Theory of the neuronal circuity of the brain and analytical thinking

ISBN 978-3-00-037458-6
ISBN 978-3-00-042153-2

Monograph of Dr. rer. nat. Andreas Heinrich Malczan

Part 2.13. Cerebellar reverberation - Basis for learning signal sequences 

It took us a lot of effort to develop a reasonably reasonable information theoretical explanation for the work of the Cerebellum.

We have discovered that in the cerebellum, different types of neurons function as self-signal detectors, others as foreign signal detectors. We noticed that the intrinsic signal detectors inhibited the output of the extraneous signal detectors. We explained the task of the primary climbing fiber signal and its origin in the striosome system. We realized that the olive serves the central or sequential distribution of the primary climbing fiber signal to the secondary climbing fiber signals. We exposed the magnocellular climbing fibers of the striosome system as neuronal write commands, which stored the imprinting signals of the moss fibers in the neurons and interneurons of the cerebellum, so that these could become the own signals of the Purkinj groups.

But we also recognized that another class of climbing fiber signals is effective in the cerebellum. The matrix system generates an associated climbing fiber signal for each cortex signal. The exciting question was where the parvocellular climbing fiber axon docks and what it does. It was shown that the parvocellular climbing fiber signal of a signal neuron of the secondary cortex docks to its generating Purkinje group and displaces the striosomal, magnocellular climbing fiber signal. We identified the nucleus olivaris inferior as the site for the substitution of the magnocellular climbing fibers by the associated parvocellular ones.

This made the minting of Purkinj groups possible. Thus, a cerebellar learning process could be continued even after the first imprint of a Purkinje cell. The re-imprinting allowed the learning process to continue during the entire lifetime of the individual.

We recognized that several or even many Purkinje cells are grouped together to form a Purkinje group which is terminated at its end by a Golgi cell. The Purkinje cells analyzed the primary echoes caused by the lower propagation speed on the parallel fibers. The basket cells integrated the primary echoes into a secondary echo and suppressed the Purkinje cells of the Purkinje group during its presence. Thus the output was stretched in time. Thus, short input signals resulted in significantly longer output signals. This enabled the cerebellum to stretch the output to short stimuli over a formative period of time. This enabled the cerebellum to further process its own output to short signals itself, because the embossing time could be guaranteed despite short input duration.

We found that the input signals arriving quickly via the moss fibers could propagate on the parallel fibers if no climbing fiber signal was present at that time. Thus, after a signal pause, signals were already present on the parallel fibers before the incoming climbing fiber signal could prevent propagation by exciting the Golgi cells. Thus, these start signals, which were already present as echoes, could be coined into a start signal sequence with the following signal.

We realized that very short input could also achieve a formable length due to the slower propagation on the parallel fibers if the input from the indirect parallel fibers was also used.

It seems that our theory can completely describe the work of Cerebellum. However, appearances are deceiving.

One aspect has never been properly analysed in the theory to date. As we know, the excitation of the golgi cells causes the double inhibition of the moss fibre signals. If a Purkinj group recognizes its own signal, the Golgi cell at the end of the group stops the further propagation of the moss fiber signals along the moss fibers and the accessible parallel fibers, because it is a self-signal detector.

If our Purkinje group is the tenth Purkinje group in the series of Purkinje groups, then in the case of recognition, the moss fibres from the tenth group on would become signalless as long as the own signal of the tenth Purkinje group is active. If this signal is active for the time period ∆t and is not suppressed by external signals, then all granule cells before this tenth Purkinj group would receive input from the moss fibres belonging to this signal. This is because none of the Purkinj groups from 1 to 9 has this signal as an inherent signal. Therefore none of the Purkinj groups 1 to 9 recognizes this signal, so that each of its Purkinje cells is strongly excited. Each Purkinje cell of groups 1 to 9 therefore inhibits the Golgi cell at the end of the group with its axon collaterals. Therefore, the Golgi cell cannot stop the signal transmission on the moss fibres. (If it did, the tenth group would not get its own signal!)

As long as the respective signal of the tenth group is active, its action potential rises at the granule cell laxons of the Purkinj groups 1 to 9 and spreads to the corresponding parallel fibers. These are - with respect to the tenth group - all indirect parallel fibres.

The effect of the input of the indirect parallel fibres, whose granule cells are located in front of the active (here tenth) Purkinj group, has not yet been analysed at all. We want to call this type of signal the indirect echo of an own signal. 

