Intersensorial Summation As a Nonlinear Contribution to Cerebral Excitation

Intersensorial Summation as a Nonlinear Contribution to Cerebral Excitation

Isabel Gonzalo1 andMiguelA.Porras2

1 Departamento de Optica. Facultad de Ciencias F´ısicas. Universidad Complutense de Madrid. Ciudad Universitaria s/n. 28040-Madrid. Spain E-mail: igonzal@ﬁs.ucm.es 2 Departamento de F´ısica Aplicada. ETSIM. Universidad Polit´ecnica de Madrid. Rios Rosas 21. 28003-Madrid. Spain

Abstract. Certain aspects of the J. Gonzalo’s research on inverted vision and intersensorial summation (facilitation or reinforcement) in patients with brain damage, are formulated and interpreted from a macroscopic time-dispersive model of cerebral dynamics. We suggest that cerebral excitation from intersensorial summation is essentially nonlinear with stimuli.

1 Introduction

J. Gonzalo characterized the central syndrome associated to a unilateral lesion in the parieto-occipital cortex, equidistant from the visual, tactile and auditory projection areas (central lesion) [8]-[13]. A central lesion produces a deficit in the cerebral excitability, and a diminution in the reaction (response) velocity of the cerebral system. The corresponding central syndrome allows the dynamics of the cerebral cortex to be investigated since the cerebral system keeps the same organization plan and the same physiological laws as in the normal case, but in a smaller excitabilityscale [9]: All sensorysystems are involved, in all their functions and with symmetric bilaterality, suffering an allometric dissociation or desynchronization of sensoryqualities (united in normal perception) according to their excitabilitydemands. In the visual system,for instance, when the illumi- nation of a vertical white arrow is diminishing, the perception of the arrow is at first upright and well-defined; next the arrow is perceived to be more and more rotated, becoming at the same time smaller, and losing its form and colors in a well defined order. The sensorial perception thus splits into components. One of them is the direction function, which gives place to the striking phenomenon of the inverted vision: about 160 degrees in patient “M” under low stimulation [8, 9]. The investigations performed byJ. Gonzalo were connected with those of other authors [7], [14]-[17], [22, 23, 25], taken into account in other works [1]-[4], [19]-[21] and can be related to other approaches in cerebral dynamics [5, 18, 24]. In a previous work [13], we introduced a linear time-dispersive model that describes some manifestations of the central syndrome, including temporal summation. Basic macroscopic concepts as the excitabilityand reaction velocitywere

J. Mira and J.R. Alvarez´ (Eds.): IWANN 2003, LNCS 2686, pp. 94-101, 2003. c Springer-Verlag Berlin Heidelberg 2003 Intersensorial Summation as a Nonlinear Contribution to Cerebral Excitation 95 there introduced to characterize the cerebral system and a sensory function. In the present paper we deal with intersensorial summation (facilitation or reinforcement). This phenomenon occurs when the perception of a sensoryfunction, stimulated bya certain stimulus S, is improved byother type of stimulus. Inter- sensorial summation is verynoticeable in central syndrome. For example, strong muscular contraction improves significantlyvisual perception. Unlike summation byiteration, intersensorial summation modifies the cerebral systemessentially, becoming more rapid and excitable, i.e., it supplies in part the neural mass lost in the central lesion [8, 9]. This effect is greater as the deficit (the central lesion) is greater, being null in a normal case. It is the aim of this work to show that intersensorial summation can be described as a nonlinear perturbation to the linear time-dispersion model introduced in [13].

