neurotransmitters around the brain or in big regions, and that are not synaptic specific. At the same time, there are hormones that are spread all around the blood.

Following our model, it is most probable that these kinds of interactions do not have local computations, or influence in what is called the local context. It would be very hard to spread some kind of very specific information with particles travelling with freedom on the brain. But, these particles can have a strong effect in the global context. It means that for example, thinking in the basic principles of learning from behaviorism, when you have a positive reinforcement, the temporary network generated at this time has to be reinforced and its probability to appears again have to be augmented.

To do that, the brain could send neurotransmitters all around, or for big regions. These neurotransmitters would change the A parameter in order to answers better to a determined context, maximizing the activation of the already active or recently active neurons. At the same time, when a negative reinforcement comes, the Brain could send these neurotransmitters changing the answer to the global context, and making the network less active, or more aleatory. In order to it, make a reorganization again with the local computations.

A second way of computation that can happens is related to the change in the own global context by sending neurotransmitters to regions of neurons, temporary networks or all the Brain. The change could be temporary, and it can elicit some kind of behaviors by activating some kind of mechanisms. It means, for example, it would be possible to change the global context to a state of angry. What would disturb a few the temporary network, and guarantee an isolation of this temporary network in order to answers properly to the instinct of angry. With the current context and knowledge related. For example:

* A temporary network M is activated by the context of a task t.

* The cognitive instinct mechanisms of the Brain perceives a threat. Neurotransmitters and/or hormones are spread towards the temporary network M giving a shift in the context C parameters of the neurons.

* The shift permits the temporary network to isolate local contexts in order to do computations related to the state of angry, but maintaining the information in the already specialized network to the task t. Or even activating a new temporary network.

In order to demonstrate it mathematically, we would have to expand our model to add the dynamics of the gates and pumps. The pumps can work as clocks that give a reference to each neuron about time. The gates have a dynamic that goes around the threshold, they can augment or diminishes the membrane potential of neurons, depending on the kind of gate. Therefore, if the gates that augment the membrane potential surpass the gates that diminish the membrane potential, we can generate an action potential. There are also possible to influence the parameter A of the neurons.

Therefore, the influence in the parameter A would be in order to alter the gate's dynamic, for maximizing the probability of have an action potential or diminishes this probability. While the pumps determine how much time the neuron can fires in a period of time, and when it will fire, in a more continuous way. In order to understand the global context, we can abstract the working of pumps, and assumes that we have a determined function b(X) that takes the neuron back to homeostasis.

Thus, as we defined before we can say that $A = f(local\_plasticity)$, therefore, we can define how $f(local\_plasticity)$ works in a more detailed manner. As this we do:

$$f(local\_plasticity) = f(positive\_gates) + f(negative\_gates) \qquad (37)$$

Where the positive gates, and the negative gates can be changed by the interaction with the local context, or with the global context. It is, in the case where we have a positive reinforcement we have:
$$ positive\_gates < -positive\_gates + fp(global\_context) \quad (38) $$

$$ negative\_gates < -negative\_gates + fp(global\_context) \quad (39) $$

The function *fp* can act augmenting the gradient quantity (The force that will move the ions towards the channel) in the positive gates channels, and diminishing the gradient in the negative gates channels. It means that the probability of have an action potential will be increased. The negative reinforcement acts exactly in the contrary direction. Therefore, the *f(positive_gates)* and *f(negative_gates)* can be viewed as the proper composition and properties of the channels, as for example the capacitance.

Thus, we can prove by induction that the temporary network will be activated in the long term, if it is heavily reinforced, and another variables are considered constant in a determined task. We should take in account that the context to a same task variate ever in a closed set T for each neuron and that the local context does not change. Let's now consider a brain network M with n neurons.

* H0: In each small window of time t of the task, the neurons K of the network M are activated. It means that these neurons, to this task surpass the threshold in the time t.

* Hn: Given that the task t gives a positive reinforcement, the A variable from the function F is augmented. Therefore, the neurons have an increase in the probability to be activated in the task t again when it happens.

