Differential Convolutions and Brain Energetics

In the study of neuroscience, both researchers and clinicians alike appreciate the notion that function follows form. This notion is a fundamental principle in differential geometry. After that, the behavior of a topology is not determined by how large it is but is ultimately defined through its morphology. Any seemingly minuscule change to a topology's structure will manifest into a corresponding radical shift in its behavior. At this point, a person may be wondering how these prior statements map to the topic of this article. Here, we will transition towards discussing the implications and underlying mathematics emulating the concepts of neural manifolds and neural circuits.

In as much, the mapping of metabolic and vascular events within the brain is most applicable when such a mapping is brought about using Functional Magnetic Resonance Imaging techniques (i.e., fMRI). Therein, the terms of events are purposefully emphasized to indicate to the reader that such a set of events is of cardinality-1. This means that although a certain stimulus when performing some computational task, the brain, accordingly, will have many metabolic and vascular events for a said task where each of these events, respectively, will be treated in a singular fashion; a cardinality of 1 provides an indication of such a subject of cardinality-1 being that of a "singleton." Accordingly, to provide an illustration of an f-MRI, please look below for a visual exhibit:

Whereupon, the above illustration provides multi-axis views of a person performing a specific series of tasks where specific white matter tracts, under the gauze of "action-potentials" between neurons, communicate. With this, how does fMRI perform such a marvelous task of visualizing real-time-neurologic-dynamics? The fundamental component of mathematics to inscribing such a visualization on neurologic matter is in lieu of the Laplace Transform.

Specifically, it is now appropriate to detail the importance of the contribution of the Laplace Transform with reference to the functionality of fMRI. To detail, the purpose of performing Laplace Transforms is to fabricate a tangible mapping between the spatial and time domains, respectively. To provide context to the application of the Laplace Transform onto f-MRI, the spatial domain with reference to f-MRI is the specific morphology for which the given clinician or researcher desires to investigate. Here then, brain matter can be relegated into functional categories to which a particular piece of neurologic anatomy is mapped to functionality via morphology. Therefore, in a sense, the conceptualization of morphology is to perform a subjugation neuro-anatomy to that of polymorphism. That being said, the utility of polymorphism is that a given "entity" (i.e., a specific region of brain tissue) may reserve numerous potential behaviors, depending on the given morphology for which it presently has surgical intervention of brain structure or traumatic brain injury, for example, will alter brain-structure-morphology, thus implicating a change in mapped function of respective regions of brain tissue. Therefore, this is to say that neurologic plasticity is both dynamic and distributive. This means that neurons placed within a given segment of neurologic morphology are designed to take on a myriad of potential tasks as a means of taking into account traumatic neurologic-morphology-alterations.

Here, then, a person may wonder what the implications are to neurons and the leveraging of the conceptualization of the Laplace transform. Hence, one can recognize this seemingly subtle relationship between the Laplace Transform and morphologic activity. To detail, take, for example, the Laplace Transform translation from the spatial domain to the time domain accordingly, where if some function "G" is parameterized under the spatial domain by some variable "s" and some function "f" is parameterized by some variable tau to provide reference to the parameterization under the time domain, we have that:

Therein, a person may be interested as to the underlying importance of having an embedded integral within the braces of the Laplace Transformation, written above. Here, the significance of the given integral is under the application of an "action potential spike train." Now, what is an action potential spike train? The answer to this is that when understanding the analog, real-time activity of dynamic action potentials amongst neurons, a neuroscientist or certified neurologic-clinician (i.e., neuro-surgeon) will attach a series of electrode-anode-cathode-pairings within a brain and observe the brain energetics (i.e., neurologic event-stimuli) whilst the brain performs specific computational tasks. Now, to provide a more precise understanding of the statements mentioned above, have a look at the illustrative exhibit below:

Each analog schematic listed above is not simply a neuron firing an action potential, but a neural circuit. The composition or 'set' of all neural circuits involved in performing a specific neurologic-computational task is actually the associative 'neurologic manifold' of that task. If we assert the presupposition that function follows form for neurologic computational modeling, the concept of 'manifold' is the physical representation of the functionality of a paired 'morphology '. In other words, a morphology can be seen as the 'verb' that is paired with a specific 'noun', a metaphor that is tied to the concept of a manifold.

Therefore, to summarize the importance of the embedded integral within the illustrated Laplace Transformation, as indicated in this article, the integral is to consider all discrete neurologic circuits within some set mapped to a specific neurologic activity (i.e., a morphology). Therein, a set of neural circuits that is mapped to a particular neurologic computation is the weighting average of electrical-brain activity that is being recorded. The integral part of the Laplace Transformation translation is to take into account all neural circuits within the set (i.e., a given neural manifold).