recursive features of circular receptive fields

Biol. Cybern. 43,199-208 (t982) Biological Cybernetics �9 Springer-Verlag 1982

Recursive Features of Circular Receptive Fields

Georg Hartmann

Fachbereich Elektrotechnik-Elektronik, Universitgt-Gesamthochschule-Paderborn, Paderborn, FRG

Abstract. The distribution of excitability in retinal receptive fields may be well approximated by functions with, recursive features. Physiological data do not exclude an implementation of recursive structures in the visual system. It is the most remarkable advantage of a recursive visual system, that cortical receptive fields tuned to different spatial frequencies will have an identical neuronal circuitry. Structural consequences for retina, LGN and visual cortex are discussed.

Introduction

Retinal receptive fields show the outstanding feature, that the weighted sum of 19 overlapping distributions of excitability resulting from adjacent fields has the shape of just a double sized circular receptive field. This may be sheer accident, of course. On the other hand, a visual system with recursive features would be very advantageous. A recursive system would provide equivalent information to the visual cortex as a set of n virtual retinas with receptive fields of the 2"-fold diameter. Hence the different spatial frequency channels (Braddick et al., 1978; Maffei, 1978) of the visual cortex would receive information from different virtual retinas, the neuronal circuitry, however, would be identical.

It is very easy to see the advantages of a recursive visual system, but it is not at all easy, to show, that a recursive structure may be implemented by weighted summation of retinal or LGN spike rates. The proof must be restricted to the simple case of moving a spot of light with constant luminance within homo- geneously iluminated receptive fields. A complete discussion of the spatio temporal features of the recursion is out of the scope of this paper.

Only two assumptions about the spatial arrangement of retinal receptive fields are necessary. It is assumed, that there exist pairs of equilocalized on-

center and off-center receptive fields, and that they are arranged at intersections of a triangular grid. These conditions seem very artificial, on the other hand they are hardly contradictory to experimental results. Both conditions, however, are natural consequences of a simple model of retinal receptive fields, which will also be presented. Finally the question will be discussed, whether a recursive structure is only possible or really implemented in the visual system. A recursive structure, however, will have some consequences, especially for the behaviour of the LGN and the visual cortex but also for the retinal structure. Experiments are discussed which will possibly verify these features and decide, whether the visual system is recursively organized.

A Class of Functions with Recursive Features

A class of functions will be investigated, which has the same shape as the distribution of excitability of retinal receptive fields. It is very essential, however, not to confuse pure mathematical results with models of the visual system. Hence the question, whether the recursive features of these functions are used in the visual system, the structural consequences, and the benefits shall be discussed later. At this moment only the recursive features of functions with shape E(Q) are investigated and not the properties of a distribution of excitability, which are described by a nonlinear spatio- temporal function E(Q, Lo, t), depending also on average luminance L o and time t.

It is the purpose of this chapter, to show that E*(Q*), the weighted sum of 19 overlapping functions E(Q), has the same shape as E(Q), but double the size, so that E*(Q*)=E(Q) for Q* =2Q. In the sequel it will be convenient, to describe a point Q by a vector Q, pointing from the origin to point Q instead of using cartesian or polar coordinates (details of nomenclature see Appendix 1).

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200

(a) (a)

E'(~")

EI(~")= 1

(b} /-- E ' ~ l ) = - 6

(b)

J Ell}

[ E(~)= I

- - U~v-'v- E{~)=/3 -.

! , / EI~)=-6

(c)

Fig. la-c. A linear superposition of seven overlapping disc shaped functions F(O) with centers at X and Uv is called E(Q). The coordinates of a point may be expressed in the coordinate system of E(Q) by a vector Q or in the coordinate system F(Q) by vectors ~x or O,v. A top view a and a section b along U - U explain the arrangement. The result of the superposition e is rotary symmetric in the central part of E(Q) with 0=<IQI__<2R. In the peripheral area, section U - U and V - V are only slightly different

Though the well known shape of E(Q) is very simple (Fig. lc), it would be very complicated to analyze the superposition E* of 19 functions E. Fortunately E(Q) may be well approximated by a superposition of 7 disc shaped functions F(Q) (Fig. 1).

