## Figure P110

" HH membrane patch ODE for 1st Node of Ranvier (the SGL) .

dv1 = A*(Jin0 - Jk1 - Jnal - JL1)/Ch + (-vi + V11)*3.333E-4/Ch " mV/ms

" 3.333E-4 is milliSiemens so current will be microA. " v1 is in mV. HH convention has v1 < 0 if it is depolarization. " A is area of N of R in cm^2.

" Ionic current densities are in microamps/cm^2.

Jin0 is a negative input current pulse needed to initiate the nerve spike at the SGL (v1) node. A is an area scaling factor: A = 3 x 10-5 cm2. All J

turns are current densities in |A/cm2. Thus, multiplication by A converts dv1/dt to mV/ms. The right-hand term gives the |A current leaving the v0 node. Take Ch = 1 x 10-4 (mF). The series resitance between the v1 HH node and the first TL (V11) node is 3 Mohms. If one were dealing in volts, amps, ohms, farads, and seconds, the conductance used in the v1 node equation would be 3.3333 x 10-7 S. However, all voltages are in mV, and the current must be in |A to be compatible with the HH equation. Thus the conductance must be multiplied by 106 and divided by 103, giving a scaled conductance of 3.333 x 10-4 in the HH node equation. The rest of the HH format is standard, given below:

dn1 = - n1*(an1 + bnl) + an1 dml = - m1 * (am1 + bm1) + am1 dh1 = - h1*(ah1 + bh1) + ah1

" K+ activation parameter.

" Na+ activation parameter.

" Na+ inactivation parameter

" HH VOLTAGE-DEPENDENT PARAMETER FUNCTIONS for v0 node:

am1 = . 100* (v1 + 2 5) / (exp (0.1 * v1 + 2.5) - 1)

The (first) V11 node equation of the myelin TL can be written:

Cm*dV11/dt + V11* (gm + 1/6M + 1/3M) + v1/3M - V12/6M = 0 " Amps or dV11 = -V11* (gm + 3/6M)/Cm - (v1/3M

V12/6M)/Cm

From Plonsey, gm = 1.37931 x 10-9 S, Cm (time-scaled so ODE has ms time) = 6.40 x 10-10 F. The total series resistance between nodes of Ranvier is 30 Mohms. Thus, each of the five RCG TL sections is separated by 6 Mohms, with 3 M on the ends. Hence the five node equations for myelin bead 1 are dV11 = -V11*7.83405E2 + v1*5.20833E2 + V12*2.6042E2

dV12 = -V12*5.22989E2 + V11*2.6042E2 + V13*2.6042E2

dV13 = -V13*5.22989E2 + V12*2.6042E2 + V14*2.6042E2

dV14 = -V14*5.22989E2 + V13*2.6042E2 + V15*2.6042E2

dV15 = -V15*7.83405E2 + V14*2.6042E2 + v2*5.20833E2

v2 is the node voltage at the second node of Ranvier. The node equations for the second HH node are

" HH membrane patch ODE. (Node #2 = v2 node) .

dv2 = -A* (Jk2 + JNa2 + JL2)/Cm + (-2*v2 + V21 + V15) *3.333E-4/Cm

" Ionic current densities in microamps/cm^2.

dn2 = - n2*(an2 + bn2) + an2 " K+ activation parameter.

dm2 = - m2*(am2 + bm2) + am2 " Na+ activation parameter.

dh2 = - h2* (ah2 + bh2) + ah2 " Na+ inactivation parameter

" VOLTAGE-DEPENDENT PARAMETER FUNCTIONS: an2 = . 010* (v2 + 10) / (exp (0. 1*v2 + 1) - 1) bn2 = .125*exp(v2/80)

Note that both adjacent myelin beads figure in the v2 node equation. The five node equations for the second bead are dV21 = -V21*7.83405E2 + v2*5.20833E2 + V22*2.6042E2

dV22 = -V2 2*5.22989E2 + V21*2.6042E2 + V23*2.6042E2

dV23 = -V2 3*5.22989E2 + V22*2.6042E2 + V24*2.6042E2

dV24 = -V2 4*5.22989E2 + V23*2.6042E2 + V25*2.6042E2

dV25 = -V25*7.83405E2 + V24*2.6042E2 + v3*5.20833E2

Note that v3 is the voltage at the third node of Ranvier, etc. One can now write the 45 ODEs describing the model. Note that each of the HH systems has the same initial conditions, vital for correct simulation. In general, for k = 1 to 5:

mk:0.052932 nk:0.31768 hk:0.59 612

a. Write the complete Simnon program to simulate the five-node, five-myelin-bead model.

b. Using Simnon with Euler integration with delT = 0.0001 ms, with Cm = 1.E-4, and Jfao = -300 (|A), run the simulation and observe the five depolarization voltages at the nodes of Ranvier. Note that they are negative. Use a vertical axis from -130 to +20 mV, and a timescale from 0 to 10 ms. (To observe the true transmembrane potentials, compute and plot Vmk = Vmo - vk, Vmo = -70 mV.) See how the time between vk peaks changes with different Cm values. Plot and observe the TL node voltages, Vjk , for the model (j = 0, 1, . 4, k = 1, 2, . 5). Make a three-dimensional plot of the five vks vs. time in ms (0 to 10 ms) and distance along the axon (0 to 10 mm). 