**Comp Neuroscience**

Neural processes are dynamic phenomena, which means that they
change in time. These temporal variations are extremely
important; indeed, many sensory stimuli are coded according tho
firing * rates* of the neurons and not their absolute
membrane potentials. The most accepted models of memory and
learning depend on the rates of * change* of the neurons,
that is, the correllation between the activities of the post-
and pre-synaptic cells. Recent evidence has pointed to the
importance of 40 Hz oscillations in binding diverse properties
of visual and olfactory stimuli. Dynamic phenomena play an
obvious role in motor activity as well. Locomotion, whether
stereotypical, such as trotting of horses and grinding of the
lobster stomato-gastric system, or driven by feedback, as in
navigation of an obstacle course, depends on precise temporal
relations between the limbs and the various components of the
locomotor event.
Autonomic processes
such as breathing, hormonal secretion, circadian cycles, and
others also depend on temporal processes such as regular
rhythms and more complex phemonena, e.g. spike bursts and
irregular activity.

Many pathologies are due to temporal difficulties in neural systems; notable among these are epilepsy, Parkinsonian seizures, and various EEG abnormalities. Indeed, the EEG is nothing more than a time series of the lumped activity of many active neurons.

The language of dynamic phenomena is differential equations. A
differential equation is an equation that relates the rate of
change of some process to other processes that are changing in
time. The simplest example and one that will play a role in
neurobiology is the decay to rest of the membrane potential. A
passive membrane can be modeled by a capacitor with capacitance
**C** and a resistor with resistance **R** and battery with
potential in parallel, Fig 1

**Figure 1:** Passive Membrane Model

The quantity of interest is the voltage across the capacitor, This voltage slowly leaks out of the capacitor across the resistor. Elementary circuit theory tells us that the rate of change of the voltage is proportional to minus the voltage difference. The rate of change of a quantity, with respect to time is of course the derivative of that quantity with respect to time. Thus, our statement above can be translated into mathematics to read as follows:

where is the constant of proportionality. For the
circuit here, the constant is , where **R** is the
resistance in ohms and **C** is the capacitance in farads. Note
that the product of one ohm with one farad has the value of one
second. The quantity, **RC** is called the membrane time-constant.
Thus, a membrane with resistance and
capacity would have a time constant of 10 ms. The
objective of the theory of differential equations is to try to
understand the behavior of systems like (1) and to use this
knowledge to predict the behavior of experiments. There are at
least 5 different levels at which one can study (1):

- The most satisfactory may seem to be an explicit solution to the equation in terms of all of the parameters and time. This is usually impossible (and certainly impossible for all but the most elementary models of neural processes). Also, it is often fairly useless as anyone who has spent a few minutes with a symbolic algebra program will attest. For (1) an exact solution is easy to find.
- Prove that a solution of the desired type exists. This is difficult for non-mathematicians to understand: a proof is true and always true. For equation (1), one might try to prove that positive solutions are always decreasing. (You mathematicians should have no problem with this)
- Qualitative analysis of the equation. This means that
rather than obtaining a precise solution to the equation, we
attempt to analyse the behavior by using properties of the
equation. This methodology is ideally suited to biological
systems, where precise formulas for the various quantities of
interest are not known. For example, in a negative resistance
preparation such as various vertebrate neurons with excitatory
transmitters applied, the shape of the
**I-V**relation is only experimentally known and no exact formula for the shape can be given. Qualitative methods (which we will emphasize) are very good for this purpose (see the Chapter 5 in the Koch-Segev book) - Numerical solutions. This is the best known way of
studying models that depend on differential equations. All
simulation tools such as Genesis, the Hinds simulator, and
XPP, numerically solve differential equations as a means
of understanding their behavior. Together with qualitative
methods, this tool provides a very complete combination for
studying differential equations.
- Approximation methods. In many biological systems, there are many different temporal regimes varying from milliseconds to months. In many cases it is possible to hold some variables constant (those that slowly change) or assume that their rapid variation can be averaged (for those that are changing quickly). When this is done, one can often obtain a simpler set of equations that can be explicitly solved or analyzed. This is the technique we have used to analyze the lamprey CPG.

Equation (1) is called a first order differential equation (ODE) and in order to solve it, we must specify one more condition. To see why, suppose I tell you that someone is driving at 50 MPH down the turnpike. After one hour, how many miles down the pike is he? To answer this, you must know what milepost he started at. That is, you must be given the initial position. In general, you must specify an initial condition for each first order differential equation. Thus, we must give the initial voltage in the capacitor in order to solve (1). Equation (1) is of the following form:

which can be solved by integration:

so, we get:

For the above equation and hence:

Inverting this equation, we obtain:

The voltage decays exponentially from its initial value. The larger the resistance, the slower it decays.

HOMEWORK

- Suppose that the membrane has a resistence of and
a capacitance of 120 picofarads. The initial potential is 100
mV and the battery is
**-60mV**(a) What is the membrane time constant. (b) What is the potential drop after 100 msec. (c) After how long has the potential dropped to 25 mV. - A simple model for a periodically varying
calcium conductance is:
(a) Assuming that , what is the potential as a function of time. (b) What are the dimensions of all the parameters. (c) As , what does the membrane potential tend to? (d) If , does the voltage ever change?

