Abstract
Artificial neural networks are a trending topic in computer science and machine
learning. Rooted in today’s current understanding of how the brain works, artificial
neural networks allow machines the capability to essentially learn and decide for
themselves. Improvements in technology have provided the means for a proposition
between finite automata and neural activity to be realized. Extensive research has been
conducted to understand the true potential of this realization, and the results of such
research show great promise. To better understand artificial neural networks, their
fundamental properties were explored and applied to an existing solution to a problem on
OpenAI’s website called CartPole-v0.

Table of Contents
Understanding the Nervous System………………………………………………………1

Foundation of Artificial Neural Networks………………………………………………...4

Teaching with Mathematics…………………………………………………………….....8

OpenAI and “Cartpole-v0”………………………………………………………………..9

“Cartpole-v0” Solution in Python………………………………………………………..11

Solution Code and Conclusion…………………………………………………………...13

Code Appendix…………………………………………………………………………..14

References……………………………………………………………………………….19

Law 1

Understanding the Nervous System
Inside our head is the most complex system known to mankind. Suspended
comfortably inside our skull, our brain is currently processing vast amounts of
information from various internal and external sources to keep us conscious. It defines
who each of us is as a person and what we are as a species. It controls everything about
us and is responsible for what has been done, what will be done, and what will continue
to be done as long as humans are around. A mass of tissue the size of approximately two
fists, weighing anywhere between two and a half to three pounds is everything.
The brain is an instrumental component of the nervous system. The human
nervous system is an incredibly elaborate collection of nerves stemming from our brain
that expands to every part of the body [1]. The central nervous systems primary function
is to send signals from one part of the body to another and receive some sort of feedback.
The way the nervous system does this is through special cells known as neurons. Neurons
are unique from other cells in a variety of ways, but the most noteworthy difference is the
ability to communicate through a space called a synapse. The synapse is a structure that
allows neurons to pass signals amongst each other via electrical currents or with
chemicals called neurotransmitters [1]. The ability of these cells to communicate using
electrical signals was a momentous discovery made by observing the structure of nerves
underneath a microscope.
Figure 1 is a diagram illustrating the key features of a standard neuron. Emanating
from the cell body the dendrites are seen; long finger-like structures that branch out
creating dendritic trees. This dendritic tree is responsible for receiving signals sent from

Law 2

the axons of other neurons and determining whether the neuron should send a signal
down its own axon.
Figure 1. Standard Neuron

Neurons are very efficient at transferring signals and processing feedback. Every
neuron comes equipped with a membrane that maintains a voltage gradient. This charged
membrane possessed by the neuron is electrically excitable and thus is capable of being
influenced by action potential.
A neuron is analogous to a battery. By itself, a battery is nothing more than a
collection of chemicals holding stored energy just waiting to be used. In the region
surrounding the neuron, positive sodium ions are present while positive potassium ions
are enclosed inside the neuron. The presence of more sodium ions outside the neuron
than potassium ions inside the neuron means the net charge of the neuron is negative. A
neuron in this state is known as polarized [2]. Figure 2 shows a polarized neuron that is
electrically charged by proteins called the sodium-potassium pumps. They ride the
membrane of the neuron pumping an unequal amount of sodium and potassium ions to
and from the neuron. This uneven distribution of ions creates the electrical gradient. To

Law 3

even out this gradient, ions must be able to pass from the

Figure 2. Sodium-Potassium Pumps

environment into the neuron through channels in the
membrane. These channels open from various stimuli
depending on the primary function of the neuron. To
send a significant signal down an axon, there must be a
powerful trigger that opens the numerous voltage-gated
channels. If the trigger crosses a certain threshold, then
the action potential is realized, and ions flood the neuron rapidly depolarizing it. That
local change in current travels down the axon to some designated neuron that will
respond accordingly [2]. The neuron will eventually return to a polarized state via the
sodium-potassium pumps and the cycle can be repeated.
Studies of the human brain have estimated that there are over 86 billion neurons
present [3]. The amount of interconnectivity amongst neurons in the typical human
creates a network of unimaginable complexity. This network of communication is where
the true power of the neuron lies. Between each junction of axon and dendrite among
connected neurons exists an ever so minute gap known as a synapse. Anywhere between
20 to 40 nanometers across, the synapse is a fundamental element of neurotransmission
[3]. Communication is achieved when a presynaptic neuron passes a signal across the
synapse to a postsynaptic neuron. Chemical synapses use neurotransmitters located in the
presynaptic membrane of the neuron to bind to receptors located on the membrane of the
postsynaptic neuron. Chemical synapses have a variety of classifications depending on
the type of neurons at play. Neurotransmitters themselves are vastly complex and their

