A Partial Mathematical and Conceptual Theory

1. Neural Nodes, Connections, and Associations
In this framework, we conceptualize the brain as a system of nodes (somas) connected by dendrites and axons, forming a vast network of associations. The process of convergence and divergence is fundamental to neural activity:
- Convergence: Multiple signals from different sources combine onto a single node.
- Divergence: A single node distributes signals outward to multiple targets.
Information Flow in the Brain
Unlike conventional network models, neurons do not merely relay information passively. They store information and then distribute it either outward (efferent) or inward (afferent), depending on function and context.
- Sensory information is an input signal that the brain receives and processes.
- Imagination functions both as input and output, integrating information across different cortical areas.
For example, a mental image of a candle can trigger the verbal recall of the word "candle." This occurs through signal transmission from the visual cortex to the auditory cortex, forming a cross-modal association.
Association Strength and Frequency
The connection between sensory input and conceptual understanding depends on association frequency, which determines the strength of neural links between concepts.
Association Laws:
Sound-Visual Association:
The frequency (λS) of a sound occurring alongside a visual (Vi) strengthens the sound association (Sa).
Visual-Visual Association:
The frequency of a visual input () determines the strength of its visual association (), linking sensory input to conceptual output.
Signal Transmission and Connectivity
Neural activity consists of both incoming (afferent) signals and outgoing (efferent) signals:
- Input signals correspond to afferent neurons, carrying sensory data toward a node.
- Output signals correspond to efferent neurons, transmitting information away from a node.
Connection Laws:
Efferent Connections (Outputs per Node):
μ(O<)n=ξ−ζnThe number of outgoing signals per node depends on the available inactive efferent neuron connections ().
Afferent Connections (Inputs per Node):
The number of incoming signals per node depends on the available inactive afferent neuron connections.
Activation Law:
- Probability of Activation:
s(stimuli+association)=pἄ
The probability of a node activating depends on the stimulus strength and the association strength of the input.
2. Reductionism and Gestalt Integration
Neurons store and retrieve concepts in hierarchical structures, with categories functioning as central nodes. Each category can be decomposed into subcomponents, reflecting the Gestalt principle of whole-part relationships.
For example, the concept of a candle consists of:
- A candle holder
- A wax candle stick
- A wick
Each component forms a part-node, while the entire concept functions as a whole-node in the network.
Structuralism and Reduction Laws:
Whole-Part Relationship:
A whole () consists of a number () of parts ().
Connection Integration:
The number of connections between a whole and its parts remains consistent with the part-reductionism principle.
Properties as Descriptors:
A property () can define either a part () or the whole ().
Phenomenal Properties and Perception
Every object can be analyzed through 15 fundamental properties, forming the basis of perceptual representation:
- Color
- Shape
- Velocity
- Size/Measurement
- State (solid/liquid/emotion)
- Texture
- Part-whole relationship
- Language association
- Aroma
- Temperature
- Pleasure/Pain association
- Emotional value
- Need/Desire relevance
- Nutritional/Health value
- Contextual meaning
Each property has a corresponding gate-node responsible for detecting and processing that specific type of information.
Property Reductionism:
Total Property Description:
This equation represents a complete property-based description of an object.
Thought-Property Mapping:
At any given moment, a thought () is focused on one of the 15 possible properties ().
Hierarchical Storage and Retrieval in Neural Networks
The human brain stores and retrieves concepts in a hierarchical manner, meaning that each category acts as a central node in a network of related subcomponents. This structure follows the principles of Gestalt psychology, which asserts that perception is organized according to whole-part relationships.
At any given moment, a concept may be activated at different levels of granularity, depending on context, familiarity, and cognitive demand.
For example, the concept of a candle is stored in memory as a whole, but can also be decomposed into its parts:
- A candle holder (support structure)
- A wax candle stick (fuel source)
- A wick (combustion facilitator)
Each of these subcomponents is represented as a part-node, while the entire concept of the candle functions as a whole-node in the network.
