Vertebrate brain theory

3.10 Theory of neuronal signal inversion

We first turn to the mean nuclei, which do their work in several segments of the segmented Bilateria. So far we assumed that the signal averages of the different modalities are used to control life functions. For example, the mean brightness controls the circadian rhythm of the living being (inner clock). The mean value of motor activity says something about the associated energy consumption and the necessary energy supply. The mean value of olfactory excitation allows an assessment of whether there is edible prey in the vicinity. Many other processes are also involved in control loops that require mean signal values.

Furthermore, we turn to the inhibiting neurons. First, they are used as inhibitory interneurons in the sensory and motor centers of the rope ladder system for lateral inhibition, which enhances the contrast between strong and weaker signals.

In neural tube systems they fulfil the same function in the different segments and levels. Here the inhibition is often relative. A strongly excited inhibitory neuron inhibits strongly, a weakly excited one only weakly. A total inhibition independent of the current rate of fire would be useless for lateral inhibition. The inhibition effect depends on the current rate of fire and increases with it.

We now combine averaging with relative inhibition to invert a signal S. For reasons that will only be explained in more detail in chapter 4 of this monograph, we form an inverted signal f* from a signal with the rate of fire f and an average signal with the rate of fire fm.

                            (1.8.1)

is enough. It follows directly from this

                 (1.8.2)

The product of a fire rate and its inverse fire rate is equal to the square of the fire rate of the mean.

If we assume that receptors and neurons have an exponential characteristic, logarithmizing leads to the original quantities from which the excitation is derived. In this respect it is not surprising that between a rate of fire f and its inverse f* the relation

  (1.8.3)

exists. In this respect, the signal inversion can be described as logarithmic mean value mirroring.

Theorem of signal inversion

A signal with fire rate f is inverted by dividing the square of the mean fire rate fm by the fire rate f.

The signal inversion is realized by first switching the signal with the firing rate f to an inhibitory transmitter - if it is not self-inhibiting - and feeding it to an output neuron, which is simultaneously excited by an average signal with the firing rate fm. Due to the non-linear, exponential characteristic of this output neuron, the superposition of excitation and inhibition results in the inverted signal f*.

A circuit that inverts a signal is called a (neural) inversion circuit.

If the signal S has a minimum at one point, the inverted signal S* has a maximum there. The monotony is reversed by the inversion transformation.

Theorem of signal inversion of minimum coded signals

A minimum coded signal S is converted by an inversion circuit into a maximum coded signal.

For the more precise identification of neurons in inversion circuits we introduce the following designations. The input neuron provides the excitation signal that is to be inverted. The following, GABAergic neuron is called a switch neuron, because the transmitter is switched from glutamate to GABA. The output neuron is called an inversion neuron because it carries out the signal inversion using the supplied signal average value and the inhibiting switching neuron. A mean value neuron supplies the inversion neuron with the necessary signal mean value so that an inversion can take place at all.

The use of other inhibiting transmitters is quite possible.

A signal inversion transforms a strong signal into a weak one and vice versa, a monotony in the signal path is reversed.

Frequently, the signals of many neurons are inverted, all of which are distributed on a line or an area. In these cases, not each of the associated inversion neurons has to occupy its own, separate mean neuron, but a mean neuron can distribute its signal mean over many inversion neurons simultaneously by branching its axon correspondingly often. The signal mean can also be tapped from a more distant mean nucleus.

With a real signal inversion, non-linear deviations from the ideal formula certainly occur. This is insignificant as long as the monotony inversion occurs, i.e. minima become maxima and maxima become minima. Therefore, the extreme value inversion is primarily necessary. This could be realized by many different neuronal characteristics of inversion neurons.

With signal inversion, the inhibiting neurons were given a new task: the relative inhibition of mean signals to transform a signal into its inverted signal. This transformation becomes the working principle of the vestibulocerebellum that is being formed.