There are usually many differential equations in a model system.
Consider the following psychological example. Suppose that
Harry is a fickle suitor and Sally is the woman who he is
interested in. The rate at which her love for him changes
depends on his love for her. Harry on the other hand is
interested only when she is not and loses interest as soon as
she finds him attractive. Let **x** denote the amount that Harry
is attracted to Sally and let **y** denote the amount that Sally
is attracted to Harry. Then the equations are:

If and then it is simple to verify that the solution to (3) is:

The point is not that this equation is solvable, but rather that
it typifies the interactions of ``excitatory'' and
``inhibitory'' processes. The variable **y** ``inhibits'' **x** and
**x** ``excites'' **y** as in Fig 2.

**Figure 2:** Typical negative feedback interaction

This type of interaction often leads to oscillations and is a form of delayed inhibition. Another way to induce a delayed inhibition is to put it in directly:

This is an example of a delay-differential equation. We will
not study these too much since they are very difficult to solve.
However, solution to * this* problem is We will see
later that this notion of delayed inhibitory feedback is responsible for
most if not all oscillatory behavior in neurons.

Now consider a general system of two linear differential equations:

The general solution to this equation (except for some special cases) is:

where **A,B,C,D** are constants (perhaps complex) that depend on
the initial conditions and the parameters **a,b,c,d** and
are the * eigenvalues* of the 2x2 matrix:

Recall that the eigenvalues of a matrix **M** are the roots of the *
characteristic polynomial* which is

where **I** is the identity matrix of all zeros except the 1's along the
diagonals.
For the present example,

This second degree polynomial has two roots. The quantity is
the * trace* of the matrix **M** and the quantity **ad-bc** is the
determinant of the matrix. The roots of the polynomial can be
either real or complex. If the real parts are positive, then it is
clear that the solutions will grow exponentially fast as **
t**
increases. Thus, the solutuion will not be bounded. On the other
hand, if all the real parts are less than or equal to zero, then the
solutions will remain bounded as As an example, the
Harry-Sally problem has So that the
eigenvalue equation is:

The roots of this are where Since we see that we recover the originally found solutions.

HOMEWORK

- (a) Find the eigenvalues if
**a=1,b=-2,c=2,d=1**. Do solutions grow exponentially or do they stay bounded. (b) How about if - Answer (1a) for the following cases: (i)
**a=2,b=3,c=4,d=0**(ii)**a=-1,b=2,c=-3,d=-2**(iii)**a=1,b=-3,c=2,d=-1**

A short aside in linear algebra
Linear differential equations, which are ultimately very important
since many nonlinear systems are approximated by them near equilibria,
are solved by using techniques from linear algebra. The most crucial
ideas are the notions of eigenvalues and eigenvectors. I will assume
that you know how to multiply matrices together and that you can find
the transpose of a matrix and the inner product of two vectors.
The * norm of a vector* is the
square root of the sum of the squares of each element. The
* inner product* of two vectors is the sum of the products of
each of the elements. Just
for notational sake, **A** is an matrix means that **A** has
**n** rows and **m** columns. The * matrix norm* of **A** is the maximum
over all rows of **A** of the sums of the absolute values of the elements
in each row. A * row vector* is a matrix and a
* column vector* is a matrix. Matrices are
multiplied in the usual manner . To multiply an matrix
by an matrix, you must have **n2 = n3** and the result
is an matrix. The **ij** entry of the product of two
matrices takes the row of the first times the column
of the second (* ie* the inner product of a row from the first with
a column from the second.)
This is why the number of columns in the first must
equal the rows in the second. A square matrix has an * inverse* if there
is a square matrix **B** such that **AB=BA=I** where **I** is the square
matrix with 1 along the diagonal and 0 everywhere else. A matrix is
invertible if and only if the determinant of that matrix is nonzero.

EXAMPLES

I define 4 matrices,

Note that **D** is a row vector. You can multiply **BA** but not **AB**
since **A** has 3 columns and **B** has 2 rows so they are not compatible.
If you multiply **AC** you get a matrix,

but if you multiply
**CA** you get a matrix

Only square matrices have inverses. It is easy to show that

The norm of the row vector, **D** is The matrix norm of **A**
is 6. The transpose of **B** is itself. We say such a matrix is *
symmetric.* Symmetric matrices play an important role in the theory of
``neural nets.'' The transpose of **D** is a column vector.

Eigenvalues

Let **A** be a square matrix. Often we want to find vectors **v** such that multiplication by **A** is equivalent to scalar multiplication:

where is a complex or real scalar.
If we can find pairs
such that equation (6) holds, then we say that is
an eigenvalue and **v** is an eigenvector for How do we solve
this problem. Subtracting, we must have:

This is a linear system of equations. One solution is that **v=0**
If the matrix, has an inverse,
then this is the only solution. Thus, we must find values of
such that **A** is not invertible. Recall that a matrix is
noninvertible if it has a zero determinant. Thus, we take the
determinant To do this, we take the determinant
of
and set it to zero.
This results in a degree polynomial, * the characteristic
polynomial* that has **n** roots.
Thus, a general matrix has **n** eigenvalues.

Eigenvalues are either real or complex. A matrix which satisfies
that is it is its own transpose (i.e. symmetric) * always*
has real eigenvalues.
EXAMPLE

The matrix **A** from the above examples is symmetric. Since **A** is
, it follows that the characteristic polynomial is given by
(5) so the eigenvalues satisfy:

whose solutions are or

HOMEWORK

- Use the matrices,
**A,B,C,D**defined in the examples above. Compute the following quantities: - Compute
**AC**and find its eigenvalues

Sat Jan 3 15:01:05 EST 1998