32nd Annual International Conference of the IEEE EMBS
`Buenos Aires, Argentina, August 31 - September 4, 2010
`
`lnpu t~Ou tpu t
`
`o
`
`0
`
`Binary
`variable
`A
`
`Under
`threshold
`pulses
`
`;, SL
`;,JL
`-----------------------
`i3 JL
`Above
`i,JL
`threshold
`pulses
`
`978-1-4244-4124-2/10/$25.00 ©2010 IEEE
`
`4829
`
`
`
`The Excitation Functional for Magnetic Stimulation of Fibers
`D. Suárez-Bagnasco, R. Armentano-Feijoo and R. Suárez-Ántola
`
`Abstract—Threshold problems in electric stimulation of
`nerve and muscle fibers have been studied from a theoretical
`standpoint using the excitation functional. Here the excitation
`functional is extended to magnetic stimulation of excitable
`nerve and muscle fibers. A unified derivation of the functional
`is done, for (non myelinated) nerve and muscle fibers, by means
`of the nonlinear cable equation with a Fitzhugh-Nagumo
`membrane model and a generalized Rattay’s activating
`function. The identification problem of the excitation functional
`for magnetic stimulation, from strength-duration experimental
`data, is briefly considered.
`
`T
`
`INTRODUCTION
`I.
`HE goal of electric and magnetic stimulation of
`excitable cells is to produce (or to block) action
`potentials in suitable locations. From the standpoint of a
`black box approach, the stimulation process may be
`described by a correspondence between each applied electric
`current history and a binary variable Λ that takes the value 0
`if stimulation fails and 1 if it succeeds (Fig.1).
`
`Fig.1. Black box approach to electric and magnetic stimulation
`
`From the standpoint of the stimulation equipment, the black
`box comprises the electrodes (and their leads) or the
`magnetic coils, the volume conductor of the tissues and the
`target elements (nerve or muscle fibers, etc.). So, this black
`box may be considered as an electric load seen by the
`stimulating equipment. The binary variable may be obtained
`through an electric measurement (detection of action
`potential by
`recording electrodes) or by external
`manifestations (like muscle twitches, function inhibitions,
`etc).
`For the electrical stimulation, there is already a theoretical
`
`D. Suárez-Bagnasco is with the School of Medicine, CLAEH, Punta del
`Este, Uruguay (corresponding author, phone/fax: +598 2 3096456; e-mail:
`dsuarezb@adinet.com.uy).
`R. Armentano-Feijoo is the Dean of the Engineering School of Favaloro
`University, Buenos Aires, Argentina.
`R Suárez-Ántola is with the School of Engineering of the Catholic
`University of Uruguay, Montevideo, Uruguay
`
`tool, the excitation functional, introduced by R. Suárez-
`Antola [1], [2]. It allows both a description and a prediction
`of the output given by the binary variable in the black box
`approach. As each system composed by the electrodes, the
`tissues and the target elements is unique (due to different
`spatial, temporal, electrical and in general, physiological
`properties), the black box must be duly identified from
`suitable experimental data.
`The experimental strength-duration curves for a given
`system can be used to obtain the excitation functional for the
`electrical stimulation of the just mentioned system [3].
`The excitation functional opens a third way between a pure
`experimental approach and a pure computational approach
`(working with nonlinear cable equation for electrical
`stimulation). As an analytical tool (which parameters can be
`adjusted
`from
`real experimental data)
`it allows a
`mathematical formulation of threshold related problems of
`interest for biomedical engineering. For example, to find
`optimal pulse shapes given an optimization criterion [2]. So,
`it could be of certain interest to try to extend the excitation
`functional to magnetic stimulation.
`The purpose of this paper is threefold: (a) to extend the
`excitation functional to magnetic stimulation produced by
`external coils, (b) to present a unified derivation of the
`functional both for electric and magnetic stimulation, and (c)
`to briefly discuss the identification problem of the functional
`from experimental data (from strength-duration curves).
`A unified derivation is possible because (from the non linear
`cable equation with a generalized activating function [4] [5])
`it is the external electric field parallel to the fibers the
`responsible of both magnetic and electric stimulation. One of
`the consequences of this extension is to have a tool that
`allows to characterize the system and to predict the outcome
`of the binary variable given a certain current history in the
`stimulating coil.
