How to handle dynamic data with chaotic neural networks?

In data science, we see the emergence of the chaotic nature of the environment. This environment consists of data, layers, mathematics, and a lot of things. It is always seen in normal practices that we use standard neural networks that are static in front of the problem of dynamic or chaotic behaviour. Chaotic neural networks are something that is specialized in dealing with the dynamic and chaotic nature of the environment and data. In this article, we are going to discuss the chaotic neural network. The major points to be discussed in the article are listed below.

Table of content  

1. What is a chaotic neural network?
2. Mathematics behind chaotic responses
3. Modeling the chaotic responses 
4. Where are the chaotic neural networks used?
5. The architecture of the chaotic neural network

Let’s start with introducing chaotic neural networks.

What is a chaotic neural network?

In general English, the word chaotic can be explained as the state of complete confusion and disorder. In technology and the industrial world, we can see that the implementation of the chaotic phenomenon is everywhere. In data science, we can say that if a model can have complex dynamics and abundant chaotic pattern detection ability then it can be applied to implementation in neurocomputing much better. 

We can clarify the significance of chaotic phenomena in neural networks by taking an example of an artificial neural network where we can use it to measure the dynamic characteristics of the artificial neural networks. 

Just like another network this network also has mathematical aspects or we can say these are also a kind of mathematical model. Talking about the chaotic dynamics we can find one example of this in the nerve membranes. Many experiments determine that real neuron membranes in the resting state respond to periodic pulse stimulation not just synchronously, but also chaotically, depending on the intensity and time of the stimulating pulses. 

There are equations like the hodgkin-Huxley equation and Fitzhugh-Nagumo equation that can be used to analyze neurocomputing. We can say the this network is the network that can be used in the analysis of the chaotic nature of the network and data. Let’s take a look at the simple networks that can generate chaotic responses.

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Mathematics behind chaotic responses

In this section, we will look at how the simple neural network generates chaotic responses and how we can model them to make them synchronous. As we discussed the Fitzhugh-Nagumo equation which is a method to model chaotic responses. This equation assumes that the increment in the chaotic nature of past data increases the chaotic nature in present and can be given by the following equation:

x(t+1)=u(A(t)- d=0tkdx(t-d)-)    

Where, 

u = unit step function 

A(t) = variation in magnitude of strength at time t

k = damping factor 

= threshold

Here we can define a new variable using the following formula 

y(t+1)=A(t)-  d=0tkdx(t-d)-)

Here we can see how we can calculate the chaotic nature of the new variable. Now let’s see how we can model.

Modeling the chaotic responses 

In the above explanation, we have seen that the chaotic nature of neurons of the network can be considered as part of the neural networks and can be called a chaotics neural network. Such a network has two kinds of inputs, one is feedback input and the second is externally applied input. The M chaotic neuron of the neural network can be modeled in terms of the dynamics as follows:

This equation has three properties:

• It is a continuous output function 
• Can perform the sum of Spatio-temporal nature of both types of input 
• The relative refractoriness 

Expanding the above-given equation can help us in dealing with the chaotic nature of neurons. With a few expansions, this network will seem like a discrete-time neural network and after that that will be converted into a back-propagation network which means in the whole scenario, we will be able to model chaotic neurons using a natural extension of neural networks. 

Where are the chaotic neural networks used?

In the last section, we discussed how chaotic dynamics can be generated in the neurons of neural networks and that can be used to model the chaotic nature of data. Some of the usages of the this network can be found in the following fields 

• We can utilize such networks in the motion control systems where the change in the position of any object can be determined by the motion function and the calculation of the function can be extracted from the idea of this network.
• These networks can be used in combinatorial optimization problems. Since the standard neural networks have the nature to be trapped in local minima, the chaotic nature of neurons can help them in escaping from local minima.
• Adding dynamic nature to the neurons makes the network promising against weak input signals or we can say against fluctuating input. In a variety of cases like fluctuating time series and audio data, these networks are very helpful in making inferences.
• Internal parameters of the standard neural networks are not capable of adjusting according to the outside system. Such a system can facilitate the biomedical applications of chaotic resonance. 
• These networks can be used in solving problems like travelling salesman problems, traffic detection problems, etc.

The architecture of the chaotic neural network

As discussed above we can say that the architecture of such a network can be similar to the standard neural network but the main difference between them is that in the layers of a chaotic neural network we find chaotic maps that can follow the rule of generalized Luroth series. Applying such maps to the network enhances the capability of the network for compression, cryptography, and computing XOR and other logical operations. The below picture can be a representation of the this network.

Image source

In this article, we are focusing on the chaotic neural network so we only need to understand the middle portion of the above picture that represents the architecture of the GLS chaotic neural network that can be utilized for classification tasks. In this kind of architecture firing of GLS becomes chaotic and neurons halt while their activity value starts from q which is an initial neural activity that reaches on stimulus.  

Final words

In this article, we discussed chaotic neural networks that control the chaotic nature of the environment. Along with this, we have also gone through the mathematics behind this type of modelling and its use cases. We have also discussed its architecture based on a network that was designed to deal with classification problems.