Mathematical ideas can relate to mental space {mathematics and mental space}. Neuron assemblies can represent mathematical objects and mathematical operations.
number
Over a one-millisecond interval, neurons have (1) or do not have (0) an impulse, so neuron series can represent binary numbers. Over a one-second interval, one neuron's series of 0s and 1s can represent a binary number with 1000 digits.
Over a one-second interval, single-neuron axon-impulse number or released-neurotransmitter-packet number can represent a whole number. Neurons have impulse frequencies up to 800 Hz, so one neuron can represent numbers from, say, 1 to 800.
Neuron series can use positional notation to represent larger numbers. For example, one neuron can represent numbers from 0 to 99, and the other can represent numbers from 0000 to 9900, so neuron pairs can represent numbers from 0 to 9999.
number: integer
In neuron series, one neuron can represent sign, so neuron series can represent integers.
number: rational
In neuron series, one neuron can represent decimal point, so neuron series can represent rational numbers.
number: real
Real numbers have rational-number approximations, so neuron series can represent real numbers.
number: imaginary
In neuron series, one neuron can represent square root of -1, so neuron series can represent imaginary numbers.
number: complex
Complex numbers add real number and imaginary number, so two neuron series can represent complex numbers.
ratio
Neurons can compare receptive-field center input to surround input to measure stimulus-intensity ratio. Opponent processes compare inputs from two neurons to find ratio. Ratios are dimensionless, because dividing cancels units.
ratio: metrics
Comparing current and memorized ratios builds standard relative lengths, angles, and other measurement units (standardized metrics).
addition
To add two numbers, neuron series can receive input from two neuron series that represent numbers. To subtract, one input is negative.
Single neurons can accumulate membrane potential or neurotransmitter over time to represent simple summation.
addition: tables
If tables are available, arithmetic operations can use table lookup. First number is in first column, second is in second column, and answer is in third column. Neuron arrays can store number tables. Using indexes allows table lookup.
multiplication
To multiply two numbers, neuron series can receive input from two neuron series that represent numbers.
multiplication: amplification
Single neurons can amplify input. Cell body priming can cause inputs to dendrites to make more membrane voltage. Axon gating near synapse can cause synapse to release more neurotransmitter. Amplification is like multiplication.
multiplication: logarithm
Neuron series can store bases and exponents, so three neuron series can represent exponentials and logarithms. Neuron-series sets can add logarithms to perform multiplications. Logarithms are smaller than original number. For example, if number is 100, logarithm is 2: 100 = 10^2.
multiplication: power and root
Powers are multiplication series: a^3 = a*a*a. Roots are multiples of reciprocals: a^0.5 = (1/a) * (1/a). Neuron-series sets can repeat multiplications and divisions to find powers and roots.
symbol
Alphabet letters and punctuation symbols can have number representations. Neuron series can represent numbers and so letters, symbols, and variables.
mathematical term
Mathematical terms are constants times variables raised to powers: a*x^b. Neuron series can represent symbols and can use powers and multiply, so five neuron series can represent mathematical terms.
polynomial
Polynomials are mathematical-term sums. Neuron-series arrays can represent mathematical terms, so neuron-series-array series can represent mathematical-term sums. For infinite polynomials, higher terms have negligibly small values, so finite polynomials can approximate infinite polynomials.
polynomial: functions
Over space, time, or numeric intervals, polynomials can represent functions, so neuron-series-array series can represent functions. Polynomials can represent periodic, trigonometric, and wave functions: sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ..., and cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ... Polynomials can represent exponential functions: e^a = 1 + a + a^2/2! + a^3/3! + ..., and e^(i*a) = cos(a) + i*sin(a).
polynomial: factoring
Polynomials can have smaller polynomials that divide evenly into the polynomial. For example, a^2 + 2*a*b + b^2 = (a + b)^2, so (a^2 + 2*a*b + b^2)/(a + b) = (a + b). Neuron-series-array arrays can factor.
equation
Equations set two functions equal to each other: 3*x + 2 = 2*x + 3. Neuron assemblies can represent functions and the equals operation, so neuron assemblies can represent equations. Because they can subtract, factor, and divide, neuron assemblies can solve linear equations. Linear equations can approximate other equations.
equation: inequality and relation
Neuron assemblies can represent equations, so neuron assemblies can represent inequalities. Inequalities can indicate relations: more, same, and less, or before and after.