`This tool can be used also to pose and solve certain
`engineering problems related to the system, like how to
`shape an input current pulse in the stimulating coil that is
`both
`threshold and optimum from
`the standpoint of
`minimizing the energy dissipated per pulse in the tissues [6].
`For
`the purposes of
`the present work, a suitable
`background in electric and magnetic stimulation may be
`found in [7]. Further information about electric stimulation
`can be found in [8] to [10], and in [11] to [16] for magnetic
`stimulation.
`
`LUMENIS EX1044
`Page 1
`
`

`

`THRESHOLD
`
`~
`
`4830
`
`
`
`
`II. EXTENSION OF THE EXCITATION FUNCTIONAL TO THE
`MAGNETIC CASE
`
`du
`
`
`
`The derivative of the coil current appears in (3) related to the
`fact that the electric field in a given point of the tissues is the
`( )
`t
`diC
`product of
`and a vector function of the position of the
`dt
`considered point in the volume conductor [4], [5], [6], [16].
`In a second step, the maximum of iM(t) is taken from t=0
`onwards and compared with a threshold current ITh,0 (Fig. 3).
`
`
`
`Fig.3. Under threshold, just threshold and above threshold time evolution of
`iM(t)
`
`
`=
`
`I
`
`. (4)
`
`A history of current in the coil is just threshold when:
`
`
`{
`( )}
`máx
`t
`i
`M
`Th
`0,
`≥
`t
`0
`ITh,0 (a characteristic parameter of the system) allows
`again the definition of a digital variable, in this case ΛM. It
`can be obtained from experimental strength-duration curves,
`as shown in section IV.
`
`A. Formulation for the electrode case
`The simplest formulation of
`the excitation functional
`corresponds to a single active electrode or a bipolar
`electrode and may be applied both to cathodic make
`excitation and to anodic break excitation [2]. The excitation
`functional when the excitable membrane is at rest prior to
`t=0 can be written in two steps. First we obtain a function
`qE(t), making the convolution of a history of injected current
`iE(t) with a non dimensional impulse response function GE(t)
`characteristic of the system (black box).
`t
`( )
`(
` )⋅−
`
`( )
`. (1)
`=
`⋅
`tq
`uiutG
`∫
`E
`E
`E
`0
`In this case, by definition GE(0)=1. The injected current
`appears in (1) related to the fact that the electric field in a
`given point of the tissues is the product of iE(t) and a vector
`function of the position of the considered point in the
`volume conductor [4].
`In a second step, the maximum of qE(t) is taken from t=0
`onwards and compared with a threshold charge QTh,0. This
`threshold charge is not the charge injected by the electrode
`neither the charge that crosses the excitable membrane of the
`target fiber.
`A history of applied current is just threshold if and only if
`
`máx
`
`=
`
`. (2)
`
`{
`
`( )}
`tq
`Q
`E
`Th
`0,
`≥
`t
`0
`Fig.2 shows three possible outcomes of the convolution
`
`
`(
`∂
`s,tE
`e
`∂
`s
`s
`. (5b)
`
`λ
`
`2m
`
`−
`
`2
`
`v
`2
`
`∂∂
`
`)
`⋅−
`+
`λα
`w
`
`2m
`
`2
`
`⋅−
`vc
`
`3
`
`)
`
`. (5a)
`
`III. A UNIFIED DERIVATION OF THE EXCITATION
`FUNCTIONAL
`The construction of the impulse response function and the
`threshold charge for electric stimulation, and the impulse
`response function and the threshold current for the magnetic
`stimulation of a (non myelinated) fiber will follow a
`common procedure. This allows an easier grasp of the
`similarities and differences between the two situations.
`Let us begin with the nonlinear cable equation with
`Rattay’s activating function generalized to take into account
`both electric and magnetic stimulation [4], [5], [6], [16]. To
`simplify the derivation we use the well known Fitzhugh-
`Nagumo model for the unit membrane [17]. If the (possibly
`bended) target fiber is represented by a curve of directed arc
`length s measured from a suitable fixed point of the fiber,
`
`( )stv ,
`
`is membrane voltage field and ( )stw ,
`is the recovery
`variable field, both relative to their rest values, we obtain the
`following set of equations [6]:
`
`(
`⋅+−=
`vbv
`
`( )wv
`−
`γ
`
`=
`
`tv
`∂∂
`
`tw
`∂∂
`
`τ
`m
`
`τ
`w
`
`
`Fig.2. Under threshold, just threshold and above threshold time evolution of
`qE(t)
`
`QTh,0 , which is a characteristic parameter of the system,
`allows the definition of the digital variable ΛE that was
`presented in the introduction. As will be seen in section D, in
`this case QTh,0 coincides with the well known limit threshold
`charge that can be obtained from strength-duration curves.