2 The Model

We recall [13] that a system is said to be time-dispersive if its response at a time t depends not onlyon the stimulus at that time, but also at previous times, t ≤ t. If we consider an stimulus S(t) acting on the cerebral system, the excitation E(t) produced at time t in the cerebral system is ∞ E(t)= χ(τ)S(t − τ)dτ , (1) 0 where τ = t − t, and the excitation permeability χ is related to the capability of the system to be excited. An approximate form is χ(τ)=χ(0)e−aτ ,whichis the response to an impulse stimulus (a delta function stimulus). The constant a characterizes the response velocityof the system.Introducing the expression of χ(τ) into (1), and assuming a constant stimulus S during the time interval [0,t], (1) yields t E(t)=χ(0)S e−aτ dτ =(χ(0)/a)S(1 − e−at) . (2) 0 Let us assume [13] that the threshold of the cerebral excitation E necessaryto perceive the minimum sensation of a particular sensorial function is the same for the normal man and for a patient with central lesion. It was then showed [13] that the excitability χ(0), the reaction velocity a and the quotient χ(0)/a are smaller in a central lesion patient than in normal man, but the necessary stimulus S is higher [as can also be seen from (2)]. It can be said that the cerebral system of the normal man works like a sat- urated system, in the sense that a very low stimulus induces cerebral excitation enough to perceive not onlythe simplest sensorial functions but also the most complex ones in a synchronized way. A luminous sensation (simple function) of a white arrow, for instance, is perceived together with its color, localization, direction and recognition (more complex functions). In the case of central lesion, however, a low stimulus produces a cerebral excitation in deﬁcit with respect to the normal man. The most complex, excitation-demanding sensorial functions, are then lost or retarded, leading to the dissociation phenomena. 96 Isabel Gonzalo and Miguel A. Porras

We consider now intersensorial summation. In contrast to temporal summation, several stimuli act at one time on different receptors of the cortex. It was found [8, 9] that patients with central syndrome are very sensitive to intersensorial summation or facilitation. For example, strong muscular contraction is veryefficient at improving the perception of all sensorysystems,for example, it straightness instantlya test arrow perceived inclined, dilates the visual field, etc. Other types of facilitation come from binocular summation, tactile and acoustic stimuli. Facilitation phenomena are more noticeable as the deficit in the cerebral excitation is greater, and are null in a normal case[8, 9]. In more formal terms, the perception of a sensoryfunction, say Fi,isnot onlydetermined bythe stimulus Si on the cerebral receptor i, but also byother stimuli Sj, j = i, acting on other cerebral receptors j. In our simple dispersive model, the total cerebral excitation Ei for sensorial function Fi is assumed to be the sum of different excitations of the type (1) originated by the different stimuli, ∞ Ei(t)= χi,j (τ)Sj (t − τ)dτ , (3) j=1,2,... 0 (0) (0) −ai,j τ where, as above, χi,j (τ)=χi,j e ,andwhereχi,j is the excitabilityof the cerebral system associated to the sensory function Fi when the stimulus Sj is acting, and ai,j is the corresponding reaction velocity. In the case of constant stimuli Sj during the time interval [0,t], (3) becomes (0) −ai,j t Ei(t)= (χi,j /ai,j)Sj 1 − e , (4) j=1,2,... and in the stationarysituation, in which all stimuli are acting for large time enough (t →∞) so that there is no changes in perception, (4) further simplifies (0) to Ei = j=1,2,···(χi,j /ai,j)Sj , or in matrix form,

   (0) (0)    E1 χ1,1/a1,1 χ1,2/a1,2 ·· S1    (0) (0)    E2 χ /a χ /a ·· S2   =  2,1 2,1 2,2 2,2    . (5)  ·   ··  ·  · ·· ·

The simplest possible case of intersensorial summation takes place when, in ad- dition to stimulus S1, a secondaryone S2 supplies excitation enough to improve the perception of sensoryfunction F1. (5) then reduces to

(0) (0) E1 =(χ1,1/a1,1)S1 +(χ1,2/a1,2)S2 . (6) Given the peculiarities of the facilitation phenomenon, it is difficult to make (0) hypotheses on the nature of the constants χ1,2 and a1,2 that characterize this phenomenon. However, as in similar situations of several interacting subsystems, the usual wayto proceed is to consider the excitation to be given bya single term of the form E χ(0)/a S , 1 =( eff eff) 1 (7) Intersensorial Summation as a Nonlinear Contribution to Cerebral Excitation 97

χ(0) a where eff and eff are effective excitabilityand reaction velocityparameters, respectively, that take into account the effects of the other stimuli. In fact, the intersensorial summation or facilitation appears to modifyin some essential way χ(0) the cerebral system, which becomes more excitable (characterized by large eff ) a and faster (large eff), as was shown in several experiences [8, 9, 13]. To write χ(0)/a down an expression of eff eff that reflects the experimental observations [8, 9], χ(0)/a S we first note, from (7), that eff eff is the total excitation per unit stimulus 1. Second, this quotient has been shown in Ref. [13] to increase with the facilitation provided bythe stimulus S2, that is, to approach that of a normal man. Thus, assuming a growth-type law for the restoration of the deficit of excitation, we can write, as a first approximation,

