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c Allerton Press, Inc., 2008. ISSN 0027-1330, Moscow University Mechanics Bulletin, 2008, Vol. 63, No. 6, pp. 139–145. c V.A. Sadovnichii, V.V. Aleksandrov, T.B. Aleksandrova, R. Vega, Castillo Quir´ Original Russian Text oz, M. Reyes Romero, E. Soto, N.E. Shulenina, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 6, pp. 55–60.

A Mathematical Model for the Generation of Output Information in a Gravitoinertial Mechanoreceptor when Moving in a Sagittal Plane V. A. Sadovnichii a , V. V. Aleksandrov a , T. B. Aleksandrova a , R. Vega b , Castillo Quir´ oz b , M. Reyes Romero b , E. Soto b , and N. E. Shulenina a a

Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119899 Russia b Autonomous University of Puebla, Institute of Physiology, Puebla, Pue. 7200, Mexico Received March 26, 2008; in final form, August 29, 2008

Abstract—A mathematical model for the generation of primary and secondary output information in a gravitoinertial mechanoreceptor is discussed. The numerical results obtained from this model are compared with the results of the physiological experiments performed in the Laboratory of Neurophysiology, Institute of Physiology, Autonomous University of Puebla, Mexico.

DOI: 10.3103/S0027133008060022 This paper continues the studies discussed in [1], where the notion of a gravitoinertial mechanoreceptor is introduced. In accordance with the definition of a secondary mechanoreceptor adopted in physiology, this notion is made more accurate by the introduction of primary and secondary output information. A mathematical model for the generation of these output information processes is proposed. The numerical results obtained from this model are compared with the results of the physiological experiments performed on a rotary table in the Laboratory of Neurophysiology, Institute of Physiology, Autonomous University of Puebla, Mexico. 1 Let us consider the situation when a person under test falls in a sagittal plane. At the initial stage of about 100 ms, it is possible for this person to stabilize the vertical pose. In [2] it is shown that the fastest response of hair cells to a mechanical stimulus causing the fall is observed in the cells situated along the sensitivity axis of the saccular macula (at the initial instant of time, this axis is orthogonal to the local vertical, see Fig. 1). Like the utricle, the saccule is a multidimensional accelerometer and allows one to gain information on the apparent acceleration of an otolith membrane in many directions of sensitivity. Fig. 1. Scheme of the saccular macuWe are interested in the study of only the direction mentioned la with the sensitivity directions of hair above. In the further discussion, therefore, we do not consider a cells. mathematical model of the otolith membrane dynamics in a plane parallel to the macula plane and do not discuss the response of many hair cells and afferent primary neurons to this mechanical stimulus; here we study only the dynamics along the above sensitivity axis. The following hair cells are situated along this axis (Fig. 1): the hair cells whose positive direction coincides with the direction of forward motion (they are situated ahead of the striola) and the hair cells whose positive direction coincides with the backward motion (they are situated behind the striola). In our study, hence, we take into account the responses of two hair cells with opposite positive directions of sensitivity. According to [3], the combination of a hair cell and an afferent primary neuron is called a vestibular mechanoreceptor. This mechanoreceptor is a base element of all sensory systems of the vestibular apparatus. The gravitoinertial mechanoreceptor under consideration used to stabilize the vertical pose in a sagittal plane is described by the following three mathematical models. The first one describes the otolith membrane dynamics along the above sensitivity axis, whereas the second and third ones describe the responses of 139

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the vestibular mechanoreceptors with opposite positive directions to the displacements of the otolith membrane (Fig. 2). For simplicity, only one vestibular mechanoreceptor is illustrated in this figure; the second mechanoreceptor has a similar structure. The mathematical model of the vestibular mechanoreceptor consists of the following four blocks: the first block corresponds to the mechanism of mechanoelectrical transduction and is combined with the second one relating to the dynamics of the total ionic current and to the dynamics of the hair cell membrane potential under the integral adaptation feedback; the third block corresponds to the mechanism of synaptic transmission; and the fourth block describes the activity of the afferent primary neuron.

Fig. 2. Scheme of a gravitoinertial mechanoreceptor.

