A MATHEMATICAL MODEL FOR THE ...

Moscow University

Vestnik Moskovskogo

Mechanics Bulletin

Universiteta. Mekhanika

Vol. 57, No. 6, pp. 13{21 , 2002

UDC 517.98

A MATHEMATICAL MODEL FOR THE MECHANORECEPTOR OF ANGULAR ACCELERATIONS V. A. Sadovnichii, V. V. Alexandrov, T. B. Alexandrova, A. Almanza, T. G. Astakhova, R. Vega, N. V. Kulikovskaya, E. Soto, and N. E. Shulenina A mathematical model describing the response of semicircular canals in the vestibular system to angular accelerations of the animal's head is proposed.

INTRODUCTION In this work we propose a mathematical model describing the response of semicircular canals in the vestibular system to the sensory stimulation in the form of angular accelerations of the animal's head. The receptor potential and the total ionic current of hair cells are understood as the end result of this model. The dependence of the receptor potential on the arrangement of hair cells, the generation of aerent pulses, and the eect of eerent signals on the dynamics of ionic currents are not considered. The pioneering results in this direction of research were published in 1, 2].

1. A FUNCTIONAL SCHEME OF THE MECHANORECEPTOR A vestibular canal can be considered as a thin curved duct of semicircular shape lled with the endolymph (a viscous incompressible uid). Both ends of the duct are open and lowered into a reservoir (the utricle). One of these ends is sharply expanded in the immediate neighborhood of the utricle, forming an ampulla and narrowing down to the initial diameter. The cupula (an elastic jelly-like partition) whose speci c weight is close to that of the endolymph resides inside the ampulla. As is known, the weight and the viscous characteristics of the endolymph are similar to those of water. However, the geometric characteristics of vestibular canals are dierent for various animals. In this paper, some experimental results obtained for the axolotl (Ambystoma tigrinum) vestibular system at the Institute of Physiology (Autonomous University of Puebla, Mexico) 3] are presented. The morphological parameters characterizing an axolotl lateral vestibular (semicircular) canal are given in Table 1 (a1 and R are the inner and outer radii of the canal, k = a0 =a1 is the widening of the ampulla, l is the length of the utricle, and L is the length of the canal). The cupula is attached to the ampullary walls on the crista (a sensory base) in such a way that it is capable of de ecting in response to the motion of the endolymph within the canal. In our model we assume that a uid interlayer ranging in thickness from 2 to 5 m exists between the cupula and the crista. See 4] for a more detailed description of the canal's anatomy. The semicircular canal, the endolymph, and the cupula taken toTable 1 gether form the cupula{endolymph system. The rst block of the funcLateral canal tional scheme for the mechanoreceptor (see Figure 1) simulates the Parameters
0:000852 mm2 =ms dynamics of this system under the action of mechanical stimuli in the  0:0001 mg=ms2 form of angular accelerations. The rotation of the semicircular canal a 0:155 mm 1 with acceleration directed along the central axis of the canal (the axis m 0:35 mg of sensitivity) leads to the displacement of the endolymph and, as a R 1 :35 mm consequence, to the de ection of the cupula. Thus, the output of the k 2:3 rst block is the de ection of the cupula. l 3 : 84 mm This de ection of the cupula relative to the crista causes the deL 4 : 6 mm formation of cupula sensitive hairs being a constituent of receptor cells these hairs were called the stereocilia and the kinocilia. The kinocilium