(0) (0) χ /a χ /a D − e−ρ1,2S2 , eff eff =( 1,1 1,1)+ 1 (8) where D is the excitation deficit per unit stimulus S1, due to the lesion, and ρ1,2 (0) − e−ρS2 χ /a is a constant. The functional form 1 accounts for saturation of eff eff (0) (0) at the highest value χ1,1/a1,1 + D equal to the value χ1,1/a1,1 of normal man, if facilitation is able to restore the excitation deficit asymptotically. On the other hand, proportionalityto D in (8) accounts for the experimental fact [8] that restoration capabilityis greater as the deficit D is greater. Introducing (8) into (7) we obtain for the total stationaryexcitation (0) −ρ1,2S2 (0) E1 = S1 (χ1,1/a1,1)+D 1 − e (χ1,1/a1,1)S1 + Dρ1,2S1S2 (9)

−ρ S due to stimuli S1 and S2. The last linear approximation [(1 − e 1,2 2 ) ρS2] holds for small restoration of excitation bystimulus S2,asisthecaseinthe experiments [8, 9]. We note that the last equalityin (9) is independent of the exact choice of the growth curve in (8) in the small restoration approximation. Finally, comparison of (9) and (6) leads to the approximate functional form for the intersensorial coeﬃcient (0) χ1,2/a1,2 = Dρ1,2S1 . (10)

The previous expressions fit to what is known from experimentation: 1) in normal (0) man (D = 0) there is no facilitation (χ1,2/a1,2 = 0); 2) in absence of primary (0) stimulus (S1 = 0), facilitation cannot act (χ1,2/a1,2 = 0). As a third consequence, we see in (9) that facilitation is essentiallya nonlinear phenomenon. From certain experiences [8], it seems that more complex nonlinearities could appear for high (0) excitation, as a decrease of D and presumablyof χ1,1/a1,1 with increasing S1. Finallywe must consider the perception P1 of the sensoryfunction F1.Sen- soryperception is assumed in manycases to depend on the external physical stimulus I1 according to a logarithmic-type law (Fechner law) in a certain range of values. This law has been much-discussed and other functional dependences have been proposed, but there is no a definitelyaccepted one [6, 22]. We will assume for simplicitythe validityof this law, at least in a certain range. Since 98 Isabel Gonzalo and Miguel A. Porras the cerebral excitation E1 is, as a first approximation, proportional to the cerebral stimulus S1, and this one can be considered proportional to the external physical stimulus I1, we shall consider, in the framework of the Fechner law, that the perception is related to the logarithm of the cerebral excitation E1 in the form P1 = Z log E1, (Z a constant) . (11)

3 Perception Curves

Let us analyze within the previous model some of the data of the visual system obtained byGonzalo [8, 9] from patient M (verypronounced central syndrome case), right eye. First we consider the case of no intersensorial summation (inactive case), analyzed in Ref. [13], to be compared with cases of intersensorial summation. An upright white 10 cm size arrow on black background, at distance of 40 cm from the patient was illuminated with light intensity I. The corresponding stimulus S1 on the cerebral system is assumed to be proportional to the intensity S1 = KI. For verylow intensitythe perception of the arrow was almost inverted and constricted, while for higher intensities the perception improves until the arrow is perceived almost upright. The reinversion of the arrow made bythe cerebral system was called direction function, which reaches its maximum value (180 degrees) when the image is seen upright. Each value of the stimulus S1 is maintained constant during long time enough until the perception of the arrow is stationary. The function ﬁtted to the experimental data is, from (11), (0) P1 = Z log (χ1,1/a1,1)S1 = Z log(YI) , (12)

(0) where Z =54andY ≡ (χ1,1/a1,1)K = 90. The experimental data and the fitted curve (i) are shown in Fig. 1. In the following case, we consider the same conditions as in the precedent case but with reinforcement bymeans of a second constant stimulus S2 con- sisting on strong muscular contraction of the patient (40 kg held in each hand). Intersensorial summation improves now the perception of the arrow significantly. The function fitted to the experimental data is now P Z χ(0)/a S Z Y I , 1 = log ( eff eff) 1 = log( ) (13) Z Y ≡ χ(0)/a K with =40and ( eff eff) = 9000, which is shown in Fig. 1[curve (r)] together with the data. Discrepancies between curves and data can be due to the controversial logarithmic Fechner law, to measurement errors (more pronounced for high excitation under reinforcement because the image becomes veryunstable χ(0)/a χ(0)/a [8]), and to the assumption that ( eff eff)and( 1,1 1,1) are independent of S1 (onlyvalid for not veryhigh excitation values). From comparison of cases (i) and (r) in Fig. 1, we see that facilitation supplied by S2 (strong muscular contraction) allows the patient to perceive the arrow almost upright with much lower stimulus S1 (lower I) than in the inactive case. Intersensorial Summation as a Nonlinear Contribution to Cerebral Excitation 99

