 0.125.e
p
flh -~
1 ezp(3-O.1V)+l
VL) + i , , ] / c
--(ah + &). h + ah
(tigress-a)
with am
97+0.363V
= l+,~p('~-~)' ph
VN,~ = ll5[mV], a h = , ~ p ( , , ? s ) , VL = lO.6[mV], v~.., = - 8 0 [ m V ] , gK = 36[kf2-1cm-~], vL = 0 . 0 1 [ m v ] , c = l[#F/cm2], gga = 1445[kf~-lcm-2], n(0) = 0.32, c = 2 . 5 ~ F / c m 2 ] , m ( 0 ) = 0.003,
°gre,
Z"~ = ,xv( . .,...,. . ) ' flh :
15.6
VNa = llS[mV], gL = 128[kg~-tcm-2],
h(0) = 0.75
Fig. 1. Comparison of equations for membrane dynamics for unmyelinated and myelinated parts of a neuron. V denotes membrane voltage; VNa, Vr, VL voltages across the membrane, caused by different (sodium, potassium, unspecific) ionic concentrations inside and outside of the neuron; gNa, g r , gL maximum conductance of sodium, potassium and leakage per cm 2 of membrane; m, n, h probabilities for ionic membrane gating processes; a,/3 opening and closing rates for ionic channels. Note the high conductances and the missed potassium currents in the myelinated mammalian node as described by the CRRSS model.
F.Rattay/ MathematicsandComputersin Simulation39(1995)589-595
591
Lumped circuits to simulate a neuron
In order to simulate neural reactions a neuron should be parted into segments and every segment is represented by an electrical circuit. Fig. 2 shows a simplified neuron which only consists of two dendrites, a cell body and a piece of the nerve fibre. We have approximated all the structures by cylinders. The following calculations are done with the reduced voltage V with V-
Vmembrane -- Vrest.
With the spatial neural model described below we can simulate both the influence of an artificial electric field (which causes an extracellular potential Ve) as well as natural situations (Ve = 0). For simplicity we assume that segmentation length Ax is constant, every segment is approximated by a cylinder of diameter d. and the center of the nth segment represents the average inside potential V~,.. The outside potential (caused by an applied electrical field) is Ve,,,, and the voltage across the membrane is
Vn=Vi,n-Ve, n.
(1)
]
I
I
I
I
v,,°+,
MEMBRANE
I
I
Ve,n
V(,. ,=~
ff--l._l--=l_
"X
R,,-1
R,~
2
2
,Xl,q
Rn2 I R,~+12 Az
Fig. 2. A simple neuron is represented by cylinders with different diameters (top). The circle marks the place where the bottom diagram is situated. Segments with the same length but with different diameters are used. The electrical components of the nth segment are the intracellular resistances (along the axis), and the capacity and conductance of the membrane.
Ax,
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
592
The currents of the nth segment are caused by capacity of membrane (C,), membrane conductance (G n) and the axonal resistances to the neighboured segments (e.g. R J 2 + R n + 1//2): dg n Cn " - .-[- Gn " V n --[-
dt
Vi,n -- Vi,n_ 1
R J 2 + R , _ 1/2
--I-
Vi,n - V/,n+ 1
R,,/2 + R n +1/2
=0.
(2)
In order to see the influence of an extracellular electrical field we use (1) and obtain an ordinary differential equation for every segment:
dVn _Gn. Vn_ ( Vn-- Vn-1 V n - V,,+I ) Cn" dt = R,,/2 + R , _ l / 2 + R , / 2 + R,,+l/2 --I
V e , n - Ven-l__~s.2._.2 .~_
[ R J 2 + R n _ 1/2
V e , n - Ve,n+l
R J 2 + R,+ i / 2
(3)
Note: In the case of Ve = 0 (natural situation) the expression [ . . . ] in Eq. (3) vanishes. Knowing the specific capacity c and conductances g of the membrane as well as the resistivity of axoplasm pi we can approximate the following variables which depend on the number N of layers of myelin which are wrapped around the cylinders with diameter d, and length Zlx: Ax "d, "rr Cn =
G.=
N Ax . d~ "rr
N
"c,
(4)
• g,
(5)
Ax'4 R~ = d2rr "pi.
