Plexus is exceptionally bio-inspired, a revolutionary approach to the artificial neural networks
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Core Principles

These are the core principles of exceptionally bio-inspired, a revolutionary approach to the artificial neural networks:

  • Neurons must be objects not tensors between matrices.
  • Each neuron should have its own thread (ideally).
  • Network must be architecture-free (i.e. adaptive).
  • Network must have a layerless design.
  • There must be fundamentally two types of neurons: sensory neuron, interneuron.
  • Input of the network must be made of sensory neurons. Any interneuron can be picked as a motor neuron (an element of the output). There is literally no difference between an interneuron and a motor neuron except the intervene of the network for igniting the wick of learning process through the motor neurons. Any non-motor interneuron can be assumed as a cognitive neuron which collectively forms the cognition of network.
  • There can be arbitrary amount of I/O groups in a single network.
  • Forget about batch size, iteration, and epoch concepts, training examples must be fed on time basis with a manner like; learn first sample for ten seconds, OK done? then learn second sample for twenty seconds. By this approach, you can assign importance factors to your samples with maximum flexibility.
  • Network must be retrainable.
  • Network must be modular. In other words: You must be able to train a small network and then plug that network into a bigger network (we are talking about some kind of self-fusing here).
  • Neurons must exhibit the characteristics of cellular automata.
  • Number of neurons in the network can be increased or decreased (scalability).
  • There must be no need for a network-wide oscillation. Yet the execution of neurons should follow a path very similar to flow of electric current nevertheless it's not compulsory.
  • Network should use randomness and/or uncertainty principle flawlessly.
  • Most importantly, the network must and can not iterate through the whole dataset. Besides that, it's also generally impossible iterate the dataset on real life situations if the system is continuous like in robotics. Because of that the network must be designed to handle such a continuous data stream that literally endless and must be designed to handle that data stream chunk by chunk. Therefore, when you are feeding the network you should follow a path like; on my left hand there is an apple and on my right hand there is a banana, apple... banana... apple, banana, apple, banana, ... if you want to teach the difference between apple and banana and don't follow a path like; apple, apple, apple, apple, banana, banana, banana, banana, ... More technically; if you have two classes to train the network, use a diverse feed but not grouped feed (ideally repetitive like 121212121212 but not 1111111222222).

Activation function

( c : average connectivity of a neuron )

Activation function

Draw the graph of this equation to see how it's insanely suitable to make random normalizations for any values between (-∞,+∞). It's also becoming more crazy, less stable when you move away from 0. In other words, it's not completely repetitive so a neuron freely walk on this equation without being trapped.

Plexus Network is an algorithm that implemented by following the criteria listed in Core Principles and by relying on the given activation function above. Rest of the paper will explain the algorithm's itself and usage.

Implementation of this algorithm in Python programming language is publicly accessible through this link:

Reading this paper requires basic knowledge of Computer Science and Neuroscience.

You can directly skip to Application part if you are not willing to understand the mathematical and algorithmic background.


Plexus Network has only two classes; Network and Neuron. In a Plexus Network, there are many instances of Neuron class but there is only one instance of Network class.

When you crate a new Plexus Network you give these five parameters to the Network class: size of the network, input dimension, output dimension, connectivity rate, precision. The network accordingly builds itself.

Network init function

  • size is literally equal to total number of neurons in the network. All neurons are referenced in an instance variable called Network.neurons
  • input dimension specifies the number of sensory neurons. Sensory neurons are randomly selected from neurons.
  • output dimension specifies the number of motor neurons. Motor neurons are randomly selected from non-sensory neurons.
  • number of neurons multiplied by connectivity rate gives the average number of subscriptions made by a single neuron.


  • precision simply defines the precision of the all calculations will be made by neurons (how many digits after the decimal point).

After the network has been successfully created. It will ignite itself automatically. Ignition in simple terms, no matter if you have plugged in some data or not, it will fire the neurons with using some mechanism very similar to flow of electric current (will be explained later on this paper).

Anatomy of a Single Neuron

A single neuron in a Plexus Network, takes the network as the only parameter and stores these seven very important information (in it's instance variables): subscriptions, publications, potential, desired_potential, loss and type

Neuron init function

There are eventually there types of neurons:

  • Neuron.type = 1 means it's a sensory neuron.
  • Neuron.type = 2 means it's a motor neuron.
  • Neuron.type = 0 means it's neither a sensory nor a motor neuron. It means it's an cognitive interneuron (or just cognitive neuron).