Definition: Indirect echo of an inherent signal

If a Purkinje group, which is not the start group, receives its own signal via the moss fibre system, the further propagation of the own signal is prevented by the inhibitory activity of the Golgi cell at the end of the group to the groups in the direction of the end group. The moss fibre signals are not inhibited in all Purkinje groups which are located before this Purkinje group in the direction of the start group. Therefore, the signal can propagate unhindered on all groups before the recognizing Purkinje group as long as this signal is active. Due to the finite speed of propagation, these signals will remain on the parallel fibers for some time after the signal is terminated. The signal propagating on the parallel fibers after signal termination is called the indirect echo of the intrinsic signal.

Probably this indirect echo is attenuated during its propagation. The longer the distance travelled, the greater the attenuation could be.

The indirect echoes spread out - beginning with the start group of the Purkinj groups - in the direction of the group that had recognized the corresponding intrinsic signal. The echo of the start group reaches the second Purkinje group first. There it has already become weaker due to damping. But the second group has its own granule cells for these moss fibers, which transmit these signals to the corresponding parallel fibers. Thus, a parallel fiber coming from the first group receives a second parallel fiber from the same moss fiber in the second group. In the kth Purkinje-group there are already k parallel fibres tapping the same moss fibre. And none of the signals are prevented from propagating on the moss fiber until the recognizing Purkinj group is reached. Therefore, a possible distance-dependent attenuation is more than compensated by the parallel fiber added to the same moss fiber in each additional group.

What happens to this signal stream?

For the Purkinj groups already formed - in the example 1 to 9 - the indirect signals represent external signals. They are the own signal of the 10th Purkinje group and therefore foreign signals of the groups 1 to 9. They cause a proportional excitation of the Purkinje cells of these groups.

As long as the imprinting signal 10 is active, this is not a tragedy, because Purkinje cells are by default highly excited when not detected.

However, if a signal suddenly appears which is an inherent signal of one of the 9 groups, the signal still running along the indirect parallel fibers acts as an interference signal. In a signal sequence consisting of 10 different own signals, which should be the first ten own signals in the Purkinje chain, the number of indirect interfering signals increases from signal to signal. The last of the ten own signals is already superimposed by 9 indirect echoes of the previous signals. The external signal component is then so high that the tenth Purkinje cell can no longer recognize its own signal. It is similar for us. We can only remember signal sequences with a maximum length. The short-term memory is a memory that can only store a limited number of different objects. The indirect own signal echoes are disturbing foreign signals for the other Purkinj groups, which make recognition more difficult.

Theorem 2.34: The indirect echo of an own signal as an interfering signal

For all Purkinj groups in front of the recognizing Purkinj group, the indirect echo of their own signal is an external signal and therefore a disturbing signal.

As a result, the number of different interfering signals increases with the number of Purkinj groups supplied by the same parallel fibre pair.  If k is the number of the last Purkinj group - i.e. the end group - then a signal sequence which successively passes through all the intrinsic signals of groups 1 to k would produce exactly k-1 interfering signals in the last group. The cause of the interference signals, however, is not the presence of many Purkinj groups, but the fact that the parallel fibres reach so many Purkinj groups. A reduction of interfering indirect echoes can therefore be achieved by shortening the length of the parallel fibers. It does not make sense to lengthen the parallel fibers after a certain length, because the increasing number of indirect echoes as interfering signals makes the recognition of an own signal completely impossible. Therefore the following theorem applies.

Theorem 2.35: Relationship between parallel fiber length and interference signals by indirect echoes

The longer the parallel fibers are and the more Purkinj groups they reach, the stronger the influence of the indirect echoes as interfering signals. This limits the lengthening of the parallel fibres upwards.

Attenuation counteracts this, but if the length is too great, the attenuation becomes so great that the parallel fiber ends only offer the zero signal. This does not lead to any gain in knowledge.

But are there also positive aspects? Does the reverberating indirect echoes of your own signals perhaps create a kind of short-term memory?

Can this explain the learning of signal sequences in Cerebellum?

Here the author repeats his call to all interested parties to participate in the development of an exact and comprehensive theory of the imprinting and re-minting of the cerebellum. The surprising results that such a theory could bring about are illustrated by a concrete example.

We now imagine two independent and imprintable complex signals K1 and K2. They may enter a Cerebellum cluster via the moss fibres in immediate succession. The signal K1 acts on the moss fibres in the time interval <t0, t1> and then again in the time interval <t1,t2>, whereas the signal K2 acts on the moss fibres in the following time interval <t2,t3>. The length of each signal exceeds the required minimum imprinting time.

In the associated cerebellum cluster, no Purkinj group has yet been imprinted with an intrinsic signal. Therefore, the first Purkinje group, the start group, is forcibly imprinted by the first incoming imprintable signal. This is the signal K1 in the first time interval <t0.t1>. The signal K1 thus becomes the start group's own signal.

