Vertebrate brain theory

ISBN 978-3-00-064888-5

Monograph of Dr. rer. nat. Andreas Heinrich Malczan

4.2 The second expansion phase of the original spinocerebellum

The spinocerebellum is often regarded as the highest integration system of the simpler vertebrates. Its development was at times erratic, as new signalling pathways were opened up.

In this monograph we postulate a second expansion phase of the spinocerebellum.

This second expansion phase began with the principle of signal redundancy. Signals were no longer transmitted on single axons, but on axon bundles. Thus, the failure of individual neurons could not significantly impede signal transmission. First, a signal was distributed by signal divergence to several axons of an axon bundle, ran on them to the target structure and was reunited there by signal convergence.

The formation of signal divergence in the nucleus olivaris

Figure 22- The formation of signal divergence in the nucleus olivaris

 

Thus, redundant signal transmission required both a divergence structure and a convergence structure. In the cerebellum, the nucleus olivaris developed into the divergence structure, while the nucleus ruber formed the convergence structure. The development of the redundancy principle created the basis for the later intelligence of vertebrates.

In this monograph we will refer to the divergence structure as a (neural) divergence grid.

Signal divergence in the nucleus olivaris required that the input axons were ordered by signal pairs. This was based on the pairing of muscles. One muscle and its motor opponent formed a pair of muscles belonging to a joint. The signals of their muscle spindles were signal-related and formed a pair of complementary signals, because one signal could be formed from the other by inversion, which was the purpose of the spinocerebellum. The same is true for the signals of the associated tendon organs, which were able to measure muscle tension.

Signal divergence only occurred between complementary signals. It assumed that the number of trunk muscles increased more strongly. If the number of trunk muscles was very low and the opponents were usually located on the contralateral half of the body, this changed with their increase in number, especially since swivel joints were created in addition to pure flexural joints. Here, each muscle was close to its opponent, at least at the point of attack of the traction forces.

The muscle spindle signal of a muscle was distributed in the nucleus olivaris to a whole series of output neurons, which simultaneously received the muscle spindle signal of the antagonist muscle.

Divergence grid in the nucleus olivaris - schematic diagram

Figure > 23- Divergence grid in the nucleus olivaris - schematic diagram

Theorem of signal divergence in the nucleus olivaris

The muscle spindle signal of one muscle and that of its motor counterpart reached the nucleus olivaris via the nucleus ruber as a complementary signal pair. The same applies to the receptor signals of the tendon organs. Each of these signal pairs projected onto more than two output neurons, so that the redundancy that occurred could compensate for the failure of individual output neurons. The number of output neurons per input signal pair grew steadily during the course of evolution. Each of the two signals projected onto the same set of output neurons of the nucleus olivaris, which were arranged on the connecting straight line above the two neurons. This caused a signal divergence of both signals.

For the two input neurons involved, we define a special name.

Definition: Generating input neurons of the nucleus olivaris

The two input neurons of the nucleus olivaris, whose signals are distributed via a divergence lattice to n output neurons of the nucleus, are called generating input neurons.

Definition: Degree of divergence

The degree of divergence is equal to the number of output neurons of a divergence grating.

Two input neurons each distributed their excitation to n output neurons. These were distributed approximately equally along the connecting path between them. What might have been the reasons for this? It is also conceivable that the excitation was distributed in a circle around the two input neurons. A spherical distribution was ruled out because the nucleus olivaris represents a neuron surface that was only curved like a bag.

Here we remember the directional theorem of the neural tube nervous system. Each axon species chose - at least in the early stages of evolution - one of three directions that were orthogonal to each other. Therefore, the connecting axons between the input and output neurons in the nucleus olivaris were all parallel to each other. If you think of the living being horizontally oriented like a fish, the nucleus olivaris had approximately the same shape as the body and also ran longitudinally from front to back. The connecting axons between input and output neurons were perpendicular to this direction - all parallel to each other. More precisely, these axons ran along circular lines that corresponded to the circumference of the body. This is the reason why the nucleus olivaris represented the parallel connection of linear divergence grids and did not have flat divergence grids. 

The second expansion phase of the cerebellum began gradually and parallel to the first and third. There is some evidence that it was never completed. Apparently, vertebrates with a higher stage of development went through this phase with higher intensity. There was a simple reason for the beginning of this phase: receptors and neurons did not have an unlimited lifespan. When important receptors and neurons died due to a shorter lifespan, injury or disease, the functioning of the entire organism was impaired, which affected its ability to survive. Redundant signal transmission could ensure that signals reached their destination even when individual nerve cells and their axons and dendrites died.

