Brain Research, 510 (1990) 343-345 Elsevier
343
BRES 23994
Modeling the repetitive firing of retinal ganglion cells J.E Fohlmeister, P.A. Coleman and R.F. Miller Department of Physiology, University of Minnesota, Minneapolis, MN 55455 (U. S.A.)
(Accepted 21 November 1989) Key words: Retina; Ganglion cell; Repetitive firing; Computer modelling; Ion channel; Channel kinetics
A kinetic model for the repetitive firing of retinal ganglion cells was synthesized from voltage-clamp data and evaluated by comparison with whole cell recordings from ganglion cells in the intact tiger salamander retina. Five distinct channels were included in the model and were sufficient to describe the physiologically observed frequency/current relationship in response to various levels of cell depolarization. The ganglion cells of the vertebrate retina form the only pathway by which the retina communicates with the brain. These output neurons convert the graded postsynaptic potentials into a pattern of impulses whose frequency is modulated by the synaptic current. Hodgkin and Huxley first demonstrated that the impulse of the squid giant axon can be adequately represented by two voltage-dependent channels, a Na ÷ and K ÷ channel 4. Voltage clamp studies in ganglion cells of the tiger salamander 6 and rat 5 suggest that at least 5 different ionic currents may be present among these neurons. Presumably, contributions from each of these channels contribute to the shape of the impulse, the interspike trajectory and the relationship between membrane polarization and impulse frequency, We have analyzed the contribution of 5 non-linear channels by developing a detailed model of their currents, simulated through a Cyber computer. These simulations were compared with direct physiological results obtained from ganglion cells (n = 15) using whole-cell recording techniques in the perfused retinaeyecup preparation of the tiger salamander 3. The kinetic model was constructed on the basis of previously reported retinal ganglion cell voltage clamp data 5'6. The five currents included were: Na ÷, Ca 2+, non-inactivating K ÷ (delayed rectifier), inactivating K ÷ (A type), and Ca 2÷ activated K ÷. Four of these currents were modelled as voltage-gated, whereas a fifth, the gK,ca, is gated by [Ca2+]i and modelled on that basis, The basic mathematical structure for voltage-gating was based on Hodgkin and Huxley 4. The differential equation for membrane potential (Kirchoff's law) is given by:
CmdE/dt gNa m3h( E - Eya) gca C3( E - Eca) - gKn4( E -- EK) -gA a3hA( E -- EK) - gK,¢a (E - EK) + Istim =
--
(l)
The inactivation state-variables h, with or without subscript, appear in two terms: the TTX-sensitive Na ÷current and the 4-amino-pyridine sensitive A-current carried by K+-ions. The gating of IK,ca was modelled as ([Ca2+]i/Ca2+diss)J gK,ca ~--- gK,ca 1 + ([Ca2+]i/Ca2+ctiss)J
(2)
where the Ca2+-dissociation constant, Ca2+diss, was taken to be 10 -6 M 1'3. The simulations given in the figure employed j = 2, although other small integers, including j = 1, may also be used. The [Ca2+]i was allowed to vary in response to/ca. We assumed that the inward flowing Ca 2÷ ions are distributed uniformly throughout the cell and that the free [Ca2+]i above a residual level, [Ca2+]res = 10 -7 M, are actively removed from the cell, or otherwise sequestered with a time-constant (r). Thus, the [Ca2+]i follows the equation: d[Ca2+]i/dt = - 3 1 C a / 2 F r - ( [ C a 2 + ] i - [Ca2+]res)/r ,
(3)
where F = Faraday, 3/r is the ratio of surface to volume of the spherical cell soma, and the factor of 2 on F is the valency. For numerical integration this equation reduces to: d[Ca2+]i/dt=-0.000015 Ica--(0.02) ([Ca2+]i-0.000l), (3a)
Correspondence: J.F. Fohlmeister, Physiology Department, 6-255 Millard Hall, University of Minnesota, 435 Delaware St., SE, Minneapolis, MN 55455, U.S.A.
