399 - A diffusion model for the latency of an electroretinogram

BioeZe&r+hemistzy and Bioenergetice. 8 (1981) 26-274 A section of J. Electraanal. Cfiem., and constituting Vol. 128 (1981) EIsevier Sequoia SA., Lausanne -Printed in The Netheriands 269 Short communication 399 - A DIFFUSION MODEL FOR THE LATENCY ELECTROREI’INOGRAM OF AN MIR. M. QASIM HOSSEINI Inditut fiir Normale und Pafhologische Physiologic der Uniuersit&t Kcln (G.F.R.) WOLFGANG Institut fir SCHMICKLER Physikalische Chemie der Universit&t Bonn, Wegelerstr. 12. D-5300 Bonrt 1 (G.F.R.) (M anuscriptreceived March 2% 1980) SUMMARY The diffusion model of Cone for the latency of the b-wape in the electroretino@am is disIt is shown that Cone’s original equation is valid only if the time of illumination is short compared to the latency. A general relation between the intensity of illumination and the latency is derived which holds for arbitrary illumination times. This theoretical result is compared with recent experimental data for the electroretinogram of the frog. By fitting two parameters, good agreement between theory and experiment is achieved. From these parameters an estimate is obtained for the diffusion length of the transmitter substance. c-. INTRODUCTLON When a light pulse is incident on the retina, it causes a change in the electrostatic potential difference between the two sides of the receptor membrane. The time resolution of this response to a light pulse, which is known as an electioretinogram, has been the subject of intensive research (for a review see Ref. 1). An electroretinogram consists of several wtzues,the socazled b-wave being the most prominent. There is a @me-lag between the onset of the light pulse and the beginning of the b-wave. This latency of the b-wave is a function both of the intensity and the duration of the light pulse. Cone [2] suggested the following model for the generation of the b-wave. The light quanti which are absorbed by the photoreceptors of the retina induce a photochemical reaction, in which a ~smitter substance is generated. The transmitter diffuses away, and its concentration c(x, t), as a function of space and time cOordinates;is govetied by the usual diffusion equation. When the concentration of the L ‘tt-er at the place of generation of the electxoretinogram surpasses a certain threshold c,, the b-wave is triggered. The ratecletermining step in this reaction sequence is the diffusion of the 0302-4598/81!00~0-0000/$02.50, .. ,: @ 1981, Envier SequoiaSA. .' 270 For a description of this process, Cone uses the following equ& hansmier. tion: c(x, t) = kIt-"2exp (+?) (1) where k and R are constants and I the intensity of the light pulse. However, ‘;his equation is valid only for an infinitesimally short light pulse, i.e. a pulse with a duration that is short compared to the latency L. This condition, is not fulfilled in the usual experimental procedure_ Therefore, in this paper we shall give an exact solution of the diffusion equation which takes the finite duration of the light pulse into account. Following the ideas of Cone outlined above, we shall use this to derive an equation relating the latency L to the intensity i and the duration 2” of the light pulse. This result will be compared with recent experimental data for the latency of the b-wave in the electioretinogram of the hgSolution of the diffusion equation We shall consider an idealized onedimensional system in which the transmitt&z is generated at the origin of the coordinate system and diffuses along the x-axis. At points away from the origin, the concentration c(x, t) of the transmitter obeys the usual diffusion equation: ac_Da2c af- (Xf ax2 0) where D denotes the diffusion coefficient of the transmitter. At the origin this equation must be supplemented by a term accounting for the generation of the transmitter. If R is the rate of generation and if the light pulse isswitched on at t = 0 and switched off at f = T, the production of the transmitter is described by r(le, t) = M(x) e(r) 8(T - t) (3) where 6 denotes the Dirac delta function, and 8 the Hekiside step function: e(t) = 1 fort20 0 fort<0 ( (4) Incorporating this term into the diffusion equation gives: ac -= at D $$+ R~(K) e(t) e(T-- t) (5) The partial differential equation (5) is supplemented by the boundary conditions: c(~,t)=O lim c(z,t)= t4m fort< 0 0 These two boundary conditions determine a unique solution of the diffusion (6) (7) equation. The derivation of this solution is given in the Appendix. (8) where e&(x) denotes the complement of the