2 Let us discuss our tenth-order model of the gravitoinertial mechanoreceptor: m+ x¨s + k0 x˙ s + ks xs = m− gs (ϕ) − Ws (ϕ) ,

(1)

where m+ = V0 (ρ0 + βρe ),

m− = V0 (ρ0 − ρe );

τad s˙ + s = k(ITr − ITr0 ),

(2)

where ITr = gTr (x, s)(V1 − ETr ),

gTr = g¯Tr p(x, s),

Cm1

x = ±xs ,

p(x, s) =

1 ; x + s − x0 1 + exp − s1

dV1 = −ITr − Itotal − IL1 , dt

(3)

where Itotal = gt m3 (h1 + h2 )(V1 − Et ),

dm = mst (V1 ) − m; dt

(4)

τh1 (V1 )

dh1 = q1 hst (V1 ) − h1 ; dt

(5)

τh2 (V1 )

dh2 = q2 hst (V1 ) − h2 ; dt

(6)

τm (V1 )

Cm2

IL1 = gL1 V1 ;

dV2 = −Isyn (V1 ) − INa − IK − IL2 , dt

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where 3 INa = gNa m∞ (V2 ) C(V2 ) − n (V2 − VNa ),

IK = gK n4 h3 (V2 − VK ),

IL2 = gL2 (V2 − VL );

dn = n∞ (V2 ) − n; dt dh3 τh3 (V2 ) = h3∞ (V2 ) − h3 . dt τn (V2 )

(8) (9)

Here mst (V1 ) = 0.37 +

hst (V1 ) = 0.73 +

m∞ (V2 ) =

1 − 0.37 ; −(V1 + 25.36) 1 + exp 15.06

1 − 0.73 ; V1 + 9.82 1 + exp 21.96

1 ; −(V2 + 33.8) 1 + exp 5.2

h3∞ (V2 ) =

τh1 (V1 ) = 0.82V1 + 55.86;

h∞ (V2 ) =

1 ; V2 + 60.5 1 + exp 9.9

0.96408 − 0.7329 + 0.7329; V2 + 33.87968) 1 + exp 10.24986

τh3 (V2 ) =

τm (V1 ) = 6.55 +

τn (V2 ) =

1250 + 500; 15 + V2 25 + V2 exp + exp −15 10

77.58 − 6.55 ; V1 + 52.23 1 + exp 15.68 τh2 (V1 ) = 1.26V1 + 282.38;

n∞ (V2 ) =

1 ; −(V2 + 35 1 + exp 5

68 ; 25 + V2 30 + V2 exp + exp −15 20 C(V2 ) = n∞ (V2 ) + h∞ (V2 ).

In the above relations, the following notation is introduced: gs (t) is the acceleration of gravity; Ws (t) is the linear head acceleration; Itotal is the total ionic current; INa is the sodium current; IK is the potassium current; IL1 and IL2 are the leakage currents; m and n are the parameters of current activation; h1 , h2 , and h3 are the parameters of current inactivation; gtotal , g¯Tr , gK , gNa , gL1 , and gL2 are the maximum conductances; h1 and h2 are the parameters corresponding to the potassium channels with fast and slow inactivation time constants; τad , τm , τh1 , τh2 , τn , and τh3 are the time constants; s is the adaptation variable; k is the adaptation coefficient; ITr is the transduction current; ITr0 is the steady-state transduction current; p(x, s) is the probability of channel opening; Cm1 and Cm2 are the membrane capacities of the corresponding hair cells; Etotal , ETr , ENa , and EK are the inversion potentials; and Isyn is the synaptic current. Equation (1) describes the otolith membrane displacement xs along the above sensitivity axis of the saccule (V0 is the volume of the otolith membrane, whereas ρ0 and ρe are the densities of the otolith membrane and the endolymph, respectively). Equation (2) describes the adaptation process in a hair bundle under the assumption that its dynamics can be ignored (x = +xs or x = −xs , where x is the displacement of the hair bundle tip of the cell with the positive forward or backward direction). The output result of the first block is the transduction current ITr . Equations (3)–(6) describe the dynamics of the total ionic current Itotal and the dynamics of the membrane potential V1 in the hair cell. The variations of this potential constitute the primary output information on a mechanical stimulus (note that here we consider the second-type hair cells). In order to transmit this priFig. 3. The dependence of the synaptic current on the membrane potential of an mary information to the central nerve system, it is represented in amphibian hair cell. the form of changing the frequency of relaxation self-oscillations, i.e., in the form of pulses forming at the boundary of unmyelinated and myelinated segments in the dendrite of a bipolar neuron. Equations (7)–(9) represent the modified Hodgkin–Huxley model discussed in [4] and