Law 4

effects on the post synaptic neuron are intricate. Chemical synapses are responsible for a
major portion of activity in a typical biological neural network.
Electrical synapses behave in a more straightforward manner. Thus, these
electrical synapses are inherently simpler than chemical synapses. The exclusion of
neurotransmitters from the communication process means that electrical synapses are less
varied and more resistant to external influences. Electrical synapses transmit signals
almost instantly, making them perfect for scenarios in which a rapid response is required.
Transmission speeds are so fast that neurons can even fire synchronously. Electrical
synapse responses occur quickly, are bidirectional if need be, and produce simple
behavior [4].
The rapid development of technology has unearthed a striking link between neural
behavior and the computational theory of finite automata. Could a machine be able to
model neurological activity? To answer that question, artificial neural networks were
conceptualized by Walter Pitts and Warren McCulloch in 1943.
Foundations of Artificial Neural Networks
The evolution of artificial neural networks traces its roots to a paper written by
Warren McCulloch and Walter Pitts. “A Logical Calculus of the Ideas Immanent in
Nervous Activity” laid the foundations from which neural networks were built.
McCulloch and Pitts demonstrated that “neural events and the relations among them can
be treated by means of propositional logic [5].” They found that the behavior of every net
without circles can be defined in their logical calculus, but certain key assumptions

Law 5

needed to be made. These are the following assumptions made on neural nets that do not
cycle (without circles):
1. The activity of the neuron is an “all-or-none” process.
2. A certain fixed number of synapses must be excited within the period of latent
addition in order to excite a neuron at any time, and this number is independent of
previous activity and position on the neuron.
3. The only significant delay within the nervous system is synaptic delay.
4. The activity of any inhibitory synapse absolutely prevents excitation of the neuron
at that time.
5. The structure of the net does not change with time.
To summarize the following assumptions, artificial neural nets are made up of an
arbitrary amount of nodes (neurons) whose network structure is immutable. The only
transmission delay between nodes is synaptic, which is the relative distance between
nodes. The activity of an individual node is all or nothing. A node that has been triggered
by the network either transmits or prohibits subsequent signals. There are no partial
transmissions or partial blockages; all activity after excitation is absolute.From these
cardinal assumptions, Pitts and McCulloch used extensive calculus to translate neural
activity into some relatively straightforward mathematics. Many papers have been

Figure 3. Artificial Neuron

Law 6

published and much research has been done regarding different network models,
mathematical properties, and neural anomalies since the Pitts-McCulloch paper.
However, the interest of this paper lies solely in understanding and applying a basic
artificial neural network with only the essential mathematics needed to solve a problem.
Figure 3 illustrates the best way to understand the model and the mathematics used in an
artificial neural network.
Figure 3 is the depiction of a standard node/neuron in an artificial neural network.
Neurons begin with an arbitrary number of inputs represented by x1 through xn. The origin
of the inputs varies depending on the location of the neuron within the network. If the
neuron is part of the input layer, or the beginning layer of the network, the input is
typically provided by a user of the network. If the neuron is not part of the input layer, the
inputs are provided or inhibited by neurons in a prior layer. The values of these inputs are
then multiplied by a weight represented by w1 through wn in Figure 3. Weights will be
discussed in further detail later. After all the inputs have been multiplied by their
respective weights, the determined values of all inputs will then be summed (Summer in
Figure 3). To this sum, a bias will also be added. Like weights, a bias will be discussed
later. The first assumption established earlier was that the activity of a neuron is all or
nothing. When all the input values are determined and summed along with the bias, that
subsequent value could be any possible number. To realize this assumption, the summed
value will be passed through an activation function labeled Threshold unit in Figure 3.
This activation function will determine if the neuron fires or not.