(6)
Eqs. (3)-(6) are suited to simulate the dendrite and the cell body of a neuron. The situation in the nerve fibre (axon) is more complicated: The nodes of Ranvier only have a length L of 1 lxm, and they are very short compared to the internOde which is up to 2 mm long. Even for qualitative results it is important to use a good model for the ionic currents, because they become the dominant driving force in the excitation process when threshold is reached. Secondly, even in the passive case (small distortion of the resting potential) the electric properties of the node are of importance, because it is not insulated by the many fatty layers of myelin. When the axon is segmented (e.g. 10 elements from node to node), the segments which include a node have to be calculated with Eqs. (4a) and (5a). Ax. d n • rr
Cn = G~ =
N
"c + Ldn~-C,
(4a)
Ax . d, • rc N "g + iiLd'rc"
(5a)
For every segment the ionic current density ii in (5a) can be calculated with the help of a set of differential equations as it was first described by Hodgkin and Huxley [5]. For the simulation of the mammalian node the CRRSS model of Box 2 [1,14] or one of the models described in [9] or [11] should be used.
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
593
For subthreshold analysis we can approximate the membrane conductance of the node by constant g. Of special interest is the case of a fibre with constant diameter. In this case we obtain Eq. (7) from Eqs. (3)-(6), for segments with active membranes Eq. (7a) has to be used.
dV,, d .N c" d----f- = -g.V~ + 4 ( a x ) 2 p i ( V . _ , - 2 V n
d .N + V,,+l) + 4(Ax)2pi[Ve,n_l-2Ve,n
q-Ve,n+l] (7)
L N ) dV n -g,Vn LN d" N c 1 + ~x d--t- = -i~--~x + 4(ax)2p,
- 2V,, + V,,+l )
d.N +
2Ve, n + Ve,n+l].
(7a)
The influence of the extracellularly applied electrical field on the nth segment of a nerve or muscle fibre is represented by that term in Eqs. (7) and (7a) which contains Ve. It is called the activating function f and it is proportional to the second difference quotient of Ve, which is a function of the fibre's length coordinate:
d ' N Ve,,,_I-- 2Ve,,, + V~,,,+I f = 40~-
(AX)2
(8)
In order to obtain an action potential it is necessary to reach threshold voltage in a segment with an active membrane. This is possible for segments where f is positive. The upper panel of Fig. 3 shows the effect of the activating function for a nonmyelinated fibre, computed with fine segmentation. A point electrode above the central part of the fibre produces an electric field by a negative current pulse. A positive electrode current would cause a result with inverse polarity, i.e. the fibre would be excited at two positions, however, this excitation is much weaker, which gives an explanation why cathodic stimulation is easier.
Discussion
Of course, it is possible to simulate parts of the neuron (e.g. the nerve fibre) with even more accurate models, see e.g. [3]. However, it seems that deviations from experimental findings rather come from neglecting the electrical properties of myelin. With our preliminary results regarding subthreshold reactions we have obtained thresholds which only have about half of the value expected from the McNeal model, which was generally used. It must be expected that many computed results in the field of electrical stimulation have to be recalculated. There exist only few data considering the ionic channel dynamics in different neural regions [6,7,12]. Especially in the input area there is a lack of information how to model the membrane behaviour. Nevertheless, many authors have obtained results with membrane models of the
594
F. Rattay ~Mathematics and Computers in Simulation 39 (1995) 589-595
0
0
1
2
3
4
5 6 7 $ 9 10 NODE Fig. 3. Influence of myelin. Top panel: The excitation of an unmyelinated fibre at the end of a subthreshold negative 100 I~s current pulse is a picture of the activating function. Comparison of the computed subthreshold reaction of a 11 node axon with an unmyelinated fibre of same diameter shows that myelinated axons are much more excitable. The curve of the top panel is also included as the smallest one in the lower panel. The excitability increases considerably for myelinated fibres, even if it is assumed that the internode is a perfect insulator (second smallest result). Further increase is seen when N grows up ( N = 100, 200, 300).