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describe the relaxation self-oscillations of the membrane potential V2 caused by the activity of an afferent primary neuron. The periodicity expressed by the distance between pulses is the secondary output information on the mechanical stimulus obtained from the gravitoinertial mechanoreceptor. A hair cell is connected to a bipolar neuron by the synaptic transmission mechanism. In this paper this mechanism is modeled by the curve in Fig. 3; this curve illustrates the dependence of the synaptic current on the hair cell membrane potential [5] (note that the synaptic current influences the primary neuron activity). This result was obtained for the case of amphibian hair cells. That is why the distinctions between the steady-state values of the membrane potential and the total ionic current of a hair cell for the axolotl (Ambystoma tigrinum) are analyzed in [6]; the resulting data are compared in [3] with similar data obtained for rats. Some results of this comparison are representFig. 4. The comparative analysis of the steady-state values for the ed in Fig. 4 with consideration of the total ionic current Itotal and the membrane potential in the cases of confidence intervals obtained during exaxolotl hair cells (black circles) and rat hair cells (white circles). perimental data processing. The curvilinear quadrangles of steady-state values differ little from each other; this fact gives grounds to use the above-mentioned curve in Fig. 3 (see also [5]) in the mathematical model (1)–(9); for this model, all numerical data (see the table) were obtained experimentally in the Laboratory of Neurophysiology, Institute of Physiology, Autonomous University of Puebla, Mexico. The corresponding experiments were performed in the case of hair cells and bipolar neurons of rats at room temperature (22–25◦C) with the use of patch-clamp technique in the whole-cell configuration. Numerical values of the parameters used in the model Parameters m+ m− k0 ks g¯Tr τad s1 k ITr0 x0 ETr Cm1 gL1 gtotal Etotal q1 q2 Cm2 VNa VK VL gNa gK gL2

Numerical values 1.413 0.628 1.635 1.3086 1.4 100 0.2 0.03 −14.4 0.3 0 11.26 232 77.84 −79 1/2 1/2 1 52 −84 −63 2.3 2.4 0.03

Dimension

Confidence intervals

mg mg mg/ms mg/ms nS ms μm pA μm pF nS nS mV

6.34–16.18 1.84–2.8 56.92–98.76 72–86

μF/cm2 mV mV mV mS/cm2 mS/cm2 mS/cm2

2–8 1–2.6 0.02–0.16


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3 The above dynamic experiments were performed on a rotary table to compare experimental results with the available numerical results obtained during our computer analysis of output information in the gravitoinertial mechanoreceptor under study. Axolotls were used in these experiments. A specimen was placed into a recording chamber and was constantly perfused by Ringer’s solution. In order to register the electrical activity of afferent neurons in the saccule, a suction electrode filled with this solution was used; this allows one to register the activity of several afferent neurons simultaneously. This approach has the following advantage: the specimen under study can be mechanically stimulated. The electrical activity registered by the electrode was fed into an AC amplifier (P-15 Grass). The amplifier output was connected to an oscillograph (WPI model 121) converting the afferent nerve discharges into TTL pulses. These pulses were transmitted to a computer to measure their number per unit time. In order to perform the mechanical stimulation, the specimen was placed together with the microelectrodes and the amplifier on a rotary table whose motion was controlled by a servomotor. This allows one to incline the table through a required angle. The afferent activity was registered under rest conditions and when the table was inclined in the rostral and caudal directions through different angles. The results obtained for the rostral direction are illustrated in Figs. 5a and 5b when the inclination angle is 30◦ . The frequency variations in the membrane potential pulsation are indicated along the y-axis for the primary neurons in hair cells with opposite sensitivity directions. The corresponding experimental data are given in a minute range. The mathematical model represented by (1)–(9) is developed for a range of seconds, which is enough to describe the stabilization process for the vertical pose in the case of uncontrolled fall. That is why this model does not contain some elements of dynamics used for a range of minutes (e.g., the slow adaptation illustrated in Fig. 5a).

Fig. 5. Experimental results obtained with the use of a rotary table. The pulse frequency variation for the primary neurons in the case of hair cells with the oppositely directed sensitivity vectors when the rotary table is inclined through an angle of 30◦ : (a) the excitation response and (b) the braking response.