Law 7

A typical activation function is a sigmoid function. The sigmoid function is
bounded, differentiable, and is defined for all real input values. Expressed as

1
1+𝑒 −𝑥

the sigmoid function will take the value of the neuron’s sum, x, and produce a number
between 0 and 1. This function works extremely well as an activation function since no
matter what real value the summation produces, once it’s passed through the sigmoid
function, only a value between 0 and 1 will be produced. This translates accordingly to
the neuron firing or not.
Nodes of an artificial neural network are then grouped into layers. These layers
are known as the input layer, one or more hidden layers, and an output layer. The nodes
in each layer are connected to each node in a subsequent layer creating a network such as
Figure 4.

Figure 4. Artificial Neural Network

Law 8

Teaching with Mathematics
Perhaps the most charming aspect of artificial neural networks is their capability
to “learn”. However, the term “learn” is rather misleading. With the increased amount of
automation in today’s workforce, giving machines the ability to learn can be a concerning
thought. Fortunately for us, training an artificial neural network is more akin to teaching a
machine to produce correct results rather than it consciously learning on its own. This is
accomplished by some extremely clever calculus and linear algebra.
Teaching an artificial neural network is fundamentally a challenge in
optimization. The network starts with input data and some randomly generated weights
and biases. These random weights and biases will at first produce results that are
unpredictable and largely incorrect. To tweak these weights and biases, a cost function is
defined. The purpose of the cost function is to measure how well the network is doing
with the current weights and biases. The cost function will take the current weights and
biases as its input and produce a single output value. This single output value is an
indication of how well the network is performing. When this output is large, the current
values of the weights and biases are causing the network to perform poorly. Lowering the
output value of the cost function is done by minimizing the cost function. By minimizing
the cost function, an artificial neural network learns.
Minimizing the cost function is accomplished by gradient descent. If a multivariable function F(x), in our case the cost function, is differentiable at some point i.e. the
random weights and biases defined, then F(x) will decrease fastest in the direction of the
negative gradient of F [6]. What is essentially transpiring is the slope of the cost function
is determined at some instance and the goal is to find out what direction to travel to get

Law 9

that slope as flat as possible. Given the current output of the cost function, adjustments
are made to the weights and biases and another output is produced. That output is then
analyzed, the weights and biases are adjusted further, and the direction to travel is
determined again. Repeating this process hundreds, thousands, even millions of times
refines the weights and biases to optimal values; thus, an artificial neural network is
trained.
Another important topic is how the gradient of the cost function is computed. This
is known as backpropagation. In 1986, “Learning representations by back-propagating
errors” by David Rumelhart, Geoffrey Hinton, and Ronald Williams produced a learning
procedure coined backpropagation that illustrated how quickly the output of the cost
function changes with respect to altering the weights and biases of a network [7].
Alterations to weights and biases have profound effects on the network that ultimately
ripple out and change aspects of the network as a whole. A minor adjustment in one area
will indefinitely alter other areas. The underlying calculus behind it is well beyond the
scope of this paper, but it deals with the partial derivative of the cost function with
respect to the weights and biases. The important aspect to take away from
backpropagation is that certain weights and biases are more influential on the network
than others.
OpenAI and “CartPole-v0”
In 2015, Elon Musk, the CEO of SpaceX, along with a handful of other investors
pledged over $1 billion in funds towards the development of artificial intelligence (AI).
This manifested itself into a non-profit organization known as OpenAI. Musk believed at
the time that AI was humanity’s greatest threat, so the goal of OpenAI was simple;

Law 10

provide a free, community-based sanctuary where AI could be developed and researched
with a focus on “positive human impact” [8]. He has since left the organization due to
possible conflicts of interest, but OpenAI is still thriving. In 2016, OpenAI released its
first platform for machine learning called “OpenAI Gym”.
Gym is essentially a toolkit that provides users with problems called
environments. Users can then freely test and develop learning algorithms on a plethora of
environments. These environments make no assumptions about algorithms or model
structures, share a common interface allowing for general algorithms to be ported
between environments, and are compatible with any numerical computation library [9].
The environments vary in complexity, spanning from simple locomotion simulations to
full on Atari emulations like Ms. Pacman and Pitfall. All the environments have a goal
which serves as a purpose for training models. Users are then able to freely post their
solution to the OpenAI website to show off how effective their training methods were.
The environment chosen for this paper is known as “CartPole-v0”. The problem
was derived from a paper titled “Neuronlike Adaptive Elements That Can Solve Difficult
Learning Control Problems” written by researchers of the Institute of Electrical and
Electronics Engineers (IEEE) [10]. It begins with a pole attached to a cart which moves
along a frictionless path. The pole begins in the upright position and is bound to the cart
by an un-actuated joint i.e. it won’t move unless a force is acted upon it. The goal is
simple; prevent the pole from falling over. If the pole is more than 15º from vertical or
the cart has moved more than 2.4 units from the center, then the game is over.