An early form of redundant signal transmission developed in the spinal cord. Signals were transmitted on several axons. If one axon failed, the remaining ones transmitted the signals. There are considerably more axon lines in the spinal cord than, for example, the incoming ganglion cells of the spinal ganglia. But the available space in the spinal canal was limited. Much more space was available at the level of the entrance level of the primordial brain, where the neural tube could already inflate, leaving more space available. The signal divergence first developed in the nucleus olivaris.

Due to the strong increase in the projection neurons of the nucleus olivaris, this structure became strongly inflated, becoming increasingly sack-like at the same length and forming folds to accommodate the ever-increasing surface of this structure in the available space. This can still be seen today in sectional images at the height of this nucleus. In the following figure, which was provided by Professor Leo Gerbilsky from Kiel, the described structure is shown in green.

The nucleus olivaris and its structure

Figure 24- The nucleus olivaris and its structure

If one imagines the body of the creature to be worm-shaped, the two olive stones together also form a worm-shaped structure in principle, but this has been inflated by the strong signal divergence and therefore has the same length from head to tail, but a much larger diameter. The inflation therefore only occurs in one dimension. The space requirement led to the folding of the surface.

The object, which is called nucleus olivaris here, was (simplified) a surface that had approximately the shape of a zeppelin. It therefore had only two dimensions as a surface. Each half served one side of the body.

The neurons of the nucleus olivaris projected in a point-to-point image as climbing fibres into the Purkinje cells, both types of neurons were (at that time) equally numerous. Therefore, the Cerebellum bark, in which the huge amount of Purkinje cells had to be accommodated, inflated equally strongly.Here, too, wrinkles were formed - analogous to the changes in the nucleus olivaris - in order to make the best possible use of the limited space available.

Theorem of the enormous increase of Purkinje cells in the spinocerebellum

The signal divergence in the nucleus olivaris resulted in an enormous increase in the number of Purkinje cells, the number of output neurons of the cerebellar nuclei also increased.

Since at this evolutionary stage each output neuron of the nucleus olivaris projected into exactly one Purkinje cell, the cerebellum cortex formed a neuron model of the nucleus olivaris. It also expanded and formed a wrinkled structure. The two halves of the cerebellum together form a body model of the animal. In the simple, worm-like creature, this would be a kind of tube, which would have approximately the same shape as the body, but which was considerably enlarged in diameter due to the strong signal divergence. The direction of the parallel fibres corresponded to the body axis from the head to the tail, the plane of the Purkinje cells was oriented perpendicular to it. The moss fibers, which formed later, represented the same direction as the body axis in this model, although the cerebellum was rotated by about 90 degrees to the cortex. 

Theorem of the body model of the spinocerebellum

Both cerebellar shells joined together to form a body model, which was also wormlike in worm-like animals. The body axis in the model was parallel to the parallel fibres. Due to the strong signal divergence in the nucleus olivaris, the diameter of this body model was greatly increased compared to the state before the signal divergence. The space requirement led to the folding of the structure, the direction of the folds in the model was parallel to the body axis in the model. Bilateralism led to the splitting into two cerebellar halves.

In living beings with extremities, lateral cylinders representing the neurons of the extremities were added, these also showed a strong growth in diameter due to the signal divergence and also caused wrinkling due to the lack of space.

Signal divergence in the nucleus olivaris and cerebellum

Figure 25- Signal divergence in the nucleus olivaris and cerebellum

Definition: Elementary line

Those Purkinje cells whose climbing fibre input is derived via a divergence lattice from two generating input neurons of the nucleus olivaris are called elementary line Purkinje cells. At degree of divergence n it consists of n Purkinje cells.

The Purkinje cells of an elementary line are arranged in the cerebellum in the same plane as the relatively flat dendrite trees of these cells. The Purkinje cells of the elementary rows arranged one behind the other to the different generating input signals of the nucleus olivaris form small windings in the cerebellum, the parallel fibres run at right angles to them.

After signal divergence, the cerebellar nucleus of the spinocerebellum also contained a much larger number of output axons of this muscle pair, which in turn arrived in the nucleus ruber of the opposite side. The increase in size of this output nucleus was also inevitable. There the signal divergence was reversed. The axon bundle now projected again onto exactly two output neurons, which in turn moved to a pair of motor neurons. These were assigned to the original muscle pair.

Theorem of the skin input of the spinocerebellum

The main input of the spinocerebellum after the formation of the signal divergence in the nucleus olivaris consisted of the climbing fiber signals supplied by the nucleus olivaris.

Theorem of signal convergence in the nucleus ruber

The signal divergence caused in the nucleus olivaris was reversed in the nucleus ruber.

With the development of the signal divergence in the nucleus olivaris and the reversal of this signal divergence in the nucleus ruber, the output neurons in the nucleus ruber formed larger and larger dendrite trees, they developed into magnocellular neurons. In the course of evolution, these neurons formed their own core part.