Fig. 2a and b. A linear superposition of 19 functions E(Q) with centers at X*, }1", U*, and V* is called E*(Q*). The coordinates of a point may be expressed by a vector Q*, related to the coordinate system E*(Q*), or by vectors Qx, Qyv, Q,v, and Q,~, related to the individual systems of E(Q). The distances between X*, Y*, U*, V*, W*, and Z* are 2R (half the scale as Fig. 1) and each of these point is also center of a function F(Q), building up E(Q). The shape b of this superposition E*(Q*) is very similar to that of E(Q) and the values of E*(Q) at Q* =X*, Y*, U*, and V* are identical with the values of E(Q) at Q = X, Y, U, and V

This approximation simplifies the discussion of E* drasticly. Of course the shape of E(Q) will depend on the shape of its components F(Q). There exists a family of functions F(Q) with the surprising common feature, that E(Q) shows identical sections along the axis U - U and V - V in the central area with [Q[<2R. One member of the family F(Q) continues this symmetry between section U - U and V - V also into the peripheral area of E(Q) (Fig. 3c). A complete derivation is given in the Appendix 2.

The above mentioned recursive features of this special shaped functions E(Q) can easily be exhibited, if the 19 overlapping functions are arranged in an equidistant triangular grid. The intersection points of this grid have a distance of 2R and are labeled by X*,

201

E , E "

V e u h g f jZ i wtm n k

I / / .~----~-._ (c)

4R 6R

(e}

j , Q.Q"

Fig. 3a--e. Sections U- U (closed line) and V- V(broken line) of five different functions E(Q) show, that only one shape of E(Q) extends rotary symmetry also to peripheral areas e. The recursions E*(Q*) are also plotted against E(Q) with a scale factor Q*=2Q Best

Table 1

Number of functions Position Weight Symbol

1 X* x E., (Q) 6 Y* y E~. (Q) 6 U* u E,, (Q) 6 V* v E, (Q)

Y*, U*, V*, W*, and Z* (Fig. 2), The 19 overlapping functions E(Q) have the following positions and weighting factors (Table 1). Each function E(Q), however, may be decomposed into seven disc shaped building blocks F(Q), and the centers of these discs will also be located at the positions X*, Y*, U*, V*, W*, and Z*. As an example, the central disc of E~ will be at position X*, the peripheral discs of E~ at the positions Y*, overlapping with the central disc of Ey, while the peripheral discs of Ey will have the positions X*, Y*, V*, U*, V*, and Y*. The positions W* and Z* are also filled by peripheral discs of E, and E~. By this superposition the discs fill up a circular area, which has just double the diameter of E(Q) and the shape E*(Q*), as shown by Fig. 2.

The coinciding discs F(Q*) of adjacent functions E(Q*) may be easily balanced now at the points X*, Y*, U*, V*, W*, and Z*, providing 19 values of the function E*(Q*) at the coordinates Q* =X*, Y*, U*, V*, W*, and Z*. Intermediate values of E*(Q*) may also be calculated, however up to four discs F(Q*) may overlap at an arbitrary coordinate Q*. Detailed calculations are reported in the Appendix 3.

The values of E*(Q*), depending on the weighting factors and the shape of E, will change with variations of these factors x, y, u, and v. That means, the weighted sum E* of 19 overlapping functions will only have the shape of the functions E for one special set of weighting factors. The recursion condition E*(Q*)=E(Q) with Q* =2Q implies four equations

E*(X*) = - 6x + 6y = - 6 = E(X), (1)

E*(Y*) = x - 4y + u + 2v = c( = E(Y), (2)

E*(U*) = y - 6u + 2v = 1 = E(U), (3)

E*(V*) = 2y + 2 u - 6v = {/= E(V) (4)

from which the weighting factors x, y, u, and v may be calculated (see Appendix 3).

This is already a remarkable result, as (1), (2), (3), and (4) force an exact equivalence E*(Q*)= E(Q) at 19 coordinates. The values of e,/3 and hence the values of x, y, u, and v will depend on the shape of the selected

correspondence between E(Q) and E*(Q*) is just achieved by the function with optimal symmetry e. Functions 1-3, 5, and 6 in Fig. 8 correspond to e, d, e, b, and a

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disc function (Fig. 8) but any member of the family F(Q) will provide a solution.

These different solutions, however, will not produce recursions of equivalent quality. The deviations appear, if E*(Q*) is compared with E(Q) at coordinates Q*=W*, Q*=Z* and at arbitrary intermediate coordinates Q* between centers of discs. All weighting factors are already fixed by (1), (2), (3), and (4) and there are no adjustable parameters in addition to u, v, x, y, if a member of the family F(Q) is selected. Fortunately just one member of F(Q), however, minimizes the above mentioned discrepances (Fig. 3), and this special F(Q) is just the same, which provides maximal symmetry in E(Q) itself.

As a summary one can say, there exists one special shaped function E(Q), which is identical with its double sized superposition E*(Q*) at 19 coordinates X*, Y*, U*, V* and which is very well approximated at all the other coordinates.

