Dynamical Neuroscience: the Critical Brain Hypothesis using Rings, Torus' and Neural Circuits & Manifolds using Fibonacci Sequences

The Fibonacci sequence, a marvel in the history of mathematics, is a prime example of a sequence of numbers that elegantly reveals a clear pattern for generating the preceding numbers at any given position within the sequence. Here, the term 'point' signifies a specific and distinct integer within the series. The Fibonacci sequence, with its unique 'geometric' structure, holds significant allure. To fully appreciate this geometric structure, it's imperative to delve into the concept of 'recursion.'??

The practical application of the Fibonacci sequence's geometric structure in algorithmic recursion is a fascinating area of study. This can be best understood by examining the specific scenario in which the series was first identified. The Fibonacci sequence was discovered in mathematics through a hypothetical scenario, with careful assessment of its details. To summarize, in the case study of rabbit breeding, the condition is that none of the rabbits now breeding will die. Therefore, each pair of breeding rabbits will consist of one male and one female. Thus, these rabbits above will engage in mating once every month, commencing the procedure when they reach the age of one month. Thus, the female rabbit will produce a single pair of offspring. Once again, this couple consists of a male rabbit and a female rabbit. Subsequently, for every previous pair of rabbits that are born, the female rabbit in that specific pair will give birth to its pair of rabbits, with each pair consisting of one male and one female. This process continues seemingly indefinitely. The practical implications of maintaining a geometric attribute for the Fibonacci sequence are vast, serving as a metaphor for describing systems in mathematics. In the context of the Fibonacci sequence, applied mathematicians often think of using data structures, specifically algorithmic 'tree' data structures, and associative search algorithms, which are also based on tree structures.??? ?

In this context, the Fibonacci sequence has a similar characteristic to the feature of geometric structure, where multi-dimensional information is kept. Just as a 3-dimensional structure holds data in distinct 1-2-and-3-dimensional planes, so does the Fibonacci sequence. The framework of algorithms can be described and utilized to depict a description. Within algorithm design, the geometric structure embodies the concept of recursion. However, regarding systems and models, two main components are crucial for physically examining these embodiments. When analyzing systems, it is essential to focus on the structure and behavior of the studied model. Although disguised inside the field of software engineering, recursion is highly relevant in today's context. Additionally, the concept of polymorphism also significantly influences the behavior of a geometric structure constructed by recursion. The critical aspect of polymorphism in the context of recursive behavior, as demonstrated by the Fibonacci sequence, is that the performance of search algorithms varies when searching for specific numbers (referred to as "nodes"). The performance of a search algorithm is measured by the time it takes to complete the search. Each algorithmic tree-search algorithm is ranked differently based on its time utilization in finding the desired "node" within the given algorithmic tree. However, it is essential to note that this method of measuring time usage is the foundation of the software engineering principle known as "Big-O Notation." The relationship between Big-O Notation and the geometric structure in the Fibonacci sequence can be compared; Big-O maps discrete search methods to time-utilization functions. In this case, one tree-search method may have a Big-O Notation described as "logarithmic," whereas another search technique may have a Big-O classification indicating "linear" time complexity.?? ?

Considering the concept of geometric structure and its ability to maintain multi-dimensional characteristics across several layers of dimensionality, it can concluded that the structure itself exhibits the most advanced level of topological behavior. An applied mathematician assumes that the high-level structure of an algorithmic search tree reveals the recursion process. The property of recursion allows the given structure to perform search-based operations to identify nodes within the structure. The described search function's behavior cannot be ranked based on a specific performance indicator. Although the Fibonacci sequence is known for its "high level" of behavior, it also exhibits recursive functionality about the discrete composition of dimensions. For example, a 3-dimensional system can be seen as an aggregate of the discrete levels of dimensionality about the first, second, and third dimensions.??

The intricate interplay of recursion within the geometric structure of the Fibonacci sequence leads to a deeper understanding of topological rings. In this context, the incorporation of topological rings in the study of the geometric properties of the Fibonacci sequence reveals that, through recursion, every node in the algorithmic tree structure is linked to every other node, thereby upholding the principle of 'continuity' in the Fibonacci sequence. Therefore, connections between every node indicate that this structure, when represented on a plane, forms a ring. Furthermore, rings symbolize a concept in a set theory known as a 'closed set.' When applied to real-life situations, the Fibonacci sequence demonstrates to applied mathematicians that the continuous branching of previous nodes at each level of the algorithmic tree, which represents the Fibonacci sequence, does not continue indefinitely. This understanding is crucial in the context of population expansion, enlightening us about the relationship between Fibonacci sequences and topological rings.??

The concept of approaching a metaphysical 'ceiling' in terms of population increase is well recognized in applied mathematics. Factors such as competition for resources, political upheaval, and climatic change significantly impact slowing down population growth. Within a closed system, the suppression of factors that contribute to population increase also leads to a decrease in the population growth rate at a specific moment in time; currently, the population has reached its metaphysical limit. The Fibonacci sequence, with its recursive and geometric structure, helps model this population growth dampening, introducing the concept of 'dampening' and its significance in the context of the sequence.??

We can establish a continuous structure by combining the concept of recursion with a structure with interconnected nodes, where each node is connected to every other node. This highlights the significance of convolution operations about the Fibonacci sequence. Therefore, due to the self-contained nature of Fibonacci sequences within their assumed topological structure, the features that promote growth inside the system are also the properties that hinder growth within the same system. This pertains to the problem of population expansion and how a population's ability to control its growth is affected within a closed system. An applied mathematician can conclude that using convolutions in a closed system, similar to the geometric structure of the Fibonacci sequence, gives the system a "gravitational" property of returning to an optimal average. This is similar to applying a Nash Equilibrium in a closed system with competition for essential resources.?

Pairing these themes aforementioned, neural circuits are adjoint to topological rings, where these rings operate in 2-dimensional space. Given this, as can be demonstrated below:

Now, interpreting the above depiction, each discrete plot on the given graphs, discretized by color-coding, indicates a specific discrete neural circuit for a given computational task that the brain for which was being measured had to perform. That being said, in the performance of this computational task, multiple discrete neural circuits operate in parallel. Thus, when a neural circuit is analogized to a ring, and this ring exists within the space of 2-dimensionality because neuronal circuits act in parallel to perform a given computational task, the aggregation of the set of all neural circuits used to accomplish that computational task may be superimposed on top of one another, thereby manifesting a volumetric structure (i.e., a neural manifold).

The state of dynamical neuroscience predicates itself on the relationship of the statements stated in this article with the concept of the golden ratio. Here, under the Critical Brain Hypothesis, the operation of the brain is assumed to be operating under a state of "criticality" (i.e., a phase transition between two unique discrete states). That being said, because the golden ratio relative to the subject of hydrodynamics is quantified by phi = 1.618, then, under the golden ratio and Luca Pacioli of the year 1835, his "golden mean" concerning his findings of what he identifies as the "Divine Proportion" we can say that discrete dynamical systems can be used for geometric situations. Thus, because the lobes of a brain can be reduced to symbolic "topological spaces," thus mapping these concepts to the golden ratio depicted below:

Each discrete square of the Fibonacci series shown above may be mapped to a given unique lobe of the brain, defining a "Poisson bracket" of some brain lobe and some "box" within the Fibonacci series (listed above). Thus, in conclusion, the Critical Brain Hypothesis holds true for dynamical neuroscience principles.