Theorem of the magnocellular nuclear fraction in the nucleus ruber

The reversal of the signal divergence in the nucleus olivaris required the formation of magnocellular neurons in the nucleus ruber, which formed their own nuclear part there.

An input neuron of the nucleus ruber now received a whole series of axons from the cerebellar nucleus, which belonged to a pair of muscles and could therefore only contact two output neurons. As a result, each output neuron of the nucleus ruber received a larger number of axons instead of one. These neurons needed a much larger dendrite tree and thus also a larger cell body. They became magnocellular. With the transition to signal divergence in the nucleus olivaris, a large-cell (magnocellular) nucleus part was formed in the nucleus ruber, which is also present in modern vertebrates.

The following illustration shows the path for two complementary signals, for example, for the muscle tension receptors of the two muscles of a flexor joint. The signals fA and fB of the two complementary receptors reach - descending from the motor cortex - the nucleus ruber, which projects to the nucleus olivaris of the opposite side. This nucleus is where the signal divergence takes place. In our illustration, the two input signals are exemplarily distributed to 6 projection neurons, which project to 6 Purkinje cells of the bark of the spinocerebellum. These switch the signals to the inhibitory transmitter GABA and project inhibitively into 6 neurons of the cerebellum nucleus.

These are - all six of them - permanently excited by an average signal from the reticular formatio, so that the inhibition by the Purkinje cells leads to a signal inversion.

The six inverted signals reach the parvocellular part of the nucleus ruber, where strong lateral inhibition by interneurons causes only one output axon to fire actively while the others are inactive. But each of the six output neurons projects to exactly two output neurons in the magnocellular part of the nucleus ruber. This is where the inversion of the signal divergence takes place. The output of the nucleus ruber (more precisely its magnocellular part) consists of two signals that are now used to control the antagonist muscles. The muscle that generated the signal fA with its muscle tension receptor receives the inverted signal fD from the nucleus ruber, the other receives the signal fC. The mathematical correlations are derived in the following in this chapter.

We must note that the spatial arrangement of the neuron nuclei involved here in this diagram does not play any role at all, only their sequence. In our schematic representation we choose a symbolic signal sequence from top to bottom.

Divergence and convergence in the vertebrate brain

Figure 26- Divergence and convergence in the vertebrate brain

The signal divergence of the nucleus olivaris continues in the cerebellum and in the cerebellar nucleus. It is only cancelled in the nucleus ruber.

For the later mathematical description of the signal changes associated with divergence and convergence, the fire rates of the neurons are given next to the corresponding axons. They are double Indexed. The first Index indicates the level, the second the number of the neuron in the corresponding level.

The algorithm of signal processing in the following circuit can be described by 5 vectors

1.     Input vector:                           i = (fA; fB)

2.      First plane vector:      v1 = (f1.1; f1.2; f1.3; f1.4; f1.5; f1.6)

3.      Second plane vector: v2 = (f2.1; f2.2; f2.3; f2.4; f2.5; f2.6)

4. Vector of the third plane:      v3 = (f3,1; f3,2; f3,3; f3,4; f3,5; f3,6) 5.    

5.    Output vector:                         o = (fC; fD).

We now derive the mathematical algorithms for these vectors. The most important side result of the signal divergence was the distance dependent signal attenuation.

Theorem of signal attenuation in the nucleus olivaris

The nucleus olivaris is a nucleus of the grey matter, the axons of the input neurons lose the myelin sheath in it. Therefore the signal propagation in this nucleus is according to the cable equation for non-myelinated fibers.

The cable equation for non-markless axons describes the decrease in excitation along an axon, where E0 is the initial excitation at the start of the axon and E is the excitation value measured at distance x from the start of the axon:

           formula.                                              (2.2.1)

The quantity λ is called the longitudinal constant of the neuron.

The excitation function can be shown graphically in a diagram, where the decrease in excitation can be clearly seen as the distance increases.

 

Cable equation for non-markless axons

 

Figure 27- Cable equation for non-markless axons

It is assumed that the axon is unmyelinated and the excitation propagation is subliminal, i.e. no action potentials occur. This form of excitation propagation can be observed in local, non-spiking interneuronsthat realize signal transmission with graduated potentials [12]. The absence of action potentials can be explained by the absence of voltage-dependent ion channels in the cell membrane along the axons of the interneurons.

When an interneuron is excited by an input neuron via chemical synapses, causing its membrane voltage to rise, this voltage rise is passed on to the connected output neuron. The increase in membrane voltage in the output neuron is caused by an ion exchange between the interneuron and the output neuron, which takes place via transmembrane proteins that form the electrical synapses, also known as gap junctions. The molecules known as connexins occur in vertebrates in a wide variety of cells, including the nervous system. They enable an exchange of substances between neighbouring cells, without which there would be no multicellular life. Ekrem Dere describes the occurrence of connexins in the brain of vertebrates in his work "Gap Junctions in the Brain: Physiological and Pathological Roles" [114].