How to Realize Recursive Structures in the Visual System

There is a surprising similarity between the shape of the function E(Q) and the well known distribution of excitability in retinal receptive fields. This similarity may be a pure accident, but it can not be excluded, that the distribution of excitability has just this special form, to make use of the recursive features of E(Q). So it seems worth to ask, how recursive structures could be realized in the visual system.

One can show, that the weighted summation of spike rates from retinal ganglion cells is equivalent to a weighted superposition of the distributions of excitability belonging to corresponding receptive fields. This is also valid with some additional remarks for the weighted summation of LGN spike rates in cortical cells.

The distribution of excitability E(Q) shall be de- termined for all the 19 overlapping retinal receptive fields Ex, Ey, Eu, and E v. E(Q) relates the variation of spike rate dS to the position Q of a spot of area dA with increased luminance L o + dL(Q), moving through the homogenously (Lo) illuminated receptive field and E(Q) shall be defined by the following equation

dS = dL(Q)- E(Q). dA. (5)

As the 19 overlapping fields are located in a very small retinal area, they will have the same size and shape. The change of spike rates, however, caused by the same dL(Q*) will be different for E~, Eye, Eu~, and Ev~ (v= 1 ... 6), as their centers have different locations. The change dS* of the weighted sum of spike rates from 19 retinal ganglion cells, caused by a spot dA*

with increased luminance dL(Q*) will be

[ ~ dS*=dL(Q*). X.Ex(Q*)+ ~ (y.Eyv(Q*)

u. E,~(Q*) + v. Ev~(Q*)) ]dA* (6a) +

with

6

dS* = x. dSx + ~ (ydSyv + udS,~ + vdSvv). (6b) v = l

Each term in (6a) is related to the corresponding term in (6b) by (5), and one can easily see now, that dS* is equivalently expressed by a weighted sum of spike rates (6b) or by a superposition of distributions of excitability (6a).

If the shape and the spatial arrangement of the retinal receptive fields satisfies the conditions of the preceding chapter, the sum over the distributions of excitability in (6a) may be substituted by the recursion E*(Q*), so that

dS* = dL(Q*). E*(Q*). dA*. (6c)

As E*(Q*) has the same shape and double the size of E(Q), the change dS* of the weighted sum of spike rates is equivalent to that of a double sized retinal receptive field. At this moment, there is only an assumption about the spatial arrangement of overlapping retinal receptive fields, which will be discussed at the end of this paper. Thus a weighted summation of spike rates from on-center neurons would reproduce a double sized on-center field and summation of spike rates from off-center units would be equivalent with a double sized off-center field. This model however is too simple, as the nonlinear relations between dS and E(Q) are not yet taken into account.

The problem is caused by the assymmetric character of the spike rate S relative to the maintained discharge S 0. E*(Q*) is a linear superposition with a linear balance between positive and negative values of overlapping functions E(Q*) at point Q*. In the case of overlapping receptive fields, positive changes of spike rate dS must be balanced by negative changes, which may be cut off, as the absolute spike rate may only be reduced to zero.

This difficulty disappears, if the whole information of the antagonistic system is processed. At this point a second assumption about the retinal structure is necessary. It is required that there exist antagonistic pairs of retinal receptive fields with coinciding centers. The distributions of excitability shall be called E+(Q) for the on-center field and E_(Q) for the off-center field. Both distributions E+ and E of an antagonistic pair will have a similar shape and opposite sign. A spot of

light with area dA and increased luminance dL(Q) will cause spike rate changes dS+ and dS_ of opposite sign at both the ganglion cells of an antagonistic pair of receptive fields. The difference

dS = dS + - dS_ = dL(O) - I-E + (Q) - E _ (O)]dA

= dL(Q). E(Q)dA (7)

has the nice feature, that it is symmetric for positive and negative light increments dL(Q).

[E+(Q)-E_(Q)] is the average shape between E_(Q) and E+(Q), called E(Q) in (7).

Now one can say more precisely, that neither on- center nor off-center fields must have a distribution of excitability with the correct shape of the discussed recursive function, but only the difference E+(Q)-E_(Q). This difference is intentionally called E(Q) again and also dS+ - d S _ is again symbolized by dS in (7). With this nomenclature (5), (6a), (6b), and (6c) remain unchanged, one must only keep in mind that in the sequel dS and E(Q) describe properties of an above defined antagonistic pair, and that the relation (5) is symmetric now. Thus the weighted sum of spike rates from overlapping antagonistic pairs with average shape E(Q) will provide the equivalent information as the spike rate of a double sized antagonistic pair of retinal receptive fields,

The question, where this lntbrmation will be processed, shall be treated in the chapter about structural consequences. Another fact, however, concerning the concept of antagonistic pairs, shall be discussed. If only the weighted sum of spike rates dS+ or dS_ would be processed, the signal to noise ratio would deteriorate with each recursion, as the maintained discharge would cumulate. The spike rates from antagonistic pairs, however, only contain the difference between the maintained discharge rates of the on-center and the off-center units.

