Artificial Neural Systems: Principle and Practice

Neurons

Pierre Lorrentz

Abstract

The aim of this chapter is to explain what a natural biological neuron is, and what an artificial neuron is. To this end, the first section introduces the biological neuron, explains its structure and its information transmission methods. The second section explains how an artificial neuron may be obtained from a corresponding biological neuron. The resources for the artificial neuron may be purely electrical in nature and the behaviour of the resulting electric circuit is expected to be similar to that of information transmission of a biological neuron.

Keywords: Active transport, Axon, Calcium, Conduction, Conductance, Central nervous system, Dendrites , Diffusion, Depolarization, Electrogenesis, Ganglia, Hyperpolarization, Motor, Myelin sheath, Neurotransmitter, Neuron, Potassium, PRVP, Sodium, Soma, Sheath.

A Biological Neuron

Neurons form the fundamental components of the central nervous system (CNS) and the ganglia of the Peripheral nervous system (PNS). Neurons are also found in other locations which may accord them a corresponding name e.g. sensory neurons, motor neurons, and interneurons.

As shown in Fig. (1), a normal neuron has a soma (cell body), dendrites, and an axon. The term neurite refers to an axon, any dendrite, or other protrusions from the soma of the neuron without paying attention to their differences. Axon emerges from the soma at a base called the axon hillock and usually extends a longer distance than any dendrite of the neuron. Neurons do not undergo cell division but are generated by stem cells. Biology and Bio-scientific researchers have confirmed that the main features that distinguish a neuron are: (1) electrical excitability, and (2) the presence of synapses which are complicated junctions that permit signals to travel to other cells.

Fig. (1))

A natural biological neuron.

Dendrites normally branched profusely from both the soma and the axon. Every neuron has only one axon which maintains the same approximate diameter throughout its length. The myelin sheath provides a protective coating around the axon. The myelin sheath allows the action potential to propagate faster than it would have been if compared with another axon of equal diameter. Neurons performs various specialized functions depending on their location and event received, Events are received by communication which is effected in two ways. One is by the release/absorption of neurotransmitter from the surrounding; this is a partly chemical process called neurotransmission. The second is the synaptic transmission. These modes of communication and the associated energy required for the communication is common to all natural biological neurons.

Synaptic Transmission

Synaptic signal is either excitatory or inhibitory. If the net signal excitation exceeds certain threshold and is sufficiently large, it generates a brief electrical pulse called action potential which originates at the soma. The action potential propagates down the axon as follows.

There are pores not covered by myelin sheath (see Fig. 2) through which ion exchanges occur between the axon and the extrinsic fluid; these pores are known as nodes of Ranvier. The ion exchanges are responsible for the production of action potential. The action potential at one node is most often sufficient to initiate another action potential at a nearby node. A signal thus travels discretely rather than continuously along an axon. This mode of transmission along the axon is termed saltatory conduction.

Fig. (2))

A section of axon showing saltatory conduction.

Transmission across synapses

A presynaptic action potential propels the calcium Ca²+ ions through the voltage-gated calcium channel.

As depicted in Fig. (3), a Presynaptic Releasable Vesicle Pool (PRVP) constitutes the active synaptic region of the dendritic terminal ends. The concentration of the Ca²+ causes the PRVP vesicles to fuse with the membrane and release the neurotransmitters into the synaptic region. The neurotransmitters move by diffusion and binds with postsynaptic current (PSC). The electrical current IN (t) that is released from a unit amount of neurotransmitter at t ≥ ts is given by:

Fig. (3))

Synaptic transmission by discharge of neurotransmitter.

Where

V(t) = postsynaptic membrane potential;

E(t) = reversal potential of ion channel;

and the activities of the neurotransmitters and other effects may be the conductance change gN (t).

Because the conductance of the synapse that connects one neuron to another neuron is very important, several experiments were performed by several eminent researchers [1-4]. Some results of the conductivity at synaptic junctions are:

Equations (2) and (3) are obtained by modelling experimental data of natural biological neuron e.g. the axon and soma of a giant squid. The movement of Calcium Ca²+ ion and other ligands in the soma or axon of the giant squid may be confirmed by injection of fluorescent dyes into the substrate before or during the experiment which is often performed at low temperature.