Law 11

“Cartpole-v0” Solution in Python
Harrison Kinsley, who will be referred to from now on by his online pseudonym
Sentdex, is a self-taught programmer and entrepreneur that has created numerous
websites and YouTube videos dedicated to educating others on a variety of topics
regarding the popular programming language Python. Sentdex’s resources on the topic of
artificial neural networks have been invaluable throughout this research process and
combing through his code was a delight. His code served as a foundation from which this
solution was built, and the many hours spent watching his videos and reading his website
has been a prime inspiration for this solution. The code is separated into three distinct
methods; one for collecting training data, one for defining the neural network model, and
one for training the model. All of this is made possible by three Python libraries; gym,
numpy, and tflearn [11].
The first method training_games() creates a set amount of games, plays the
games with random action input, and stores all the necessary data collected from each
game session. This method utilizes gym to create the “CartPole-v0” environment and
manage the data collection. A score_requirement is created so that information is
only collected from games with optimal scores. This is necessary to ensure the neural
network is being trained with the best possible data collected. Another important note
about this method is that it converts the collected game data into what is known as a onehot format. The cart in this environment can only move left or right. If it chooses to move
right, then that choice is ‘hot’. It’s very similar to binary in which a one is ‘hot’ and a
zero is ‘cold’. The data is encoded in this manner so that it can be easily translated by the
neural network later.

Law 12

The next method, neural_network_model, takes one parameter,
input_size, which is used to create the input layer for the neural network. tflearn
utilizes two methods, fully_connected and dropout to define and create the
entire network instead of manually creating this data structure from scratch. Each line of
fully_connected is passed the size (number of nodes in the layer) and the type of
activation function used. dropout is passed the object which invokes
fully_connected and then the keep probability. The keep probability is a threshold
which each single node must cross to activate. The final fully_connected
establishes the output layer and a part of the cost function. All of this is then passed to a
DNN method which wraps up creation of the network and stores it in an object called
model.
The final method train_model is admittedly convoluted, but it uses the
numpy library to perform matrix manipulation on the training data and essentially
performs reinforcement training. With two lists, one X, one Y, train_model acts like
a cost function by comparing the two lists, one being what was observed, the other being
what should have been observed, and performing alterations to the weights and biases
based on the cost of the differences.
By calling all three of these methods, the network is built, provided data, and
subsequently trained. All that is left is to see how well it performs by invoking the
environment again and allowing the network to predict what actions to take. “CartPolev0” is considered solved if the average score is over 195 across 100 trials. With samples

Law 13

from 100,000 training games and a score requirement of 100, this program averages a
score of 200 and thus is considered solved.
Solution Code and Conclusion
Now that technology has finally caught up to theory, artificial neural networks are
becoming a fascinating paradigm to follow. A team at OpenAI created a bot for the
popular game Dota 2 that trained entirely against itself for roughly three months. It was
then released upon professional players in a one-on-one scenario and crushed each and
every one of them [12]. A bot had developed greater skill in three months than these
players had developed over the course of years. MNIST (Modified National Institute of
Standards and Technology) created a substantial database of pixel data from which
networks have trained from in order to recognize handwritten digits. There are even
reports of networks being able to steer vehicles, recognize faces, and even diagnose
certain cancers based on cell shape information.
In a world where information and data are everything, artificial neural networks
will indefinitely find a niche in future computing. Although artificial neural networks
have been criticized for the inability to solve computationally difficult problems and the
need for massive amounts of data/computing power to train, they should not be
underestimated or discarded. It appears artificial neural networks are simply a piece of
the much larger puzzle of technological enlightenment. OpenAI’s Dota 2 bot has shown
that artificial neural networks can easily surpass the capabilities of a human. The future
of artificial neural networks is as mystifying as it is potentially horrifying, but it certainly
is incredible what technology is capable of.