Hodgkin Huxley type. Different software is used to simulate single neurons and neural nets, see e.g. [4,13]. Some of the authors, e.g. [2], reduce the reaction of a single neuron to a combination of a simplified threshold process with a refractory function, and they do not look at the detailed membrane reactions because of the many elements involved.
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
595
References [1] S.Y. Chiu, J.M. Ritchie, R.B. Rogart and D. Stagg, A quantitative description of membrane currents in rabbit myelinated nerve, J. Physiol. 292 (1979) 149-166. [2] W. Gerstner and J.L. van Hemmen, Associative memory in a network of 'spiking' neurons, Network 3 (1992) 139-164. [3] J.A. Halter and J.W. Clark, A distributed parameter model of the myelinated nerve fiber, J. Theor. Biol. 148 (1991) 345-382. [4] M. Hines, A program for simulation of nerve equations with branching connections, Int. J. Biomed. Comput. 24 (1989) 55-68. [5] A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952) 500-544. [6] C. Koch and I. Segev, eds., Methods in Neural Modeling (MIT Press, Cambridge, MA, 1989). [7] R.J. MacGregor, Neural and Brain Modeling. (Academic Press, San Diego, CA, 1987). [8] D.R. McNeal, Analysis of a model for excitation of myelinated nerve, IEEE Trans. Biomed. Eng. 23 (1976) 329-337. [9] F. Rattay, Simulation of artificial neural reactions produced with electric fields, Simulation Practice Theory 1 (1993) 137-152. [10] F. Rattay, Analysis of models for extracellular fiber stimulation, IEEE Trans. Biomed. Eng. 36 (1989) 676-682. [ll] F. Rattay, Electrical Nerve Stimulation (Springer, Wien, 1990). [12] G.M. Shepherd, ed., The Synaptic Organization of the Brain (Oxford Univ. Press, 1990). [13] R.G. Smith, Neuron C: a computational language for investigating functional architecture of neural circuits, J. Neuroscience Methods 43 (1992) 83-108. [14] J.D. Sweeney, J.T. Mortimer and D. Durand, Modeling of mammalian myelinated nerve for functional neuromuscular electrostimulation, IEEE 9th Ann. Conf. Eng. Med. Biol. Soc. Boston. (1987) 1577-1578.
Mathematics and Computers in Simulation 39 (1995) 589-595
Modelling of the excitation and the propagation of nerve impulses by natural and artificial stimulations Frank Rattay Technical University Vienna, A-1040 l,Tenna, Austria
Abstract
The functional properties of a neuron can be simulated by electrical circuits. This technique is of use for natural as well as for electrical stimulation. In particular, a new model is proposed which shows the influence of the myelinated parts of the neuron.