Functionality of a neuron is relative to its type.

subscriptions is neuron's indirect data feed. Each non-sensory neuron subscribes to some other neurons of any type. For sensory neurons subscriptions are completely meaningless and empty by default because it gets its data feed from outside world by assignments of the network. It is literally the Plexus Network equivalent of Dendrites in biological neurons. subscriptions is a dictionary that holds Neuron(reference) as key and Weight as value.

publications holds literally the mirror data of subscriptions in the target neurons. In other words; any subscription creates also a publication reference in the target neuron. Similarly, publications is the Plexus Network equivalent of Axons in biological neurons.

potential is the overall total potential value of all subscriptions multiplied by the corresponding weights. Only in sensory neurons, it is directly assigned by the network. Value of potential may only be updated by the neuron's itself and its being calculated by this simple formula each time when the neuron is fired:

Total potential

Apply activation

desired_potential is the ideal value of the neuron's potential that is desired to eventually reach. For sensory neurons, it is meaningless. For motor neurons, it is assigned by the network. If it's None then neuron don't learn anything and just calculates potential when it's fired.

loss is calculated not just at the output but in every neuron except sensory ones and it is equal to absolute difference (distance) between desired potential and current potential.

Calculation of loss

All numerical values inside a neuron are floating point numbers and all the calculations obey to precision that given at start.

Sensory and Motor Neurons

Input Layer in classical neural networks renamed as Sensory Neurons in Plexus networks, and Target/Output Layer renamed as Motor Neurons. This naming convention is necessary cause the built of the relevance of artificial neural networks with biological neural networks and Neuroscience.

The difference of sensory neurons from the cognitive neurons (that neither sensory nor motor ones) is, they do not actually fire. They just stand still for the data load. They do not have any subscriptions to the other neurons (literally no subscriptions). But they can be subscribed by the other neurons, including motor ones. They do not learn, they do not consume any CPU resources. They just stored in the memory. You can assign an image, a frame of a video, or a chunk of an audio to a group of sensory neurons.

The difference of motor neurons form the other neurons is, they are only responsible to the network. They act as the fuse of the learning and calculation of the loss. The network dictates a desired potential on each motor neuron. The motor neuron calculates its potential, compares it with desired potential, calculates the loss then tries to update its weights randomly many times and if it fails, it blames its subscriptions. So just like the network, motor neurons are also able to dictate a desired potential on the other non-motor neurons. This is why any neuron holds an additional potential variable called desired_potential.

Partially Subscribe

On the second phase of the network initiation, any non-sensory neurons are forced to subscribe to some non-motor neurons which are selected by random sampling. Length of this sample is also selected by a random sampling (rounds to nearest integer) is done from a normal distribution. Such a normal distribution that, the average number of subscriptions is the mean, and square root of the mean is the standard deviation. (e.g. if neurons on average has 100 subscriptions then the mean is 100 and the standard deviation is 10)

Normal distribution


Even so the Python implementation of Plexus Network is easy to understand, it will be helpful for readers to explain the algorithm in pseudocode.


procedure initiate the network is
    connectivity ← size * connectivity_rate;
    connectivity_sqrt ← sqrt(connectivity);
    connectivity_sqrt_sqrt ← sqrt(connectivity_sqrt);
    for item in size, do
        create neuron;
    pick sensory neurons randomly;
    pick motor neurons randomly;
    determine non-sensory neurons;
    determine non-motor neurons;
    initiate subscriptions;
    initiate instance variables;
    ignite the network;

Initiation is nothing more than a make the assignments for once phase until the ignition. The final step (ignition) never stops but can be paused (if user wants).

Initiate Subscriptions

procedure initiate subscriptions is
    for neuron in neurons, do
        if neuron is not a sensory neuron, then
            call neuron.partially_subscribe();
    return True;

Partially Subscribe

procedure partially subscribe is
    sample ← randomly sample approximately "connectivity" units of a neuron from within all non-motor neurons;
    for neuron in sample, do
        if neuron is not self, then
            establish a subscription;    // weight is randomly assigned
            establish a publication;
    return True;

The time complexity of the procedure initiate subscriptions is O(n2), so this may take a while if the size of the network and connectivity is big.