After being stamped to the start group's own signal, signal K1 occurs again in the time interval <t1,t2>. The start group recognizes its own signal, and the Golgi cell at the end of the group prevents the signal from spreading to the other Purkinj groups on the corresponding moss fibers.

But while the complex signal K1 acts in the period ∆t1 = t2 - t1, its action potentials rise from the granule cells of the starting group upwards to the parallel fibres and spread out there. In this way they also reach the second Purkinje group. Because of the low propagation speed, they are still travelling on these parallel fibres when the signal K2 arrives. Yes, even up to the end of the signal K2, the parallel fibers belonging to K1 are active. Because of their great length they also cross the second, still undefined Purkinj group and excite it. At the same time, the signal K2 enters the Cerebellum cluster via the moss fibres for a period of time t2 = t3 - t2. It is not the intrinsic signal of the start group. Therefore it is not suppressed by the start group. Thus it reaches the second Purkinje group via the moss fibres and excites a pair of parallel fibres belonging to it. At the same time, the indirect echo of the signal K1 also excites this second Purkinje group over the same period of time. And thirdly, the magno-cellular climbing fiber signal arrives at the second Purkinje group, since it is formed in the cortex cluster from the signal averages of the input.

Thus the second Purkinje group is forcibly imprinted. But its imprinting signal is the signal sum of signal K1 and signal K2, because each parallel fiber belonging to signal K1 was as active as each parallel fiber of signal K2 because of the indirect echo. Therefore our Cerebellum cluster has learned a signal sequence. It is the signal sequence K1, K2.

Here K1 is the start signal and K2 is the following signal. If we mark the older signal with a dash (as, for example, the derivative in differential calculus), the following applies

            K = K1* K2 = (K1; K2)

Whenever the signal sequence K = K1* K2 occurs in the cerebellum and each of the two partial signals is not too short, the second Purkinj group imprinted with this signal will recognize this signal as its own signal and report this to the cortex. For example, K1 could correspond to sound "a", while K2 corresponds to sound "m". Then the complex signal K would be the word (or syllable) "am". A signal sequence thus combines signals that are active at different times: first the "a" is pronounced, immediately followed by the "m". This is, for example, the basis for understanding speech.

But the first Purkinje group also reports, because it recognizes the beginning of the signal combination, the signal K1.

Therefore the cortex receives two recognition messages via the responsible thalamus:

-        K1 is detected first

-        then the signal combination K1* K2 is recognized.

Whether the second Purkinje group responds to signal K2 if signal K1 did not advance depends on the external signals present. If the K1 excitation is missing, then (statistically) only half of the self-signal excitation is supplied. This is either too little, because the external signals are stronger, or it is just enough, and the detection response of the positive nuclear neuron is very weak.

Theorem 2.36: The imprinting of temporal signal sequences in the cerebellum 

The indirect echo of an already learnt own signal S1 can be stored by a free Purkinje cell with a previously unstamped but stampable signal S2 as complex signal S1*S2 = (S1; S2) in the next free Purkinje cell. The only prerequisite is that the signal S1, after its embossing, appears as a start signal, which is immediately followed by the signal S2.

There is neither time nor space for a precise analysis of the different ways in which temporal signal sequences can be stored in the cerebellum.

But what does this simple example teach us? For the learning of temporal signal sequences, a longer length of the parallel fibers is quite favorable. This gives the indirect echoes of the intrinsic signals a second, third or fourth chance to be imprinted. With each new chance of embossing, the number of signals combined in the signal sequence increases.

On the other hand, of course, many existing indirect echoes of the previously recognized eigen-signals interfere with the recognition of current and independent eigen-signals, because they act as external signal components and therefore excite the Purkinj groups more strongly. They thus reduce the sensitivity of the system. Nature has therefore found ways and possibilities to develop further algorithms for learning and recognizing temporal signal sequences.

Others may clarify the extent to which repercussion influences the learning of temporal signal sequences.

The planned third and fourth parts of this monograph will show in which other subsystems of the brain temporal signal sequences can be learned. The imprinting of short signal sequences in the cerebellum actually only enables the development of short-term memory. Here, the reader can either be patient or try to find a solution for long-term memory himself. In the planned further parts of this monograph, we will also see that, in addition to the imprinting of signal sequences, there is also a substitution of signals which results in the clearing of previously occupied memory.

ISBN 978-3-00-037458-6
ISBN 978-3-00-042153-2

Monografie von Dr. rer. nat. Andreas Heinrich Malczan