Law 14

Code Appendix
import gym
#Toolkit for learning algorithms
https://github.com/openai/gym
import random
#Pseudo-random number module
https://docs.python.org/3.4/library/random.html
import numpy as np
#Package for scientific computing
http://www.numpy.org/
import tflearn
#Deep learning library from Tensorflow
https://github.com/tflearn/tflearn
from
from
from
from

tflearn.layers.core import input_data, dropout, fully_connected
tflearn.layers.estimator import regression
statistics import mean, median
collections import Counter

#Learning rate of neural network
LR = 1e-3
#Define our environment for the network, initialize it
environment = gym.make('CartPole-v0')
environment.reset()
#Timesteps for environment
time_steps = 500
#Keep the data from games with this score or higher
score_requirement = 70
#Number of initial games played for training data
initial_games = 1000
#Target number of timesteps
goal_steps = 500
#Play some games with random behavior for training data
def training_games():
#[Observations, Moves]
training_data = []
#Every score achieved in training games
scores = []
#Scores that met our requirements
accepted_scores = []
#Begin simulating training games
for _ in range(initial_games):
score = 0
#[Cart Position, Cart Velocity, Pole Angle, Pole Velocity at
Tip]
game_memory = []

Law 15
#List containing each value of the previous observation
prev_observation = []
#environment.render()
#For each time step, do a random action
for _ in range(time_steps):
#Actions are 0 (push left) or 1 (push right)
action = environment.action_space.sample()
'''
Step returns 4 values: observation(object), reward(float)
done(boolean), and info(dictionary). This is the agentenvironment
loop. Each time step, the agent chooses and action and the
environment returns an observation and a reward
'''
observation, reward, done, info = environment.step(action)
#If our previous observation was successful
if len(prev_observation) > 0:
#Append to game_memory the previous observation
#and the action taken by the agent
game_memory.append([prev_observation, action])
prev_observation = observation
score+=reward
#If done returns true, the game has finished
if done:
break
#If a score reached our score requirement
if score >= score_requirement:
#Add it to the list of accepted scores
accepted_scores.append(score)
#Convert game_memory to one-hot format for output layer
for data in game_memory:
if data[1] == 1:
#Action agent took (right)
output = [0,1]
elif data[1] == 0:
#Action agent took (left)
output = [1,0]
#Store data from training games into training_data list
training_data.append([data[0], output])

Law 16

environment.reset()
scores.append(score)
'''
Use to view stats about training data
print('Highest score:', max(scores))
print('Average accepted score:', mean(accepted_scores))
print('Median score for accepted scores:', median(accepted_scores))
print(Counter(accepted_scores))
'''
return training_data
training_games()
#Create model for neural network
def neural_network_model(input_size):
network = input_data(shape=[None, input_size, 1], name='input')
network = fully_connected(network, 64, activation='relu')
network = dropout(network, 0.8)
network = fully_connected(network, 128, activation='relu')
network = dropout(network, 0.8)
network = fully_connected(network, 256, activation='relu')
network = dropout(network, 0.8)
network = fully_connected(network, 128, activation='relu')
network = dropout(network, 0.8)
network = fully_connected(network, 64, activation='relu')
network = dropout(network, 0.8)
network = fully_connected(network, 2, activation='softmax')
network = regression(network, optimizer='adam', learning_rate=LR,
loss='categorical_crossentropy', name='targets')
model = tflearn.DNN(network, tensorboard_dir='log')
return model

def train_model(training_data, model=False):
#Reshaping training data with numpy (not really sure)
X = np.array([i[0] for i in training_data]).reshape(1,len(training_data[0][0]),1)
#Filling Y with target data
Y = [i[1] for i in training_data]
if not model:
model = neural_network_model(input_size = len(X[0]))

