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Abstract

As an introduction, this paper explores our attribution of intelligence to living organisms that rely on neural networks to produce behavior. It then describes an early attempt to implement a biologically realistic computational model capable of recreating key principles of biological nervous systems by growing connections instead of working with predetermined connections. Present work focuses on analyzing the growth of connections that depends on neural activity as a mechanism for learning during development and adulthood.

The main idea focuses on creating an array of neurons where every single neuron has the possibility to establish a direct connection with every other neuron, including itself. In this manner, present limiting factors in computational neurosciences that rely solely on adjusting synaptic strengths (weights) can be surpassed, resulting in a massively parallel, highly interdependent, scalable, robust, reactive and versatile processing architecture.

The long-term goal of this project is a VLSI implementation of the computational model for use as an artificial nervous system for autonomous robots, but the model and the future hardware themselves are expected to serve as powerful research platforms for neurosciences in general as well as artificial intelligence and artificial life.

So far, only living organisms have been able to display truly intelligent behavior.   There are thousands of species on earth and every single one displays intelligence, that is if we consider intelligence as the ability of an individual to behave appropriately relative to its environment (i.e., avoid obstacles, evade predators, find food and mates, etc). [3]   

However, no one would consider a bacteria or a microbe as a truly intelligent being, they are very adequate to their environment but their size and lifestyle is so different from ours that a direct comparison is not practical; also, adequacy is usually achieved through generations and not by the adaptation of a particular individual.    Instead of talking about intelligence it is better to talk about our perception of intelligence.   In this frame of reference we will only attribute intelligence to organisms that share a similar spatial scale as we do and to which we can relate.  

As we climb up the phylogenetic tree of life we can begin to perceive intelligence at the second level, that is the animal kingdom Eukaryotes/ Metazoa (Animals), and it is no coincidence that every single branch of the animal kingdom, except for the sponges Porifera, has a nervous system composed of specialized cells.   These specialized nervous cells or neurons interconnect to one another forming information networks that transmit electrochemical messages.   The simplest nervous system from the Cnidaria (i.e., jelly fish, coral, hydra) is composed of a nerve network that branches throughout the body with very little organization but still proves sufficient for the degree or information processing that such simple beings require [8].    As the complexity of an organism increases so does the complexity of its nervous system and the degree of specialization of its neurons, all of which relate directly to our perception of intelligence of that particular class of organism.

The human nervous system is the most complex of all species on earth; it provides us with superior adaptation abilities and places us as the dominating species, but we must not forget that, every single action we perform, every idea we conceive, every emotion we feel and even our sense of being special are all a direct consequence of neural activity.   Just like simple invertebrates our intelligence and cognition are the products of a neural network. [3,4]

Nervous Systems Process Intensities

So far we have analyzed that our attribution of intelligence to organisms is directly proportional to the complexity of their nervous systems.   Nevertheless nervous systems alone cannot produce intelligent behavior since they do not deal with the real world directly, either as input or output.

As inputs, the messages a nervous system deals with an encoded representation of the environment and the immediate state of the individual.   The proper encoding is the responsibility of the senses, composed of specialized receptors, which reflect the intensity and time of stimulus.   The nature and localization of such stimulus is determined by the networks wiring rather than by the particular receptor being stimulated.    In other words, a pressure receptor, responsible for tactile sensing, will increase its firing rate when submitted to higher pressure the same way a photoreceptive rod will increase its firing rate when exposed to brighter light.   There is no encoding by either receptor that differentiates a message as touch or sight.

On the output end of a nervous system are actuators or muscles, and their function is to convert messages from the nervous system into mechanical force.    Their response is also based on the intensity of the message being received.   The higher the firing rate a muscle fiber receives the harder it will attempt to contract.

Since inputs and outputs of any nervous system are based on intensity, there is no reason to believe that the in-between processing works differently.   There is no evidence of the production of any form of representation or model of the perceived world within the nervous system other than the routing of sensory intensities to motor intensities, creating much higher levels of cognition in between.   This is easy to see in the simple wiring of involuntary muscle reflexes (see figure) but at present is practically impossible for higher cognitive functions, where the number of connections is so vast that present analysis techniques can only distinguish areas of activity or at best firing patterns from a small group of neurons.

Stepping on a tack initiates a flexor reflex, exiting the appropriate muscles to raise the punctured foot and provide additional support on the opposite leg.  [5]

Artificial Neural Networks and Computational Neurosciences work with Predetermined Connections

Neurons are the main processing unit of any nervous system.   There are many shapes and flavors of neurons even in a single organism but they all share a common theme, their function is to process and relay messages [4].