Connexins are usually named after their molecular weights, e.g. Cx36 is the connexin protein of 36 kDa. According to [114], Cx36-positive neurons have been detected in the retina, the dentate gyrus, in the regions CA1, CA3 and CA4 in the hippocampus, in the cerebral and periform cortex, in the amygdala, in the cerebellum, in the mesencephalon, in the suprachiasmatic nucleus, in the thalamus, hypothalamus and in various cranial nerve nuclei. Gap junction with Cx36 were also found in the olivocerebellar system. In [114] many classes of connexins are analysed with regard to their occurrence in the brain. We can therefore assume that such electrical synapses built from connexins exist in the nucleus olivaris in particular.

A sufficient neuronal excitation of such interneurons can lead in downstream neurons via local and temporal summation to the exceeding of the triggering threshold for action potentials. For example, in the retina, the action potentials of the ganglion cells are caused by graduated potentials of the retinal interneurons. We postulate here an analogous excitation propagation in the olive nucleus.

From the cable equation for non-markless fibers, the fire rate f can be inferred directly. This fire rate f has a neuron that is at a distance x from an input neuron with the fire rate f0 and is connected to it via a non-markless axon. Here, the excitation E is replaced by the fire rate f.

formula           (2.2.2)

The graphic representation of the fire rate function as a function of distance x is shown in the following figure.

Fire rate for signal propagation on non-markless axons

 

Figure 28- Fire rate for signal propagation on non-markless axons

An output neuron can be connected to two input neurons. And finally, a group of output neurons can be connected to two common input neurons. Then we can talk about signal divergence. The number of input neurons is always significantly smaller than the number of output neurons.

We first analyze the functioning of the divergence grating, which is the first substructure shown in Figure 28.

Divergence grid in the nucleus olivaris - schematic diagram

Figure 29- Divergence grid in the nucleus olivaris - schematic diagram

We can combine the input fire rates into one input vector:  

i = (fA; fB)                                                    (2.2.3)

We can also combine the output fire rates into an output vector, where n represents the number of output neurons:

                                   v1 = (f1.1; f1.2; f1.3; f1.4; ..., f1.n)                    (2.2.4)

We can now use the cable equation to determine the excitation of the individual output neurons, with the fire rate of the input suppliers as a factor in the excitation transfer. This gives us intermediate values for the excitation function at the points where the output neurons receive their excitation.

Divergence Grid - Derivation of the Fire Rate

Figure 30- Divergence Grid - Derivation of the Fire Rate  

The excitation f(x) of the output neurons of a divergence grating can be described by a functional equation, which contains as parameters the fire rates fA and fB and the constant quantities λ and D and in which the variable x indicates the distance of the output neuron from the left input neuron. We will call this function f(x) the excitation function of the (linear) divergence lattice:

formula                              (2.2.5)

The fire rates of the output neurons are not only theoretically calculable, but also exist in real life. They could, for example, be verified by measurements on site.

We can justifiably call equation (14.4) a virtual excitation function of a linear divergence grating. This is because it allows us to determine the rate of fire that an output neuron would have if it were at position x in the lattice.

The use of the virtual excitation function allows to abstract from the real existing dendrites and axons of the neurons. We consider the excitation propagation in a continuous substitute model. This is comparable to the sound propagation of the sound of several sound sources or to the heat propagation of several heat sources. This model is also comparable with the electric field strength of several existing electric charges or even with the gravitational field of celestial bodies and the forces acting between them. This shows the weakness of the numerical simulation of neuronal circuits through the simulation of their axons, dendrites and synapses in the European Human Brain Project. There are models that do not need synapses at all, but which can still explain signal processing by limiting themselves to signal propagation.

The output of the divergence system is obtained by the scanning neurons taking over the excitation values of the virtual excitation function.

The output neurons quasi interpolate the excitation function f(x) at the intermediate points where they are located. We call such a circuit a divergence grating. It is a neuron grid that interpolates the excitation function f(x) and delivers the function values at the sampling points as output. The output neurons are called sampling neurons. Since the sampling takes place along a line (more precisely a distance), it is a linear divergence grid.

If D represents the distance between both input neurons, n their number and

formula

their distance, the following equation applies to the excitation fk of the neuron Nk

 

formula             (2.2.6)

This allows the type of signal processing in a divergence grating to be reduced to a mathematical transformation Δformula which assigns a divergence vector of n components to the two input fire rates fA and fB and the numbers n and λ. 

Theorem of divergence transformation

The divergence transformation

formula

with

formula

and

formula

assigns to each input vector i = (fA; fB) a clearly defined n-digit output vector. The neuronal realization takes place in a divergence grid. The values of the vector components correspond to the sample values of an excitation function at the sampling points.