Introduction In contrast to other cells neurons are remarkable because of their shapes showing a very complicated fine structure and because of the electrical properties of the membrane which separates the ionic components of the intracellular and the extracellular fluids. In the resting state the neuron's inside potential is constant with a value of about - 7 5 mV compared to that of the outside. In general, however, this membrane voltage essentially depends on location. As an example, we assume that a neuron generates a spike train consisting of 250 spikes/s. These action potentials are the answer to synaptic activities in the dendritic region and at the cell body. An action potential with a duration of 1 ms will propagate with a velocity of 50 m / s ( = 50 m m / m s ) along the nerve fibre (axon) which may have a diameter of 10 ~ m and a length of 50 cm. Thus, 5 cm intervals of excited membrane regions which are separated by 15 cm regions with resting state voltages are traveling along the nerve fibre from the cell body towards the branching part of the nerve fibre (output region). Of course, membrane voltage is also a function of space within the dendritic area and in the different branches in the terminal region. The main elements of a neuron are the soma (cell body), the dendrites and the axon. Dendrites and soma are usually covered with synapses from other neurons (input zone), whereas the task of the axon is the transport of neural information via action potentials into distant regions. The mammalian axons with diameters from 1-20 ~ m are usually myelinated, i.e. only in the nodes of 'Ranvier', which have a length of about 1 ~ m the membrane is active. The internode is covered with many layers of membranes of Schwann cells which are tightly wrapped around the axon. The internode has a length of about 100 times the fibre diameter. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4754(95)00122-7
590
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
The insulating properties of the internode allow a 20 ~ m mammalian fibre to obtain a higher velocity of propagation as it is reached in the unmyelinated giant axons of a squid with diameters up to 1 mm. The purpose of this paper is to discuss several models which finally can be used to predict the input-output relations of a single neuron.
Membrane models A patch of the membrane of a neuron can be described by an electrical circuit consisting of a voltage source, which accounts for the resting voltage, a capacitance (1-3 ~ F / c m 2) and a resistance. The conductance of the active membrane changes within a large range depending on the open/closed status of the ionic channels. The membrane models which are mostly used are the Hodgkin-Huxley model for unmyelinated (parts of) neurons (Box 1, see Fig. 1) and the CRRSS model for the nodal membrane of myelinated mammalian nerve fibres (Box 2, see Fig. 1). Further discussions about membrane models can be found, e.g. in [9-11].
Boxl I
HODGKIN-HUXLEY MODEL
= [-9~omsh(V
- VNo) - g K ~ ( V
- VK)
(HH-1) (HH-2) (HH-3) (HH-4)
--gL(V -- VL) + i,t]/c
rh = [ - ( a m + tim)" m + am]- k h = [ - ( a n + ft,). n + an]" k = [--(ah + fib)" h + ah]" k
Box 2 I
CRRSS MODEL
SWEENEY el al. (1987) transformed the original data ofOHIU, RITCHIE, ROGEKT & STAGG (1979) from experimental temperature of T = 14°C to T = 37°C. We name the following equations after the investigators the CRRSS model.
with the coefficient k for temperature T (in °C) k = 3 °1T-°63 (HH-5
= [-gNom2h(Y
- YNo) - gL(V -
CI:LKSS-1) CRRSS-2)
,'h = - ( a . ~ + Z . , ) - m + a m /~ =
and am
~
2.5-0.1V exp(2.5-O.1V)-I 1-0.1V
'
a~ = lo.(~p0_o.tv)_o , = 0.07.
V~e,t = -70[mV],
Vg =
-12[mY],
INs = 120[kf2-lcm-2], gL = 0 . 3 [ k ~ - I c m - 2 ] , m(0) = 0.05, h(0) = 0.6
e = = 4.,
p
z . = 0.125.e
p
flh -~
1 ezp(3-O.1V)+l
VL) + i , , ] / c
--(ah + &). h + ah
(tigress-a)
with am
97+0.363V
= l+,~p('~-~)' ph
VN,~ = ll5[mV], a h = , ~ p ( , , ? s ) , VL = lO.6[mV], v~.., = - 8 0 [ m V ] , gK = 36[kf2-1cm-~], vL = 0 . 0 1 [ m v ] , c = l[#F/cm2], gga = 1445[kf~-lcm-2], n(0) = 0.32, c = 2 . 5 ~ F / c m 2 ] , m ( 0 ) = 0.003,
°gre,
Z"~ = ,xv( . .,...,. . ) ' flh :
15.6
VNa = llS[mV], gL = 128[kg~-tcm-2],
h(0) = 0.75
Fig. 1. Comparison of equations for membrane dynamics for unmyelinated and myelinated parts of a neuron. V denotes membrane voltage; VNa, Vr, VL voltages across the membrane, caused by different (sodium, potassium, unspecific) ionic concentrations inside and outside of the neuron; gNa, g r , gL maximum conductance of sodium, potassium and leakage per cm 2 of membrane; m, n, h probabilities for ionic membrane gating processes; a,/3 opening and closing rates for ionic channels. Note the high conductances and the missed potassium currents in the myelinated mammalian node as described by the CRRSS model.