Whole-Part Structuralism and Reduction Laws
Reductionism seeks to break down complex systems into their fundamental components. However, Gestalt psychology emphasizes that the whole is greater than the sum of its parts—a principle that must be accounted for in any neural framework.
The interaction between whole-part relationships can be expressed mathematically as follows:
1. Whole-Part Relationship
W=nPWhere:
- represents the whole concept,
- represents individual parts,
- represents the number of parts contributing to the whole.
This equation indicates that each whole is composed of a discrete number of elements, but it does not imply that understanding the parts alone fully explains the whole. The interactions, emergent properties, and contextual meaning of the whole concept extend beyond mere summation.
For example, a car is made up of many components:
Although these parts are necessary to form the whole concept of a car, they do not function independently as a car. The interaction between these parts—how they fit together and serve a functional role—determines the emergent perception of "car-ness."
2. Connection Integration
Where:
- represents the number of direct connections between parts,
- The right-hand side represents the total number of components contributing to the whole.
This equation suggests that the strength of a whole-node’s representation is not merely about having parts, but also about the degree to which these parts are interconnected.
For instance, in the case of the candle:
- If the wax is removed, the candle can no longer function as intended, even if the wick and holder remain.
- If the wick is missing, the candle is incomplete, but might still be recognized as a candle due to prior knowledge and Gestalt completion principles.
Thus, connection strength plays a critical role in neural representation.
3. Properties as Descriptors
Where:
- represents descriptive properties (e.g., color, texture, function),
- n is the number of properties assigned to a particular part () or whole (W).
This principle states that an object or its components can be defined by a set of unique properties, and each property acts as a descriptor within the neural network.
For example, a red apple can be broken down into:
- Color: Red
- Shape: Round
- Texture: Smooth
- Taste: Sweet/tart
- Function: Edible
Each property is assigned a neural representation, which collectively allows for object recognition, categorization, and retrieval.
Gestalt Perception and Neural Integration
Gestalt principles describe how the brain organizes sensory input into meaningful patterns, even when information is incomplete. These principles play a crucial role in perception, imagination, and memory retrieval.
1. Principle of Closure
- Even if a stimulus is incomplete, the brain fills in the gaps based on previous knowledge.
- Example: Seeing a partially obscured candle and still recognizing it as a candle.
2. Principle of Continuity
- The brain prefers continuous patterns over abrupt changes.
- Example: If a candle is melting and distorting, we still perceive it as the same object.
3. Principle of Similarity
- Similar items are grouped together in perception.
- Example: Different types of candles (scented, pillar, birthday candles) are categorized under a single conceptual node.
4. Principle of Figure-Ground Separation
- Objects are perceived as distinct from their background.
- Example: A candle in a dark room is easily distinguishable from its surroundings due to its light emission.
These principles explain why object recognition remains stable despite variations in input, such as changes in lighting, angle, or partial obstruction.
Functional Implications of Whole-Part Representation
The whole-part distinction is crucial in problem-solving, creativity, and decision-making.
Problem-Solving:
- Breaking a complex problem into smaller subproblems (parts) before integrating them into a solution (whole).
- Example: Learning to cook a meal involves mastering individual techniques (chopping, seasoning) before combining them into a complete dish.
Creativity:
- Reassembling existing knowledge in new configurations (whole from new parts).
- Example: An artist visualizing a hybrid object, like a candle fused with a sculpture.
Memory Recall:
- A partial cue activates a whole memory representation through associative links.
- Example: Seeing a wick might remind someone of a candle, even in the absence of wax.
Neural Efficiency and Predictive Processing
The brain optimizes cognitive resources by activating only the necessary level of detail for a given situation.
Global Activation (Whole-Level Processing):
- Rapid recognition of familiar objects using minimal details.
- Example: Seeing a flickering light and immediately recognizing it as a candle.