procedure ignite subscriptions is
    create an empty ban_list;
    while network is not frozen, do
        if next_queue is empty, then
            get the output of network and print it;
            increase the wave_counter;
            if first_queue is empty, then
                for neuron in sensory neurons, do
                    for target_neuron in neuron.publications, do
                        append target_neuron to first_queue;
                copy first_queue to next_queue;
        copy next_queue to current_queue;
        empty next_queue;
        for neuron in ban_list, do
            if neuron.ban_counter > connectivity_sqrt_sqrt, then
                remove the neuron from current_queue;
        while current_queue is not empty, do
            neuron ← select a random neuron from current_queue;
            remove the neuron from current_queue;
            if neuron.ban_counter <= connectivity_sqrt_sqrt, then
                append the neuron to ban_list;
                increase neuron.ban_counter;
                for target_neuron in neuron.publications, do
                    append target_neuron to next_queue;

Procedure ignite regulates the firing order of neurons and creates an effect very similar to flow of electric current, network wide. It continuously runs until the network frozen, nothing else can stop it. It fires the neurons step by step through adding them to a queue.

It generates its first queue from the publications of sensory neurons. Time complexity of if next_queue is empty, then block is O(n2) but it can be ignored (unless there are too many sensory neurons) because it runs once per wave.

It eliminates banned neurons with for neuron in ban_list, do block. Function of ban_counter is giving neurons connectivity_sqrt_sqrt amount of chance after they added to ban_list. Then it fires the neurons inside current_queue, one by one, choosing them randomly.

After a neuron fired, it adds the fired neuron to ban_list and lastly copies the publications of that neuron to next_queue so execution(firing process) can follow the path through the connections.

Each execution from first sensory neuron to last motor neuron symbolizes one wave. Every time a wave finished, procedure falls into if next_queue is empty, then block so wave starts over from the sensory neurons.

ban_counter and connectivity_sqrt_sqrt comparison creates execution loops inside cognitive neurons and these loops act like memory units which is a pretty important concept. Because loops create the relation between currently fed data and previously learned data. Without these loops the network fails on both classification and regression problems.

Because has a time complexity of O(n2), each turn inside while network is not frozen, do block, has a time complexity of O(n4). But don't worry because it will approximate to O(n3) because of the probabilistic nature of fire function and the network will fire more than a million of neurons per minute. By the way, while network is not frozen, do block is ignored because it's an endless loop under normal conditions.


procedure fire is
    if self is not a sensory neuron, then
        potential ← calculate potential;
        increase fire counter;
        if desired_potential is not None, then

            loss ← calculate loss;
            if loss = 0, then
                desired_potential ← None;
                return True;
            if blame_lock is not empty, then
                if (wave_counter - blame_lock) < connectivity, then
                    return True;
                    blame_lock ← None;

            try connectivity times:
                generate new weights randomly;
                calculate new potential and new loss according to these weights;
                if loss_new < loss_current, then return True;

            try sqrt(connectivity) times:
                generate hypothetical potentials for neurons in subscriptions randomly;
                calculate new potential and new loss according to these hypothetical potentials;
                if loss_new < loss_current, then
                    apply these hypothetical potentials as "desired_potential"s;
                    return True;

            if (still) not improved, then
                either create some new subscriptions;
                or break some of the subscriptions;
                return True;


Procedure fire handles all feedforwarding, backpropagation and learning process by itself. fire function is an instance method of Neuron class. This procedure is by far the most important one in the Plexus Network. It's basically the core function and CPU spends most of its time to execute fire functions again and again.

If desired_potential is not assigned to a value, then it just calculates the potential and finishes.

If desired_potential is assigned to a value, then first it calculates the loss. If loss is equal to zero, then the current state of the neuron is perfectly well and there is nothing to learn.

If blame_lock is not empty, then it will pass this function connectivity times with this control statement: if blame_lock is not empty, then.

Neuron improved

It tries to improve the current state of the neuron by updating its weights randomly, connectivity times. If it's improved, then break.

It tries to improve the current state of the neuron by dictating randomly generated hypothetical potentials over the subscriptions, square root of connectivity times. If it's improved, then break.

If it still is not improved, then it either creates some new subscriptions or breaks some of the subscriptions it currently has and hopes it will lead the neuron to new improvements in the future.

On the first wave, the fire function is only meaningful for motor neurons but after the first wave desired_potential dictation will spread throughout the cognitive neurons.


procedure load (input, output) is
    if output is None, then
        for neuron in motor neurons, do
            neuron.desired_potential ← None;
    if (number of sensory neurons is not equal to input length), then
        raise an error but do not interrupt the network;
        for neuron in sensory neurons, do
            neuron.potential ← load from the input;
    if (number of motor neurons is not equal to output length), then
        raise an error but do not interrupt the network;
        for neuron in motor neurons, do
            neuron.desired_potential ← load from the output;

Procedure load is the only method that you can feed your data to the network. You should call that function and load your data in real time. Also you should do it periodically and continuously, like every 3 seconds. If you leave second parameter empty then this procedure will automatically assume that you are testing the network, so it will replace desired_potential values of motor neurons with None. Otherwise, it means you are training the network so it will load the input data to sensory neurons and it will load the output data to desired_potential values of motor neurons.