F.Rattay/ MathematicsandComputersin Simulation39(1995)589-595
591
Lumped circuits to simulate a neuron
In order to simulate neural reactions a neuron should be parted into segments and every segment is represented by an electrical circuit. Fig. 2 shows a simplified neuron which only consists of two dendrites, a cell body and a piece of the nerve fibre. We have approximated all the structures by cylinders. The following calculations are done with the reduced voltage V with V-
Vmembrane -- Vrest.
With the spatial neural model described below we can simulate both the influence of an artificial electric field (which causes an extracellular potential Ve) as well as natural situations (Ve = 0). For simplicity we assume that segmentation length Ax is constant, every segment is approximated by a cylinder of diameter d. and the center of the nth segment represents the average inside potential V~,.. The outside potential (caused by an applied electrical field) is Ve,,,, and the voltage across the membrane is
Vn=Vi,n-Ve, n.
(1)
]
I
I
I
I
v,,°+,
MEMBRANE
I
I
Ve,n
V(,. ,=~
ff--l._l--=l_
"X
R,,-1
R,~
2
2
,Xl,q
Rn2 I R,~+12 Az
Fig. 2. A simple neuron is represented by cylinders with different diameters (top). The circle marks the place where the bottom diagram is situated. Segments with the same length but with different diameters are used. The electrical components of the nth segment are the intracellular resistances (along the axis), and the capacity and conductance of the membrane.
Ax,
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
592
The currents of the nth segment are caused by capacity of membrane (C,), membrane conductance (G n) and the axonal resistances to the neighboured segments (e.g. R J 2 + R n + 1//2): dg n Cn " - .-[- Gn " V n --[-
dt
Vi,n -- Vi,n_ 1
R J 2 + R , _ 1/2
--I-
Vi,n - V/,n+ 1
R,,/2 + R n +1/2
=0.
(2)
In order to see the influence of an extracellular electrical field we use (1) and obtain an ordinary differential equation for every segment:
dVn _Gn. Vn_ ( Vn-- Vn-1 V n - V,,+I ) Cn" dt = R,,/2 + R , _ l / 2 + R , / 2 + R,,+l/2 --I
V e , n - Ven-l__~s.2._.2 .~_
[ R J 2 + R n _ 1/2
V e , n - Ve,n+l
R J 2 + R,+ i / 2
(3)
Note: In the case of Ve = 0 (natural situation) the expression [ . . . ] in Eq. (3) vanishes. Knowing the specific capacity c and conductances g of the membrane as well as the resistivity of axoplasm pi we can approximate the following variables which depend on the number N of layers of myelin which are wrapped around the cylinders with diameter d, and length Zlx: Ax "d, "rr Cn =
G.=
N Ax . d~ "rr
N
"c,
(4)
• g,
(5)
Ax'4 R~ = d2rr "pi.
