Mathematical approach to the study of a cerebral cortex

J. Theoret. Biol. (1969) 24, 307-316

Mathematical Approach to the Study of a Cerebral Cortex HI. The Columnar Organization in the Cortex and the Spread of Activity in the Model F. E. LAURIA

Laboratorio di Cibernetica del CNR, Arco Felice, Napoli, ltaly (Received 8 February 1969) The mathematical model of an aspecific portion of cerebral cortex, which was previously studied, has been simulated on a computer. The results thus obtained show that the model exhibits a 'columnar' functional organization, similar to the one typical of the cerebral cortex; the agreement with known neurophysiological and anotomical data seems quite satisfactory. 1. Introduction The present paper has two aims. The first is to verify that the so-called 'columnar' functional organization, already found in the mammalian cerebral cortex, also exists in the mathematical model previously presented (Braitenberg & Lauria, 1960; Lauria, 1965, 1969), the theoretical and experimental values being in good agreement. The second aim is to show that the 'principle of economy' of the maximum excitation proposed earlier (Lauria, 1969), seems to offer a convenient way of calculating the spread of activity in the model. Let us recall that this principle allows us to write down a set of equations which formalize an obvious observation, namely, that the cortex, and thus also the model, is a system that, in a normally functioning state, can interact with the external world only via the natural channels, i.e. only via the input-output nervous fibre systems. It seems useful to review briefly the essential characteristics of the mathematical model under study; a more detailed and complete discussion of the subject is presented in the above-mentioned papers. We recall that the excitation has been defined (Braitenberg & Lauria, 1960) as a measure of the influence of each neuron on all of the others. The dendritic, or axonal, field of the neuron h is that region in which the probability of finding dendritic, or axonal, ramifications of the neuron h is non-zero. The nerve cell will then be described as an ordered couple of functions (q)h(X,XO, y, yo, Z, Zo), ~lh(X, "~0, Y, YO, Z, ZO))'- these represent the 3O7

308

F.E.

LAURIA

TABLE 1 Type

Dendritic fields

Axional fields I]/1

P~ PIII

Pv

Radii o f the base sphere and half sphere of the first layer Zo = 0"2 m m r = 0"05 m m Zo = 1.8 m m

r = 0'25 m m

Rl(zo) = 0-025 + 0-125 go The centre of the base sphere is in Zo The centre of the half sphere of the first layer is in Z = 0 Horizontal semi-axis ax and vertical semi-axis bx of the central ellipsoid Zo = 1 m m a~ = 0-075 m m bl = 0"325 m m Zo= l'8mm a=0"15mm b = 0"65 m m Rl(Z0) = - 0"25 + 1'25 zo The centre of the central ellipsoid is in G.(Zo) = 0-4375 Zo + 0-0875 17111 =

1

Dl12

AI

~

1

1;

17/22

=

11123 =

0

1

2

Dlr

4x104 15"87 4'76

1"5×104 6'33 3"48

Yl, 5a,

3

mas =

ma3 = 0

Aa = 4 × 10 -3 d = 0"16

4

1"2×10 ~ 1'5× 4"7 6'3! 7"55 5"5

Horizontal semi-axis A and vertical semi-axis gz of the ellipsoid f2 = 0"25 m m g~ = 0"025 m m The centre of the ellipsoid is in G a ( z , ) = z l - - 0 " 2 B 2 = 2 × I0 -2a R2 = s2 = 1 m m Cz = 0"156 Da = 4"5 × 104; Y2 = 10"5; 52 = 12"9 I/18

~3

1;

r

I//2

Sphere with centre in Zo radius pa = Ra = 0-075 m m maz ~

10 -9

1

1H13 =

A2 = 5 x 10 -4 d2 = 0.17

F~v

1"5 ×

cx = 2"6

3"5 X 10 -7 dl = 0"63

Sphere with centre in zo, radius p2 = R2 = 0-025 m m 11121 =

BI =

=

~2 G

Horizontal semi-axis and vertical semi-axis # of the ellipsoid Z1 = 0-2 m m f l = 0.075 m m ,qx = 0-04 m m Z I = 1.8 m m f = 0.525 m m g = 0"28 m m Rl(zl) = 0'25 + 3"75 zl The centre o f the ellipsoid is in Ga(zl) = 1 '25 zl -F 0"2

Horizontal semi-axis f3 and vertical semi-axis g3 o f the ellipsoid f3=0'25mm ga=0'15mm The centre o f the ellipsoid is in G3(z~) = zl--0"2 B3 = --1 "5 × 10 -20 R3=sa=lmm ca=0"156 Da = 4'5 × 10a; Ya = 10'5; 53 = 12"9

CEREBRAL

CORTEX

309

TABLE I ( c o n t i n u e d ) Type

D e n d r i t i c fields

A x i o n a l fields I/1"4

~4

F~

Base sphere, centre in zo, r a d i u s Zo = 1.85 m m r = 0"25 m m Zo = 2'75 m m r = 0'2 mm R4(zo)P = 0'354 -- 0'056zo H o r i z o n t a l semi-axis a a n d vertical semi-ax is b of the e l l i p s o i d Zo = 1.85 m m a4 = 0-09 m m b4 = 0"14 m m Zo = 2"75 m m a = 0'128 m m b = 0"2 m m R'4(zo) = 0.11 -- 0'47zo T h e centre o f the e l l i p s o i d is in C4(zo) ----- 1 "38zo - - 1 '608 rn41=l; ma2=l; rn,3=0

Sphere w i t h c e nt re in

G4(zO = zt -- 0.55, radius r Z1 = 1"85 m m r = 0"5 m m Z1 = 2 " 7 5 m m r=0.25mm R4(zl) = 0.85 -- 0 ' 2 7 7 5 z l B 4 = 3 " 5 × 10 -6 c4~3 r

1

2

3

4

D4T 74,.

1.6×104 7.56 16"3

1'4×104 7.56 18.6

0 0

0 0

0

0

~4~

A 4 = 3 X 10 - 7

d~=l

1if5

M

Sphere, centre in Zo, r a d i u s P5 = R5 = 0"25 m m m51=1; msa=msa=0 A5 = 7 × 10 -2 ,/5 = 0"2

Sphere w i t h c e nt re in Z = 0.25 m m ; ss = R6 = 0-25 m m B5=9 × 10-4; cs=0-58 r Ds~ YsT 6~,

~6

Specific afferent

m6z

=

m82

=

1

2

1'6×10 3 1"6×10 3 7"56 7"56 16'3 18"6

3

4

0 0 0

0 0 0

~6 me3

=

0

H o r i z o n t a l s e mi -a xi s f6 a n d ve rt i c a l semi-axis #6 o f t he e l l i p s o i d =0-1mm #6=0"7 C e n t r e in Z = 1 "05 m m ; s6 = R s = 1 m m D6 = 3 × 104; 78 = 31.2; ~8 = 32.81 B8 = 3 × 10 -6 c6 = 0"3

310

F . E . LAURIA

,,

o

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,,

t ~

/\

t,)

t,}

',

'

,

~

i

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o o I

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2 -6

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*

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2.8

3-0 ZO

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Jill

3 4 5x 104 NO.of cells/ram3

FiG. 1. At left are reported the curves assumed to represent the densities (number of nuclei of nerve cells/mm a, on logarithmic scale) vs. the depth of the cortex. At right, the straight lines represent the dendritic fields of some of the neurons which we simulated in the model presented. The dotted lines represent the axonal field. For technical reason, as can be easily observed from the left-hand graphs, the density of the nervous cells has been appropriately reduced (from Lauria, 1969). distrubition o f the density o f the excitation in the dendritic a n d in the axonal field, respectively, o f the n e u r o n o f type/7. The same n e u r o n has the cellular b o d y co-ordinates (Xo, Yo, Zo) with respect to a system o f o r t h o g o n a l axes having the origin on the external surface o f the cortex and the z-axis oriented t o w a r d the white matter. F r o m n o w on, for convenience, we will consider only the dendritic field o f the n e u r o n o f type h, having (Xo, Yo, Zo) as co-ordinates o f the cellular body, and only the axonal field o f the n e u r o n o f type k, having (xl, yl, zl) as co-ordinates o f the cellular body.

CEREBRAL

311

CORTEX

The mutual interactions of the neurons obviously depend on the dendritic and axonal arborization. From Sholl's (1956) anatomical data, we know that the dendritic arborizations decrease exponentially as function of the distance. Then we can take as analytical expressions of the ~0h and Sk the following: tp,, = A,,(mhl exp ( -- dh 2[(x -- Xo)2 + (y -- yo) 2 + (z - Z o ) 2 - - R2(zo)]) + m h 2 e x p [ _ d ~ 2 ( [ ( X _ X o ) 2 + ( y - Yo)2lab-2 bh2 + [ z - Ch(zo)] 2 -- R;,2(z0))] + mh3 exp (-- d~- 2[(x - Xo)2 + (y - yo) 2 + z 2 -- R~(zo)])}, qJk = Bk exp { -- c~-2([(x-- xt) 2 + (y -- y,)2]A-2g~ + [z -- a,(zl)] 2 --R~(zl))}. It is useful to point out that this particular expression has been chosen simply for its remarkable analytical properties; any other fitting the experimental data with the same accuracy, could be chosen. The density of the cellular bodies, as function of depth z, is proportional to Dk(Z1)

~-

n E Dkr exp --[yk, z l t

(~kr]2,

where the constant of proportionality n is one if the density is given in number of cells/mm a. In Table 1 appear all the constants used in the previous expressions. In Fig. 1 the graphs of Dk(zt), for each k, and the shapes of the dendritic and axonal fields, following the dimensions given in Table l, are reported as functions of depth. Finally we recall that the coupling coefficient between a neuron of type k and a neuron of type h, is given by:

a,,k=f ~f tPhOkdxdydz. --00

2. The Columnar Functional Organization in the Cortex

On the basis of anatomical observation (McCulloch, 1947), it was suggested long ago that in the elaboration of the information inside the cortex the vertical connections, i.e. those between elements situated at different depths, are more important than the horizontal ones, i.e. those between elements situated at the same depth. Such observations were confirmed by Szentagothai (1965, 1967) with degenerative techniques, i.e. with a method which permits obtaining quantitative data. More precisely, from the histological evidence (Scheibel & Scheibel, 1958) it has been inferred that from a functional point of view the grey matter was arranged in 'columns'. The activity of an input fibre can at most excite the nerve ceils contained in a cylinder having the same height as the thickness of the cortex, obtained by projecting, orthogonally, on the white substance or on the pia mater, the region in which the active fibre has branched. From the work of Sperry and collaborators, it can T.n.

21

312

F. E. L A U R I A

be considered as proved that the horizontal spreading of the nervous activity is negligible. After having cut into small squares the sensory-somatic and motor cortex of a monkey, the cuts affecting the entire griseum thickness, it was not possible to observe somatic-sensory deficiencies or motor defects (Sperry, 1947). Analogously (Sperry, Miner & Myers, 1955), no difference was noted in the visual discrimination and in the behaviour of cats after operations of this sort were performed on the visual cortex, Brodman's areas 17, 18 and 19, or after tantalium wires were implanted horizontally in the cortex, or after (Sperry & Miner, 1955) mica plates were implanted vertically so as to isolate small portions of the grey matter. Apparently it has been possible to ascertain, by means of neurophysiological techniques, the existence of columns as functional units o f the cortex in the somatic-sensory (Mountcastle, 1957; Mountcastle & Rose, 1959), visual (particularly remarkable and complete are the works of Hubel & Wiesel, 1962, 1963a,b, 1965, 1968), and motor areas (Landgren, Phillips & Porter, 1962; Asanuma & Sakata, 1967; Asanuma et al., 1968). However, there seems to be no clear evidence of functional organization in the acoustic area (Evans & Whitfield, 1964; Evans, Ross & Whitfield, 1965; Whitfield & Evans, 1965; Katsuki, 1966; Evans, 1968). This can be due either to structural reasons (e.g. see Colonnier, 1966, p. 19), since in this area stellate cells of exceptional dimensions are present, or to the fact that it is not yet clear which parameters are to be taken into account. The functional properties which characterize the organization in columns vary from area to area. Thus there are columns of neurons, in the somaticsensory area, which respond to skin stimulations and columns of neurons responding to stimulations of deeper tissues. In the visual area, there are columns of neurons which respond to the movement of a figure in a certain direction. And finally, the neurons of the motor area, which stimulate monosynaptically a motoneuron of the spinal cord, belong to one and the same column. Although it is not yet clear which is the law by which the columns are arranged (Hubel & Wiesel, 1963a), it seems probable (Hubel & Wiesel, 1965, 1968) that there exist more than one system, each depending on a different parameter, e.g. ocular dominance, movement direction, etc. The exact relation between this functional partition of the cortex and the afore-mentioned partition on anatomical bases seems to be quite well proved both by the recordings made on newly-born kittens, i.e. without any visual experience (Hubel & Wiesel, 1963b), and from the minimum dimensions which the various authors, in good agreement, ascribe to the columns. These dimensions, in the sensory areas, coincide with those of the cellular bodies (Mountcastle, 1957; Hubel & Wiesel, 1962, 1963a, 1965, 1968), and in the motor area they are of about 1 mm 2 (Landgren et al., 1962; Asanuma &

CEREBRAL CORTEX

313

Sakata, 1967). Since in the first case they are recorded with micro-electrodes in the cell bodies, and therefore the dendritic ramifications must be taken into account, it is tempting to conclude that we can identify the same entity both from a functional and structural point of view.

3. The Columnar Functional Organization in the Model Let us now consider the quantity of excitation, Eh, received by the dendritic field of a type h neuron of depth z o when all the neurons are active, i.e. the maximum quantity of excitation which can be received by any neuron. This quantity is given by

f;f

Eh=~Ahk=~ Dk(Zl)ahkdXtdyldzl. -oo

The exact expressions of the E h, easily obtainable from those of the qh,, ffk and Dk(z~), are given in the Appendix of a previous work (Lauria, 1969). The values of E h, calculated with the aid of a computer are, for each h as functions of the depth Zo, represented by the leftmost graphs in Fig. 2. It is possible to obtain the same values in the following way. Let us consider the neurons arranged, for each k, in the vertices of a cubic lattice. F o r each k and zl, the sides of the lattice are chosen so as to represent the mean separation of the cell bodies: such a datum is given in the last column of Table 1 of the previously quoted works (Braitenberg & Lauria, 1960; Lauria, 1965). Thus the mean values of the density in the model and in the cortex will be the same. Now it is possible to calculate, as function of r o, the quantity of excitation received by the dendritic field of the neuron (0, 0, Zo), when the neurons (xl,y 1, z 0 contained in a cylinder of radius r o having the same height as the thickness of the cortex for which x l~+2y 2 ~< ro2, are supposed active. Such a quantity is obviously given by the sum of the quantities of excitation that each of the latter can transmit to the former, i.e. it is sufficient for each h to calculate the sum 6

2"75

Z z l2= 0

k=l

ro

Y,

xl,yl=--ro

with xt, Yt and zt varying on the vertices of the cubic lattice previously defined. When ro increases, the value of the sum approximates, better and better, the value of Eh, for the same h and Zo. Two times the ro for which the sum is equal to Eh, is the diameter of the cylinder which, in the model, is the equivalent of a column in the cortex. The results of the calculation are represented in the rightmost graphs of Fig. 2. The values chosen for Zo are reported near each curve, and those for ro are successively 0-1, 0-2, 0-3, 0.4,

314

F . E . LAURIA

Zo ~

' E~o,Z°I HO-6 ,0-~' -o2 i0-o

Subpiol surfoce

0.2

(~1, ¢zO

~o-~

0.4 - (~ z,]~z) ......

• ~•~0~•

(¢'3,¢'3)......... 0.6 -(~4,¢t4) (¢5,¢I5)

~•~

•

~I 0 - " - 0 " 4

10-7 -5 ,,0_°_06

........

10-7

......

-5 ,Ig-,-o.8

0.8

10 - 7 10-5

, 0-6-1,0

1.0

,o-7 :::::::::::::::::::::

~o

•.....•....*•,..,.•.*...•...:

•.....•.....•-....•.....•.....

I-2 10__4

~ .-O- .--O-.--O----O---

o~,O--O--O--O--

I-4

~o

1.6

/

/ O~

2.2 2.4

i I

i

i

iI

jI

10-2

17=

° i,8 110-6-2, 0

T

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10-'-22 _ i0-5 o-~'"

-0- -O--O-'

10-7 10-65

o~O~.O---@

--O-"

~o.-o,-o'-

I -3 I00 - 4 - 2 - 4 ~ ..- - - e---o.--o--

i

r

-5

10-~ -~.u i0-~ ,tO-~

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• ~'•-

10-~-1.15 I0-~1 10-=-1.25 i0-~

-3 I0 4 ~

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2"0

10-~-1.05

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1.8

Io-.3

e.

2 ~

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I°_,_2.4 10-7

-3

1~-4 -2-6 ~0.-0--o-10-5 e-~e"

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I

10-t10-2]0-310-410-510 -6 --,---Em~,

I

I

I

l

[

0-1 0.20.3 0.40.50.6 ro~

I

f

I

1

[

0-1 0.20.30.40.5 0,6 ro---~--

FIG. 2. The leftmost curves represent for each h the threshold of the neurons with depth zo. The rightmost curves, for the various depths indicated, give the value of the maximum excitation that could impinge upon each neuron vs. the values, indicated by the abscissa, of the radius of the columns in the model (for further explanation see text). 0.5, 0-6 mm. As it is easily seen, each curve at the right tends to the asymptotic value, represented by the abscissa o f the corresponding curve at the left, for the same value o f %. It is interesting to notice that, for the great majority o f neurons, the asymptotic value is practically attained for 0.3 -%
CEREBRAL CORTEX

315

4. Conclusions

It seems appropriate to stress here the importance of having been able to recognize, anatomically and functionally, an aggregate of neurons which quite probably is the operating unit of the cortex. As already observed by others (Colonnier, 1966), it is wise to recall that the columns should be arranged in an intertwined manner. Neurophysiology (Asanuma & Sakata, 1967) confirms that the intersection of different columns is not empty: they should be compared to the tree tops of a thick wood where the branches of each tree touch not only the branches of the neighbouring trees but also of those farther away, rather than to the tesserae of a mosaic placed side by side. We now wonder what can be the average density of the columns per mm 2. This value is obviously given by the square of cubic root of the density per mm 3, of the neurons in the grey matter; estimating such values as 1.5 x 104, which is certainly an under-estimation of the real value (e.g. see Pakkenberg, 1966), we deduce that there is an average of about 6x 102 columns/mm 2. This must be taken into account, otherwise the simple description of the cortex as an aggregate of columns, more or less homogeneous functionally, would mean reproposing t o u t c o u r t the academic and detailed parcellation which is the result of cytoarchitechtonics. It is interesting to notice that having obtained the parameter contained in the Eh, with Ba < 0, permits the computer simulation of a column, with the hope of obtaining some suggestions about the input-output relationships of one such cortical subunity. REFERENCES ASANUMA,FI., STONEYJR., S. D. • ABZUG, C. (1968). J. NeurophysioL 31, 670. ASANUMA,H. & SAKATA,H. (1967). J. Neurophysiol. 30, 34. BRAITENBERG,V. & LAURIA,F. (1960). Nuovo Cim., Suppl. 2, 18, 149. COLONNIER, M. L. (1966). In "Brain and Conscious Experience" (J. C. Eccles, ed.). Berlin: Springer Verlag. EVANS, E. P. & WHITFIELD,I. C. (1964). J. Physiol., Lond. 171, 476. EVANS, E. P., ROSS, H. F. & WHrrEtELD, I. C. (1965). J. Physiol., Lond. 179, 238. EVANS, E. F. (1968). In "Neural Network" (E. R. Caianiello, ed.). Berlin: Springer Verlag. HUBEL, D. H. & WIESEL,T. N. (1962). J. Physiol., Lond. 160, 106. HUBEL, D. H. & WIESEL,T. N. (1963a). at. Physiol., Lond. 165, 559. HUBEL, D. H. & WIESEL,T. N. (1963b). J. Neurophysiol. 26, 994. HUBEL, D. H. & WIESEL,T. N. (1965). J. Neurophysiol. 28, 229. HUBEL, D. H. & WIESEL,T. N. (1968). J. Physiol., Lond. 195, 215. KATSUKI,Y. (1966). In "The Thalamus" (D. P. Purpura and M. D. Yahr, eds.). New York: Columbia University Press. LANDGREN, S., PHILLIPS,C. C. & PORTER, R. (1962). 3". Physiol., Lond. 161, 112. LAURIA, F. E. (1965). J. Theoret. Biol. 8, 54. LAURIA,F. E. (1969). 3". Theoret. Biol. 23, 72. McCULLOCH, W. S. (1947). Fedn Proc. Fedn Am. Socs exp. Biol. 6, 448. MOUNt'CASTLE,V. B. (1957). or. NeurophysioL 20, 408.

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MOUNTCASTLE,V. B. & ROSE,J. E. (1959). bl "Handbook of Physiology," Vol. I (J. Field, ed.). Washington, D.C.: American Physiological Society. PAKKENBERG,H. (1966). J. comp. Neurol. 128, 17. SCHEIBEL, M. E. & SCHEmEL, A. B. (1958). In "Reticular Formation of the Brain" (H. H. Jasper, L. D. Proctor, R. S. Knighton, W. C. Noshay, R. T Costello, eds.). Boston and Toronto: Little Brown. SPERRY, R. W. (1947). J. NeurophysioL 10, 275. SPERRY, R. W., MINER, N. & MYERS, R. E. (1955). J. comp. physiol. Psychol. 48, 50. SPERRY, R. W. & MINER, N. (1955b). J. comp. physiol. Psychol. 48, 463. SZENTAGOTHAI,J. (1965). Syrup. Biol. Hung. 5, 251. SZENTAGOTHAI,J. (1967). bt "Recent Development of Neurobiology in Hungary," Vol. I (K. Lissak, ed.). Budapest: Akademiai Kiado. WmTFIELD, I. C. & EVANS, E. F. (1965). J. Neurophysiol. 28, 655.