Article Publishing History Article Received on : 15/06/2019 Article Accepted on : 11/09/2019 Article Published : 13 Sep 2019
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ABSTRACT:
In 1943 Machculloch and Pitts create the formal neuron where many input signals are linearly composed with different weights on the neuron soma. When the soma electrical signal goes over a specific threshold an output is produced. The main topic in this model is that the response is the same response as in a Boolean function used a lot for the digital computer. Logic functions can be simplified with the formal neuron. But there is the big problem for which not all logic functions, as XOR , cannot be designed in the formal neuron. After a long time the back propagation and many other neural models overcame the big problem in some cases but not in all cases creating a lot of uncertainty. The model proposed does not consider the formal neuron but the natural network controlled by a set of differential equations for neural channels that model the current and voltage on the neuron surface. The steady state of the probabilities is the activation state continuous function whose maximum and minimum are the values of the Boolean function associated with the activation time of spikes of the neuron. With this method the activation function can be designed when the Boolean functions are known. Moreover the neuron differential equation can be designed in order to realize the wanted Boolean function in the neuron itself. The activation function theory permits to compute the neural parameters in agreement with the intention.
KEYWORDS:
Activation Function; Boolean Function; Digital and Continuous Computation; Differential Equation for Neuron Channel; Intention Implemented into the Brain Parameters; Natural Neuron
Copy the following to cite this article:
Resconi G. Boolean Models Guide Intentionally Continuous Information and Computation Inside the Brain. Orient. J. Comp. Sci. and Technol;12(3).
Copy the following to cite this URL:
Resconi G. Boolean Models Guide Intentionally Continuous Information and Computation Inside the Brain. Orient. J. Comp. Sci. and Technol;12(3). Available from: https://bit.ly/2lZSmdy
Introduction
In this paper we
present a method to transform every Boolean function into one dimension
continuous function denoted implication Boolean function or activation
function. The values of the Boolean functions are represented by the maximum
value as one (true) and the minimum value as zero (false); between true and
false there are all possible degrees of truth. Channels in one neuron with
electrical ionic can be modelled to realize the wanted implication Boolean
function. Now we can transmit this function in space time to other neurons by a
spike transformation of the implication Boolean function. Synaptic degrees of
the superposition of the spikes can be computed by a special code to generate
the wanted function. With this code every Boolean function can be generated by
other input Boolean functions. We can use multiple channel network to obtain
any type of complex Boolean functions in the steady state condition. We remark
that discrete and continuous logic can be implemented in the brain to represent
the intention of the agent in a physical way.
Neural Solution of Boolean contradictory function by dependent function k’(x) [5,6,7,8]
Graph of the dependent
function k’(x) for AND
Figure 1: AND operation by one dependent function in (1)
We know
that each channel in figure 10 is a graduate or fuzzy switch that depends on
different elements that can open or close the channels by active process or
passive process. In figure 10 we can see the channel mutual dependences that
can change the electrical charges in the membrane to generate activation
potential or other complex changes in the membrane electrical potential Each
channel controls the inside and outside movement of the ions as K+ or Na+.
Mathematical representation of the spike
process by the Hodgkin-Huxley model
The Hodgkin-Huxley
model assumes that the electrical activity of the squid giant axon is mainly
due to the movement of Na+ and K+ ions across the membrane. Thus, in the model,
the neuronal membrane contains Na+ channels, K+ channels, and a leakage channel
through which various other ionic species, such as chloride Cl−, can pass. The
equivalent circuit diagram corresponding to the Hodgkin-Huxley model is shown
in Figure 19
Figure 11: Circuit Diagram for the Hodgkin-Huxley Model of the Squid Giant Axon.
Hodgkin, Huxley neuron electronic [13] image by the equation
Where the rate constant b is the activation rate constant for which the ionic channel from closed state becomes open and the ion moves from internal to external part of the neuron. The rate a, b are coupled with electronic system of the neuron by the voltages on the membrane. Channel process is given by the following system (figure 12) ba
Figure 12: model of the ion movement from internal to external of the cell by the voltages control
Boolean function and activation function by electronic systems and channels
Each
channel controls the inside and outside movement of the ions as K+ or Na+. The
Channel can be represented in a very simple way by the graph in figure (12)
that we show in this system
Figure 14: simple system of outside inside channel
In conclusion we can see that the Boolean functions x and NOT x that we write in this way
Are translated into the two implicit Boolean
functions (activation functions ) shown in figures 16 and 17. Digital Boolean
function whose values are 1 and 0 is the constraint in which we build the two
continuous functions that define the parameters of the differential equation.
In a graphic way we have
Figure 18: input current and RC= T parameters function of p = V.
we have the
graphic image of the implication or dependent Boolean function in (17)
Intention inside the neuron and logic [12]
To introduce intension
inside the neuron and in the neural network we must associate with the
intention a logic proposition in a classical and in many valued logic as
activation function. In this way the neuron is constrained by intention so from
conceptual space we realise intention by physical device. So for example given
the two Boolean functions
As shown in
figures (16), (17) they can be represented by differential equation in figure
(14) of the neuron. Intention to obtain the two Boolean functions is realised
in the physical world by the system in figure (14).
Figure 19: The connection between the input and the soma of the neuron is given by weighted element denoted synapse shown in this figure.
We know that formal neuron given by Maculloch and Pitts [12] was a very important step to understand the logic of the brain. To solve the problems of formal neuron we come back to natural neuron where the channel dynamics controls the neurons behavior. With the new revised study of natural neural network we try to model the activation state and the activation time in the open and closed channel process. Steady state in neuron and neural network is defined a priori to model intention represented by logic processes as AND, OR, IF and other types of Boolean function. Now each Boolean function is a set of spikes or truth that we can represent by a continuous function which has the max and min values in agreement with the logic function. Max value is one or true and Min value is zero or false. In this way when Boolean discrete or digital function is transformed into the continuous function of time and space we can use the natural neural differential equation whose steady state is the intention or continuous function described before. The continuous dependence function that is created in the soma can be codified by set of spikes to transmit by the axon the message to other neurons to give other possible aggregation to build more and more complex functions for more complex intention to be implemented in the physical world. We begin with a very simple channel master equation and after we move to a complex graph of channel mutual interaction to give a stronger instrument to obtain a physical image of the original conceptual intention.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
Conflict of Interest
The authors do not have any conflict of interest.
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