Until now the recursive features of retinal receptive fields are only discussed with regard to the response on a spot of light on a homogenous illuminated back- ground. The question, however, whether the response of an antagonistic pair of retinal receptive fields to an extended stimulus and the response of a recursion to a stimulus of double the size will be equal, can not easily be answered. An explicit treatment of nonlinear features of retinal receptive fields, including spatial summation, would be necessary to answer this question. This applies also to the question, how spatio-temporal properties of retinal receptive fields will be reproduced, e.g. whether the response of an antagonistic pair of retinal fields is the same as the response of a recursion to a grid of half the spatial frequency.

The tenet of this chapter, however, will be, that in spite of these open questions a recursive structure can

203

not be excluded. Hence the structural consequences for the visual cortex and for the spatial arrangement of the retina shall be discussed.

Structural Consequences and Advantages of a Recursive System

In a recursively structured visual cortex, the same information will be available, as if each point of the retinal image would be transformed by an antagonistic pair of retinal receptive fields with a diameter d o and additionally by a set of virtual fields with diameters d, =2".d0. Of course d o and hence d, depend on the excentricity of the retinal location. With this information, available in a recursive cortex, it is very easy to build up high order cortical filters like simple, complex and hypercomplex receptive fields. High order cortical filters tuned to the highest spatial frequency are assumed to receive information from the real retinal receptive fields with diameter d o . Cortical filters tuned to half the frequency would process only information, available from the first recursion, which is equivalent with information from virtual retinal receptive fields of diameter 2d o and cortical filters tuned to the 1/2"-fold spatial frequency would receive information from recursion n. This concept would provide a very effective cortical structure, sketched in Fig. 4, with an identical circuitry for each recursion R, and with a second type of circuitry for the high order cortical filters Fn, again identical for all the spatial frequencies. The circuitry of the cortical filters F, shall be presented in a separate paper, here only the building blocks processed by the recursive circuitries R, shall be discussed.

The fine structure of the recursive circuitry is sketched in Fig. 5. The 19 overlapping antagonistic pairs of receptive fields Ex, E : E,, and E~ are symbolized by (for clearity non overlapping) circles, the corresponding pairs of retinal ganglion cells by small circles and dots for on-center and of center units respectively. At this moment, the LGN shall be assumed not to change relation (7), so that the change of spike rates of an antagonistic pair of retinal ganglion cells is proportional to the change of spike rates of the corresponding pair of LGN neurons. The weighted summation of the spike rates from 19 pairs of LGN neurons shall be executed in the cortical neuron C+. All pulse rates Sx+, Sy+, S,+, and S~+, originating from an on center retinal receptive field will act as excitatory inputs to C+ with gain factors x, y, u, and v respectively, pulse rates Sx_, Sy_, Su_, and S~_ will inhibit C+ with corresponding gain factors.

Without a stimulus, the 19 antagonistic pairs of retinal receptive fields will only produce approximately equal rates of maintained discharge at 19 excitatory

204

i

q

i

r

Fig. 4. Antagonistic pairs of receptive fields in retina (Ro) and antagonistic pairs of recursions (R1,R2,R3) are symbolized by rectangular boxes. For clearness not an area but only one row is displayed without overlap and with only one symbolic output for each antagonistic pair. The 19 pairs of inputs to the recursive network are sketched by five inputs from five adjacent antagonistic pairs, symbolizing U, Y, X, Y, U of section U - U in Fig. 2. Orientation selective high order cortical fields (simple, complex, hypercomplex), tuned to different spatial frequency channels, are represented by the boxes Fo, F1, F2, and F 3. The highest spatial frequency channel receives information from the retinal receptive fields (Ro), while the lower spatial frequency channels use the circular receptive fields of the virtual retinas R1, R2, R 3 as basic building blocks

Retina

LGN

Cortical Cell C+

Fig. 5. The information from 19 antagonistic pairs of retinal receptive fields (for clearity symbolized by non-overlapping circles) is processed by the cortical cell C+. The spike rates S+ from the on- center fields (small open circles) are excitatory, the spike rates S_ from the off-center fields (small filled circles) are inhibitory to C+. The influence of the LGN is discussed in the text

and 19 inhibitory inputs of C+. Wi thout detailed statistical considerations one can say, that the average excitation of C+ will be zero, and that there will be statistical fluctuations between inhibition and excitation due to the r andom character of the input rates. Hence the output of C+ will also be a small r andom spike rate (Levick, 1973).