Traditionally neurons have been modeled as simple integrators of input signals.   Once the integrated potential reaches a certain threshold the neuron fires an output signal that propagates to other neurons.   In biological neurons the integration is an analog process while the firing is an all or none digital quantity.    Action potentials encode the time and the intensity of the inputs.    Since action potentials are an all or none quantity, with very little amplitude variation, the intensity is encoded in the frequency of such potentials or firing rate.   Recently, neural models based on membrane-ion-channel activity have been developed.   They are far more consistent with biological neurons than the simple integrator model and are capable of simulating much more complex neural behaviors like transmitter depletion, potentiation, depression and facilitation.

In both traditional neural-networks and recent computational-neuroscience approaches neurons are arranged in a matrix and connections between neurons arranged in layers.   The strength of each connection is represented by a weight that must be selected carefully for every single connection since the entire functionality of the networks depends on the appropriate set of weights.   There are many algorithms for choosing the appropriate weights, a process usually referred to as training that reflects the abilities of the network to learn.   However, no matter how carefully the weights are chosen, the limiting factor of any network will be, apart from the number of neurons, the way they are connected.

 Some neural models rely only on unidirectional (feed-forward) connections extending from an input layer to an output layer, sometimes with hidden layers in between.  Other more sophisticated models have lateral (inter-layer) and even bidirectional connections [7].    Regardless of the direction, the breadth of the connection tree and its reach is also a determining factor.    Some models connect a neuron on a given layer with all neurons on the next layer, while other models just establish connections among neighboring neurons.   Nevertheless traditional and recent approaches all share a common similarity inconsistent with biological systems; all the connections within a model network are always predetermined.

Probably the most remarkable feature of biological nervous systems is their ability to program themselves [2]. In biological systems the growth and trimming of connections is essential during development and learning in the adult.    A model capable of dynamically adjusting its connections would provide an additional level of plasticity far superior from that of models with predetermined connections.

Biological Mechanisms Responsible for Growing Connections

In search for a computational model capable of growing connections in consistency with biology we must first analyze the mechanisms through which connections are established in biologic nervous systems and attempt to extract a reasonable algorithm.

Biological nervous systems do not grow connections randomly. The growth of connections depends on two key factors: genetics and neural activity produced by experience.   During development an immature neuron will need to grow dendritic and axonal branches to other neurons as well as muscles and receptors.   The direction in which a neuron extends its branches is guided by two main mechanisms: molecular adhesion and concentration gradients.    Guidance through molecular adhesion occurs as molecules in the growth cone of a neurite bind to external molecules in the surrounding tissue.  Concentration gradients rely on guidance molecules, which may be either soluble molecules secreted from a distant target or be present at the surfaces along which the growth cone makes its journey.   There is a wide variety of guidance molecules including neurotransmitters and secretions from non-neural tissue like muscles and receptors.    Diffusion of an attractant molecule from a remote source will cause a growth cone to turn and head in the direction of the source while a gradient of repulsive molecules will cause the growth cone to turn away from the source.    It should be noted that a single guidance molecule can be an attractant for one cell and a repellant for another.   Moreover the same molecule can be both attractive and repulsive to different parts of the same neuron or to the same neuron at different times.   This is because apart from molecular adhesion and guidance molecules neurite growth is also regulated by growth factors and neural activity.   It is presumed that normal growth occurs only when the calcium concentration in the growth cone, which is highly dependant on that cell's activity, is within an appropriate range [4].

Living creatures are born with a set of genetically predetermined set of connections or rough wiring diagram.  These connections are established during development and account for behaviors usually called instincts, but in more advanced species, like humans, the total number of connections is so vast that only a small percentage of the wiring diagram is encoded in the genome, the rest is connected by learning through experience.    Having this in mind, a biological or artificial individual must only have a rough wiring diagram and the necessary rules for learning encoded in it genome; most of its behavior will be determined by its experience with the environment.

The Computational Model (so far)

In order to succeed in creating an artificial nervous system it is necessary to have a computational model capable of representing a neuron with enough detail to account for biologic neurons but with enough abstraction to simplify computation, a development mechanism to grow connections from one neuron to another and become a neural network, and finally a proper learning mechanism to regulate connection strength and trimming of unnecessary connections.  However, being the initial goal of this research to explore the growth of connections based on activity rather than single-neuron models and learning mechanisms, a simple neural model was chosen as a starting point.