The main fundamental structure and function of a single natural neuron has been described, so also its connections to other neighbouring neurons. They are common to all biological neurons. There are also very many neurons in the CNS.

It is noteworthy that equations (2) and (3) are also solutions of a second-order damped wave oscillator given by:

An Artificial Neuron

In this section, we would like to design an artificial neuron from a natural biological neuron of section 1. Basic resources such as resistance, capacitance, voltage sources, and basic electric circuit analysis are employed in this design. Inside the soma and axon are called the intracellular medium. The intracellular medium is higher in sodium (Na+) and potassium (K+) ion concentration as compared to extracellular fluid. Other ions present include, but not limited to, chlorine (Cl-), Phosphate (Ph0-4), Magnesium (Mg²+). Delimiting the neuron from the surrounding is the cell membrane which consist mainly lipids. The cell membrane may be impermeable to water and ions but permeate ion only at the ion channels and pumps. Because each channel is selectively permeable, when positive ions are concentrated on one side of a membrane as a result, it induces a corresponding negative charge on the opposite side which is the behaviour of a capacitor. For this reason, the neuron cell membrane shall be represented by capacitance. Charged particles in the intracellular fluid do not accelerate despite the field potential, but moves with certain average velocity. This is due to frequent collision with other element which obstructs their movement. Also at the ion pumps, energy is supplied by the hydrolysis of Adenosine triphosphate (ATP), in a process called Electrogenesis, to Adenosine diphosphate (ADP). The sodium-potassium exchanger is an example of ionic pumps that pushes K+ into the intracellular fluid against its concentration gradient.

Because energy is supplied and resistances are present in the intracellular fluid, the electrical representation of the intracellular fluid is shown in Fig. (4). This is similar for other active pump and ion channels. A schematic representation of one section of a neuron is shown in Fig. (5).

Fig. (4))

Resistance and voltage source as electric model of ionic pump and active conduction.

Fig. (5))

A simple electric model of a neurite.

By Kirchhoff’s current law [5] the algebraic sum of current at a junction is given by:

Where

And

Substituting equations (7) and (6) into (5) and re-arranging gives

The equation (10) is a first-order Ordinary Differential Equation (ODE) of the membrane potential V. This equation is valid for an isolated section of part of a neuron. Following the standard method of solution to first-order ODE, the solution to equation (10) may be represented as:

This is a rise and fall exponential solution. The initial increase of V(t) from resting potential is known as depolarization. The product RmCm is called the time constant of the membrane. When t→∞ the steady state value of V (t) is given by:

When the membrane re-charges its capacitance to regain the resting potential, it is termed repolarization. By injecting current or voltage from an external source, it is always possible to drive the membrane below the resting potential; this phenomenon is known as hyperpolarization. Fig. (6) shows an extension of Fig. (5) to make a complete neuron in an extracellular fluid with both continuity and boundary conditions included.

Fig. (6))

A complete electric model, with boundary condition, of a section of a neuron.

These cases will now be considered. The axial resistance of cross-section of an axon is proportional to its length l and inversely proportional to cylindrical crosssectional area . Specific axial resistivity (in Ωcm) is denoted by Ra so that axial resistance R is calculated as follows.

Recall that resistivity pis defined by

Also, the membrane current Ia now flows both to the left and to the right; the sum is given by:

We have now included voltages from other membrane sections and indexed them by j as show in equation (17). Modifying equations (8) by substituting equation (17) into it we have;

The surface area a of a cylindrical axon is πdl. Dividing (18) by πdl gives;

Equation (19) is a second-order difference equation making it suitable for numerical integration. To derive a continuous version of equation (19) replace the length l by x δ x and evaluate it in the limit δx→0.

Substitute (21) into (20)

The equation (22) gives a more accurate description of an artificial neuron than equation (10). This is the first example of an artificial neuron obtained by modelling natural neuron directly. This method whereby an attempt is made to produce a morphological, and structural equivalent of a neuron, and watch for the same behavioural pattern, is termed neuromorphic. When Equation (22) is constructed as a neuromorphic neuron, it may be verified if it possesses equivalent information-transmission characteristics by checking against that of biological neuron data. Additional design issue may be the choice of capacitances, variable resistance ranges, and initial calibration. Equation (22) is usually refers to as the cable [6] equation and also bear much semblance to wave equation.