If any point Q* of the assembly of 19 pairs of retinal receptive fields is stimulated by a spot of light with increased luminance dL(Q*), the weighted sum of spike rates at C § will cause an excitation of C+ if the weighted sum of spike rates at the exciting inputs exceeds the weighted sum of spike rates at the inhibit- ing inputs. The change of spike rate dS*+ of this cortical cell C+ will be propor t ional to this excitation,

and

[ 6 ] dS*+ ~ x'dSx+ + ~ (y.dSy,+ +u.dSu~ + +v.dSv,+)

v = l

- x.dSx_+ ~ (y.dSrv_+u.dS~,_+v.dS~_).(8)

The spot of light, however, will inhibit the cortical cell C+, if the weighted sum of inhibitory spike rates at C+ exceeds the excitatory spike rates, and dS*+ will be suppressed. Thus the response of C+ is just the same, as that from a retinal on center neuron of a double sized receptive field.

A second cortical cell C_, with the same inputs as C § in Fig. 5, will be antagonistic to C +, if all inhibitory

205

and excitatory inputs are interchanged. C_ will just be excited and respond with a rate S*_ if the antagonistic cortical cell C+ is inhibited and S*+ is suppressed. Thus the response of C_ is equivalent to that of a retinal off-center field and C+ and C_ will form an antagonistic pair of circular cortical receptive fields.

The antagonistic pairs C+, C_ in the visual cortex would represent the first recursion R~ in Fig. 4 of the antagonistic pairs of retinal cells in Ro. Hence Rt may also be seen as a virtual retina with pairs of cortical cells C +, C equivalent to virtual retinal ganglion cells of virtual receptive fields with double the size. At this point the recursion is completed and a second recursion can start, leading to a second virtual retina R 2 with receptive fields of double the size as in R~ and of quadruple size as in the real retina R o (Fig. 4). The fine structure in the second recursion is the same as in Fig. 5 when retina is replaced by virtual retina and LGN is omitted.

If n recursions are implemented, this last recursion will produce a virtual retina R, which will provide the building blocks for the high order cortical filters, tuned to the lowest spatial frequency. Each cell in R,, however has 19 pairs of inputs from virtual retina R,_ t, and each cell of R,_ 1 again 19 pairs from R._ 2 and thus in the last step, a circular area of retinal fields will be involved. Additional retinal inputs to this network would be redundant for the filters, formed at R,, less retinal inputs, however, would provide incomplete information to R,. Complete networks of this kind should be prefered by nature and hence a columnar cortical structure would be the result of this optimization.

A recursive structure of the visual system would also have consequences to the spatial arrangement of the retina itself. It was already necessary in the last chapter, to assume, that retinal fields overlap in the same way as it was assumed for the recursive functions E(Q), and also that there exist equi-localized pairs of antagonistic retinal receptive fields. These assumptions look very artificial at the first moment. Both, however, would be fullfilled in a very natural way, if the retinal receptive fields would be really composed by disc shaped overlapping retinal areas with a distribution of excitability F(Q). It shall be remembered, that a distribution of excitability E(Q) of a retinal receptive field may be well approximated by seven discs F(Q) (Siminoff, 1980).

The minimum size of a disc shaped retinal area with a distribution of excitability like F(Q) would be a cone with maximum sensitivity in the center. There will be still some sensitivity also beyond the radius of the cone resulting from stray light, and the distributions of excitability would indeed overlap (Fig. 6a). Foveal receptive fields could be formed by these minimum size

(a)

~b)

I> 3 (c)

F(?)

/ 'd' . Ir~l

Fig. 6a-d. A disc-shaped function of excitability d, may be achieved by a single cone a, by small b and by large pools e of cones. Stray light effects and electrical coupling (only symbolized !) are responsive for the correct recursive shape

discs, while peripheral retinal fields would be formed by pools of cones (Fig. 6b and c). In the case of a pool, the cross structure of the disc shaped distribution of excitability E(Q) would be the result of electrical coupling between adjacent cones. Without calculations one can say that the indirect inputs by electrical coupling will decrease with the distance from the considered cone, leading to a doughnut shaped distribution. Also in this case, where the disc is formed by a pool of cones, stray light effects will smooth the shape of the distribution of excitability.