The excitation function f(x) has two maxima, on the left at x = 0 and on the right at x = D. Between them it has a minimum. This can be calculated using the first derivative of (2.2.5), which must then be set to zero to obtain the condition for the minimum.

formula                         (2.2.7)

As a result one gets the realization that the excitation function f(x) has a local minimum at the position

formula                                  (2.2.8)

formula                                                                          (2.2.9)

has. Depending on the relationship between the input fire rates fA and fB, the minimum is assumed at another position x*. A multiplication of both firing rates fA and fB by any factor a > 0 does not change the position of the minimum, the factor is shortened in (14.6). Therefore, it is advisable to restrict oneself to the fire rate quotient q, which is defined below:

formula                                                                               

 

This relationship can be resolved according to the parameter q, then an equation results, which we will call the coding function q(x):

                                   formula                                   (2.2.10)

It can be used to calculate which signal strength ratio q = fA/fB must be present so that in a divergence grating the excitation minimum at the position x*    is assumed. The signal strength ratio thus encodes the location x* where the minimum is. The output vector of such a divergence grating is also minimum codedbecause it contains the fire rate values at the sampling points where the output neurons are located.

Because of the great importance of the fire rate quotient, we summarize the results in a separate theorem.

Theorem of the relationship between the fire rate quotient and the distance of the input neurons in the divergence lattice

Between the fire rate quotient q, the minimum point x* and the distance of the input neurons in the divergence lattice the following relationship applies

formula .                                            (2.2.11)

 

This also means that

formula                                                      (2.2.12)

As is well known, a divergence grating requires two input fire rates fA and fB. These input firing rates can be supplied by receptors. This is exactly the case we are now investigating. We need two receptors RA and RB, whose input may feed a divergence grating with n scanning neurons. Both receptors may belong to the same modality, RA may be on-type, RB may be off-type.

Theorem of signal transformation of receptor input by a divergence grating

A receptor RA measures a physical, chemical or other quantity, the measured value h lies in the interval < hA; hB>, the rate of fire in the middle of the interval is fm, hm is half the width of the interval according to

formula .                          (2.2.13)

  and it provides the on fire rate when the size h changes in the interval < hA; hB> the on fire rate

formula .                          (2.2.14)

The receptor RB delivers the off-fire rate for changes of the same size h

formula             (2.2.15)

  This means that fA and fB are complementary to each other, because

formula                                         (2.2.16)

and

formula                       (2.2.17)

and

formula                (2.2.18)

 

The virtual excitation function of the divergence grating with the input values fA and fB according

formula                  (2.2.19)

has a minimum excitation at position x* according to

formula                                     (2.2.20)

 

Inserting (2.2.17) in (2.2.20) gives the equation

formula                                (2.2.21)          

and

formula                                    (2.2.22)

 

From the minimum value x* the value of the original magnitude h can be determined unambiguously.

The divergence transformation not only maps the original interval to the interval of the fire rates of the excitation function, but the minimum of the excitation function encodes the signal value of the original quantity. Thus the divergence grating serves to code the extreme value of a great magnitude.

The input of a divergence grating consists of two signals that are inverted to each other. Two receptors or their associated ganglion cells, which deliver signals inverted to each other, are called complementary receptors or ganglion cells. The signal of one receptor can be inverted into the signal of the second receptor for any value of the detected original quantity. As a generic term for the receptor we also use the term detector.

Theorem of minimum coding in linear divergence grids

In a linear divergence grating the signal strength ratio of the two input signals is transformed into a minimum coded signal vector. The value of the original quantity, which provides the input via two complementary receptors, is minimum coded in the output signals.

Linear divergence grids mathematically represent the one-dimensional case. Here, the excitation propagates (idealized) along a path that connects the two input neurons. In the two-dimensional case, the excitation propagation would take place in the plane, with at least three, but also four, sometimes even six input neurons participating. In the same way, a spatial propagation of the input excitation could also take place, whereby the input neurons would then be distributed in an associated spatial area. In the following figure a linear divergence grid is compared to a plane divergence grid, whereby the latter has four input neurons.

Linear and plane divergence grid in the olivaric nucleus

Figure  31 - Linear and plane divergence grid in the olivaric nucleus

However, a minimum coded signal vector is often unsuitable because the coding neuron is minimally excited. Such a signal is useless for motor control of a muscle. Therefore, a conversion to a maximum coded representation is useful and is done in neuronal reality.

Theorem of signal inversion of minimum coded signal vectors

Each minimum coded output vector of the divergence lattice in the nucleus olivaris is transformed by signal inversion in the spinocerebellum into a maximum coded signal vector of the nucleus interpositus. A lateral inhibition between the output neurons of this nucleus transforms the minimum coded output vector approximately into an output vector with sparse coding.