To implement the simulation three basic types of elements were chosen: Neurons, Routing Element and I/O Elements.  Neurons (large circles) are arranged in a two-dimensional matrix surrounded by input/output elements (squares) and physically interconnected through routing elements (small circles) as shown in the figure.   The simulation was programmed in C++ where proper evaluation mechanisms were implemented to simulate parallel and asynchronous processing.   Communication between elements (neurons, routing elements and I/O elements) occurs only through physical connections in the form of data and instruction signals.   There are two basic types of signals: singlecast and multicast.

A firing neuron will transmit a multicast signal indicating the magnitude of activity and the position of the active growth cone to every other neuron in order to attract/repel other growing neural processes.   Singlecast signals have a single specified target and are used to transmit connection instructions as well as activity data between already connected neurons.

A Neuron

The present neural model is a simple pseudo-energy point of view abstraction rather than an implementation of ion-channel, neurotransmitter and membrane potential activity.  The pseudo-energy considered here does not relate directly to the electrochemical energy responsible for neural processes.   It is more closely related to the flow of information/activity, which to a certain extent must relate to electrochemical energy but no formal analysis has been performed.   It was chosen as a simple representation of neural activity and will further be referred to only as Energy.

An action potential is considered to transmit a given amount of energy in its signal.   The signals arriving at the soma are summed over the different dendrites and over time, and thus the soma acts as an energy tank.   The energy stored at the soma is subject to decay in a manner that resembles passive ion-channel pumps or leak currents, but if the amount of energy at the soma exceeds a given threshold then the neuron fires an action potential and energy is transmitted through its axon to the connected dendrites and the tank is drained.

summarizes the behavior of a neuron where E refers to the energy stored at the soma, d to the incoming energy from dendrites and decay to a positive real number less than 1.   It is important to mention that this initial model does not account for connection strengths or weights.  Instead the energy transmitted within an action potential is arbitrarily chosen at some level bellow threshold to prevent receiving neurons from firing after only one reception.

Implementation of a more realistic neural model is pending.

The Neuron class has two subclasses: axon and dendrites.  A neuron has only one axon but can have many dendrites.    Both axons and dendrites are considered as neural processes and obey the same rules regarding growth.

The growth of connections attempts to recreate biological neurons and thus axonal and dendritic neural processes are grown.   Since the long-term goal is to implement the computational model in silicon where connections cannot be grown at will after manufacturing, neurons and supporting elements are physically connected through a predetermined routing architecture and the growth of connections refers to virtual connections.    Also, the first biological guidance mechanism, adhesion molecules, is not relevant due to the lack of surrounding tissue.   However concentration gradients with attractive and repulsive influences are essential.   

Instead of creating a layer of medium to diffuse guidance molecules, attraction/repulsion of neural processes is estimated as a force proportional to the activity and inversely proportional to the square of the distance between processes.    Although at first glance this might seem inconsistent with biology it is a simple way of approximating a gradient with inverse square concentration decay as well as provide a very simple and computationally inexpensive method to calculate a force vector given by:

where Aj is the activity of the emitting process and Ai of the receiving process.  Ai is considered at a constant level of 1, otherwise it would be necessary to include another growth mechanism since at the beginning there would be no activity at any neuron and thus no growth.   Xs and Ys refer to the present location of the neurons active processes.   Moreover, activity has so far been considered an all or none binary quantity directly related to action potentials, in other words, when a given neuron fires it not only transmits an action potential to the connected neurons but also transmits a signal to every other neuron indicating the location of the firing process's growth cone.   Further refinement of the neural model will make possible to estimate activity (A) from Ca++ concentrations and transmitter release.   It is important to note that transmitting processes are not limited to axons and receivers to dendrites.  Dendrites can also become active and attract axons.   Retrograde messengers will explain this in below.

Growth is slow so there is no need to introduce dynamics, the velocity of a growth cone can be ignored and so there is no need to calculate its previous trajectory or inertia.   A growth vector (G) for a given process can then be directly proportional to the force vector scaled by a growth factor

G = F * growth factor

which is simply added to the receiving process's position vector to simulate growth.   The growth factor for dendrites is a positive number and a negative number is used for axons.

For this experiment two types of guidance molecules were used and have so far been represented by activity (A) previously discussed.   One acts as an attractor for dendrites and repellant for axons, the other one acts conversely.   They both originate at I/O elements simulating either receptors or muscles and can propagate in the form of activity throughout the forming network.   The first one is referred to as positive and the latter as negative to establish an analogy between the energy/information being contributed to the system from receptors and drained from effectors.    These positive and negative molecules combined with the signed growth factor determine attraction/repulsion of axons and dendrites.