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A retinal receptive field would be a combination of seven discs. The central disc would have a positive weighting factor in an on center field and a negative weighting factor in an off center field, while the peripheral discs would have opposite sign. This structure would necessarily lead to antagonistic pairs of retinal receptive fields with coinciding center and also to the correct spatial arrangement, where the central discs of adjacent fields coincide with peripheral discs of the considered field.

Discussion

It is a mathematical fact, that functions with a shape E(Q) are recursive and it is a result of physiological experiments, that the distribution of excitability of retinal receptive fields has a very similar shape. This similarity may be a pure accident, it may not be excluded, however, that just the recursive features of this distribution of excitability are used in the visual system.

It turned out, that a recursive structure of the visual system may be established on the base of well known neuronal structures. Only two assumptions about the spatial arrangement of retinal receptive fields are necessary, and both would be consequences of a proposed retinal structure. But also a retina with another structure, and with the same spatial arrangement of receptive fields would support a recursive visual system.

The advantages of a recursive visual system with a set of virtual retinas in the visual cortex is evident. High order cortical receptive fields tuned to different spatial frequencies would use different sized circular receptive fields as building blocks, which are provided by recursion in the different virtual retinas of the cortex. An identical structure for all filters of different spatial frequency and a columnar structure would be the consequence.

The question, however, whether this advantageous structure is really used in the visual system, can not be answered at this moment. Some consequences of a recursive visual structure, however, could be verified by experiments:

A recursive structure would be very likely, if in a small cortical area, e.g. a column, a set of circular antagonistic receptive fields would be discovered. This set should contain receptive fields with diameters 2". d o compared with the diameter d o of the corresponding retinal receptive fields. This experiment will not be very easy, as the number of circular cortical fields should decrease by a factor 1/4 from each virtual retina to the next one, and only one circular field of 16d o would be found within an area covered by 256 retinal fields of diameter d o .

On the other hand, if no recursive structures are used in the visual system, large sized cortical receptive

fields should have tenthousands of inputs from LGN, and cortical neurons with tenthousands of synapses should be found.

it would be also a very interesting hint, if a retinal structure could be verified, where the retinal receptive fields are composed by seven disc shaped units.

After all, one should not forget, that the existence of the antagonistic system itself is a necessary condition for a recursive structure.

The most conclusive proof, however could come from an investigation of the LGN. Until now its influence within the recursive system was neglected. The LGN, however, could play a very essential role under the condition of scotopic vision. In this case, the distribution of excitability of retinal receptive fields changes drasticly its shape, and may no longer be described by the recursive function E(Q). Hence the cortical recursions would collapse under scotopic conditions. There is only the chance, that the correct shape is reconstructed by a corresponding superposition of distributions from adjacent retinal receptive fields. This superposition, however, is known to be equivalent with a weighted sum of spike rates. As LGN neurons have indead excitatory and inhibitory inputs from adjacent retinal receptive fields, it could be just the main function of the LGN, to reconstruct the distorted shape of retinal receptive fields under scotopic conditions (Singer et al., 1972; Mallei and Fiorentini, 1972). Experiments, verifying a recursive shape of LGN receptive fields under scotopic conditions would be a very conclusive proof for the existence of a recursive structure of the visual system.

Appendix

I. The Coordinate Systems

Disc shaped functions F(Q) are the building blocks for the functions E(Q). The weighted sum of 19 functions E(Q) will be a function with the same shape as E(Q) but double the size and shall be called E*(Q*).

The superposition implies three coordinate systems. The first one is related to the center of F(Q), with a vector e pointing from this center to the coordinates of point ~o. The second coordinate system is related to the center of E(Q), with a vector Q pointing from this center to the coordinates of point Q. The third system is related to the center of E*(Q*), with a vector Q* pointing from this center to the coordinates of point Q* (Figs. la and 2a).

2. Synthesis of E(Q)

The equidistant intersection points of a triangular grid shall be labeled X, Y~, U~, V~; v = 1, 2 ... 6 (Fig. 1). X shall be the center of a coordinate system related to E(Q) and hence these points will have the coordinates X = 0 ; Yv with LY~I=R, U, with IU,I=2R and V~

with IV,I = R ~/3. The disc shaped functions F(Q) are located at X and U,. They are

limited to 0=<~<2R, normalized to F(Q)=I at [Q]=0 and rotary symmetrical (Fig. 6). Seven disc functions, forming E(Q), are located at X and U v. The point Q in the system related to E(Q) has different distances Q~, Q,~ to the disc centers X and U v respectively, with Q~ = Q - X = Q and Q,v=Q-Uv. Thus one can either write different

functions F~(Q), F,,(Q) depending on the same coordinate Q related to the system of E(Q), or equal functions F(0~), F(s with different coordinates, related to the individual systems of the seven discs.