The signal inversion in spinocerebellum has already been described in detail. The output of the nucleus olivaris reaches the Purkinje cells of the spinocerebellum via the climbing fibre axons and leads to their excitation. Each Purkinje cell takes over the fire rate of the corresponding neuron in the nucleus olivaris. With this rate of fire, it inhibits the associated output neuron in the nucleus of the cerebellum, which is, however, permanently excited. This inverts the signal. The spinocerebellum remains true to itself in its mode of operation: every input from the olive is inverted and represents the excited output of the spinocerebellum.

Let us denote with fD the firing rate of the tonic excitation in the cerebellum nucleus and the output vector of the divergence lattice in the nucleus olivaris with  

formula                                                                   (2.2.23)

the inverted output vector k from the cerebellar nucleus is given by the relation

formula                                    (2.2.24)

 
Here a signal minimum in vector o is transformed into a signal maximum in vector k.

With a signal inversion the monotony is reversed, a minimum becomes a maximum. The maximum excited output neuron can now cause a motor reaction via the connected motor neuron, which was excluded in the minimum coding.

The following diagram shows an example of the superposition of two excitation functions in a divergence grating. The blue function represents the firing rate of the left input neuron, whose excitation is passed along an axon to the right, where exponential attenuation occurs according to the cable equation for non-markless fibers. The brown function represents the firing rate of the right input neuron, whose excitation is propagated to the left along the axon and is also attenuated depending on distance.

The function shown in yellow results from the pure addition of these two input excitations and is a minimum coded function. By signal inversion - i.e. the inhibition of a continuous signal, which itself has a constant rate of fire - the function shown in grey results, which is a maximum coded function.

The excitation function resulting from superposition is shown in grey, the excitation function obtained by inversion is shown in yellow. The inverted function has a maximum exactly where the original function has a minimum.

Divergence Grid and Signal Inversion

Figure  32 - Divergence Grid and Signal Inversion

In the diagram above the interpolation is shown with continuous functions. In reality the inversion is done point by point, each time by the neurons involved.

The inverted signal curve of a divergence grating can therefore be displayed as a bar chart. In the following, we select the value of 250 as the signal mean value, which is relatively inhibited by the output of the divergence grating, so that the following representation is obtained:

Inverted output of a divergence grating

Figure 33 - Inverted output of a divergence grating

You can see that in addition to the extreme value, there are also second-line stocks that are weaker. Exactly these values must be eliminated by a following neural circuit, so that only one output neuron is maximally excited, but the others are completely inactive. This can be achieved by a strong lateral neighbor inhibition.  

Theorem of extreme value selection

If a maximum coded output vector is transferred to a group of output neurons of a nucleus in which there is strong lateral inhibition, this output vector is transformed into a digital vector that has a maximum rate of fire at only one position, while at the remaining positions the rate of fire is zero.

We call such a vector an elementary vector. Only at exactly one vector position the rate of fire is greater than zero, at the other positions the values are zero.

For example, in the diagram in Figure 20, only one output neuron remains active after an extreme value selection due to strong lateral inhibition, as the following diagram illustrates.  

Output Divergence Grid after Extreme Value Selection

Figure  34 - Output Divergence Grid after Extreme Value Selection

This transformation can rightly be called digitization.

The minimum coded vector from (2.2.23) could look like this:

o = (257,0; 254,2; 252,0; 250,5; 249,5; 249,2; 249,5; 250,5; 252,0; 254,2; 257,0) (2.2.25)

  If it is inverted on a continuous signal with the firing rate of fD = 280, the maximum coded vector is obtained

o* = (305; 308,4; 311,1; 313,6; 314,2; 314,6; 314,2; 312,9; 311,1; 308,4; 305)            (2.2.26)

This vector has a maximum at the sixth component, it has the value 314.6.

An extreme value selection would ensure that all other components whose value is less than the maximum would be totally suppressed and given the value 0. The result vector o** would then be the following after the extreme value selection:

o** = (0; 0; 0; 0; 0; 314,6; 0; 0; 0; 0; 0)                               (2.2.27)

Of all eleven output neurons that represent this vector with their fire rates, only the sixth in the row would have a fire rate of 314.6, while all other neurons would be totally inhibited by it and have a zero fire rate. And its firing rate would be significantly higher than the mean firing rate of the mean signal, since the square of the mean firing rate has to be divided by the firing rate of the minimum from the minimum coded vector to obtain the maximum firing rate of the now inverted signal. Such a coding form is called sparse coding.  

sparse coding theorem

The extreme value selection converts a minimum coded signal vector into a signal vector with sparse coding.

Extreme selection requires a special type of lateral inhibitionthat can only be achieved with fast conducting axons that are either thicker or have a myelin sheath. The neuron that shows the strongest increase in membrane tension fires its action potential first. This almost instantaneously excites the connected inhibitory interneurons, which in turn immediately inhibit neighbouring neurons, including those that are further away. For this purpose, these inhibitory interneurons must have fast-conducting or myelinated axons. They form an inhibition layer below the projection neurons, in which the horizontally running axons form a white network if they are myelinated. The white colour is caused by the myelin sheaths of the axons.