As a dendrite approaches an active axon or vice versa the distance between them becomes very small and thus the growth vector can become larger than the distance separating both processes and cause overgrowth.   To prevent this a simple rule comparing the magnitude of the distance and growth vectors checks for overgrowth and if such occurs it makes a connection instead of adding both vectors.  

To establish a connection the receptive process sends a signal to the transmitting process.   Both neurons add each others IDs to their connection list.  In the case of axons it simply means that the next time it is exited it will send an action potential to the newly connected neuron. Its growth will continue undisturbed.   In the case of dendrites, in order to implement a simple mechanism to simulate branching, whenever a dendrite is connected its growth factor becomes cero and thus stops growing.    However a new dendrite is created and its position vector is set to that of its parent neuron.

An interesting outcome of initial experiments revealed the need to implement retrograde messenger mechanisms.   Sensory neurons (neurons that connect directly to inputs) have no problem growing their dendrites towards exited input elements. Once they were connected these neurons were exited by the inputs and propagated information efficiently when firing action potentials, which in turn attracted neighboring neuron's dendrites to connect to their axons and further propagate the network forming process.   On the motor neuron side (neurons that connect directly to outputs) axons would be attracted and form connections to active outputs.   However, lacking an activity-propagation mechanism from outputs to inputs, the network forming process would not continue to propagate and thus motor neurons had to wait until neighboring neurons formed connections (directly or indirectly) to sensory neurons to extend their dendrites and close the input/output pathway.

An intuitive solution was to implement a retrograde-messenger mechanism identical to the neural model used to process energy but in opposite direction.  In other words, another information channel was created where neurons receive inputs through the axon, integrate them and transmit action potentials through their dendrites.   In consequence this provides a mechanism where motor neuron dendrites can attract neighboring cell axons and facilitate the formation of the input/output pathway.

Neuroscientists suspected the existence of some type of retrograde messaging in biological neurons but its functions and mechanisms are not yet fully understood [4].  

Once the model was properly implemented it was presented with five types of stimuli in order to analyze its behavior to increasingly complex situations.    Within the figures presented ahead axons are drawn in red and dendrites in green.   Neurons and I/O elements are color filled to indicate their status.  A red colored neuron denotes a high value of positive energy while a blue one denotes a high value of negative energy.   The brighter the color is, the nearer a neuron is to firing.     Routing elements and physical connections discussed previously are not shown for clarity.

This type of stimulation consisted on setting a single Input element to a constant value.   One trial was performed with a positive value and another with a negative value.   The results, after one thousand iterations with a moderate growth factor, are shown in the figure.  

The positive value trial shows how as the network filled up with positive energy axons were repelled away from the network.  The negative value trial shows dendrites being repelled by the accumulated negative energy but a great concentration of axons at the center, this is due to the increasingly attractive negative energy and to the fact that axons continue to grow no matter how many connections they have established.   In both cases proper and predictable growth occurred.

All Input Constant Stimulation

The next type of stimuli presented to the network consisted of setting all Input elements to a constant and equal value.   One trial was performed with a positive value and another with a negative value.   Results shown, after one thousand iterations with a moderate growth factor, are similar to the previous trial except these trials show a higher degree of symmetry as expected. 

All Input Random Stimulation

This experiment attempted to recreate the stimulation of the network with sensorimotor information in a rudimental but similar manner to a nervous system.  Here, all inputs where randomly chosen to become either positive or negative and their values randomly generated each iteration.    The figure shows a trial with a 4 to 1 relation of positive to negative inputs after one thousand and two thousand iterations.  

Remarkably, even though the system's stimulation was predominantly positive, not as many axons escaped as with the previous positive constant value experiment, the small amount of negative stimulation provided enough attraction to group axons toward the center.   Moreover, the formation of an elliptical cluster populated mainly by axons and surrounded by a stronger population of dendrites suggests a structural differentiation similar to gray and white matter in biological nervous systems!

Opposite Constant Stimulation

This is an exploration to see how an information pathway from inputs to outputs is formed.    Opposing inputs where stimulated with constant values of opposite sign.  The figures show a sequence taken at one and two thousand iterations.

Such a simple task as connecting an input to an output would only require a single neuron to extend a dendrite to the input and its axon to the output.   However, since all neuron's processes are attracted to the active inputs, competition arises.   Additionally, once a neuron establishes a connection to an input it becomes a greater focus of attraction than the original input so a communication chain begins to form.    As interneurons connect to sensory or motor neurons these become slightly active but the reception of a single action potential is not enough to excite them so they need to grow additional branches to neighboring neurons before becoming exited.  