(6)
Eqs. (3)-(6) are suited to simulate the dendrite and the cell body of a neuron. The situation in the nerve fibre (axon) is more complicated: The nodes of Ranvier only have a length L of 1 lxm, and they are very short compared to the internOde which is up to 2 mm long. Even for qualitative results it is important to use a good model for the ionic currents, because they become the dominant driving force in the excitation process when threshold is reached. Secondly, even in the passive case (small distortion of the resting potential) the electric properties of the node are of importance, because it is not insulated by the many fatty layers of myelin. When the axon is segmented (e.g. 10 elements from node to node), the segments which include a node have to be calculated with Eqs. (4a) and (5a). Ax. d n • rr
Cn = G~ =
N
"c + Ldn~-C,
(4a)
Ax . d, • rc N "g + iiLd'rc"
(5a)
For every segment the ionic current density ii in (5a) can be calculated with the help of a set of differential equations as it was first described by Hodgkin and Huxley [5]. For the simulation of the mammalian node the CRRSS model of Box 2 [1,14] or one of the models described in [9] or [11] should be used.
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
593
For subthreshold analysis we can approximate the membrane conductance of the node by constant g. Of special interest is the case of a fibre with constant diameter. In this case we obtain Eq. (7) from Eqs. (3)-(6), for segments with active membranes Eq. (7a) has to be used.
dV,, d .N c" d----f- = -g.V~ + 4 ( a x ) 2 p i ( V . _ , - 2 V n
d .N + V,,+l) + 4(Ax)2pi[Ve,n_l-2Ve,n
q-Ve,n+l] (7)
L N ) dV n -g,Vn LN d" N c 1 + ~x d--t- = -i~--~x + 4(ax)2p,
- 2V,, + V,,+l )
d.N +
2Ve, n + Ve,n+l].
(7a)
The influence of the extracellularly applied electrical field on the nth segment of a nerve or muscle fibre is represented by that term in Eqs. (7) and (7a) which contains Ve. It is called the activating function f and it is proportional to the second difference quotient of Ve, which is a function of the fibre's length coordinate:
d ' N Ve,,,_I-- 2Ve,,, + V~,,,+I f = 40~-
(AX)2
(8)
In order to obtain an action potential it is necessary to reach threshold voltage in a segment with an active membrane. This is possible for segments where f is positive. The upper panel of Fig. 3 shows the effect of the activating function for a nonmyelinated fibre, computed with fine segmentation. A point electrode above the central part of the fibre produces an electric field by a negative current pulse. A positive electrode current would cause a result with inverse polarity, i.e. the fibre would be excited at two positions, however, this excitation is much weaker, which gives an explanation why cathodic stimulation is easier.
Discussion
Of course, it is possible to simulate parts of the neuron (e.g. the nerve fibre) with even more accurate models, see e.g. [3]. However, it seems that deviations from experimental findings rather come from neglecting the electrical properties of myelin. With our preliminary results regarding subthreshold reactions we have obtained thresholds which only have about half of the value expected from the McNeal model, which was generally used. It must be expected that many computed results in the field of electrical stimulation have to be recalculated. There exist only few data considering the ionic channel dynamics in different neural regions [6,7,12]. Especially in the input area there is a lack of information how to model the membrane behaviour. Nevertheless, many authors have obtained results with membrane models of the
594
F. Rattay ~Mathematics and Computers in Simulation 39 (1995) 589-595
0
0
1
2
3
4
5 6 7 $ 9 10 NODE Fig. 3. Influence of myelin. Top panel: The excitation of an unmyelinated fibre at the end of a subthreshold negative 100 I~s current pulse is a picture of the activating function. Comparison of the computed subthreshold reaction of a 11 node axon with an unmyelinated fibre of same diameter shows that myelinated axons are much more excitable. The curve of the top panel is also included as the smallest one in the lower panel. The excitability increases considerably for myelinated fibres, even if it is assumed that the internode is a perfect insulator (second smallest result). Further increase is seen when N grows up ( N = 100, 200, 300).
Hodgkin Huxley type. Different software is used to simulate single neurons and neural nets, see e.g. [4,13]. Some of the authors, e.g. [2], reduce the reaction of a single neuron to a combination of a simplified threshold process with a refractory function, and they do not look at the detailed membrane reactions because of the many elements involved.
F. Rattay / Mathematics and Computers in Simulation 39 (1995) 589-595
595
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