* Hn+1: when the task comes again the An+1 variable of each neuron, in each time t of the task will gave us An+1*(X + C), as An+1 is higher, therefore the chance of activation is greater and the temporary network is maintained. As here, X and C are considered positive constants. , we can prove the same to the dissipation of a temporary network towards the negative reinforcement. Q.E.D. (conclusion of the proof).

Therefore, we proved that the brain also can use the global context, with plasticity, to reserve or not regions of the brain to perform computations in face of a task. It means that brain regions could have regions specialized in doing specific computations. This process is also helped by what is called sustained fire (Wang, 1999), as the context maintains itself, the neuron will ever fires when it comes back to the homeostasis, after the pumps put it back to an active state.

The second way of interfere as said is directly in the parameter C, it means, a changing in the global context of the region. It can make a shift in the function answer, moving neurons to the zone of deactivation, and another to the zone of activation. It can be showed by definition, if the C from a neuron changes due to external interference, therefore we have the following possibilities:

* The neuron continues to surpass the threshold, but his answer diminishes. It can provoke a cascade effect on the network because the X of the post-synaptic neuron will be lowered or somehow different. For example, it could be that the calcium delivered in the synaptic vesicles will be highest with a highest answer.

* The neuron surpass the threshold with a stronger answer.

* The neuron approaches the threshold, but yet, does not surpass it.

The interference in the context is important because probably the Binaural/Monaural beats are able to make this kind of interference in the context. While the influence in the plasticity is important to understand, if the beat could cause some permanent effect in the Brain. One could state, for example, that a plus high context activation generate a stronger A plasticity, generating a stronger temporary network and therefore improving attention.
This kind of computations also could be found in the EEG, at least with abductions. By associating the EEG signal to the phases of the task, studying the possible generators of influences in the context (for example the substantia nigra responsible to inhibits broad regions, or the regions responsible to delivery dopamine in also broad regions (Ott, 2019)). We could therefore see an influence in the global context by studying the clusters of the EEG. For example, a global excitation could increase the electromagnetic potential globally, and along the time. It could be studied observing the learning time, when the plasticity happens. In the learned task, it is possible that the values will respect a determined interval, without very much changes.

As the same way, a global inhibition could decrease the electromagnetic potential globally. It is important to see that the mechanisms of attention can have a limit. It means, when a determined amount of effort is made, the regions start to be deactivated even to a positively reinforced task. This mechanism can be detected by the alpha pattern of the rest state, and the beta pattern of the attention state. The global inhibition is therefore also somehow related to this possible phenomena. And a stronger temporary network can be more resistant to this kind of inhibition. It means that the Binaural beats could have an effect related to: Augment the window where a positive reinforcement can happen, increasing the force of the temporary network. And influence in the own answer to a global context, making a more cohesive temporary network.

## F.5 An example of local computation using this model

The local computations are the most complex of all, because they carry the details of computations. There are infinite possibilities to it be made, and here we will explore just some examples to illustrate this richness of a connectionist learning system. In order to do that we can again change a few of our model, this time we need something to change the output of each neuron messages to the post-synaptic neurons. These mechanisms can be found exactly in the synapses, with the vesicles and calcium dynamics. Therefore, we could add a function $Sy$ that answers to the threshold of each neuron, for each one of its synapses:

$$Sy(K) = R * K \qquad (40)$$

$$Sy(s(F(X)) = R * s(F(X)) \qquad (41)$$

This is, depending on the situation the neuron could have an inhibitory synapse, with R negative. An excitatory synapse, with R positive. And could represent information or do computations with the parameter R. Let's say that, for example, as the work (Ott, 2019) propose there are two neurons representing two different rules: Click if it is a target or do not click as it is not a target. A post-synaptic neuron, that integrates both the information from these two neurons, would be able to take the decision according to the situation and the reinforcements. In order to do that, we will include the threshold of the calcium in the synapses:

$$Tsy(Sy(K)) \qquad (42)$$

That is a mathematically analogous threshold as the one from the action potential, in our model. Therefore, the post-synaptic neuron would learn with plasticity to increase or decrease the R constant in a way that the threshold would be surpassed in the contexts where there is a target, and do not would be surpassed in the contexts where we do not have a target. The local context generated by both the previous neurons representing the rules would happen to decide what is the better R parameter, while the global context is responsible to reserve this network to this kind of task and context.