The superposition may be written as 6

E ( Q ) = - 6 F ~ ( Q ) + ~ F , ~ ( Q ) = - 6 F ( s ~ F(0,~). (10) v 1 v - 1

The weighting factor 6 provides a zero integral over E(Q), inde- pendent Of the shape of F(s This feature will be necessary if E(Q) shall approximate the distribution of excitability of a retinal receptive field.

For the discussion of the recursive features, E(Q) must be evaluated at Q =X, Yv, U~, and V~. Without calculations one can see from Fig, 1 and from (10), that

E(X) = - 6F(O) = - 6,

E(Y,) = - 6F(R) + F(R) + 2F(R [/3) = a,

E ( U , , ) = V(O) = l ,

E(V~) = - 6F(R ~/3) + 2F(R) =/3.

( l la)

( l lb)

( l lc)

( l ld)

The values of E(Y~)=c~ and E(V~)=fl depend on the shape of the discs F(0). This shape will be optimal, if the sections U - U and V - V through E(Q) are congruent. If Q ,_ , is a point of section U - U and Q~_~ is a point of section V-V, a family of functions F(s will provide E(Q,_ , )= E(Q~_ ~) for

o=<IQ,_,I = I Q o _ J < 2 R .

Figure 7 shows, that Fx, F.~, F. ._ 1, and F.~+ 1 overlap at Q ._ . and that IQ.I=IQ._J, 1 0 . J = I Q . _ . - U J , le .~+, I=IQ.- . -U~+~I , and 10.=_~I=JQ._ -u~_~l . Only three functions F., F.~, and F. ._ l overlap at Q~,_., with ]0x[=JQ~_~l, [s ~ - U J , and 10.~-xl = [Qo-~-U~_ 11. As F is rotary symmetric, only the absolute values of the coordinates are of interest and these distances may be expressed in dependence on the variable ~o.~ = 0 = I Q . _ ~ - U J . Thus according to (10), E(Q._.) and E(Q~_.) may be calculated

E(Q._.) = - 6F(IQ._.J) + F(JQ._. - U J) + F(]Q. _. - U~ _ t I)

+ F([Qu_~-U~+~I)

= - 6F(2R - ~o) + F(~) + 2F( ]f4R 2 _ 2R0 + Q~) (12a)

E(Q, _ ~) = - 6F(JQ,_ ~l) + F([Q,_~ - U J) + F([Q~_~ - U~ _~ I)

= - 6F(2R - 0) + 2F( 4 R ~ - ] / 3 ) - 2RQ(2- ~/3) + ~2). (12b)

This result (12) together with the condition of symmetry E(Q,_,) = E(Q~_v) provides a relation

F(~o) + 2F( [ / ~ 2 _ 2R0 + 02)

= 2 F ( ~ 1 / 3 ) - 2R~o(2 - ~/3)+ 02), (13)

which allows a numerical calculation of F(0). An explicite form of F(0) could not be achieved. Six members of the family of functions F(e) with different values for ~ and fi are presented in Fig. 8.

Symmetrie is forced by (13) only for ]Q.-.I = ]Q~-~[ =<2R, where E(Q) is defined by (12a) and (12b). In the peripheral region with 2R_-< ]Q,,-.[ = ]Q~-~[, there may be differences between E(Q._.) and E(Q._~), depending on the shape of F(0). This symmetrie is con- tinued as far as possible to peripheral regions by one special F(Q) with ~ = -2 .37 and fl=0.324 (Fig. 3c).

3. The Recursivity of E(Q)

The result E*(Q*) o fa superposition of 19 functions E~(Q*).Er.(Q* ), E~.(Q*), and E~(Q*) will be

E*(Q*) = x-Ex(Q* ) + ~ [y . E,~(Q*) + u-E,~(Q*) + v. E~(Q*)]

= x . E ( Q ~ ) + ~ [y.E(Qy~)+u.E(Q,~)+v.E(Q~)]. (14) v = l

207

4"....\//~- ~ ~""

Fig. 7. For the calculation of E(Q) at a point Q = Q . _ . of the section U - U, the values from four disc functions F(0) with s 0.1, 0.2, and 0.3 must be added. A point Q = Q ~ - v of section V - V involves only three functions with s 0.5, and 0u6 in this example

I -2.20 -0.03 2 -2.30 0.18 3 -2.37 0.32

-2.41 0.43 5 -2./.9 0.57 6 -2.5/. 0.70

3

---~ Fig. 8. Six numerically evaluated members of the family of functions F(s are plotted for different parameters of ~ and/3