`B. Formulation for the coil case
`In the magnetic case, assuming that the membrane is at
`rest prior to t=0, we propose to convolve time derivative of
`( )
`t
`diC
` with a non dimensional
`the electric current in the coil
`dt
`impulse response function GM(t) in order to obtain the time
`function iM(t). Here, by definition, GM(0)=1.
`
`( )
`di
`u
`C
`du
`
` )⋅−
`(
`
`utG
`M
`
`t
`∫
`0
`
`( )
`t
`
`=
`
`i
`
`M
`
`
`
`⋅
`
`du
`
`. (3)
`
`
`The time constant of the membrane is mτ , and wτ is the time
`constant of the recovery variable (under physiological
`conditions at least an order of magnitude greater than mτ ).
`The fiber’s space constant is mλ . The parameters of the unit
`γα,
`membrane model
`are positive.
`,cb
`,
`
`LUMENIS EX1044
`Page 2
`
`

`

`f
`f
`f
`
`f
`f
`f
`
`4831
`
`
`
`point reaches the barrier, an action potential emerges [6],
`[19].
`The under threshold behavior can be described with
`enough accuracy, up to the threshold barrier, by the linear
`system:
`
`
`τ
`m
`
`τ
`w
`
`1
`
`1
`
`dA
`dt
`dB
`dt
`
`2
`
`−
` A )
`1
`
`λα
`B
` +
`1
`m
`
`2
`
`⋅
`
`
`M ⋅1,
`F
`
`
`
`λ
`−
`π
`m2
` + (1
` =
`
`2
`(cid:65)
`)1
`
`=
`
`(
`A
`
`1
`
`⋅−
`γ
`
`B
`
`.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`d
`dt
`
`
`( )t
`
`.
`
`i
`
`C
`
`
`
`
`
`
`
` (8)
`
`
`This approach is the equivalent, for a fiber with a non-
`uniformly polarized membrane, to the classical discussion of
`Fitz-Hugh for a unit membrane [19]. Fig 4 shows the
` )11, BA
`simplified dynamics in the (
`
` state space. The results
`of the digital simulation suggest the introduction of a
`decaying and an amplifying set, separated by a threshold
`curve [19].
`
`
`
`
`Rattay’s generalized activating function is given by [4], [5],
`[6], [12], [16]:
`
`
`⋅
`
`⋅
`
`( )
`⋅
`sF
`E
`( )
`sF
`M
`
`i
`⋅
`
`→
`( )
`t
`
`C
`
`i
`
`electric
`→
`
`magnetic
`
`. (6)
`
`( )
`t
`E
`d
`⎪⎩
`dt
`
`( )sFM
`( )sFE
`The functions
`give the spatial distribution
` and
`of the perturbation produced on the excitable membrane by
`the electric field due to the electrodes or induced by the time
`varying magnetic flux due to the coil, and may be called
`geometric form factors for electric and magnetic stimulation,
`respectively. Now, the stimulation is always a localized
`phenomenon. As a consequence, outside a certain interval of
`influence along the fiber, of length (cid:65) , the form factors may
`be neglected [6], [18], [19], [20]. If we assume that until the
`fiber reaches a threshold state, the fields of membrane
`voltage and recovery variable take their rest values at the end
`points of this interval of influence, it is possible to make a
`nonlinear modal analysis of the cable equations. The value
`of (cid:65) depends of the form factor. If s is measured from the
`mid-point of the interval of influence, we may use the
`Fourier development:
`
`
`λλ
`
`2m
`
`2m
`
`⎪⎨⎧
`
`∂
`
`)
`
`(
`s,tE
`e
`∂
`s
`
`=
`
`λ
`
`2m
`
`−
`
`. (7)
`
`
`
`...
