**ISBN
978-3-00-064888-5**

**
Monograph**
of Dr. rer.
nat. Andreas Heinrich
Malczan

In the visual cortex V1 of vertebrates a directional selectivity of the so-called orientation columns is observed. These are groups of neurons that run through all six layers of the visual cortex and form small cylindrical columns. Their activity is selectively sensitive to the angle of inclination of a dark line against a light background. The neurons involved must therefore respond to dark objects, and are therefore of the dark on type.

Figure 58 - Chord length on the circle

The chord length will depend on the position of the straight lines, more precisely on the distance of the straight line from the center of the circle.

Figure 59 - Chord Length and Center Distance

A suitable variable for calculating the chord length is the distance of the chord from the centre of the circle. The known radiusFigure 60 - Calculating the chord length on a circle

So it's true:

This results in the following equation for the square of the chord length:

In general

We now consider the chord
length or its square using a coordinate system where the receptive field is not
in the coordinate origin. The center of the circle K, which should represent the
receptive field of the ganglion cell, should be the point P_{k} in the
coordinate system, to which we assign the coordinates x_{m}
and y_{m}. Then we can also represent the point P_{k} as
P_{k}(x_{m}; y_{m}).

The line g, which may intersect the circle K, has the angle φ to the x axis. Its distance from the coordinate origin P(0;0) is r.

The circle center P_{k}(x_{m};
y_{m}) may be at a distance r_{k} from the coordinate origin
P(0; 0). The line connecting the two points has the length r_{k} and
should form the angle φ_{k} with the x-axis..

Then the following equation
applies to the chord length s_{k}

The Pythagorean theorem and
the representations of the quantities x_{m}, y_{m} over the sine
and cosine of the angle difference

Figure 61 - Chord length calculation for a shifted receptive field

With reference to the above
figure:

We assume that a dark on
ganglion magnocellular cell, whose receptive field is intersected by a straight
line with the angle of rise φ and the distance r from the coordinate origin, has
a fire rate f_{k}

We now investigate the firing rates f12, f2, f3 and f4 of four retinal ganglion cells of the described type under the assumption that they are arranged in the Cartesian coordinate system exactly along the coordinate axes and have a distance r = 1 from the coordinate origin. This is sketched in the following figure.

Figure 62 - Arrangement of four visual ganglion cells

Each of the four ganglion
cells has a circular receptive field with radius r_{s}. The partial or
complete overlapping of the four receptive fields is important. The straight
line has again the angle of ascent φ.

This simplifies
the formulas for the rate of fire

These four fire rates travel
from the retina via the Corpus geniculatum laterale to the visual cortex or
visual cranial turning loop from which it emerged. All following considerations
now refer to the signal propagation of these four input signals in the visual
cortex. The topological arrangement of the four input signals should be
preserved. Therefore there are four input neurons in the visual cortex, which in
turn form the corners of a square and whose fire rates are the values f_{1},
f_{2}, f_{3}
and f_{4} according to our formulas.

The fire rates f_{k} originate
from the described retinal ganglion cells.

We can also derive suitable
formulas for the distances r_{1}, r_{2}, r_{3}
and r_{4} of the output neuron to the four input neurons.
For this we need the following figure.

We note that the input neurons lie on the coordinate axes according to our specifications and have the distance 1 from the coordinate origin.

Figure 63- Radius vectors to a neuron at point P(x,y)

At the latest here it becomes
clear that in this case, too, we are dealing with a plane divergence grating,
for whose excitation function we have already derived the formula
(3.6)

We recapitulate
the most important requirements: The output neuron in the point **
immovable**. It

Furthermore, we
assume that the angle of rise φ of the straight line, which also overlaps the
receptive fields of the four retinal ganglion cells, also remains constant.

Mathematically,
this means that the fire rate f of the output cell under consideration in the
point

(4.2.10)

The second derivative after r gives the simple equation

The equation resulting from zeroing the first derivative after r is first simplified by using the common factor

divide. This is allowed because this factor is not equal to zero. So we get the first simplification:

(4.2.12)

The conversion into the
representation with hyperbolic functions is advantageous.

Using these relations we get the condition

(4.2.13)

Thus we have a condition for
the existence of a maximum of the excitation function f of the output neuron in
the
point

(4.2.14)

This equation of determination can be solved by the angle φ. For this we use the formula

With

we obtain the
equation

(4.2.15)

This can be transformed into

(4.2.16)

Taking into account

(4.2.17)

For this value
of φ the derivative of the rate of fire f to r is zero, the second derivative
being less than zero. Thus f assumes a local maximum. Thus, if the inclined line
separating the receptive fields of the four retinal ganglion cells involved has
this angle of rise φ and the distance r from the coordinate origin, the output
neuron fires in the visual cortex with the coordinat

We now use the possibility to represent the equation of determination as a function of x and y. Then, for example, we can display the relationship between the best angle and the location of an output neuron in Excel.

First of all,
the constant part in the diagram is

Figure 64
- The Angle Dependence of the Term T2

This shows the contribution of the second summand in the equation of determination for φ. This contribution is independent of the distance r of the straight line to the coordinate origin.

The following two diagrams show the angle φ of the total formula (18.5.3.21) as a function of the values x and y. In the left-hand diagram seen from the side and next to it from above. First we choose the value r = 0.

Figure 65 -
Display of the angle seen from the side |
igure 66 -
Viewing the angle from above |

The influence of the radius r can also be displayed graphically. In the above illustration r = 0 was given. The figures are Excel graphics, where the derived formulas for the angle φ were used to create the diagram. For a value of r = 0.2 the following graphs are obtained.

Figure 67 - The
influence of r on the directional selectivity |
Figure 68 - The
influence of r |

Here you can see that a circle-like small area of the coordinate origin responds to the angle φ = 0, while in the rest of the area the angles are arranged like a windmill, with each angle interval being assigned a different color in steps of 15 degrees. Exactly this was also done in the experiments on the angular selectivity of the orientation columns in the visual cortex V1. In this monograph, the mathematical context was presented for the first time.

Finally the graphics for r = 0.5.

Figure 69 -
Orientation Columns for Large r |
Figure 70 -
Orientation columns with large r |

You can see that the circle around the coordinate origin, in which there is a maximum at an angle of incidence of 0 degrees, becomes larger as r grows. Approximately at the value of r = 0.7, the blue area is already so large that it fills almost the entire square. At about this value, the retinal ganglion cells no longer react with meaningful output because the inclined straight line no longer darkens the receptive fields sufficiently.

calculable angle values - this is what the programmers from Excel decided. This solution is far better than issuing hundreds of error messages that would read: "Function value not defined!

herefore we have to prove at
this point why the function values are not defined in the "circle" shown in
blue. Equation (4.2.18) contains the term

The **
directional selectivity of** the

**
unnecessary**. The directional selectivity is not learned, it is present
from the very beginning. It is based on elementary laws of nature, in particular
the exponential damping of excitations when propagating in the surface.

**Monografie** von Dr. rer. nat. Andreas Heinrich Malczan