The conditions for recursivity shall be established at the 19 centers X*, Y*, U*, and V*. The coordinates Qx, Qrv, Qua, and Q~v related to these centers of the superposed functions E(Q) can only have the values 0 and U~. Hence E(Q~), E(Qyv), E(Q,v), and E(Q~v) may only have the values - 6 or 1 at these points according to (10). These values must be multiplied by the corresponding weighting factors x, y, u, and v and added over all contributing functions (Table 2). From this table the values for E*(Q*) may be written as

E*(X*) = - 6x + 6y, (15a)

E*(Y*) = x - 4 y + u+2v, (15b)

E*(U*) = y - 6u + 2v, (15c)

E*(V*) = 2y + 2 u - 6v. (15d)

Four weighting factors x, y, u, and v are defined uniquely by these four equations for arbitrary values E*(X*), E*(Y*), E*(U*), and E*(V*) and hence E* may be forced to the same shape as E, defined by (11).

E*(X*) = E(X) = - 6, (16a)

E*(Y*) = E(Y) = c~, (16b)

E*(U*) = E(U) = 1, (16c)

E*(V*) = E(V) = fl (16d)

208

Table 2

Coordinate Center of Number of Weighting Value Q* of function contributing contributing factor of of E (Q) E* (Q*) E (Q) functions E (Q) atQ =Q*

Contribution of E (Q*) to superposition E* (Q*)

X X* 1 x - 6 - 6 x Y* 6 y 1 + 6y

Y~ Y* 1 y - 6 - 6 y Y*+- 1 2 y 1 + 2y V* V*+ 1 2 v 1 + 2v U~ 1 u 1 +u X* 1 x 1 + x

U* U* 1 u - 6 - 6 u Y* 1 y 1 +y V*, V*+ t 2 v 1 + 2v

V* V~* 1 v - 6 - 6 v Y*, Y*- 1 2 y 1 + 2y U* U*_ l 2 u 1 + 2u

Acknowledgement. I am very grateful to Prof. Dr.-Ing. W. v. Seelen for his valuable comments.

Fig. 9. The results of the second recursion are evaluated at 19 points X, Y, U, and V,, where the values are exact by definition, and at additional 174 intermediate values

for X*=2X, Y*=2Y, U*=2U, and V*=2V. This, however is just the condition for recursivity at these 19 points and a combination between (15) and (16) leeds to (1), (2), (3), and (4).

At coordinates, different from these 19 points, E*(Q*) is not subject to the conditions of recursivity (16), and E*(Q*) must not be equal to E(Q) for Q* =2Q. But E*(Q*) will be as smooth a function as E(Q), and both the functions will have intermediate values at intermediate coordinates. Hence it is not surprising that E*(Q*)~E(Q). This relation was checked at additional 174 coordinates, and the result is shown for c~= - 2.37 and/~ =0.324 in Fig. 9. Results for other pairs of e and fl may be seen in Fig. 3.

R e f e r e n c e s

Braddick, O., Campbell, F.W., Atkinson, J. : Channels in vision: basic aspects. In : Handbook of sensory physiology, Vol. VIII, pp. 3-38. Berlin, Heidelberg, New York : Springer 1978

Jung, R. : Visual perception and neurophysiology. In : Handbook of sensory physiology, Vol. VII/3, pp. 1-152. Berlin, Heidelberg, New York: Springer 1973

Levick, W.R. : Maintained discharge in the visual system and its role for information processing. In: Handbook of sensory physiology, Vol. VII/3, pp. 575-598. Berlin, Heidelberg, New York: Springer 1973

Maffei, L., Fiorentini, A. : Retinogeniculate convergence and anal- ysis of contrast. J. Neuropbysiol. 35, 65-72 (1972)

Maffei, L.: Spatial Frequency channels: neural mechanismus. In: Handbook of sensory physiology, VIII, pp. 39-66. Berlin, Heidelberg, New York: Springer 1978

Siminoff, R. : Modeling of the vertebrate visual system. 1. Wiring diagram of the cone retina. J. Theor. Biol. 86, 673-708 (1980)

Siminoff, R.: Modeling of the vertebrate visual system. 2. Application to the turtle cone retina. J. Theor. Biol. 87, 307-347 (1980)

Singer, W., P6ppel, E., Creutzfeldt, O. : Inhibitory interactions in the cat's lateral geniculate nucleus. Exp. Brain Res. 14, 210-226 (1972)

Received: September 30, 1981

Dr. Georg Hartmann Fachbereich Elektrotechnik-Elektronik Universit~it Gesamthochschule Paderborn D-4790 Paderborn Federal Republic of Germany

recursive features of circular receptive fields

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