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is a supporting hair, whereas in contrast to it the stereocilia (10 { 60 in each cell) possess the conductance channels of ionic currents (mainly, of potassium ionic currents). The deformation of stereocilia causes variations in the inward potassium ionic current and, hence, causes the response of a receptor cell. In the literature, this conversion of the cupula de ection into the above electrophysiological process was called the mechanism of mechanoelectrical transduction. This mechanism is simulated in the second block of the functional scheme. The further evolution of mechanoreceptor responses is associated with the dynamics of ionic currents and with the dynamics of the membrane potential of hair cells. In response to variations of the transduction current, the changes of inward and outward ionic currents owing through the corresponding membrane channels can be observed this leads to variations of the membrane potential, i.e., to the appearance of the receptor potential equal to the dierence between the membrane potential and the rest potential. Since the linear sizes of a hair cell are small (of order 10 m), we may assume that the cell membrane is an equipotential surface. We also assume that the response dynamics of a receptor cell may be described with the aid of the membrane potential V and the total ionic current IT , where IT is the sum of the following three currents: the potassium currents, the calcium current, and the potassium current dependent on the calcium concentration. The latter assumption is reasonable if the natural oscillations of ionic currents and of the membrane potential are absent (this agrees with experimental data). Here we do not take into account the constituents of the total ionic current and consider this simpli cation as a rst step in the further analysis of our mathematical model for the mechanoreceptor of angular accelerations. Thus, the functional scheme for the mechanoreceptor consists of three blocks (Figure 1). The scalar time function !_ (the projection of the angular acceleration vector onto the axis perpendicular to the semicircular canal plane) is the input of this scheme, whereas its output is a vector function involving the membrane potential V and the total ionic current IT . In order to specify internal variables, below we describe mathematical models of the above blocks.

2. AN APPROXIMATE MATHEMATICAL MODEL OF CUPULA DYNAMICS An approximate description of the cupula dynamics is based on the following assumptions: a) the endolymph is an incompressible Newtonian uid whose viscosity  and density  are constant b) the local eect caused by the curvature Fig. 1. The functional scheme for the of the semicircular canal can be ignored, since mechanoreceptor of angular accelerations the ratio a1 =R is small c) the viscous forces in the duct can be replaced by those originating by Poiseuille's ow in a rectilinear segment of a circular pipe d) the pressure drop across the utricle is approximately equal to ;lR!_ 5] e) the endolymph moves through the narrow segment of the semicircular canal under the conditions valid for Poiseuille's ow. Let us consider the following two mechanotheoretical models of the cupula.

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Fig. 2. Two mechanotheoretical models of the cupula{endolymph system (a: the cupula as a piston, b: the cupula as a diaphragm)

1. The cupula is modeled by an elastic piston with width 2, height 2b (b < a0 is the ampullary radius), and density  equal to that of the endolymph (Figure 2, a). When the piston is displaced, there appears an elastic force proportional to the piston displacement with a coecient  . The following approximate mathematical model of the cupula dynamics was obtained in 3, 6, 7]:



  x = ; 1 1 + l !R: x + 8a2 1 + 2k4 L(aa0; b) x_ + k4  a 2L k2 L _ 0 1 1

(1)

Here !_ (t) is the projection of the angular acceleration vector (generated by the rotation of the axolotl's head) onto the axis perpendicular to the semicircular canal plane (the input of the rst block) and x is the linear displacement of the piston-like cupula from its equilibrium position (the output of the rst block). Equation (1) was derived after some simpli cations performed for the system of equations describing the endolymph and piston-like cupula dynamics in the rotating semicircular canal with the following small parameters: a1 =R is the ratio between the inner and outer radii of the narrow segment in the canal, 2=L is the relative thickness of the cupula, and (a0 ; b)=a0 is the relative thickness of the cleft. 2. The cupula is modeled by an elastic diaphragm with density , width 2, and stiness T (Figure 2, b). Let us assume that the diaphragm completely blocks the endolymph ux through the semicircular canal and that the de ection of the diaphragm is perpendicular to the plane of its equilibrium position. In the linear approximation, the local de ection of the diaphragm can be described by the equations 7 { 9]


