Matrix Analysis of the Branched Dendrite Nerve Fiber2.62001/07/102004/09/13 20:57:57.434 GMT-5DougDanielsrainking@alumni.rice.eduStevenCoxcox@rice.eduDougDanielsrainking@alumni.rice.eduStevenCoxcox@rice.edubranched dendrite fiberIn this module we will undertake a real-world example of the electrical modeling scheme we've built up in previous modules (see introductory paragraph). Modeling an actual 3-branched dendrite fiber with the Strang Quartet, we will solve the resulting system with the Backward-Euler method. 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?
PresentationAn Actual Nerve Fiber
A pyramidal neuron from the CA3
region of a rat's hippocampus, scanned at (FIX ME) X
magnification.
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
the diagram below.
3-branch Dendrite Model
Multi-compartment electrical model of a rendered dendrite
fiber.
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 y denoting the
vector of currents
y1…y27.
Using the typical value for a cell's membrane capacitance,
c1(μF/cm2),
we derive (see
variable
conventions):
Capacitance of a Single CompartmentCm2alNc
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
Capacitance of cell bodyCcbAcbcResistance of cell bodyRcbAcbρm.
Applying the Strang QuartetStep (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:
e1x1e2x1Eme3x1x2e4x2e5x2Eme6x2x3…e27x10Em
In matrix for, letting
b
denote the
vector of
batteries,
ebAxwherebEm010010010001001001001001001
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:
y1Ccbte1y2e2Rcby3e3Riy4Cmte4y5e5Rmy27e27Rm
In matrix terms, this compiles to
yGeCte,
where
Conductance matrixG00000000000000000000000000001Rcb0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri000000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm0000000000000000000000000001Ri00000000000000000000000000000000000000000000000000000001Rm
and
Capacitance matrixCCcb00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm00000000000000000000000000000000000000000000000000000000000000000000000000000000000Cm0000000000000000000000000000.Step (S3): Applying Kirchoff's Law
Our next step is to write out the equations for Kirchoff's
Current Law. We see:
i0y1y2y30y3y4y5y60y6y7y8y90y9y10y190y10y11y12y130y10y11y12y130y13y14y15y160y16y17y180y19y20y21y220y22y23y24y250y25y26y270
Since the B coefficient matrix we'd form
here is equal to
A,
we can say in matrix terms:
Ayf
where the vector f is composed of
f1i0
and
f2...270.
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
ACAtxAGAxAGbfACtb,
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,
x0X.
Observing our circuit, and letting
1RfooGfoo,
we calculate the necessary quantities to fill out 's pieces (for these
calculations, see
dendrite.m):
ACACcb0000000000Cm0000000000Cm000000000000000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000Cm0000000000CmAGAGiGcbGi00000000Gi2GiGmGi00000000Gi2GiGmGi00000000Gi3GiGi00Gi00000Gi2GiGmGi00000000Gi2GiGmGi00000000GiGiGm000000Gi0002GiGmGi00000000Gi2GiGmGi00000000GiGiGmAGbEmGcbGmGm0GmGmGmGmGmGmACtb0,
and an initial (rest) potential of
x0Em1111111111.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
ACA,
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
xBxg,
we used in earlier
Backward-Euler-ings. However, we will see that a
modest generalization to our previous form yields :
DxExg
capturing the form of our system and allowing us to solve for
xt.
We manipulate
as follows:
DxExgDx˜tx˜tdtdtEx˜tgDEdtx˜tDx˜tdtgdtx˜tDEdtDx˜tdtgdt,
where in our case
DACA,EAGA, and gAGbf.
This method is implemented in
dendrite.m with typical cell dimensions and
resistivity properties, yielding the following graph of
potentials.
Graph of Dendrite Potentials