# Connexions

You are here: Home » Content » Matrix Analysis » Nerve Fibers and the Strang Quartet

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This collection is included in aLens by: Digital Scholarship at Rice University

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

#### Also in these lenses

• Lens for Engineering

This module and collection are included inLens: Lens for Engineering
By: Sidney Burrus

Click the "Lens for Engineering" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Steven J. Cox. E-mail the author

# Nerve Fibers and the Strang Quartet

Module by: Doug Daniels. E-mail the author

Summary: This module introduces matrix algebra as a tool for solving multivariable problems. Setting up a model for a nerve cell, we use matrices to simply express the electrical properties of the nerve cell, and utilize matrix algebra to solve for the potential differences across nodes and axial and membrane current. By working several examples, we also introduce and reinforce a general problem modeling methodology, and demonstrate the usefulness of matrix algebra for realizing a solution to these problems.

## Nerve Fibers and the Strang Quartet

We wish to confirm, by example, the prefatory claim that matrix algebra is a useful means of organizing (stating and solving) multivariable problems. In our first such example we investigate the response of a nerve fiber to a constant current stimulus. Ideally, a nerve fiber is simply a cylinder of radius aa and length ll that conducts electricity both along its length and across its lateral membrane. Though we shall, in subsequent chapters, delve more deeply into the biophysics, here, in our first outing, we shall stick to its purely resistive properties. The latter are expressed via two quantities:

1. ρ i ρ i , the resistivity in cm cm of the cytoplasm that fills the cell, and
2. ρ m ρ m , the resistivity in cm 2 cm 2 of the cell's lateral membrane.

Although current surely varies from point to point along the fiber it is hoped that these variations are regular enough to be captured by a multicompartment model. By that we mean that we choose a number NN and divide the fiber into NN segments each of length lN l N . Denoting a segment's

Definition 1: axial resistance
R i = ρ i lNπa2 R i ρ i l N a 2
and
Definition 2: membrane resistance
R m = ρ m 2πalN R m ρ m 2 a l N
we arrive at the lumped circuit model of Figure 1. For a fiber in culture we may assume a constant extracellular potential, e.g., zero. We accomplish this by connecting and grounding the extracellular nodes, see Figure 2.

Figure 2 also incorporates the exogenous disturbance, a current stimulus between ground and the left end of the fiber. Our immediate goal is to compute the resulting currents through each resistor and the potential at each of the nodes. Our long--range goal is to provide a modeling methodology that can be used across the engineering and science disciplines. As an aid to computing the desired quantities we give them names. With respect to Figure 3, we label the vector of potentials x=( x 1 x 2 x 3 x 4 ) x x 1 x 2 x 3 x 4 and the vector of currents y=( y 1 y 2 y 3 y 4 y 5 y 6 ) . y y 1 y 2 y 3 y 4 y 5 y 6 . We have also (arbitrarily) assigned directions to the currents as a graphical aid in the consistent application of the basic circuit laws.

We incorporate the circuit laws in a modeling methodology that takes the form of a Strang Quartet:

• (S1) Express the voltage drops via e=(Ax) e A x .
• (S2) Express Ohm's Law via y=Ge y G e .
• (S3) Express Kirchhoff's Current Law via ATy=f A y f .
• (S4) Combine the above into ATGAx=f A G A x f .

The AA in (S1) is the node-edge adjacency matrix -- it encodes the network's connectivity. The GG in (S2) is the diagonal matrix of edge conductances -- it encodes the physics of the network. The ff in (S3) is the vector of current sources -- it encodes the network's stimuli. The culminating ATGA A G A in (S4) is the symmetric matrix whose inverse, when applied to ff, reveals the vector of potentials, xx. In order to make these ideas our own we must work many, many examples.

## Example

### Strang Quartet, Step 1

With respect to the circuit of Figure 3, in accordance with step (S1), we express the six potential differences (always tail minus head) e 1 = x 1 x 2 e 1 x 1 x 2 e 2 = x 2 e 2 x 2 e 3 = x 2 x 3 e 3 x 2 x 3 e 4 = x 3 e 4 x 3 e 5 = x 3 x 4 e 5 x 3 x 4 e 6 = x 4 e 6 x 4 Such long, tedious lists cry out for matrix representation, to wit e=(Ax) e A x where A=( -1100 0-100 0-110 00-10 00-11 000-1 ) A -1100 0-100 0-110 00-10 00-11 000-1

### Strang Quartet, Step 2

Step (S2), Ohm's Law, states:

#### Law 1: Ohm's Law

The current along an edge is equal to the potential drop across the edge divided by the resistance of the edge.