Pattern Stimulation & Learning

In this experiment alternating opposing patterns stimulates the network.  To simplify discussion the four active inputs are labeled: left = A, right = B, bottom = C and top = D.  First an opposing stimulus, like the previous experiment, is presented at inputs A and B for 600 iterations.   Then inputs are not stimulated for 200 iterations to allow accumulated energy to dissipate and decay.   Now another opposing stimulus is presented from inputs C and D for 600 iterations, and finally another rest period of 200 before repeating the sequence.

The goal was to determine if input A would form a pathway to either B or D and if input C would connect to B or D.   It was also important to determine if the formed pathways were independent of each other or if they âcrossed wires'.   Ideally since inputs A and B where active at simultaneously they would form a pathway and so would inputs C and D. 

Two trials with different growth rates were performed.   The first trial, with the smaller growth rate, established pathways among the inputs that did not fire together.   Although this result was not expected an interesting division, which to some extent resembles cerebral hemisphere separation, emerged diagonally.   The figure shows a sequence taken at two, four and six thousand iterations.

The second trial, with a slightly increased growth rate, initially displays a similar diagonal division as before but soon develops into a seemingly ineffective tangle.   However an output plot presented ahead reveals that this intricate network was able to establish suitable connections and properly separate both information pathways.   The figure shows a sequence taken at two, four and six thousand iterations.

This is an output plot obtained near the sixth thousand iteration and clearly shows that IOs A and B established a clean pathway between each other.   IOs C and D also established an appropriate pathway but propagated a very small amount of noise to IOs A and B.  In conclusion the network seemed to learn this simple task with relatively good accuracy.

Even though so far this model is very rudimentary and only sensitive to neural activity, the preliminary findings reveal a promising architecture.    The experiments performed have been extremely helpful in visualizing neural development and specially in understanding how the formation of a new connection can completely alter the growth dynamics of the network.

Regardless of the intuitive implementation of retrograde messengers without solid scientific evidence of their mechanisms, unexpected results like the emergence of gray and white matter and the formation of hemispheres indicates that there is some degree of correspondence with biological systems.    The last experiment demonstrates the capacity of the network to learn very simple patterns even though there are no implicit learning mechanisms or even weights to be adjusted.    It is expected that further research regarding retrograde messaging will provide for a better implementation of this mechanism or its substitution by more adequate theories.

It is also important to note that most experiments performed reached a stage of overgrowth where the network became overexcited and useless.   Proper mechanisms to regulate growth factors will be essential to provide fast learning during an initial development stage or childhood and the continuing ability to keep learning during adulthood.  

Future work contemplates the improvement of the neural model to provide plasticity and inhibition at this stage as well as the addition of learning mechanisms capable of adjusting connection strength and trimming of unnecessary connections.   The future neural model will attempt to capture the virtues of Ion-channel and neurotransmitter release mechanisms present in biological neurons without overcomplicating computation.   There are no foreseen complications in the implementation of synapses to the current model as well as local and global learning mechanisms.  Hebbian and some type of reinforcement learning are so far the strongest candidates.

The most exciting work ahead will be the encoding of the network properties into a genome and the inclusion of the network into an artificial agent to provide for learning through evolution and experience, hopefully even the use of the model to control an embodied robot.

[1] Ayers J., Davis J.L. and Rudolph A., Editors, Neurotechnology for Biomimetic Robots, 2002, MIT Press.

[2] Boahen Kwabena, http://www.neuroengineering.upenn.edu/boahen/

[3] Brooks R.A., Flesh and Machines: How Robots Will Change Us, February 12, 2002, Pantheon Books.

[4] Irwin B. Levitan, Leonard K. Kaczmarek , The Neuron: Cell and Molecular Biology, December 15, 2001, Oxford University Press.

[5] Figure  10-13 Reprinted from Nolte J: THE HUMAN BRAIN: AN INTRODUCTION TO ITS FUNCTIONAL ANATOMY, 5th/ed, Copyright © 2002, Mosby, Inc., with permission from Elsevier Science.

[6] Nolte J: THE HUMAN BRAIN: AN INTRODUCTION TO ITS FUNCTIONAL ANATOMY, 5th/ed, January 2002, Mosby, Inc.

[7] Randall C. O'Reilly, Yuko Munakata, James L. McClellar, Computational Explorations in Cognitive Neuroscience : Understanding the Mind by Simulating the Brain, 1st Edition , September 4, 2000, MIT Press.

[8] Tree of Life Web Project, http://tolweb.org/tree/phylogeny.html