CONFLICT OF INTEREST

The author confirm that this chapter contents have no conflict of interest.

ACKNOWLEDGEMENTS

Cited works are appreciated.

REFERENCES

Basic Neurons

Pierre Lorrentz

Abstract

The aim and objectives of this chapter is to present other types of artificial neuromorphic neurons with capability of reset and recovery. For this reason, the first section starts with the integrate-and-fire neuron, which has the propensity for reset. The second section introduces probability theory owing to the fact that many processes in the brain and central nervous system obey probability laws. The third section introduces another artificial neuromorphic neuron which employs a Poisson process and is closer in behaviour to a biological neuron.

Keywords: Bayes theorem, Binomial, Bernoulli, Charging, Depolarization, Density function, Excitatory, Expected-value, Inhibitory, Mean, Moment, ODE, Pseudo-random-number-generator, Poisson, Steady-state, Synaptic strength, Spike, Threshold potential, Uniform distribution, Variance.

The first chapter has introduced one biological neuron and one artificial neuron. One advantage of developing ANN from principle is that reproduction is assured with minimal loss of resources and a target performance may often be achieved. Since the book is more about artificial neural network systems, chapter 1 contains the last item on biological neuron. Most development throughout the book however depends, directly or indirectly, on the biological neuron so that it may be regarded as an introduction to the rest of the book.

INTEGRATE-AND-FIRE NEURON

There is another version of artificial neuron model known as integrate-and-fire model; this is a version of figure 5 chapter 1 neuron with an inclusion of spike generation and reset. It states that when the membrane potential [1, 2] reaches or exceeds a threshold potential θ, firing an action potential [3] and discharging occurs. After that, it reset and (re-)build its potential again. The charging proceeds as follows.

Multiplying (1) by Rm;

Equation (2) is a first-order ODE, whose solution is given by:

One may be interested at what frequency f (I) does the neuron fire. The neuron fire whenever the voltage V equals θ the threshold voltage or exceed it. Setting Em = 0 and V = θ in equation (3);

where I is the injected current.

In order to apply this artificial neuron to model a stereo-typical situation found in CNS of some animals, a distribution known as Poisson distribution shall be introduced. A relevant introductory probability theory is presented now.

PROBABILITY

Definition 1.1: Probability is a set function p that assigns to each datum xi in the sample space X a number p(xi) called the probability of the datum xi, such that the following properties hold:

1) p(xi) ≥ 0

2) p(X) = 1

3) If x1, x2, x3, ...are data and xi ∩ xj = Ø, i ≠ j, then

p(x1 x2 ... xk) = p(x 1 ) + p(x 2 ) + ... + p( xk ), for each positive integer k, and p(x1 x2 x3 ...)= p(x1) + p(x2) + p(x3+ ...), for an infinite, but countable number of data. For any datum xi,;

If xi and yi are any two independent databases with no data in common, then:

Otherwise;

In this case p( xi∩yi ) is defined as;

p(xi | yi ) is called conditionalprobability. p(xi | yi ) reads "the probability that xi occurs given that yi occurs". The p(xi ) and p(yi ) are examples of prior probability, while the conditional probability is an example of a posterior probability.

Bayes Theorem: Let x1,x2,x3, …xm be a partition of the database X such that xi X and xi ∩ xj = Ø,i ≠ j , by mixing database xi with yi, databases xi and yi are said to intersect. The intersection of xi and yi ( yi Y ) may be written as:

Given that:

And by the defining equation (9)

If p( xi ) ≥ 0, then

Recall that yi ⊆ Y; and xi ⊆ X; therefore,

Equation (15) is referred to as Bayes Theorem.

Probability Density Function (p.d.f.): If xi is allowed to take any value ranging from 0 to 1 inclusive, it is said to be a random variable. If it is discovered that xi follows certain pattern when assuming any value whatsoever, then this certain pattern is a distribution. Since the pattern is certain, it is representable by a function which is called a probability density function (p.d.f.) f(xi) As f(xi) moves (i.e.; assume values) in space, it trace out what is called a distribution. Let f(xi) be the p.d.f. of a the random variable xi, and let R be the space of