As soon as the neuron fires its action potential at the point of maximum excitation, the excitation of the remaining neurons, which has been built up in the meantime, is completely destroyed via the connected inhibition neurons, because inhibited neurons are virtually caused by a kind of electronic short circuit.  

Theorem of substructures with extreme value selection

An extreme value selection takes place in the nucleus ruber as well as in the thalamus nuclei and in the output nuclei of the spinocerebellum. Furthermore, an extreme value selection can take place in the output nuclei of the amygdala.  

Theorem of the digitization of an original quantity by a divergence grating

If two complementary receptors detect the value of an original quantity u in the interval <uA ; uB> which is subdivided into n subintervals T1, T2, T3, ..., Tn of equal size, a divergence lattice of n equally spaced output neurons O1, O2, O3, ..., transforms On this input tuple into an output vector of n components in such a way that the kth output neuron Ok is excited precisely when, and only when, the value of the primal quantity u lies in the kth subinterval Tk, and only minimally excited. This corresponds to a digitization.

In a downstream inversion lattice of n neurons, the kth neuron has maximum excitation if and only if the value of the original quantity u lies in the kth subinterval Tk. If a grid consisting of n neurons with extreme value selection is connected downstream, the kth neuron is maximally excited then and only then if the value of the original quantity u lies in the kth subinterval Tk, while the remaining neurons are unexcited.

A neuronal digitizing circuit, consisting of the series connection of a divergence grid, an inversion grid and a grid with extreme value selection, achieves approximately the same performance as a digital clinical thermometer, in which the measured temperature is not displayed as a number, but on a labelled number beam (similar to a metre scale), in that only exactly the one under which the relevant number is displayed lights up in a row of light-emitting diodes. This also corresponds to the number format. Neuronal digitized values do not know any digits, but are represented on a number beam which is labeled to match the measured original size and has only a finite number of numerical values.

The signal divergence in the nucleus olivaris led to a multiplication of the number of neurons. Instead of two signals each from the muscle spindles of a joint, there were now a multitude of them. However, the motor control of the two muscles required only two signals. Therefore, these multiplied signals had to be brought together again by signal convergence. This has already been shown in the theorem of signal convergence in the nucleus ruber.

We can also analyse the signal convergence mathematically. Here the output of a neuron series converges to two output neurons. We limit ourselves to the part of Figure 28 that contains the convergence grid.

Convergence Grid - Block Diagram

Figure  35 - Convergence Grid - Block Diagram

We recall that the output of a divergence grating, when it suffers a signal inversion with subsequent extreme value selection, has the peculiarity that (normally) only one neuron fires actively, while the others are totally inhibited. The position of the neuron depends only on the signal strength ratio of the input of the divergence grating. We can therefore assume that only one neuron has a firing rate fk unequal to zero. We analyze the mathematical background.

For this purpose, we first consider the excitation propagation from a single input neuron to two connected output neurons.

Convergence Grid - Derivation of the Fire Rate

Figure  36 - Convergence Grid - Derivation of the Fire Rate

Here, the input excitation provided by the input neuron with the fire rate f is distributed to the two output neurons again according to the cable equation for non-markless axons.

formula                           (16.10)

formula                               (16.11)

  This also means that

formula                               (16.12)

We remember that x here represents the value of the minimum in the divergence lattice or the maximum in the inversion lattice.

Here the fire rate quotient Q represents the reciprocal of the value q from the divergence grating according to (14.7). The output of the convergence grating is thus swapped with respect to the original input of the divergence grating, its signal strength ratio is the reciprocal of the original ratio.

Theorem of series connection of divergence grating, inversion grating, extreme value selection and convergence grating

The series connection of a divergence grating with an inversion grating, an extreme value selection and a convergence grating, in which the input consists of two signals with the firing rate fA and fB and has the signal strength ratio q according to

formula ,

 

provides as output two output signals with the fire rates fC and fD with the signal strength ratio

formula .

 

If the stronger input neuron of the divergence grating is on the right (left), the stronger output of the convergence grating is on the left (right), i.e. swapped.

The mix-up must be reversed by mirroring,

Signal divergence in the nucleus olivaris, followed by signal inversion in the spinocerebellum, extreme value selection in the nucleus ruber, signal convergence in the nucleus ruber and subsequent signal mirroring thus provides the same output as it would have been without these transformations.

Theorem of harmlessness of signal divergence and convergence

With regard to motor output, the formation of signal divergence in the olivaris nucleus and signal convergence in the ruber nucleus had no effect on the resulting output of the ruber nucleus if signal mirroring had previously taken place.