Installation of the Python Package

sudo pip install plexus

If you want to install Plexus on development mode:

git clone
cd Plexus/
pip install -e .


Basic Example

(you can alternatively run this example with python examples/ command using a pre-written script version of below commands)

Suppose you need to train the network to figure out that the elements of given arrays are bigger than 0.5 or not (like [0.9, 0.6, 1.0, 0.8] or [0.1, 0.3, 0.0, 0.4]) and suppose it's a 4-element array. So let's create a network according to your needs:

net = plexus.Network(256,4,2,0.5,1)

Because our network is automatically initiated and ignited, now all we have to do is training the network. So let's train our network with 80 samples:

for i in range(1,80):
    if (i % 2) == 0:
        net.load(get_a_bigger(), [1.0, 0.0])
        net.load(get_a_smaller(), [0.0, 1.0])

You should load your data one by one from each kind, respectively. Because it will prevent over fitting to one specific kind. You must wait a short time like 3 seconds (which is a reasonable duration in such a case), after each load.

output[0] will converge to detect bigger than 0.5 inputs.

output[1] will converge to detect smaller than 0.5 inputs.

Before the testing you should define a criteria called DOMINANCE_THRESHOLD so you can catch the decision making. Now let's test the network:

net.load([0.6, 0.9, 0.7, 1.0])
expected = [1.0, 0.0]

wave_zero = net.wave_counter
while True:
    wave_current = net.wave_counter
    if wave_current > wave_zero:
        wave_zero = wave_current
        output = net.output
        if abs(output[1] - output[0]) > DOMINANCE_THRESHOLD:
            error += abs(expected[0] - output[0])
            error += abs(expected[1] - output[1])
            error_divisor += 2

With the while loop given above, you will be able to check the output each time when a wave finished and if one element of the output dominated over the other, you will get that output as your ultimate result.

CatDog Example

(you can alternatively run this example with python examples/ command using a pre-written script version of below commands)

Suppose you need to train the network to figure out that the given image (32x32 RGB) is an image of a cat or a dog and map them to blue and red respectively. So let's create a network according to your needs:

net = plexus.Network( (32 * 32 * 3 + 3 + 256), (32 * 32 * 3), 3, 0.05, 3 )

We will plug in 32x32 RGB to the network so we need 3072 sensory neurons. 3 motor neurons for see how RGB our result is and 256 cognitive neurons to train. We need 3 digits precision because we need to store 255 different values between 0.0 and 1.0 range.

Explaining the answer of How to load CIFAR-10 dataset and use it is out of the scope of this paper but you can easily understand it by reading the code: python examples/ Once you get the numpy array of CIFAR-10 (or any other image data) just normalize it and load:

for i in range(1,80):
    if (i % 2) == 0:
        cat = random.sample(cats, 1)[0]
        cat_normalized = np.true_divide(cat, 255).flatten()
        blue_normalized = np.true_divide(blue, 255).flatten()
        cv2.imshow("Input", cat)
        dog = random.sample(dogs, 1)[0]
        dog_normalized = np.true_divide(dog, 255).flatten()
        red_normalized = np.true_divide(red, 255).flatten()
        cv2.imshow("Input", dog)

Catching the decision making is pretty much same with Basic Example. You will get an Overall error as the result very similar to Basic Example although this time the input length was 768 times bigger. This is because Plexus Network amalgamates the problems from all levels of difficulty on a single medium. It makes easy problems relatively hard, and hard problems relatively easy.

When you run this example, you will get a slightly better result when compared to flipping a coin. You will most likely get an Overall error between 0.35 - 0.45 which is the proof that the network is able to learn something.

By the way, don't forget that; Plexus Network does not iterate over the dataset and furthermore it runs in real-time. Also you have trained the network just for 4-5 minutes. Now let's see what happens if we train our network for a long period of time:


Because of this implementation of the Plexus network relies on the complex data structures and OOP of Python programming language and because CPython (the most common implementation of Python) has a headache called GIL, it is currently impossible to gain the advantage of multi-core processing. But I'm planning to implement a workaround in the future for this specific limitation.