equation: system
Two or more equations with same variables are equation systems. For example, 3*x + 2*y = 6 and 2*x + 3*y = -6. Large neuron assemblies can represent an equation system. Because they can subtract, multiply, and divide, and so substitute, neuron assemblies can solve linear-equation systems. Linear-equation systems can approximate other-equation systems.
algebra
Algebras have elements, such as integers. Algebras have operations on elements, such as addition and multiplication. Operations on elements result in existing elements. Neuron series can represent numbers and perform arithmetic operations, so neuron assemblies can represent algebra.
calculus
All differentiations and integrations use only exponentials, multiplications, and powers. Neuron series can represent logarithms, multiplication, and powers, so neuron assemblies can differentiate and integrate.
mathematical group
Mathematical groups have elements, such as triangles. Mathematical groups have one operation, such as addition or rotation. Operations map every element to the same or another group element. For example, if element is equilateral triangle, 120-degree rotations result in same element. Tables show group-operation results for all element pairs. Neuron assemblies can represent number tables and table lookup and so represent mathematical groups.
logic
Neuron series can represent letters and symbols, so neuron-series arrays can represent words and statements. Statements can use nested variable relations. Neuron assemblies can represent and understand grammar.
logic: truth value
Neurons can represent TRUE or FALSE by potential above threshold or below threshold.
logic: operations
Two or three neuron series can represent NOT, AND, and OR operations. NOT operations can change input into no output, or vice versa, using excitation or inhibition. AND operations add two inputs to pass high threshold, which neither one alone can pass. OR operations add two inputs to pass low threshold, which either input alone can pass.
logic: tables
Logic operations can use table lookup. First variable is in first column, second variable is in second column, and truth-values are in third column. Neuron assemblies can store tables and perform table lookup.
logic: conditionals
Conditional statements combine NOT and AND operators: p -> q = ~(p & ~q). Neuron assemblies can represent NOT and AND operations and so represent conditionals.
logic: reasoning
Reasoning uses statement series. Neuron-assembly series can represent statement series and so reasoning.
computation
Neuron assemblies can represent numbers and statements and perform logic operations, so complex neuron assemblies can use programming languages and compute. Neuron-assembly activity patterns can represent cellular automata, which can simulate universal Turing machines and so compute any algorithm.
geometry
Visual processing can represent geometric objects, relations, and operations [Burgess and O'Keefe, 2003] [Moscovitch et al., 1995]. Representations have same relative lengths, angles, and orientations as physical geometric objects in space.
Geometric objects are points, lines, angles, and surfaces. Geometric objects have location, extension, and shape. Geometric objects have brightness, hue, and saturation. Geometric-object relations are up, down, above, below, right, left, in, out, near, and far. Geometric operations are constructions, transformations, vector operations, topological operations, region marking, and boundary making and removing.
geometry: point
Dendritic-tree center-region input excites ON-center neurons. Surrounding-annulus input inhibits ON-center neurons. ON-center neurons can represent points [Hubel and Wiesel, 1959] [Kuffler, 1953].
geometry: line
Lines are point series, so ON-center-neuron series can represent straight and curved lines [Livingstone, 1998] [Wilson et al., 1990]. Neuron-series length can represent line length.
Lines are boundaries of regions. Distance and intensity change rates are greatest at boundaries.
geometry: surface
Surfaces are line series, so ON-center-neuron arrays can represent flat and curved surfaces. Distance and intensity change rates are small in surfaces. Neuron-array area can represent surface area. Line boundaries are surface edges and separate surfaces.
geometry: orientation
Lines and surfaces have orientation/direction. Topographic-map orientation columns, perpendicular to cortical neuron layers, detect orientation. Orientation columns are for specific space locations. Orientation columns are for specific line lengths and sizes. Therefore, orientation columns represent one space location, one orientation, and one line length [Blasdel, 1992] [Das and Gilbert, 1997] [Dow, 2002] [Hübener et al., 1997] [LeVay and Nelson, 1991].
geometry: angle
For same space location and line length, adjacent orientation columns detect orientations. Neuron assemblies calculate plane angles between two line orientations or solid angles between three line orientations. Object and body rotation movements have angle changes.
geometry: geometric figures
Neuron assemblies can represent points, lines, orientations, angles, and surfaces, so neuron assemblies can represent geometric figures, such as spheres, cylinders, and ellipsoids.
geometry: distance
Neuron-series length can represent distance between two points. Neuron series can have all orientations, so neuron series can detect distance in any direction.
Topographic-map orientation columns calculate line and surface orientations. At farther distances, concave angles appear smaller, and convex angles appear larger.