`
`⎠⎞
`
`+⎟
`
`s
`
`s
`
`⎜⎝⎛
`
`⋅
`
`( )
`tA
`2
`
`⋅
`
`⎠⎞
`
`+⎟
`
`s
`
`s
`
`⎜⎝⎛
`
`cos
`
`(
`stv
`,
`
`)
`
`=
`
`( )
`tA
`1
`
`⋅
`
`)
`
`=
`
`⋅
`
`⎟⎟ ⎠⎞
`
`+
`
`2
`
`λπ
`m2
`
`2
`(cid:65)
`
`−
`11
`τ
`m
`
`⎜⎜ ⎝⎛
`
`1
`τ
`w
`
`⎡
`
`⎢⎢⎢⎢⎢⎢
`
`⎣
`
`⎟⎟⎠⎞
`
`(cid:65)λ
`
`m
`2
`
`
`
`+
`
`π
`2
`
`⎜⎜⎝⎛
`
`
`
`
`
`=Α
`
`⎥⎥⎥ ⎦⎤
`
`BA
`⎢⎢⎢ ⎣⎡
`
`1
`
`1
`
`=
`
`x(cid:71)
`
`number
`
`π
`⋅
`(cid:65)
`π
`⋅
`(cid:65)
`π
`⋅
`(cid:65)
`
`⎜⎝⎛
`
`cos
`
`⎜⎝⎛
`
`cos
`
`...
`
`⎠⎞
`
`+⎟
`
`
`
`...
`
`⎠⎞
`
`+⎟
`
`π
`⋅
`2
`(cid:65)
`π
`⋅
`2
`(cid:65)
`π
`⋅
`2
`(cid:65)
`
`s
`
`⎜⎝⎛
`
`sin2
`(cid:65)
`sin2
`(cid:65)
`sin2
`(cid:65)
`
`⎜⎝⎛
`
`( )
`tB
`2
`
`F
`M
`
`2,
`
`⋅
`
`⎠⎞
`
`+⎟
`
`⎠⎞
`
`+⎟
`
`s
`
`2
`(cid:65)
`2
`(cid:65)
`2
`(cid:65)
`
`(
`stw
`,
`
`( )
`tB
`1
`
`( )
`sF
`M
`
`=
`
`F
`M
`
`1,
`
`⋅
`
`
`
`Substituting (7) in the nonlinear cable equation and
`eliminating the spatial dependence, we obtain a system of
`first order nonlinear ordinary differential equations in the
`( )tBk
`( )tA j
`unknown mode amplitudes
`and
`. Solving
`this
`system with suitable initial conditions, we obtain the mode
`amplitudes. The procedure is the same as already developed
`for stimulation by electrodes [18], [19]. The only difference
`between electric and magnetic stimulation in this approach is
`the forcing term. So the dynamics of the unforced fiber, after
`the end of the stimulating pulse, is the same if the length of
`the interval of influence is the same.
`Digital simulation of both cathodic make and anodic break
`electrical stimulation using up to seventeen mode amplitudes
`suggest that the first mode is the most relevant in
`determining threshold behavior of the fiber [20].
` Truncation to the first mode, uncoupled, gives in both
`cases, a set of nonlinear ordinary differential equations in the
`mode amplitudes corresponding to membrane voltage and
`the recovery variable [19].
`the
`The study of
`threshold dynamics done with
`aforementioned equations, with parameter’s values within
`the physiological ranges, shows that a threshold barrier may
` )11, BA
`be defined in phase space (
`
` such that when the phase
`
`
`Fig 4. Sketch of the dynamics of the system, simplified by the threshold
`barrier. R is the rest state. The decaying set is shown as a shaded area
`bounded by the double arrowed threshold curve.
`
`The threshold curve is composed by: half-straight line taken
`from the threshold barrier and an orbit in the decaying set
`that is just tangent to the threshold barrier. The construction
`of the threshold barrier can be seen in [17].
`The linear dynamic system (8) can be recast in a matrix
`form:
`(cid:71)
`( )
`d
`tx
`dt
`
`⋅Α=
`
`(cid:71)
`( )
`tx
`
`+
`
`λ
`⋅
`2
`m
`
`F
`M
`
`1,
`
`⋅
`
`(cid:71)
`( )
`⋅
`et
`1
`
`i
`C
`
`d
`dt
`
`.
`
`
`
` (9)
`
`. (10)
`
`⎥⎦⎤
`01
`⎢⎣⎡
`
`
`
`(cid:71)
`1e
`
`=
`
`⎤
`−
`α
`τ
`m
`
`⎥⎥⎥⎥⎥⎥
`
`−
`γ
`τ
`w
`
`⎦
`
`
`In the matrix A, the spatial properties appear in the
`−
`2
`11
`element
`
`
`through
`the non-dimensional
`τ
`m
`mλ . The other elements of the matrix A depend
`(cid:65)
`only on unit membrane properties through the parameters
`ττγα
`,
`,
`.