The signal reflection took place in the cortex. The output of the cerebellum was transferred in ascending order to the class 4 neurons on the sensory side of the frontal cortex. These transferred the signals to class 3 projection neurons, whose axons moved to the motor part of the frontal cortex. There the signals reached the downward projecting neurons of class 5, whose target neurons were the motor neurons.

The cortex surface of the frontal cortex acted on these signals more or less like a concave mirror, thus mirroring the signals. This mirroring of the signals reversed the side reversal in the divergence grating. This is described in more detail in the chapter on the origin of the frontal cortex.

So if the motor output has not been changed significantly, the question arises as to the additional benefit of these signal transformations. One benefit was redundancy, the circuit was more stable when individual neurons failed.

A much greater benefit was that all body images whose signals were subjected to signal divergence in the spinocerebellum were quasi-digitized. The joint angle, which resulted from the excitation ratio of the muscle tension sensors of two muscles working against each other, was transferred in the output of the cerebellum into a maximum coded signal vector, which was also sparsely coded by the lateral inhibition. The same applied to all other sensory variables. With increasing degree of divergence, the resolution of this transformation increased, joint angles and other variables could be detected much more finely.

If the development of these algorithms had been harmful, evolution would have put an end to this aberration. But since it was output neutral and therefore harmless, it could manifest itself. It only remains to show that this development even - in the course of further evolution - brought further considerable advantages. These new advantages resulted from the development of the Pontocerebellum, which is described in a separate chapter of this monograph. Beforehand we analyse the spatial changes in the brain systemthat occurred as a consequence of the divergence development.

It should be mentioned here that the described signal divergence in the nucleus olivaris did not only affect the spinocerebellum, but also the pontocerebellum. Let us recall: The signal class of the mean signals of neuron class 6 also took the path via the nucleus ruber to the nucleus olivaris and reached a cerebellum structure that we have named Pontocerebellum. The associated cerebellar nucleus was the nucleus dentatus. Also in this section of the olive an increasing signal divergence occurred. Their consequences are described in chapter 7, which is dedicated to the functioning of the Pontocerebellum.

Summary of the Spinocerebellum

With the development of signal divergence in the nucleus olivaris, the spinocerebellum developed into a divergence circuit that transferred the signals of a pair of related muscles arriving in pairs into a maximum-coded signal vector in which the signal maximum encoded the joint angle. Lateral inhibition between the associated output neurons caused the output vector to be sparsely coded. Thus the joint angle was in principle digitized. With increasing degree of divergence, the accuracy of this angle representation increased and allowed the living being to have a much more finely controllable motor function.

Other sensory variables represented by complementary signals could also be digitized in this way.

The inclusion of the mean value signals in this system created the conditions for the later development of the Pontocerebellum.

The spinocerebellum after the second expansion phase is called the late spinocerebellum here.

Both the late spinocerebellum and the associated nucleus olivaris and cerebellum nucleus form a body model of the living being. If one thinks of the body of the early chordates as being approximately elongated and wormlike, the spinocerebellum had approximately the shape of a zeppelin. Due to the signal divergence in the nucleus olivaris, the body model of the cerebellum had a significantly larger diameter than before the signal divergence. If one thinks of the surface of the body divided into narrow stripes, these stripes are clearly widened in the cerebellum model and form outwardly curved arcs. Each stripe represents a row of joints running from head to tail, in which two muscles working against each other in each segment supply the Purkinje cells of such an arc with their signals. The minimum excitation in each Purkinje row of an arc encodes the joint angle.

Below the Purkinje cells, the moss fibres ran in a direction parallel to the body axis. The granule cells were topologically arranged in the same way as the signal-providing receptors were distributed on or in the body. The signal divergence in the granule cell system was realized by the axons of the granule cells striving radially in all directions of the Purkinje cells, which were part of a row in an arc.

Theorem of the body model of the late spinocerebellum

After the second expansion phase, the spinocerebellum forms a two-layered body model of the living being, consisting of two bilateral halves.

The body model was a striped cortex with radial expansion and arching. An arc was assigned to each joint. Because of the almost identical structure of the body segments, the joints formed rows of joints that ran from the head segment to the tail segment. Each joint row was assigned a cerebellum arch from Purkinjereihen, which received from the nucleus olivaris the receptor signals of the corresponding muscle pairs generated by signal divergence.

The granule cells represented the receptors of the muscles as well as the tactile and pain receptors, each modality forming its own granule cell layer, in which the topology of the body's receptors was preserved.

The Purkinje cells form arcuately arranged rows, each of which belongs to a cerebellum wrap. The excitation minimum of the Purkinje cells of an arched row encoded the joint angle of the joint in the corresponding segment.


 

Monograph of Dr. rer. nat. Andreas Heinrich Malczan