2  (r t) = 2 (t) 1 ; ar 2 r 2 0 a0] 0
 8  8 T

 + a2 _ + m k4 = ; k12 1 + Ll !R _ 1 1

(2)

where  (r t) is the membrane displacement relative to its equilibrium position, r is the polar radius in the cross-sectional plane, and m1 = a21 L is the eective mass of the endolymph. In equation (2) we put T = =(8 ) and in equation (1) we neglect the expression 2k4L(aa0; b) by 0 virtue of its smallness in comparison with 1. Then, the diaphragm averaged de ection (t) can be described by equation (1). Hence, at any instant of time the displacements of the uid volume in the ampulla are approximately the same for the above mechanotheoretical models. In a macromechanical sense, thus, the

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dynamics of cupula behavior is the same in these models. It should be also noted that our assumption on the absence of the endolymph leakage through the subcupular for the cupula  importance  
areais aof little 8  0 dynamics: if this assumption is abandoned, then the term a2 2k4 L(a ; b) x_ should not be ignored in 0 1 equation (1). Introducing the angle variable ' = x=c, where c  a0 ;b, we can represent the approximate mathematical model for the rst block in the form
 8   R l ' + '_ + '=; 1 + !: _ (3)

a21

2

m1 k4

ck2

L

4

1k Since 8a1  8m a21  (see Table 1), we may come to the conclusion that the approximate model (3) is a mathematical model of an overdamped torsion pendulum or, in other words, is a Shteinhausen-type model 10] with the only fundamental dierence: all parameters of model (3) have a physiological sense and can be evaluated from experimental data.

3. A MATHEMATICAL MODEL OF MEMBRANE POTENTIAL DYNAMICS FOR HAIR CELLS In the model we propose here the total current of a hair cell is considered as the potassium current. This simpli cation is reasonable, since the sodium channels are practically absent in the hair cell membrane. Moreover, our recordings of the potential response to current injection demonstrate that even for large positive pulses (up to 1 nA) the membrane potential becomes little more than 0 mV. In addition, we may ignore the phenomena that appear in the case of positive test potentials and that are caused by the possible opening of calcium channels and by the subsequent variations in conductance of potassium channels dependent on calcium concentrations. The changes in the membrane potential V and in the total ionic current IT can be described by the following Hodgkin{Huxley type equations (the outward and inward currents are assumed to be positive and negative, respectively) 11]: Cm V_ + IT + IL = Icom IT = gT mr h (V ; ET ) IL = gL V: (4) Here IT is the total ionic current, m is the parameter that speci es the activation process, h is the parameter that speci es the inactivation process, gT is the maximal value of conductance 11], IL is the leakage current, and Icom is, under natural conditions, the current owing into a hair cell through the transduction channels situated in stereocilia (Icom = ;ITr ) or, in the experiments, is the command current. If experiments are carried out with the use of the current-clamp protocol, then Icom may be speci ed by a researcher according to the voltage-clamp protocol when the potential is changed jumpwise from one level to another, Icom is considered as a registered time variation of the total membrane current 12]. The experiments carried out at the laboratory of neurophysiology (Autonomous University of Puebla, Mexico) revealed that the inactivation parameter h has two constituents (h = h1 + h2 ) corresponding to the potassium channels with fast and slow components of damping. Here we consider the process of variation in h1 and h2 as a Markovian one with the three discrete states h1 , h2 , and 1 ; h1 ; h2 under the assumption that the corresponding transient intensities are equal: 21 = 31 and 12 = 32 . Therefore, Kolmogorov's equations for the inactivation parameters h1 and h2 (these parameters have a probabilistic sense) take the form h_ 1 = ;(12 + 13 + 31 ) h1 + 31 h_ 2 = ;(21 + 23 + 32 ) h2 + 32 or 1 h_ 1 + h1 = h1st 2 h_ 2 + h2 = h2st : (5) Here 1 and 2 are the time constants for two dierent inactivation processes, whereas h1st and h2st are stationary probabilities. The parameters 1 , 2 , h1st , and h2st are constant when the membrane potential is xed. According to the Hodgkin{Huxley model, the potential dependence can be written down in the form