`m,
`w
`The solution, beginning from the rest state, and until the
`first arrival to the threshold barrier, is:
`
`LUMENIS EX1044
`Page 3
`
`

`

`R
`
`,,~1\~T~
`n X
`p ~
`
`/
`~
`/ ,I
`n
`,I
`,.,,_ Threshold barrier
`
`..
`
`4832
`
`S
`
`( )
`ui
`C
`
`⋅
`
`(
`e
`
`(
`
` )Α−
`
`ut
`
`)
`
`⋅
`
`(cid:71)
`e
`
`1
`
`d
`du
`
`du
`
`.
`
`
`
`
`
` (11)
`
`t 0
`
`∫
`
`
`
`
`
`
`
`(cid:71)
`( )
`=
`λ
`tx
`m
`
`2
`
`⋅
`F
`1,M
`
`⋅
`
`
`From the results of digital simulation (summarized in Fig
`4) the condition that characterizes a just threshold magnetic
`stimulation is (Fig.5):
`
`} p
`{
`(cid:71)(cid:71)
`( )
`txnT
`max
`
`[∈
`+∞
`t
`,0
`
`
`=
`
`.
`
`
`
`
`
`
`
`
`
`
`
` (12)
`
`)
`
`by the length of the interval of influence (cid:65) .
`It is possible to show that an approximate formula for the
`time constant of the cathodal strength-duration curve for a
`nerve fiber is given by [21]
`. (16)
`≈
`t
`
`
`
`⎟⎟⎠⎞
`
`2
`
`(cid:65)λ
`
`m
`2
`
`2m
`πτ
`
`
`
`+
`
`1
`
`⎜⎜⎝⎛
`
`The chronaxy is proportional to tS. For a given peripheral
`nerve, in the electric stimulation (cid:65) is smaller than in the
`magnetic stimulation case. From (16) it follows that
`chronaxies for electric stimulation should be smaller than
`chronaxies for magnetic stimulation. This explains the
`experimental findings [13].
`Equation (16) is obtained from the simplest model for the
`impulse response function. A more realistic model of GM
`(given by (13)) is sketched in Fig.6.
`
`
`(cid:71)
`Fig.5. Graphical interpretation of Eq.(12). The unit vector n
`the threshold barrier.
`
`
`
`is normal to
`
`Th
`
`0,
`
`)
`
`−
`
`=
`
`
`
` (15)
`
`
`Fig.6. Schematic representation of an impulse response function taken
`from [19].
`
`
`It allows the calculation of the chronaxies both for
`cathodic and anodic stimulations. However, despite the
`analytical formulae are different, the same theoretical
`predictions about the behavior of chronaxies for electric and
`magnetic stimulation of peripheral nerves are derived from
`this more complete model.
`
`⎜⎜⎜⎜ ⎝
`
`(
`tG
`PM
`
`)
`
`∂=
`∂
`t
`
`P
`
`(cid:71)
`There are analytical formulae for the unit vector n
` and
`the distance p as functions of system’s parameters [19], [20].
`(cid:71)
`( )tx
`We put
` from (11) into (12) and define:
`(a) The impulse response function
`(cid:71)
`(cid:71)
`Α⋅
`tT
`( )
`en
`e
`
` . (13)
`=
`1
`tG
`(cid:71)(cid:71)
`M
`T
`en
`1
`(b) The threshold current
`p
` . (14)
`=
`I
`(cid:71)(cid:71)⋅
`⋅
`λ
`T
`2
`F
`en
`m
`M
`1
`1,
`Then we obtain the just threshold condition for magnetic
`stimulation in terms of the excitation functional (equivalent
`to (3) and (4) combined)
`
`max
`
`[∈
`)
`,0t
`
`
`.