In our case, y j = e j R i ,   j=1 , 3 , 5   and   y j = e j R m ,   j=2 , 4 , 6 y j e j R i ,   j 1 , 3 , 5   and   y j e j R m ,   j 2 , 4 , 6 or, in matrix notation, y=Ge y G e where G=( 1 R i 00000 01 R m 0000 001 R i 000 0001 R m 00 00001 R i 0 000001 R m ) G 1 R i 0000 0 0 1 R m 0000 00 1 R i 000 000 1 R m 00 0000 1 R i 0 0000 0 1 R m

### Strang Quartet, Step 3

Step (S3), Kirchhoff's Current Law, states:

#### Law 2: Kirchhoff's Current Law

The sum of the currents into each node must be zero.

In our case i 0 y 1 =0 i 0 y 1 0 y 1 y 2 y 3 =0 y 1 y 2 y 3 0 y 3 y 4 y 5 =0 y 3 y 4 y 5 0 y 5 y 6 =0 y 5 y 6 0 or, in matrix terms By=f B y f where B=( -100000 1-1-1000 001-1-10 00001-1 )   and   f= i 0 000 B -1000 00 1-1-10 00 001-1 -10 0000 1-1   and   f i 0 0 0 0

### Strang Quartet, Step 4

Looking back at AA: A=( -1100 0-100 0-110 00-10 00-11 000-1 ) A -1100 0-100 0-110 00-10 00-11 000-1 we recognize in BB the transpose of AA. Calling it such, we recall our main steps

• (S1) e=(Ax) e A x ,
• (S2) y=Ge y G e , and
• (S3) ATy=f A y f .
On substitution of the first two into the third we arrive, in accordance with (S4), at
ATGAx=f . A G A x f .
(1)
This is a system of four equations for the 4 unknown potentials, x 1 x 1 through x 4 x 4 . As you know, the system Equation 1 may have either 1, 0, or infinitely many solutions, depending on ff and ATGA A G A . We shall devote (FIX ME CNXN TO CHAPTER 3 AND 4) to an unraveling of the previous sentence. For now, we cross our fingers and solve' by invoking the Matlab program, fib1.m .

This program is a bit more ambitious than the above in that it allows us to specify the number of compartments and that rather than just spewing the xx and yy values it plots them as a function of distance along the fiber. We note that, as expected, everything tapers off with distance from the source and that the axial current is significantly greater than the membrane, or leakage, current.

## Example

We have seen in the previous example how a current source may produce a potential difference across a cell's membrane. We note that, even in the absence of electrical stimuli, there is always a difference in potential between the inside and outside of a living cell. In fact, this difference is the biologist's definition of living.' Life is maintained by the fact that the cell's interior is rich in potassium ions, K + K + , and poor in sodium ions, Na + Na + , while in the exterior medium it is just the opposite. These concentration differences beget potential differences under the guise of the Nernst potentials:

Definition 3: Nernst potentials
E Na =RTFlog [Na] o [Na] i   and   E K =RTFlog [K] o [K] i E Na R T F [Na] o [Na] i   and   E K R T F [K] o [K] i where RR is the gas constant, TT is temperature, and FF is the Faraday constant.
Associated with these potentials are membrane resistances ρ m , Na   and   ρ m , K ρ m , Na   and   ρ m , K that together produce the ρ m ρ m above via 1 ρ m =1 ρ m , Na +1 ρ m , K 1 ρ m 1 ρ m , Na 1 ρ m , K and produce the aforementioned rest potential E m = ρ m ( E Na ρ m , Na + E K ρ m , Na ) E m ρ m E Na ρ m , Na E K ρ m , Na

With respect to our old circuit model, each compartment now sports a battery in series with its membrane resistance, as shown in Figure 6.

Revisiting steps (S1-4) we note that in (S1) the even numbered voltage drops are now e 2 = x 2 E m e 2 x 2 E m e 4 = x 3 E m e 4 x 3 E m e 6 = x 4 E m e 6 x 4 E m We accommodate such things by generalizing (S1) to:

• (S1') Express the voltage drops as e=bAx e b A x where bb is the vector of batteries.
No changes are necessary for (S2) and (S3). The final step now reads,
• (S4') Combine (S1'), (S2), and (S3) to produce ATGAx=ATGb+f A G A x A G b f .

Returning to Figure 6, we note that b=( E m 010101)       and       ATGb= E m R m 0111 b E m 01 01 01       and       A G b E m R m 01 11 This requires only minor changes to our old code. The new program is called fib2.m and results of its use are indicated in the next two figures.

## References

1. G. Strang,. (1986). Introduction to Applied Mathematics. Wellesley-Cambridge Press.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks