User:Boolean brain
BOOLEAN BRAIN (Until now -sept-2011- I could not insert the figures and many formulas, e.g. (k2) must be (k square))
A MATHEMATICAL ANALYSIS OF SOME FUNCTIONS OF THE INFORMATION UNIT IN THE HUMAN CEREBRAL CORTEX
E. Colon, Inst.Brain Res.
Summary
A Boolean Algebra based model of the cerebral cortex is developed.
Introduction
The cerebral cortex can be described by neural models of 3-D networks (Soltesz and Stanley, 2008). As such we can describe the cerebral cortex as built up of a series of randomly connected neurons, also randomly distributed in the cortical space. It follows that the probability of connections of one neuronal element with other elements is equal to the probability of having connections of an other element with these others.
However, in random systems the probability of contact (a) of one neuron A with an other B (a: A→B) will be equal for all A and B. This is untrue in adult cerebral cortical systems. We do not have the degree of freedom of space location. Location in the brain is a stable state: one neuron located between his neighbours will always be.
But yet it is untrue to state for this particular neuron that this implies that the connectivity-probability is located in his neighbourhood: out flowing axonal systems are interconnecting different space-locations, and this will make it possible to state that, notwithstanding the fact that space-location is not random, and notwithstanding the circumscriptive and very local arborisation of the receptive neural parts (the dendrites) that at least for neuronal columns, or neuron-blocks, there is a random connectivity with all the other neuron-blocks.
There are defined infinite of these blocks in the cerebral cortex (see fig. 1).
An analysis of a single block-cortical function will be given in this study.
PART I
SET THEORETIC APPROACH OF SOME CORTICAL FUNCTIONS IN MAMMALS
Let S be defined the set ingoing axonal fibers and their branching patterns in certain cortical structure.
Then the subclasses (axons) Ai (I = 1, 2,.........n) may be represented by: Ai⊂ S ; ∑ (from i=1 til n) of Ai.S
The intersection of S1∩ S2 of two sets Si⊃Ai(1) , S2⊃Ai(2) contains the collections X p(1) = Xq(2)⊂S1∪S2(p,q∈i)
The subclasses Ai are build up by series of units (Act. Potentials) so that Ai consist the elements ωt : Ai ϶ ωi(t) (t = 1,2,3.......n); ωi(t) being the frequency of Act. Potentials on a unit time t , in Ai.
The set S of subsets Ai is finite and denumerable with cardinal number n. Let Ptot. be the set of neurons lying in the above mentioned cortical structure and Pn be the subset of n neurons (n= 1,2,3,......n).
Pn⊂Ptot. ; P being finite denumerable
Then there is a mapping α of the subset Ai into the subset Pn, Ai x Pn such that each Ai∈S occurs exactly once as the first component of a member of the subset Ai x Pn. This mapping is neither injective (A1α = A2α does not mean A1=A2) nor surjective (each p∈Pn does not necessarily occur as an image of Aiα), so the inverse α-1 does not exist for every p∈Pn⊂Ptot
The collection of subsets A1 is not a partition of S (see fig 2), for, notwithstanding the fact that S =∪(i=delta)Ai ,the subset A1 may posses identical elements ωt, so Si∩Sj for i≠j, Si∩Sj≠ɸ for i≠j
The set of ingoing axonal fibers is a groupoid, while there is defined an operation in it, which is commutative ( ωt1 . ωt2 = ωt2 . ωt1), namely the total activity at each time t brought into Pn, is a mapping of the groupoid SxS, in which SxS is defined as a mapping ∫, where the subelements ωt are taken in a definitive order, and then are associated with an other well defined subelement ωt.
This groupoid is associative [(ωt1 .ωt2) ω t3]= [(ωt1 ( ωt2 . ωt3)], so it is known as a semigroup.
We do not know if this semi-group has an identity element, while we do not know if this operation is additive or duplicative or something else, but we define it a neutral element e, for then we can describe it as a monoid.
For arbitrary subelements ωta and ωtb in this monoid, there exist element x and y in this system, such that ωa t. x = ωb t and y.ωa t= ωb t
There exist an unique solution for both x and y, and so this operation in the system has an inverse. In other words we can define the monoid as being a group, and while it is a commutative system (ω t 1 .ω t 2 = ωt2.ωt1) we can define it as an ABELIAN GROUP.
If the sets S1 and S2, each with the same operation ∫ in it, are isomorphic and we write S1≅S2 (see fig 3), in other words if there is
1. a one to one correspondence between the elements of S1 and S2
2. the operation is preserved by the correspondence. E.g. if ωa1⇔ω a2 and ωb1⇔ωb2 with ωa1,ωb1∈S1 and ωa2,ωb2∈S2 it follows that ωa1.ωb1⇔ωa1.ωb1 (ωa.ωb indicating the operation) then S1 = S2 (by definition, for we are dealing here with nervous tissue)
The group of ingoing axonal fibers which we defined before may be described now as a ring R (1). In this system namely we can find at least two binary compositions known as addition and multiplication, such that the following distributive laws hold for arbitrary elements:
A1(ωt1), A2(ωt2), A3(ωt3)∈S
A1(ωt1) {A2(ωt2) +A3(ωt3) } = A1(ωt1).A2(ωt2) +A1(ωt1). A3(ωt3)
and {A2(ωt2)) +A3(ωt3) } . A1(ωt1) = A1(ωt1).A2(ωt2) + A1(ωt1) .A3(ωt3)
This ring is not an integral domain while for arbitrary elements:
Ax(ωtx) , Ay(ωty), Az(ωtz) ∈ R(1)
Both: Ax Ay = Ax Az and Ay Ax = Az Ax does not imply Ay = Az
We point out here that this binary composition represented by the way in which the and ωt in two axonal incoming fibers Ai and Aj in S1 are worked up by the neurons of Pn and the representation of this process by the outgoing fibers of Pn. In other words we may state that this is a first order classification.
We are able now to write out the ring Rn of all nxn matrices with elements in R (1). The matrices or elements of Rn are the arrays [ Aij ] (i indicates the axon and j the neuron).
First order classification: [ Aij ] = "MATRIX Aij"
We can define Bi the axonal system incoming to the same neural receptive field as was described for Ai.
[ Bij ] = "MATRIX Bij"
When [Aij ] + [ Bij ] = [ Cij ], where Aij + Bij = Cij, we have defined in a circumscriptive neuronal field the ability of classification of Ai and Bi, not only for the individual axons but also for a pair or more axons together, and the total classification of all sets of incoming fibers till all possible orders of classification.
If necessary we can make use of the product matrix of [ Aij ] x [ Bij ] = [ Dij ] . Dij = ∑ (from n until k=1) aik . bkj (see fig 4,5,6).
PART II
TRANSMISSION PROBABILITIES IN A NERVOUS BLOCK OR COLUMN
We define the nervous unit-area as a block of nervous tissue with height and depth R, being the radius of the mean basal dendritic tree of the pyramids in this field. The axonal arborisations in this unit area will be considered as being uniformly distributed in space, and axo-dendritic contact will take place on the terminal branches of the axonal systems being also randomly distributed through the block. So the density of synapses is proportional to the density of the terminal branches of the axonal system.
What are the probabilities of contact of the central neuron of which the whole dendritic tree is located within the unit area, with this axo-dendritic- randomly distributed - synapse system.
We can consider this synapse distribution as a system of points, randomly distributed over the mentioned unit area. Are these points lying in dendritic material or very nearby, then there will be contact. For it is by definition impossible that synapse points are located in space without contact with neighbor dendrites. So we will have to consider this unit areas as being filled up with “Dendritic Space” (D.S), in such a way, that a synaptic point lying in this area is associated with the dendrites that does originate this “Dendritic Space”.
The probability of contact of one point with the exponentially distributed dendritic system (Sholl, 1953, 1953, 1956, 1956) is dependant on the location of this point in relation to the pericarial body.
If a point is located at xp, yp, zp, then the distance between the center of the pericarion and this point is equal to:
D = √ (xp 2 + yp 2+ zp2)
The probability of contact of this point with the basal dendrites of this cell is proportional to the volume of the “Dendritic Spaces” on the half-sphere surface, being equal to:
a.e -k1√(x 2 + y2 + z2) . DS . 1/ sin ɸ DS = surface of a “Dendritic Space” section
ɸ = mean angle of intersection of the dendrites with the sphere surface.
The number of points on this shell is also proportional to the surface of the sphere, for the points are distributed randomly.
So the total probability of contact for one spherical shell equals:
a.e -k1√(x 2 + y2 + z2) . DS . 1/ sin ɸ. k22 . 2 π (x2 + y2 + z2)
K2 = point density constant
P contact on one shell = a.e -k1√(x 2 + y2 + z2) . DS . 1/ sin ɸ. k22 . 2 π (x2 + y2 + z2)
So the mean total number of axo-dendritic synapses Ntot. is equal to:
Ntot = ∫∫∫ (a.e -k1√(x 2 + y2 + z2) . DS . 1/ sinɸ. k2 (x2 + y2 + z2 )) dv =
= ∫0⇒∞ a.e -k1r . DS . 1/sin ɸ . k2 . 2 πr2 dr
= ∫0⇒≈ exp (-k1 r) r2 dr . a. DS . 1/ sin ɸ .k2 .2 π
= a. DS. 1/ sin ɸ. k2 . 2 π.2 (1/ k1)3 = ξ Ntot
In this population of neurons the probability of different numbers of connections will follow a Poisson distribution, for the probability that a dendrite somewhere is connected is small, but N need not to be so. Hence the probability Px that a neuron possesses x connections will be given by the Poisson distribution:
P(x) = (e-n.(N)x) / x! (Uttley, 1954, 1955, 1956)
So we have defined now a block of nervous material with B neurons of which the statistical distribution of synapses in relation to the basal dendritic system is known.
What can we say about the input function now. Before, we assumed the synapse points being randomly distributed over the unit area. We know from quantitative studies of cerebral cortex axonal systems that the input may come from everywhere, but that there possibly is a sort of direction in axonal transport systems.
If we consider all the axons that are making contact with the basal dendritic system of the unit block neurons, the number of axons that are associated to one single cell will also be expected to follow a Poisson distribution.
P(Ax) = (e-A.(A)x) / x! A= mean number of connected axons. P(Ax) being the probability that one neuron makes contact with x axons.
The axons considered are all the axons that possibly may contact to the basal ramifications of the neurons in the block. We assume that the axonal ramification patterns are large in relation to the defined block. Also the number of axo-dendritic synapses given by one axon to one particular basal dendritic system follows such a Poisson distribution
P(Cx) = (e-C.(C)x) / x! C = the number of contacts of one axon to different neurons.
Now there arises the question about probability of contact of one indicated neuron j of the block with one special axonal system i that is penetrating in the basal system of the pyramid.
The number of terminal branches of one axon in the branching dendritic tree is proportional to the “volume” of this “dendritic tree space” . The probability of contact of axon i with neuron j equals: P(Єij) = Ci∫0⇒∞ aj.e -kjr 2 π (r √2)2. DSj. dr . 1/ (½ sphere volume)
= Ci∫0⇒∞ aj.e -kjr .2 π (r √2)2. DSj. dr. (2 ∫∫∫v (x2 + y2 + z2 ) dv )-1
= Ci . aj . 4 π. DSj. 2 . (1/kj)3. 1/(2/3 π(Rj)3)
= 12. Ci. aj. DSj . (1/(kj . Rj)3
(Ci = constant for axonal field density; Rj = dendritic tree radius.)
The mean number of contacts of neuron j with axon-system i is equal to (N/A)
So the probability of having n axonal contact points is equal to
Pn = (e -(N/A) . (N/A)n) / n!
For the previously mentioned (part I) input matrix [Aij] we now can construct a parametric representation.
Let us consider the rows A11 A12 ..................A1n representing the input transmission of axon system 1 into the n neurons of the block. The transmission ability is proportional to the number of axo-dendritic contacts. This number is to be described in the above mentioned statistical way. So the number of contacts in the rows, but also in the columns, will follow the Poisson distribution as represented before:
Pn = (e -(N/A) . (N/A)n) / n!
The effect of firing of one single axo-dendritic contact-point is exponentially decreasing in relation to the dendritic distance of this contact to the cell body.
What is the distribution of these points in space?
The total length Lbd of the basal dendrites equals:
Lbd =∫r0⇒∞ a.e -k1r . 2 π (r√2)2 dr
=∫r0⇒∞ 4. π. a. a.e -k1r. r2 dr
ro = radius of the cell body
√2 = correction for winding of the dendrites
In order to find the length x of the (half) sphere radius in which is located half the length of the total dendritic system, we can state:
½∫r0⇒∞ a.e -k1r . 2 π (r √2)2 dr = a.e -k1r . 2 π(r √2)2 dr
½∫r0⇒∞ e -k1r . r2 dr = e -k1r . r2 dr
L= ∫r0⇒∞ a.e -kr .2.π r2. 2 dr
½ L = ∫r0⇒x a.e -k.r 2.π. r2. 2 dr
∫r0⇒∞ a.e -kr . r2. dr = 2∫r0⇒x e -kr .r2. dr
∫a⇒be -kr .r2. dr = (- 1/k. e -kr. r2 /a⇒b ) + ∫a⇒b -1/k . a.e -kr. 2r dr
= (- 1/k. e -kr . r2 /a⇒b ) - 2/k { (- 1/k. e -kr. r /a⇒b ) + ( -1/k. e -kr dr ) } =
= (- 1/k. e -kr. r2 /a⇒b ) + 2/k (1/k. e -kr. r /a⇒b ) + 2/k2 (-1/k. e -kr /a⇒b ) =
= -1/k. (e -kr. r2 /a⇒b ) + 2/k2 ( e -kr. r /a⇒b ) - 2/k3 (e -kr /a⇒b ).
so
∫r0⇒∞ e -kr . r2. dr = 2 ∫r0⇒x e -kr .r2. dr
is equal to
- 1/k (e -kr. r2/r0⇒∞) + 2/k2 ( e -kr. r /r0⇒∞ ) - 2/k3 ( e -kr /r0⇒∞ ) =
= - 2/k (e -kr. r2 /r0⇒x ) + 4/k2 (e -kr .r /r0⇒x ) - 4/k3 (e -kr. r / r0⇒x )
- k2 (-e -kr0 .r0 2) + 2k (-e -kr0 .r0 ) - 2 (-e -kr0) = -2 k2 (e -kx. x2 - e -kr0. r0 2) +
+ 4k (e -kx .x - e -kr0 .r0 ) - 4 (e -kx - e -kr0 ).
k2 .e -kr0 . r02 - 2 k e -kr0 r0 + 2 e -kr0 = 2 k2 e -kx . x2 - 4k e -kx .x -4 e –kx.
ξ1 = 2 k2 e -kx . x2 - 4k e -kx . x - 4 e -k.x.
ξ1 .e kx = 2k2 x2 - 4 kx + 4
Estimates for the parameters k and ro in frontal area of cerebral human cortex:
k ≃ 0.02 r0 ≃ 15 π
so ξ .e-kx = 1,09. e kx = 2 k2 x2 - 4kx + 4
0,55 . e 0,02 x = 0,02 2 x2 - 4. 0,02 . x + 4
µ1 ≃ 40μ .
Performing the same calculation for finding the distance on which the total amount of dendritic space in two solids is the same, beginning now at a distance µ1
k2 .e -k.40 (40)2 - 2k. 40. e -k.40 + 2. e -k.40 = ξ 2
ξ2 . e kx = 2 k2 x2 - 4kx + 4
0,47 e kx = 2 k2 x2 - 4kx + 4
0,24 e kx = k2 x2 - 2kx + 2
x = µ 2 ≃ 150 µ
The total amount of “dendritic space” material between two spheres with center in the pericarion and with radius x and 00 (or: ∞) is equal to:
F (x) = ∫x⇒∞ a.e -kx . 2 π (r √2)2. DS. dr = a. DS{1/k. e -kx .x2- 2 / k2 . e -kx . x + 2/k 3. e -kx}
Differentiating toward x gives an estimate for the number of contacts on a distance x of one dendritic system with one (or all) axonal systems, for contact is correlated with the Dendritic Space volume.
F’ (x) = d/dx . DS{ 1/k. e -kx . x2- 2 /k2 . e -kx . x + 2/k 3. e -kx } =
= DS{ 2/k. e -kx . x + e -kx . x2 - 2 /k. e -kx - 2 e -kx. x + 2/k 2. e -kx } =
= DS. e -kx {x2 + (2/k -2) x + (2/k2 - 2/k)}.
The number of axonal contacts on distance x equals Kax. F’ (x).
Along the dendrites, conductivity is a decreasing function for the (e-bx.√2) energy input.
The effect of these axonal contacts on the cell when the axon is firing equals:
E = ∫0⇒∞ KAx. F’ (x) . e -bx√2 dx =
= ∫0⇒∞ KAx . e -bx√2 .d/dx. ∫0⇒∞ a.e -kr . 2 ∏ (r √2)2. DS. dr.dx
E = KA ∫0⇒∞ DS. e -kx {x2+ (2 /k -2) .x + (2/k2 - 2/k) }e -bx √2 dx =
= KA. DS∫0⇒∞ e -(b√2 + k) x {x2+ (2 /k -2) .x + (2/k2 - 2/k)} dx =
= KA. DS{ 2/ (b√2 + k ) 3+ (2 /k -2). (-1/(b√2 + k)2) + (2/k2 - 2/k) . 1/(b √2+k)}=
= KA. DS {2/ɸ3- (2 /k -2). 1/ɸ2 + (2/k2 - 2/k). 1/ɸ }
The input matrix [Aij] or [Bij] can be described now in a statistical way:
1 the number of contact of one axon to different neurons follows the distribution: Pcx = (e-c . c x ) / x! ≃ 1/ √(2 π c) . e -(x-c) 2 / 2c 2 KA is directly related in its magnitude distribution to P(cx)
so:
3 The energy transmission Aij from axon i into neuron j when axon i is firing equals E.
THE OUTPUT MATRIX
Let us consider now a system pf input matrix [Aij], a block of neurons B and an output matrix [Cij] .
A neuron in B will fire when the energy (E) intake has exceeded a minimum value. That in the neighborhood of this value noise is playing an important role is useful to mention here.
We can say that for each cell there is a “probability input wave” and a “probability output wave” of axon potentials Tt of which we are able to give measures in space and time if we have enough quantitative histological data concerning the cortex structures.
To describe this system of input and output, it is not useful to consider just only one single time-point (ѱ), for energy transmission of [Aij] t on time t = ѱ, in the “resting” block of neurons B, into [Cij] t+δ , may be a loosing process if this occurs just once in a time and never more, but will not be loosing if there is a flow of input over time.
For not reproduced energy of [Aij ] t = (ѱ) will have a higher probability now to be reproduced in [Cij] (t +1) + δ (δ = time delay).
So only if we consider an input over a time t that is long enough: ∫0⇒t f ([ Aij ]t ) dt ≤ ∫0⇒t f ([ Cij ]t+δ ) dt
If we consider the input [Aij] exactly on time ѱ, the delay (or latency) time for each individual cell to fire will be individually different.
This latency period is a stochastic function to which at least two factors contribute:
1 noise in the dendritic and pericarial system 2 propagation speed differences of energy over the axonal terminal system (input for neurons in B) and dendritic branching trees of the individual cells.
This delay time will (possibly) follow a Poisson distribution.
δij =( e-δm. (δm)Δδ )/ Δδ! = 1/√(2πδm) . e [-(Δδ - δm) 2 / 2 δm ] (*m = mean of *)
FIGURES
fig 1 fig 2 fig 3 fig 4 fig 5 fig 6
REFERENCES
Uttley, A.M. : The classification of signals in the nervous system.
Electroenc. Clin. Neuro-Physiol. , 6 , 479-494, 1954.
Uttley, A.M. :The probability of neuronal connections. Proc. Roy. Soc. B., 144, 229-240, 1955.
Uttley, A.M. : A theory of the mechanism of learning based on the computation of conditional probabilities. 1st Int. Congr. Cyb., Namur, Gauthier-Villars, Paris, 830-857, 1956.
Sholl, D.A. : Dendritic organization in the neurons of the visual and motor cortices of the cat. J. Anat., (London), 87, 387-406, 1953.
Sholl, D.A. : The measurable parameters of the cerebral cortex and their significance in its organization. Progress in Neurobiology, 324-333, J. Ariens Kappers, ed., Amsterdam , Elsevier, 1956
Sholl, D.A. : The organization of the cerebral cortex. J. Wiley and sons, N.Y. 1956.
Sholl, D.A. and A.M Uttley: Pattern discrimination in the visual cortex. Nature. 171, 387-388, 1953.
Soltesz, I. and Stanley, K.: Computational Neuroscience in Epilepsy. ISBN 78-0-12-373649-9, 2008