5.6: Supplemental - Matrix Analysis of the Branched Dendrite Nerve Fiber
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Introduction
In the prior modules on static and dynamic electrical systems, we analyzed basic, hypothetical one-branch nerve fibers using a modeling methodology we dubbed the Strang Quartet. You may be asking yourself whether this method is stout enough to handle the real fiber of our minds. Indeed, can we use our tools in a real-world setting (Figure 5.6.1)?

To answer your question, the above is a rendering of a neuron from a rat's hippocampus. The tools we have refined will enable us to model the electrical properties of a dendrite leaving the neuron's cell body. A three-branch model of such a dendrite, traced out with painstaking accuracy, appears in Figure 5.6.2.

Our multi-compartment model reveals a 3 branch, 10 node, 27 edge structure to the fiber. Note that we have included the Nernst potentials, the nervous impulse as a current source, and the additional leftmost edges depicting stimulus current shunted by the cell body.
We will continue using our previous notation, namely: Ri and Rm denoting cell body. and membrane resistances, respectively; x representing the vector of potentials x1⋯x10, and x denoting the vector of currents y1⋯y27. Using the typical value for a cell's membrane
c=1(μF/cm2)
we derive (see variable conventions):
Cm=2πalNc
This capacitance is modeled in parallel with the cell's membrane resistance. Additionally, letting Acb denote the cell body's surface area, we recall that its capacitance and resistance are
Ccb=Acbc
Ccb=Acbρm
Applying the Strang Quartet
Step (S1')--Voltage Drops
Let's begin filling out the Strang Quartet. For Step (S1'), we first observe the voltage drops in the figure. Since there are a whopping 27 of them, we include only the first six, which are slightly more than we need to cover all variations in the set:
e1=x1
e2=x1−Em
e3=x1−x2
e4=x2
e5=x2−Em
e6=x2−x3
⋯
e27=x10−Em
In matrix for, letting b denote the vector of batteries,
x=b−Axwhereb=(−Em)(010010010001001001001001001)
and
A=(−1000000000−1000000000−11000000000−1000000000−1000000000−11000000000−1000000000−1000000000−110000000001−1000000000−1000000000−1000000000−11000000000−1000000000−1000000000−11000000000−1000000000−1000000−10001000000000−1000000000−1000000000−11000000000−1000000000−1000000000−11000000000−1000000000−1)
Although our adjacency matrix A is appreciably larger than our previous examples, we have captured the same phenomena as before.
Applying (S2): Ohm's Law Augmented with Voltage-Current Law for Capacitors
Now, recalling Ohm's Law and remembering that the current through a capacitor varies proportionately with the time rate of change of the potential across it, we assemble our vector of currents. As before, we list only enough of the 27 currents to fully characterize the set:
y1=Ccbde1dt
y2=e2Rcb
y3=e3Ri
y4=Cmde4dt
y5=e5Rm
⋯
y27=e27Rm
In matrix terms, this compiles to
y=Ge+Cdedt
where
Conductance matrix
G=(00000000000000000000000000001Rcb0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Ri0000000000000000000000000001Ri0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm)
and
Capacitance matrix
C=(Ccb00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm0000000000000000000000000000)
Step (S3): Applying Kirchoff's Law
Our next step is to write out the equations for Kirchoff's Current Law. We see:
i0−y1−y2−y3=0
y3−y4−y5−y6=0
y6−y7−y8−y9=0
y9−y10−y19=0
y10−y11−y12−y13=0
y13−y14−y15−y16=0
y16−y17−y18−y19=0
y19−y20−y21−y22=0
y22−y23−y24−y25=0
y25−y26−y27=0
Since the B coefficient matrix we'd form here is equal to AT, we can say in matrix terms:
ATy=−f
where the vector f is composed of f1=i0 and f2⋯27=0
Step (S4): Stirring the Ingredients Together
Step (S4) directs us to assemble our previous toils together into a final equation, which we will then endeavor to solve. Using the process derived in the dynamic Strang module, we arrive at the equation
ATCAdxdt+ATGAx=ATGb+f+ATCdbdt
which is the general form for RC circuit potential equations. As we have mentioned, this equation presumes knowledge of the initial value of each of the potentials, x(0)=X.
Observing our circuit, and letting 1Rfoo=Gfoo, we calculate the necessary quantities to fill out Equation's pieces (for these calculations, see dendrite.m):
ATCA=(Ccb0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm)
ATGA=(Gi+Gcb−Gi00000000−Gi2Gi+Gm−Gi00000000−Gi2Gi+Gm−Gi00000000−Gi3Gi−Gi00−Gi00000−Gi2Gi+Gm−Gi00000000−Gi2Gi+Gm−Gi00000000−GiGi+Gm000000−Gi0002Gi+Gm−Gi00000000−Gi2Gi+Gm−Gi00000000−GiGi+Gm)
ATGb=Em(GcbGmGm0GmGmGmGmGmGm)
ATCdbdt=0
and an initial (rest) potential of
x(0)=Em(1111111111)
Applying the Backward-Euler Method
Since our system is so large, the Backward-Euler method is the best path to a solution. Looking at the matrix ATCA we observe that it is singular and therefore non-invertible. This singularity arises from the node connecting the three branches of the fiber and prevents us from using the simple equation x′=Bx+g, we used in earlier Backward-Euler-ings. However, we will see that a modest generalization to our previous form yields Equation:
Dx′=Ex+g
capturing the form of our system and allowing us to solve for x(t) Equation as follows:
Dx′=Ex+g
D˜x(t)−˜x(t−dt)dt=E˜x(t)+g
(D−Edt)˜x(t)=D˜x(t−dt)+gdt
˜x(t)=(D−Edt)−1(˜x(t−dt)+gdt)
where in our case
D=ATCA
E=−(ATGA)
g=ATGb+f.
This method is implemented in dendrite.m with typical cell dimensions and resistivity properties, yielding the following graph of potentials.
Graph of Dendrite Potentials