Therefore, when there is a target, $Sy(K)$ is higher than 0, triggering an action. While when it is not a target, $Sy(K)$ is lower than 0, and nothing happens. It has to be clear that it is just a simple approximation
and tool example to demonstrate the potential of local computations. These kinds of computations are very sensitive and do not could be neuromodulated by Binaural/monaural beats.

Also, we can perceive that the improvement of an answer, as the absence of a motor response when we do not have a target, can leads to a decrease in the potential of the EEG electrodes due to a lower level of activity in the network. Therefore, in order to do the analysis in the EEG it is necessary to do not associate higher activity or lower activity with a better or worst performance. What is better, is to verify a lower degree of changes combined with a higher performance of the subject, meaning that we arrived to a steady-state where the subject really learned a determined task. The changes of this steady-state, or the increase of the capacity to maintain it, due to the mechanism of inattention would be a possible neuromodulation for Binaural/monaural beats.

## F.6 A hypothesis about the Binaural/Monaural beats effects and Possible analysis with EEG

The hypothesis is that the Binaural beats can generate a constant and phasic shift in the global context, as in the second way of interfere in the example of global context computation. Depending on how it is changed, the Binaural beats would improve attention.

In the EEG it would affect the ERPs patterns, or somehow, the own temporal data pattern in a constant way. The beat could change the results in the threshold of each neuron, helping to decrease some kind of competitive context. For example, let's state that the global state of attention, in the beta frequency, generates a global context of value B in a small window of time. And the global context in the global state of rest in the alpha frequency has the value of Alpha in a determined time t.

Consider now a set M of active neurons in a determined time t, here called a temporary network. Each one has his own B_m and Alpha_m values related to the states of attention and rest, respectively. If we compose this global context with the global context T_m related to the task, we will have the global context C_m for each m active neuron from M in both situations.

Consider now a new influence from binaural beats, that generates a global context influence called Bi_m, and is added to the global context of the active neurons M, in the instant t. Now, let's visualize one neuron m from M:

$$F(X) = A * (X + (T_m + B_m + Bi_m)) \quad (43)$$

In order to improve attention, we have to observe B_m, the influence from the beta state of attention. If the neuron is sensitive to the attention state, for this task (T_m) therefore we can help to sustain the threshold active increasing the Bi_m. It can augment the time that a neuron have to be reinforced in this context, generating a stronger temporary network. And also can affect the hebbian learning, as proposed before.

It means, the attention is an interaction with the context generated by the task T_M and the cognition. In order to pay attention to it, the Brain would have to rely on a mechanism to excite this context, and split a temporary network to it. It would be made by changing the A parameter of our function from the neurons of M, in our simple representation. Or by influencing in the global context, parameter B. Therefore, it is clear that if the beat could generate the same global context of attention, so, the attention would be improved to the task T_M.

At the same time, if we have the state of rest, or inattention:

$$F(X) = A * (X + (T_m + A_m + Bi_m)) \quad (44)$$
We can directly change the parameter B = T_m + A_m + Bi_m with the Bi_m global context influence. It means that we can put the function in a state of attention, or move the membrane potential to have an action potential. Putting the brain in a state of attention with the beat.

In order to measure that on EEG it would be necessary to use a function f(x) that is able to separate the influence of a global context or in the temporal domain, or in the frequency domain. It means that it is necessary a function as:

$$f(EEG\_clusters) > global\_contexts \quad (45)$$

Fortunately, there are already some kind of functions to do that. The own brain states related to alpha, beta, and gamma bands are currently measures of some kind of global context. In this sense, a possible measure is verifying if the beta band is strongest with a beta neuromodulation from the beat.

Also, the Independent Component Analysis or related techniques give an idea about how to get the global contexts. In this sense, we could say that the inhibition context, the excitatory context and the Binaural Beats/Monaural Beats responses would generate an independent component in the EEG analysis, what would be the global context influence.

The analysis can also be more detailed and look for means, variances, and differences in the domain of time. It could be possible to build a proper technique to identify different global contexts influences from clusters of neurons in the network. A first approach, would be to do that statistically. And the second would involve the most fine approximation of the Brain computations.