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of the following Boltzmann functions:

hmax
; hi min  i = 1 2 h (6) 1 max + h2 max = 1 1 + exp V S; Vi hi i  i = 1 2: i = min i + max i
;V min (7) ; V 1 + exp S
i
i Note that in this paper the cubic approximation i (V ) = k0i + k1i V + k2i V 2 + k3i V 3 is also used along hist = hi min +

with (7). The activation parameter m is subject to the equations

m m_ + m = mst

mst (V ) = mmin +

1;
mmin  m (V ) = min + max
; min  : 1 + exp ; V S; Vac 1 + exp V S; V



ac

Fig. 3. The volt-clamp (a) and current-clamp (b) protocols for a hair cell of the semicircular canal

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Fig. 4. The graphs of the functional parameters m, h, m , 1 , 2 , and h (a: the activation and inactivation curves, b: the activation time constant, c: the inactivation time constants) Table 2 Parameters Hair cell Cm 11.1 pF gL 1.44 nS gT 65.32 nS ET ;86 mV max 9.98 ms min 0.69 ms V
;52:62 mV S
8:86 mV mmin 0.3 Vac ;32:84 mV Sac 14:16 mV kh0 39.68 kh1 0.56 kh2 0.0136 kh3 0.00005 hmin 0.55 Vh ;75:3 mV Sh 7:1 mV gTr 1.4 nS r 3 Table 3 Parameters Transduction current G1 0:75 kcal=mole Z1 10:0 kcal=mole  m G2 0:25 kcal=mole Z2 2:0 kcal=mole  m R 1:987 cal=(mole  K) T 290 K

The experiments were carried out with hair cells of the axolotl's (Ambystoma tigrinum) semicircular canal by the patchclamp method. The total cell current and the membrane potential were measured according to the voltage-clamp and currentclamp methods, respectively. The total current was activated with Vhold = ;85 mV when the sequential test potential pulses of duration 800 ms ranging from ;130 to +50 mV in steps 10 mV were applied. The membrane potential was measured with Vhold = ;70 mV a series of current (Icom ) pulses was used with the following parameters: the duration 200 ms and the amplitudes in the range between ;0:5 and +0:9 nA. Some experimental results obtained for a single crista hair cell are presented in Figure 3. A program package produced by Axon Instruments was used to process experimental data and to represent them in graphic form. The functions mst (V ), hst (V ), m (V ), 1 (V ), and 2 (V ) are illustrated in Figure 4. Since all potassium currents were combined into a single common current, in the representation of the above activation and inactivation parameters the following peculiarities can be observed: a) when the membrane potential is less than the rest potential, the activation parameter m is not equal to zero b) the parameters h1st (V ) and h2st (V ) are speci ed under the assumption that h1st +h2st = q1 hst +q2 hst = (q1 + q2 ) hst , where q1 and q2 are nonnegative constants and q1 + q2 = 1. Some numerical results obtained according to the procedure given in 12] are presented in Table 2.

4. A MATHEMATICAL MODEL FOR THE MECHANORECEPTOR OF ANGULAR ACCELERATIONS

In order to combine the model of cupula dynamics and the model describing the dynamics of the total ionic current and membrane potential, we have to construct a mathematical model describing the mechanism of mechanoelectrical transduction that realizes the conversion of mechanical energy (needed for the rotation of the axolotl's head) into the electric transduction current energy. Based on the well-known Hudspeth model 13] describing the dependence of transduction current on the hair-bundle displacement (see Table 3), we can write down the relation ITr = gTr (')  V where gTr is the conductance of the transduction current. The representation of this conductance as a function of hair-bundle displacement was empirically obtained in 13]:

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gTr (') = gTr



G ; Z c' 
G ; Z c' 2 2 1 1 1 + exp : 1 + exp RpTp Rp Tp



Here gTr is the maximal conductance corresponding to the case when all transduction channels are open, x = c' is the hair-bundle displacement, Rp is the gas constant, Tp is the absolute temperature, and G1 , Z1, G2 , and Z2 are the parameters of the Boltzmann function for the transduction current. Let us combine equations (3) { (9):