0006-8993/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)
344 TABLE I
T A B L E 1I
Rate constants Jor the jour channels with direct voltage-dependent gating
Value of ,simulation parameters used in Eqn. 1 to generate impulse trains
These constants operate in the first order kinetic equation; dx/dt = - ( a x + fix) x + %. M e m b r a n e potential is in mV; temperature was modeled at 22 °C. .
.
.
.
.
.
-0.6 (E+30) Na+-channel: am -e o , i (E+30)_l
a h = 0.4 e -(E+5°)/2°
.
.
Ca 2+-channel: a~ -
-0.3 ( E +
13)
(a)
6 e-0.1 (E+20) +
1
--
-0.02 (E+40) e o.~E+4O)_l
A_channel:aA _-0.006(E+90) e 0.~(e,9O)_l
tic = 10 e (E+38)n8
(c)
~n = 0.4 e -(E+ 5o)/8o
(d)
flA =0.1 e (E+3o)/.,
(e)
0.6 (thA= 0'04e-(E+70)/20 /~hA- e-D1(E+4o)+ 1
(f)
where the units of time are ms, [Ca2+]i in mM, and Ica in/~A/cm 2. The minus (-) sign of the first term on the right compensates for the convention that inwardly directed currents (i.e. IN, and Ic~) are negative, We employed the continually updated [Ca2+]~ in calculating the Ca2+-reversal potential Eca, which was incorporated in the computation of /Ca' Thus, Eca was highly variable, being ca. + 160 mV at rest, but dropping
40
gA gK,C, ENa EK
(b)
e-01(E+13)--i
K*-channel: ~n
1.O~F/cm:: 40.0 mS/cm 2 2.0 mS/cm 2 12 -0 mS/cm2 36.1) mS/cm 2 0.05 mS/cm 2 +35 mV - 7 5 mV
gK
flrn = 20 e
343
BRES 23994
Modeling the repetitive firing of retinal ganglion cells J.E Fohlmeister, P.A. Coleman and R.F. Miller Department of Physiology, University of Minnesota, Minneapolis, MN 55455 (U. S.A.)
(Accepted 21 November 1989) Key words: Retina; Ganglion cell; Repetitive firing; Computer modelling; Ion channel; Channel kinetics
A kinetic model for the repetitive firing of retinal ganglion cells was synthesized from voltage-clamp data and evaluated by comparison with whole cell recordings from ganglion cells in the intact tiger salamander retina. Five distinct channels were included in the model and were sufficient to describe the physiologically observed frequency/current relationship in response to various levels of cell depolarization. The ganglion cells of the vertebrate retina form the only pathway by which the retina communicates with the brain. These output neurons convert the graded postsynaptic potentials into a pattern of impulses whose frequency is modulated by the synaptic current. Hodgkin and Huxley first demonstrated that the impulse of the squid giant axon can be adequately represented by two voltage-dependent channels, a Na ÷ and K ÷ channel 4. Voltage clamp studies in ganglion cells of the tiger salamander 6 and rat 5 suggest that at least 5 different ionic currents may be present among these neurons. Presumably, contributions from each of these channels contribute to the shape of the impulse, the interspike trajectory and the relationship between membrane polarization and impulse frequency, We have analyzed the contribution of 5 non-linear channels by developing a detailed model of their currents, simulated through a Cyber computer. These simulations were compared with direct physiological results obtained from ganglion cells (n = 15) using whole-cell recording techniques in the perfused retinaeyecup preparation of the tiger salamander 3. The kinetic model was