error function. Note, that the second term contributes only for t > T, when the generation of the transmitter has stopped. The time dependence of the concentration is shown in Fig. 1 for points at various distences from the origin. For f < T, the concentration increases everywhere. For t = T, the concenkation of the origin starts to decrease immediately, while for points away from the origin it fustcontinues to increase. Only with a certain time-lag, which is the longer the greater the distance from the origin, does the concentration at these points also start to decrease. THE LAT!ZNCY OF THE B-WAVE We assume that the receptive membrane is situated at a distence d &om the origin_ When the concentration c at this membrane surpasses a threshold value c, , the b-wave is triggered_ The latency L of the b-wave is thus determined by the relation: (9) Note that the second term contributes only if the latency is greater than the Xl Y=O 0.4 0.2 0.3 0.1 0 Fig. I. C0ueentre;tion.c of the tranmlitter8s a f~u&ion of the time i for fiie v&es noxnmJiiMdistaaceE,=xA/D,m. switched off at t.= 0.2 ~3.‘ inslz.TheLightpulseis~tchedonatt=Oand J. of the 272 .. 0.4 0.6 O- 0.8 1:. t :, _' ,_: .' .: ._: t(s) -1 - -2 - -3 - Fig_ 2.ComparisonbetareenthetheoreticalrelationbetweentheintensityI L (solid line) and experimental data T%e units on the y-axi~ are arbitrary. andthelatency GneTofilhuGna tion, For the application to experimental data it is conin a slig&ly different form. The production rate R is proportional to the intensity I of illumination: R = KI, where K is a constant. Introducing the constants: cl = c,D”~/K and c2 = d2/4D, we cm express the intensity I as a function of the Iatency‘L, with cl and c2 as parameters: venient to write this equation In Fig. 2 this theoretical relation between the intensity I &nd the latency L is compared with experimental data for the electroretinogram of the frog, Under experimental conditions the time of illumination T was always greater than the latency L, so that the second term did not contribute. The experiments followed the usual procedure; details will be given elsewhere [ 3]_ The ‘Jleoretical curve was obtained by-fitting the two parameters cl and c2. The agreement between the theoretical curve and the experimental points is see& to be good. The parameter c2 can be used to derive an estimate for the diffusion length d of the transmitter. From the data in Fig. 2 we obtained c2 = (1.83 -C0.07) s. Considering the fact that the diffusion coefficients of organic species,in agueous solutions are generally of the order of IO+ cm2 s-l, the diffusion length should be of the order of 30 m. This is a reasonable value, since it i&of the same order of magnitude as the diameter of the photoreceptors of Ram escuknta, which according to Baumann [4] is about 80 /.nn. CONCLUSION The above work is an extension of the model of C&e, in which the la&&y of the b-wave is explained by the diffusion of a tram&i&r suck generam by the incjdent light pulse, Cone’s original equ+ion appli& only .to$h& spell case that the time T of illumi&tion is short compared to the la&ency L. The One of us (M.Q;H.) would like to thank Prof. Dr. W. Sickei from the Institut fiir normale und pathologische Physiologic der UniversitZt Kijln, and Prof. Dr. E. Bodenstedt from the InsMut ftir Strahlen- und Kemphysik d? UniversitZit Bonn for their support of this work and for useful discussions. APPENDIX The Green function g(x, t) = e(t)(4XDt)-“’ of the aon exp --& E equation is: 1 (Al) The solution of equation (5) is obt..ainedfrom the convolution: x2 O(t 4Q(f - t’) 11 t’) dt’ (A21 For the evaluation of this integral we distinguish the two cases T 2 f and T < f. In this case c(x, f) is independent of T. We have: t & =-S,(x, where t) (A31 (A41 withr=t- t’. By substituting u =-I./r this integral can be evaluated [S] : 274 we thus obtain the solution: CaseII: T-C t where 4D(f Substituting S2b, t, n T = = t - ,‘_n~e112 J x2 dt’ - t’) II b-1 t’ we obtain: (-1 (exp[-d]]dr=S,(x,t)-S,(x,t-T) We have thus reduced this case to Case I. The solution is: The two cases T >, t and T < t can be combined equation (8). to the general solution REFERENCES ~s_Briadtey.Phy~ologyoftheRetinaandtheVisualPa~~ay,W~~dWfIkiru,Baftimore. 196o.Chap.VI. ~~A.Cone.J.GenPb~dol..46<1963~1267;47(1964)1267:47<1964~1107. b1.e. HOSS&& in QE!ParatiOn. Cn. Baum~.PnuegasArch..280<1964~81. J.O.ZI.Bocbis.zad A.K.N. Reddy.ModemQectFochemistry.PlenumPrenn.N~Yo~.1976.~o~.1IS_Gradshte~andI~_RyrhiZr,Tab!eofLn~.Seri~andRoducts.AcademLcResp.NewYo~. 1965. given in