Local Activation (Part-Level Processing):
- Detailed analysis when precision is required.
- Example: A scientist studying the chemical composition of wax.
Predictive Efficiency:
- The brain anticipates missing details based on past experience, reducing computational load.
- Example: Recognizing a half-melted candle without needing to analyze every detail.
This model of Gestalt-structured neural networks suggests that reductionism alone is insufficient to explain cognition. The interplay of hierarchical representations, whole-part relationships, and emergent properties enables efficient categorization, recognition, and creative thought.
By combining:
- Reductionist principles (breaking objects into components),
- Gestalt principles (emphasizing holistic perception),
- Mathematical models (quantifying neural connections),
we can develop a computational neurology framework that better explains how the brain encodes, retrieves, and integrates information.
4. Neurogenesis and Learning
Principles of Neurogenesis:
- New Information & Neural Growth:
Novel properties, parts, or functions can trigger neurogenesis (the formation of new neurons). - Pre-existing Network Activation:
If a newly encountered stimulus overlaps with existing neural structures, it activates pre-existing pathways rather than generating new ones.
5. Pain, Pleasure, and Expectation
Pain and pleasure perception follows a threshold model, where specific sensory inputs activate corresponding neural clusters.
- The higher the stimulus intensity, the greater the probability of crossing the pain/pleasure threshold.
P(T)=f(S)Where:
- is the probability of crossing the pain/pleasure threshold.
- is the stimulus intensity.
- is a function representing how stimulus intensity affects threshold crossing.
A common way to represent this is using a logistic or sigmoid function, which models threshold effects in psychophysics:
Where:
- is the critical stimulus intensity at which the threshold is crossed 50% of the time.
- is a sensitivity parameter that determines how rapidly the probability changes near the threshold.
This equation reflects that as stimulus intensity S increases, the probability of exceeding the threshold approaches 1 (certainty), while at very low intensities, it remains near 0.
- Expectation arises from repeated associations, shaping the likelihood of future predictions.
6. Imagination and the Interference Problem
Imagination, while crucial for abstract thinking, introduces errors in perception and recognition.
Factors Contributing to Misidentification:
- Absence of primed neural gates (no prior exposure).
- Lack of sensory receptors (e.g., inability to perceive ultraviolet light).
- Limited temporal cohesion (inability to link cause and effect).
- Unobservable properties (e.g., atomic structure).
- Weak contrast-comparison connections (failure to distinguish differences).
By refining inhibitory processes, the brain minimizes imaginative interference, enhancing perceptual accuracy.
Factors Contributing to Misidentification
Several mechanisms contribute to errors in recognition and perception, making it difficult to distinguish imagination from reality.
1. Absence of Primed Neural Gates (No Prior Exposure)
Neural priming is the activation of pathways that facilitate recognition. If the brain has never encountered a stimulus before, it struggles to identify or interpret it accurately.
Example: Seeing an Unfamiliar Object
Imagine seeing a quantum computer for the first time. Without prior exposure to its shape, function, or context, the brain tries to associate it with familiar objects (e.g., a server rack or a high-tech appliance). This misidentification occurs because there are no primed neural gates to correctly interpret the object.
🔹 Cognitive Consequence: The brain relies on heuristics (mental shortcuts), leading to inaccurate generalizations.
🔹 Neurological Basis: The hippocampus, responsible for memory formation, is unable to cross-reference the new information with stored schemas, leading to distorted interpretation.
Real-World Implication:
- This explains why travelers in foreign cultures may misinterpret objects, customs, or symbols.
- It also accounts for why young children mislabel objects they’ve never encountered.
2. Lack of Sensory Receptors (Biological Limitations of Perception)
Humans perceive reality through sensory receptors, but these have biological constraints.
🔹 Example: Ultraviolet Light Perception
- Bees can see ultraviolet (UV) patterns on flowers, which guide them to nectar.
- Humans lack UV-sensitive photoreceptors, so we perceive flowers differently, missing critical visual information.