4 Our computer modeling of the mathematical model (1)–(9) is illustrated in Figs. 6a and 6b. Here we consider the mechanical stimulus used in the above experiment on the saccule of the axolotl. The following two types of hair cells are taken: (i) a hair cell with the forward positive direction (i.e., toward the rotation) and (ii) a hair cell with the backward positive direction. In this experiment it is impossible to exactly identify the sensitivity axis, whereas only the point estimates of numerical parameters are known in the mathematical model of the otolith membrane dynamics. That is why we assume that an otolith membrane displacement of about 1 μm corresponds to the forward inclination through an angle of 30◦ . The response of two vestibular mechanoreceptors are modeled under this assumption. MOSCOW UNIVERSITY MECHANICS BULLETIN

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Comparing the pulse frequencies obtained experimentally and numerically, we come to the following conclusions: (i) these frequencies coincide (about 20 Hz) if the mechanical stimulus is absent (the apparent acceleration projection onto the horizontal direction is equal to zero); (ii) when the mechanical stimulus is present (the forward inclination through an angle of 30◦ ), in the case of excitation response we observe that the numerically obtained frequency increases up to 40 Hz, whereas the experimentally obtained frequency increases on average up to 45 Hz on the interval of length 30 s from 1.5 to 2 minutes; the averaging procedure should be performed because of the effect of slow adaptation; (iii) in the case of braking response, for the same stimulus we observe that the numerically obtained frequency decreases down to 15 Hz, whereas the experimentally obtained frequency decreases down to 7 Hz on average. It should be noted that Figs. 6a and 6b illustrate the primary output information from the gravitoinertial mechanoreceptor, which is difficult to obtain in dynamic experiments with physiological specimens.

Fig. 6. The primary and secondary output information from the gravitoinertial mechanoreceptor of the vestibular apparatus as a result of the mechanical stimulus represented in Fig. 5: (a) the computer modeling of the excitation response and (b) the computer modeling of the braking response.

Our preliminary analysis of the above mathematical model and our comparative analysis of numerical and experimental results show that our model allows one to obtain the primary and secondary output information from the gravitoinertial mechanoreceptor of the vestibular apparatus in response to the mechanical stimulus corresponding to the fall in a sagittal plane. ACKNOWLEDGMENTS This work was performed in the framework of contract no. 02.512.11.2161 and was supported by the Russian Foundation for Basic Research (project no. 07–01–00216). REFERENCES 1. V. V. Aleksandrov, T. B. Aleksandrova, and S. S. Migunov, “A Mathematical Model of Gravitational Inertial Mechanical Receptor,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 2, 59–63 (2006) [Moscow Univ. Mech. Bull. 61 (2), 23–37 (2006)]. 2. V. A. Sadovnichii, V. V. Aleksandrov, E. Soto, et al., “A Mathematical Model of the Response of the Semicircular Canal and Otolith to Vestibular System Rotation under Gravity,” Fund. Prikl. Mat. 11 (7), 207–220 (2005) [J. of Math. Sci. 146 (3), 5938–5947 (2007)]. 3. V. V. Alexandrov, T. B. Alexandrova, R. Vega, et al., “A Mathematical Model of Information Process in Vestibular Mechanoreceptor,” in Proc. 4th WSEAS Int. Conf. on Math. Biology and Ecology, January 25–27, 2008 (Acapulco, Mexico, 2008), pp. 86–91. MOSCOW UNIVERSITY MECHANICS BULLETIN

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4. V. V. Aleksandrov, E. Yu. Mikhaleva, E. Soto, and R. Garsia-Tamayo, “Modification of the Mathematical Hodgkin– Huxley Model for Primary Neurons of the Vestibular Apparatus,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 5, 65–68 (2006) [Moscow Univ. Mech. Bull. 61 (5), 21–24 (2006)]. 5. E. C. Keen and A. J. Hudspeth, “Transfer Characteristic of the Hair Cell’s Afferent Synapse,” Proc. Natl. Acad. Sci. USA 103 (14), 5537–5542 (2006). 6. V. V. Alexandrov, A. Almanza, N. V. Kulikovskaya, et al., “A Mathematical Model of the Total Current Dynamics in Hair Cells,” in Mathematical Modeling of Complex Information Processing Systems (Mosk. Gos. Univ., Moscow, 2001), pp. 26–41.

Translated by O. Arushanyan


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