Closer regions are brighter, and farther regions are darker, so neuron excitation can estimate distance.
Closer surfaces have larger average surface-texture size and larger spatial-frequency-change gradient. Neuron assemblies can detect surface texture and spatial-frequency-change gradients and estimate distance.
Object movements and body movements occur over distances, and neuron assemblies can track trajectories.
geometry: triangulation
To find triangle lengths and angles, neuron assemblies can use trigonometry cosine rule or sine rule.
geometry: trilateralization
Trilateralization finds point coordinates, using three reference points. The four points form a tetrahedron, with four triangles. Distance from first reference point defines a sphere. Distance from second reference point defines a circle on the sphere. Distance from third reference point defines two points on the circle. Neuron assemblies can measure distances between points and angles, and can use the cosine rule or sine rule to find all triangle angles and sides.
Animals continually track distances and directions to distinctive landmarks. Animals navigate environments using maps with centroid reference points and gradient slopes [O'Keefe, 1991].
geometry: space
Brain can represent perceptual space in topographic maps [Andersen et al., 1997] [Bridgeman et al., 1997] [Gross and Graziano, 1995] [Owens, 1987] [Rizzolatti et al., 1997].
Midbrain tectum and cuneiform nucleus have multimodal neurons, whose axons envelop reticular thalamic nucleus and other thalamic nuclei to map three-dimensional space.
Vision processing derives three-dimensional images from two-dimensional ones by assigning convexity and concavity to lines and vertices and making convexities and concavities consistent.
geometry: spatial axes
Vestibular-system saccule, utricle, and semicircular canals establish vertical axis by determining gravity direction and horizontal directions by detecting body accelerations and head rotations. Three planes, one horizontal and two vertical, define vertical axis and two horizontal axes.
Animal eyes are right and left, not above and below, and establish horizontal plane that visual brain regions maintain.
Vision processing can detect vertical lines and determine height and angle above horizontal plane. Vertical gaze center near midbrain oculomotor nucleus detects up and down motions [Pelphrey et al., 2003] [Tomasello et al., 1999].
Body has right-left and front-back, and visual brain regions maintain right-left and front-back in horizontal plane. Horizontal gaze center near pons abducens nucleus detects right-to-left motion and left-to-right motion [Löwel and Singer, 1992].
Topographic-map orientation columns with same orientation align and link to establish coordinate axes, in all directions.
Sense and motor topographic maps have regularly spaced lattices of special pyramidal cells. Non-myelinated and non-branched superficial-pyramidal-cell axons travel horizontally 0.4 to 0.9 millimeters and synapse in clusters on next superficial pyramidal cells. The skipping pattern aids macrocolumn neuron-excitation synchronization [Calvin, 1995]. The regularly spaced pyramidal-cell lattice can represent topographic-map reference points and make vertical, horizontal, and other-orientation axes. Lattice helps determine spatial frequencies, distances, and lengths.
Medial entorhinal cortex has some grid cells that fire when body is at many spatial locations, which form a triangular grid [Sargolini et al., 2006].
geometry: coordinate system
Vision processing relates spatial axes to make a coordinate system. Spatial axes intersect at a coordinate origin. In spherical coordinates, space points have distance to origin, horizontal angle to horizontal axis, and azimuthal angle to vertical axis. In Cartesian coordinates, points have distances to vertical, right-left-horizontal, and front-back-horizontal axes. Brain and external three-dimensional space use the same spatial axes and coordinate system. Coordinate origin establishes an egocenter, for egocentric space.
tensor
Neuron series can represent number magnitudes and space directions, so two neuron series can represent mathematical vectors. Neuron arrays can represent vectors and motions, so they can represent spinors as rotating vectors.
Neuron arrays can represent vectors, so they can represent matrices, which can represent surfaces. Matrices can be two-dimensional tensors, which have all vector-component products as elements. For example, |x1*x2, y1*x2 / x1*y2, y1*y2| for vectors (x1, y1) and (x2, y2) has four elements. |x1*x2, y1*x2, z1*x2 / x1*y2, y1*y2 z1*y2 / x1*z2, y1*z2 z1*z2| for vectors (x1, y1, z1) and (x2, y2, z2) has nine elements.
Three-dimensional tensors have all vector-component products. Neuron arrays can represent matrices, so neuron assemblies can represent three-dimensional tensors. During eye, head, and body movements, tensors can transform egocentric-space coordinates to maintain stationary allocentric-space coordinates.
Consciousness>Consciousness>Speculations>Space>Mathematics
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Date Modified: 2022.0224