🔹 Consequences:
- The human brain compensates for missing sensory data by filling in gaps using imagination.
- This often leads to incorrect assumptions about the true nature of the world.
🔹 Neurological Basis:
- The visual cortex (V1-V5) processes sensory inputs, but gaps in sensory perception lead to extrapolated interpretations based on existing knowledge.
- This explains optical illusions, pareidolia (seeing faces in objects), and phantom limb syndrome.
Real-World Implication:
- Scientific Instrumentation: Humans rely on technology (e.g., telescopes, microscopes) to extend sensory perception, reducing errors caused by biological limitations.
3. Limited Temporal Cohesion (Inability to Link Cause and Effect Correctly)
Temporal cohesion refers to the brain’s ability to maintain logical time sequences for perception and memory.
🔹 Example: Déjà Vu
- Sometimes, a person experiences a situation and believes they’ve lived through it before.
- This occurs when the brain misaligns temporal signals, creating an illusion of familiarity.
🔹 Why This Happens:
- The entorhinal cortex and hippocampus synchronize temporal events.
- When synchronization fails, memories become misordered.
- This can cause false predictions (believing an event will unfold in a specific way when it won’t).
🔹 Cognitive Consequence:
- Limited temporal cohesion results in superstitions, conspiracy theories, and irrational cause-effect relationships.
Real-World Implication:
- Explains why eyewitness testimony is unreliable—memories of events are often reconstructed out of order.
- Suggests why false memories form—people imagine an event so vividly that they later believe it happened.
4. Unobservable Properties (Concepts Beyond Sensory Experience)
Many real-world phenomena are unobservable, requiring abstract thought to conceptualize them.
🔹 Example: Atomic Structure
- No human has ever seen an atom directly, yet we construct mental models of atoms using scientific inference.
- The imagination fills in visual details, but these may not accurately reflect reality.
🔹 Consequences:
- In the absence of direct observation, errors in conceptualization occur.
- Early models of the atom (e.g., Bohr’s planetary model) were later refined using quantum mechanics.
🔹 Neurological Basis:
- The prefrontal cortex constructs abstract models.
- The default mode network (DMN) generates hypothetical scenarios, even if they don’t align with reality.
Real-World Implication:
- Scientists must test and refine models rather than trust initial imaginative assumptions.
5. Weak Contrast-Comparison Connections (Failure to Distinguish Differences)
Recognition depends on contrast—the ability to distinguish one concept from another. When contrast is weak, errors occur.
🔹 Example: Identical Twins Confusion
- If two people look extremely similar, the brain struggles to differentiate them.
- This happens due to low feature contrast between their facial characteristics.
🔹 Why This Happens:
- The fusiform gyrus, responsible for face recognition, relies on subtle differences in spacing and proportion.
- When two stimuli overlap too much, differentiation fails, leading to misidentification.
🔹 Cognitive Consequence:
- This explains stereotyping—grouping individuals based on broad categories rather than specific traits.
- Also relates to false memory syndrome—mistaking one event or person for another due to similarity.
Real-World Implication:
- Security measures like biometric identification compensate for human recognition errors.
Refining Inhibitory Processes to Reduce Interference
The brain actively suppresses errors in perception and memory by refining inhibitory control mechanisms.
🔹 1. Neural Inhibition (Suppressing Irrelevant Information)
- The prefrontal cortex filters out imaginative distortions when focusing on factual recall.
- This prevents confabulation—filling memory gaps with false information.
🔹 2. Sensory Calibration (Comparing Perceptions to External Reality)
- When sensory data conflicts with imagination, the brain prioritizes real-world input.
- Example: Touching an object to confirm its presence when visual cues are unreliable.
🔹 3. Memory Reconsolidation (Updating Incorrect Memories)
- When new evidence contradicts a false memory, the brain modifies stored information.
- Example: Learning correct history after years of believing misinformation.