Manahel Thabet Talk on a very basic level, the entire universe runs on the rules of physics, which are mathematical concept which our reality conforms to. On a similarly basic level, our entire being is run based on our brain. Then, the need to explore the brain using the tools of our most basic reality should emerge naturally. In this report, we shall first lay down some theoretical groundwork, and then see in what ways quantum mechanics and chaos emerge in our brain.
Chaos
Differential equations
The basis of chaos is in the behaviour of a system of differential equations. A differential equation is an equation which describes the change of a variable in regard to time. Consider one of the most basic differential equations, Hook’s equation:
𝑑2𝑥𝑑𝑡2=−𝑘𝑥 (1.1)
For the sake of clarity, a first time-differential will have a single do as a superscript, and a second time-differential will have two dots as a superscript, reformulating Hook’s law as:
𝑥̇=−𝑘𝑥 (1.2)
In essence, Hook’s law describes a behaviour of a weight oscillating on a spring, where the more it is displaced from the point of origin, the greater the acceleration it will experience towards the point of origin.
A system of differential equations will be of the form:
𝑥̇=𝑓(𝑥)+𝑔(𝑦)+𝐿(𝑥,𝑦)+𝐶 𝑦̇=𝑘(𝑦)+ℎ(𝑥)+𝑃(𝑥,𝑦)+𝐷
(1.3)
In this form, you can see how the current state of x and y influence each other. Such a system will describe the behaviour of two variables. The classical example is predator vs. prey, where x describes the number of prey, and y describes the number of predators:
𝑥̇=𝑓(𝑥)−𝑔(𝑦) 𝑦̇=−𝑘(𝑦)+ℎ(𝑥)
(1.4)
Model assumptions:
•There is infinite food
•The populations of prey and predators are controlled purely but their own populationand not outside forces
•Prey meeting will produce offspring
•Predators have infinite appetite
•Predators starve and die
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