 ' = ; R 1 + l ! ' + 8a2 '_ + mk 4 ck2 L _ 1 ITr = gTr (')V Cm V_ + IT + IL + ITr = 0 IL = gL V IT = gT mr (h1 + h2 ) (V ; EK ) m (V )m_ + m = mst (V ) 1 (V )h_ 1 + h1 = q1 hst (V ) 2 (V )h_ 2 + h2 = q2 hst (V ):

(10)

Here the functional parameters gTr ('), m (V ), 1 (V ), 2 (V ), mst (V ), and hst (V ) are of the form g
Tr

 gTr (') = 1 + exp G1R; TZ1 c' 1 + exp G2R; TZ2 c' p p

p p

 m (V ) = min + max
;Vmin 1 + exp S; V

i (V ) = kh0i + kh1i V + kh2i V 2 + kh3i V 3 i = 1 2 1;
mmin  mst (V ) = mmin + 1 + exp ; V S; Vac ac hst (V ) = hmin + 1 ; hVmin; Vh : 1 + exp S h

Thus, the above mathematical model for the mechanoreceptor of angular accelerations can be represented in the form of the sixth-order system of dierential equations with the six functional parameters and with the numerical parameters given in Tables 1 { 3.

5. ANALYSIS OF THE MATHEMATICAL MODEL Fig. 5. The response of the mechanoreceptor of Let us analyze the mathematical model (10) angular accelerations to a mechanical stimulus (a: for the following two particular cases with the pathe time dependence of the mechanical stimulus !, rameters from Tables 1 { 3. b: the cupula deection, c: the membrane potential 1. Let h1  h2 . Then, system (10) becomes of fth order and has a single stationary point (for variation, d: the variation in the total ionic current) !_  0): '0  0, '_ 0  0, Vrest  = = ;55 mV, mrest 

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0  0:42, and hrest  = 0:58. This case corresponds to the state of rest with IT0  = 86:68 pA, ITr = ;7:45 pA, 0  and IL = ;79:23 pA. Using the linear numerical analysis in a neighborhood of the stationary point, we obtain the following roots of the characteristic equation for the linearized system of model (10): 1 = ;0:00006, 2 = ;0:28364, 3 = ;0:02369, 4 = ;0:36069, and 5 = ;0:17156 (1/ms). For small amplitudes of a periodic stimulus, thus, the problem takes the form of a periodic variation of the receptor potential Vr = V ; Vrest with the same period. Some examples illustrating the dynamics of model (10) are presented in Figure 5. 2. The presence of high damping in the model of cupula dynamics and the fact that the time constant m (V ) is much less than the time constant h (V ) allow us to introduce a small parameter into (10) and, hence, to simplify the model even more. Let us pass to the dimensionless variables t = T
 T , ' = !
 !, v = V
 V , and ! = "
 ", where !
= 1, T
= 17466 ms (the large time constant in the cupula{endolymph dynamics of the axolotl posterior canal), "
= !T
, and V
= 150 mV. The dimensionless and normed
variables m and h (0  m  1 and 0  h  1) have a probabilistic sense. When h1 = h2 , system (10) expressed in the dimensionless variables !, ", V , m, and h takes the form d! = "

dT
 T1 d" = ;" ; ! ; 1 1 + l R ;T T !_ (t) T
dT k2 L c 1 2
 Cm dV = ; gT m3 h V ; EK ; gL V ; V T g dT g V g (!)


where

Tr

Tr




(11)