Law 17

'''
This model.fit passes 4 parameters:
1: X input data as a dictionary
2: Y target data to train model
3: Number of epochs (backpropagation after each epoch)
4: Displays accuracy at every step
'''
model.fit({'input': X}, {'targets': Y}, n_epoch=7,
show_metric=True)
return model
#Perform training_games and subsequent training
training_data = training_games()
model = train_model(training_data)
#Lists for new scores and choices
#made by the model
scores = []
choices = []
#Functions similarly to training_games
for each_game in range(100):
score = 0
game_memory = []
prev_obs = []
environment.reset()
for _ in range(goal_steps):
#environment.render()
#Start game off with random action
if len(prev_obs) == 0:
action = random.randrange(0,2)
#Otherwise use model's prediction
else:
action = np.argmax(model.predict(prev_obs.reshape(1,len(prev_obs),1))[0])
#Save action taken by model
choices.append(action)
new_observation, reward, done, info = environment.step(action)
prev_obs = new_observation
game_memory.append([new_observation, action])
score+=reward
if done: break
scores.append(score)

Law 18

#Useful stats for post training
print('Average Score:',sum(scores)/len(scores))
print('choice 1:{} choice
0:{}'.format(choices.count(1)/len(choices),choices.count(0)/len(choices
)))

Law 19

References
[1] Kandel ER, Schwartz JH, Jessel TM, eds. (2000). "Ch. 2: Nerve cells and
behavior". Principles of Neural Science. McGraw-Hill Professional. ISBN 978-08385-7701-1.
[2] Green, Hank. (2015, March 2). Crash Course the Nervous System. The Nervous

System, Part 2 – Action! Potential!: Crash Course A&P #9. Retrieved from
https://www.youtube.com/watch?v=OZG8M_ldA1M
[3] Sukel, Kayt. (2011, March 15). The Dana Foundation. The Synapse – A Primer.
Retrieved from http://www.dana.org/News/Details.aspx?id=43512
[4] Hormuzdi, Sheriar G. et al. (2004, March). BioChimica et Biophysica Acta – (BBA)
Biomembranes. Electrical synapses: a dynamic signaling system that shapes the
activity of neuronal networks. Retrieved from
https://www.sciencedirect.com/science/article/pii/S0005273604000410
[5] Pitts, Walter. McCulloch, Warren S. (1943). Society for Mathematical Biology. A
Logical Calculus Of The Ideas Immanent In Nervous Activity. Retrieved from
https://www.cs.cmu.edu/~./epxing/Class/10715/reading/McCulloch.and.Pitts.pdf
[6] 3Blue1Brown. (2017, October 16). Neural Networks. Gradient descent, how neural
networks learn | Chapter 2, deep learning. Retrieved from
https://www.youtube.com/watch?v=IHZwWFHWa-w&t=336s
[7] Nielson, Michael. (2017, December). Neural Networks and Deep Learning. Retrieved
from http://neuralnetworksanddeeplearning.com/index.html

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[8] British Broadcasting Company. (2015, December 12). British Broadcasting
Company. Tech giants pledge $1bn for ‘altruistic AI’ venture, OpenAI. Retrieved
from http://www.bbc.com/news/technology-35082344
[9] OpenAI. (2016). OpenAI. Getting Started with Gym. Retrieved from
https://gym.openai.com/docs/
[10] Barto, AG. Sutton, RS. Anderson, CW. (1983). IEEE Transactions on Systems,
Man, and Cybernetics. Neuronlike Adaptive Elements That Can Solve Difficult
Learning Control Problem. Retrieved from
https://gym.openai.com/envs/CartPole-v0/#barto83
[11] Kinsely, Harrison. (2017, March 13). Sentdex. Using a neural network to solve
OpenAI’s CartPole balancing environment. Retrieved from
https://pythonprogramming.net/openai-cartpole-neural-network-examplemachine-learning-tutorial/
[12] OpenAI. (2017, August 6). OpenAI. More on Dota 2. Retrieved from
https://blog.openai.com/more-on-dota-2/
FIGURE 1 https://askabiologist.asu.edu/neuron-anatomy
FIGURE 2 https://socratic.org/questions/what-is-the-role-of-potassium-in-muscle-contraction
FIGURE 3 https://medium.com/@xenonstack/overview-of-artificial-neural-networks-and-itsapplications-2525c1addff7
FIGURE 4 http://www.thewindowsclub.com/deep-learning-and-neural-network

Law 21

Seth Law
Major in Computer Science, Minor in Mathematics
Committee Members: Paul Sible, Weifeng Chen, Gregg Gould
Artificial Neural Networks, Machine Learning, Python, Neurons
Title: Exploring Artificial Neural Networks with “CartPole-v0”
Solution