`
`
`
`
`
`
`
`I
`
`0,Th
`
`⎪⎬⎫
`
`⎪⎭
`
`du
`
`( )
`uid
`C
`du
`
`(
`tG
`M
`
`u
`
`t 0
`⎪⎨⎧
`
`⎪⎩
`
`∫
`
`+∞
`
`The same procedure applied to electric stimulation gives
`(cid:71)
`(cid:71)
`Α⋅
`T
`( )
`en
`e
`p
`
`
`=
`1
`tG
`(cid:71)(cid:71)
`(cid:71)(cid:71)⋅
`E
`λ
`⋅
`T
`2
`en
`F
`en
`1
`m
`E
`1
`1,
`So, if the parameters of the unit membrane are the same,
`the only difference in the impulse response function (in the
`framework of this mathematical model) is due to different
`values of (cid:65) related with differences in the form factor.
`The activation of peripheral nerves can be studied under
`the following modeling conditions: the medium can be
`considered as homogeneous, the fiber can be considered as
`straight and unbounded but the external electric field varies
`in a region along the fiber [12].
`In the framework of the present mathematical model, the
`degree of spatial localization of the perturbation of the
`peripheral nerve membrane due to the external field is given
`
`Tt
`
`Q
`Th
`
`0,
`
`=
`
`di
`ThC
`,
`dt
`
`(
`t
`
`P
`
`)
`
`=
`
`I
`
`⋅
`
`du
`
`⎟⎟ ⎠⎞
`
`i
`
`
`
`ThC,
`
`P
`
`t
`
`p
`
`⎜⎜ ⎝⎛
`
`lim
`
`I
`
`Th
`
`0,
`
`IV. THE IDENTIFICATION OF THE IMPULSE RESPONSE
`FUNCTIONS
`For electric stimulation, the identification problem of the
`excitation functional is already studied in [2], [3], [6].
`For magnetic stimulation, let us consider a linear ramp of
`electric current in the coil. From the excitation functional
`(15) it follows that for a ramp of duration tP, that produces a
`suitable depolarization of the fiber membrane in the interval
`di ,
`(
`)P
`of influence, the threshold slope
` is given by:
`ThC
`t
`dt
`. (17)
`
`Th
`0,
`Pt
`( )
`uG
`∫
`M
`0
`From (15) when the ramp duration tends to zero, we derive
`(
`)
`d
`. (18)
`=
`⋅
`t
`dt
`p
`Once ITh,0 is known, from (16) and (17) it follows
`⎛
`⎞
`. (19)
`
`⎟⎟⎟ ⎠
`
`)⎟
`
`0,
`(
`t
`
`P
`
`I
`
`Th
`
`di
`ThC
`,
`dt
`
`LUMENIS EX1044
`Page 4
`
`

`

`4833
`
`
`
`
`This means that the impulse response functions may be
`obtained from experimental strength-duration curves for
`magnetic stimulation. These experimental curves can be
`done using equipment similar to the one described in [22].
`However the derivative makes it an ill posed problem. A
`way out of this difficulty is to use analytical formulae (13)
`and (14) derived in this paper and adjust the parameters of
`(15) to the experimental data.
`
`V. FINAL COMMENTS
`The excitation functional is a systemic property. From the
`standpoint of attaining the threshold in a given target fiber in
`an external electric field, an action potential can emerge
`under four modeling circumstances: (a) the external field is
`uniform and the fiber crosses a region of fast variation of the
`electrical conductivity of the medium, (b) the external field
`is uniform and
`the medium can be considered as
`homogeneous but the fiber bends, (c) the external field is
`uniform and the medium can be considered as homogeneous
`but the fiber originates or terminates (short circuit or open
`circuit conditions), (d) the medium can be considered as
`homogeneous, the fiber can be considered as straight and
`unbounded but the external electric field varies in a region
`along the fiber. The model of the present paper applies to
`circumstances (b) and (d), because what matters is the spatial
`variation of the projection of the electric field tangential to
`the fiber and its time variation. This time variation is
`proportional to: time variation of the current in the electrode
`case, and the time derivative of the current circulating in the
`working coils. An extension to cases (a) and (c) could be
`done.
`In the magnetic case the determination of the form factor
`is a difficult problem that deserves further study.
`However, once determined GM(t) and ITh,0, the binary
`output response of a given target fiber to different pulse
`shapes in the coil could be predicted and contrasted with
`experiments, assuming that the system composed by the
`coil, volume conductor and target fiber remains unchanged.
`An extension of the present derivation of the excitation
`functional for magnetic excitation, to take into account lag
`effects in the activation of the excitation channels in fiber’s
`membrane, and myelinated fibers, both neglected in this
`paper, remains to be done.
`
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`
`LUMENIS EX1044
`Page 5
`
`