Tr



G ; Z c!  G ; Z c ! 1 1 1 + exp 2R T 2 (!) = 1 + exp Rp Tp p p m (V ) dm = ;m + m (V ) h (V ) dh = ;h + h (V ): st st T
dT T
dT 1 k 4 ), Here T1 = 3:5247 ms, T
= T2 = 17466 ms, TCgm = 4:5, TT1 = 2 (where T1 = 8a1 and T2 = 8m a21 
Tr
and 0:39  mT(V )  1:3 (where  = 10;4).
In the case of slowly changing mechanical stimuli !_ (t), thus, we may come to the case of the following degenerate system consisting of two dierential equations for !e and Ve and of three algebraic relations:
 d!e = ;!e ; 1 1 + l R ;T T !_ (T ) h (Ve ) dh~ = ;h~ + h (Ve ) st dT k2 L c 1 2 T dT " #
(12) gTr "e = ;!e m = m (Ve ): V
e  g + h~ = ; L st (!e ) gT m3st (Ve ) Ve ; EVK





Our analysis of system (12) shows that the stationary points of systems (11) and (12) coincide and that system (12) does not have a limit cycle. Thus, we may come to the conclusion that there exist at least three approximate mathematical models (see (10), (11), and (12)) for the mechanoreceptor of angular accelerations.

REFERENCES

1. C. Fernandez and J.M. Goldberg, \Physiology of peripheral neurons innervating semicircular canals of the squirrel monkey. II: Response to sinusoidal stimulation and dynamics of peripheral vestibular system", J. Neurophysiol., 34: 661{675, 1971. 2. B.N. Segal and J.S. Outerbridge, \A nonlinear model of semicircular canal primary aerents in bullfrog", J. Neurophysiol., 47, 4: 563{578, 1982.

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3. E. Soto, V.V. Alexandrov, T.B. Alexandrova, R. Cruz, R. Vega, and T.G. Astakhova, \A mechanical coupling model for the axolotl (Ambystoma tigrinum) semicircular canals", in: Mathematical Modeling of Complex Information Processing Systems, pp. 5{15, Moscow State University, 2001. 4. I.V. Orlov, The Vestibular Function (in Russian), Moscow, 1998. 5. W.C. van Buskirk, \The eect of the utricle on uid ow in the semicircular canals", Ann. Biomed. Eng., 5: 1{11, 1977. 6. T.G. Astakhova, \A mathematical model for the semicircular canal of the vestibular system as an angular acceleration sensor", Vestn. Mosk. Univ. Matem. Mekhan., 1: 69{72, 1989. 7. V.V. Alexandrov, T.B. Alexandrova, T.G. Astakhova, A.G. Yakushev, and E. Soto, \Dynamic equations for the cupula{endolymph system of semicircular canals", Dieren. Uravnenia, 35, 4: 1{5, 1999. 8. V.V. Alexandrov, T.G. Astakhova, and V.K. Trincher, \Mathematical modeling of functions of the vestibular canal", Vestn. Mosk. Univ. Matem. Mekhan., 5: 53{57, 1999. 9. V.A. Sadovnichii, V.V. Alexandrov, T.G. Astakhova, V.K. Trincher, and E.Yu. Michaleva, \Harmonic oscillations of the cupula in the vestibular canal", in: Mathematical Modeling of Complex Information Processing Systems, pp. 42{49, Moscow State University, 2001. 10. W. Shteinhausen, \U ber die Beobachtung der Cupula in den Bogengangsampullen des Labirinthes des libenden Heechts", Pug. Arch., 232: 500{512, 1933. 11. A.L. Hodgkin and A.F. Huxley, \Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo", J. Physiol., 116: 449{472, 1952. 12. V.V. Alexandrov, A. Almanza, N.V. Kulikovskaya, R. Vega, T.B. Alexandrova, N.E. Shulenina, A. Limon, and E. Soto, \A mathematical model of the total current dynamics in hair cells", in: Mathematical Modeling of Complex Information Processing Systems, pp. 26{41, Moscow State University, 2001. 13. A.J. Hudspeth and R.S. Lewis, \A model for electrical resonance and frequency tuning in saccular hair cells of bullfrog (Rana Catesbeiana)", J. Physiol., 400: 275{297, 1988. 14 June 2002

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