constructed on the basis of previously reported retinal ganglion cell voltage clamp data 5'6. The five currents included were: Na ÷, Ca 2+, non-inactivating K ÷ (delayed rectifier), inactivating K ÷ (A type), and Ca 2÷ activated K ÷. Four of these currents were modelled as voltage-gated, whereas a fifth, the gK,ca, is gated by [Ca2+]i and modelled on that basis, The basic mathematical structure for voltage-gating was based on Hodgkin and Huxley 4. The differential equation for membrane potential (Kirchoff's law) is given by:
CmdE/dt gNa m3h( E - Eya) gca C3( E - Eca) - gKn4( E -- EK) -gA a3hA( E -- EK) - gK,¢a (E - EK) + Istim =
--
(l)
The inactivation state-variables h, with or without subscript, appear in two terms: the TTX-sensitive Na ÷current and the 4-amino-pyridine sensitive A-current carried by K+-ions. The gating of IK,ca was modelled as ([Ca2+]i/Ca2+diss)J gK,ca ~--- gK,ca 1 + ([Ca2+]i/Ca2+ctiss)J
(2)
where the Ca2+-dissociation constant, Ca2+diss, was taken to be 10 -6 M 1'3. The simulations given in the figure employed j = 2, although other small integers, including j = 1, may also be used. The [Ca2+]i was allowed to vary in response to/ca. We assumed that the inward flowing Ca 2÷ ions are distributed uniformly throughout the cell and that the free [Ca2+]i above a residual level, [Ca2+]res = 10 -7 M, are actively removed from the cell, or otherwise sequestered with a time-constant (r). Thus, the [Ca2+]i follows the equation: d[Ca2+]i/dt = - 3 1 C a / 2 F r - ( [ C a 2 + ] i - [Ca2+]res)/r ,
(3)
where F = Faraday, 3/r is the ratio of surface to volume of the spherical cell soma, and the factor of 2 on F is the valency. For numerical integration this equation reduces to: d[Ca2+]i/dt=-0.000015 Ica--(0.02) ([Ca2+]i-0.000l), (3a)
Correspondence: J.F. Fohlmeister, Physiology Department, 6-255 Millard Hall, University of Minnesota, 435 Delaware St., SE, Minneapolis, MN 55455, U.S.A.
0006-8993/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)
344 TABLE I
T A B L E 1I
Rate constants Jor the jour channels with direct voltage-dependent gating
Value of ,simulation parameters used in Eqn. 1 to generate impulse trains
These constants operate in the first order kinetic equation; dx/dt = - ( a x + fix) x + %. M e m b r a n e potential is in mV; temperature was modeled at 22 °C. .
.
.
.
.
.
-0.6 (E+30) Na+-channel: am -e o , i (E+30)_l
a h = 0.4 e -(E+5°)/2°
.
.
Ca 2+-channel: a~ -
-0.3 ( E +
13)
(a)
6 e-0.1 (E+20) +
1
--
-0.02 (E+40) e o.~E+4O)_l
A_channel:aA _-0.006(E+90) e 0.~(e,9O)_l
tic = 10 e (E+38)n8
(c)
~n = 0.4 e -(E+ 5o)/8o
(d)
flA =0.1 e (E+3o)/.,
(e)
0.6 (thA= 0'04e-(E+70)/20 /~hA- e-D1(E+4o)+ 1
(f)
where the units of time are ms, [Ca2+]i in mM, and Ica in/~A/cm 2. The minus (-) sign of the first term on the right compensates for the convention that inwardly directed currents (i.e. IN, and Ic~) are negative, We employed the continually updated [Ca2+]~ in calculating the Ca2+-reversal potential Eca, which was incorporated in the computation of /Ca' Thus, Eca was highly variable, being ca. + 160 mV at rest, but dropping
40
gA gK,C, ENa EK
(b)
e-01(E+13)--i
K*-channel: ~n
1.O~F/cm:: 40.0 mS/cm 2 2.0 mS/cm 2 12 -0 mS/cm2 36.1) mS/cm 2 0.05 mS/cm 2 +35 mV - 7 5 mV
gK
flrn = 20 e