Fig. 1. Perceived direction of the upright test arrow versus light intensity for inactive case (i), and reinforced case (r) by muscular contraction (80 kg).

We analyze now the direction function versus facilitation for given stimulus S1. In this case the light intensityilluminating the arrow is constant ( I and S1 = KI constants), while the stimulus S2 (muscular contraction) varies. As in the preceding cases, the stimuli were acting during long time enough to reach an stationarystate for each measurement. The cerebral excitation E1 is given by(6) , where S2 = CR (C constant) is now the variable and the remaining quantities take ﬁxed values. The variable R is expressed in kg. For the sensorial perception of the direction of the arrow we then have, according to (11),

P1 = Z log(A + BR) , (14)

(0) (0) where A ≡ (χ1,1/a1,1)S1 and B ≡ (χ1,2/a1,2)C = Dρ1,2S1C. The ﬁtting of this function to the data yields Z = 78, A =2.43 and B =1.5. It is shown in Fig. 2 (a). The following example is the same as the previous one except that facilitation is not muscular contraction but luminous intensity Il on the left eye, while the observation of the arrow is made, as above, with the right eye only [8]. In this case S2 = KlIl (Kl constant). The constant stimulus S1 is now higher than in the previous case. The ﬁtting of P1 = Z log(A + BIl) to the data gives Z = 41, (0) A = 100 and B ≡ (χ1,2/a1,2)Kl = Dρ1,2S1Kl = 80. It is shown in Fig. 2(b). The values of A and B are higher than in the previous case in agreement with the higher value of S1 during the experience.

4 Conclusions

We have extended our previous time-dispersion formulation for cerebral excitation to describe the eﬀects of intersensorial summation. On the assumption that References excitation, excitation low of the limit to the facilitation contribution in intersensorial fact, the In from intrinsicallynonlinear. contributions are reinforcement the or additive, is excitation cerebral intensity light by (b) , traction 2. Fig. a.Ti omi nareetwt h bevdfcsta nesnoilsum- stimulus intersensorial reinforcing that the facts observed as the pronounced with more agreement is in mation is form This man. nov h ne fmr ope,strto-iennierte ihdrc and direct with nonlinearities saturation-like stimuli. obser- complex, crossed could Further more levels of model. excitation onset higher the the that support involve however, indicate, to Gonzalo appear byJ. proposed made would vations forms data functional experimental the the of with agreement reasonablygood The lesion. central eursaprimarystimulus a requires uinfo te tml aeteqartcform quadratic the take stimuli other from bution 0 IsabelGonzaloandMiguelA.Porras 100 .d jrauraJ tH´ et J. Ajuriaguerra de 1. .DlaoGarc´ Delgado Neurology 4. in Mc.D.: Progress Critchley in: 3. H.L.:“Neuro-ophthalmology” Teuber and M.B. Bender 2. .GnaoJ.: Gonzalo 8. K.: Goldstein and A. integration on Gelb vistas new 7. cortex: R.H.: visual Forgus the 6. in coding “Temporal al.: et A.K. Engel 5. n scity d pee E.A., Spiegel Ed. Psychiatry, and pathologique nfunci´ en abnormer bei Gesichtsfeldbefunde 109 Ueber Erm¨ VII F¨ Hirnverletzter: hirnpathologischer Untersuchungen Analysen “Psychologische 1920. Leipzig Barth, TINS, system”, nervous the in Telecomunicaci´ de Ingenieros de E.T.S. sis. ecie ieto ftets ro essfcltto a ymsua con- muscular by (a) facilitation versus arrow test the of direction Perceived 8-0 (1922). 387-403 , daki e ue sg igktm),Abeh .GafsAc.Ophthal., Arch. Graefes v. Albrecht Ringskotome)”, (sog. Auges des udbarkeit nd a odcoe din´ condiciones las de on netgcoe or aneadin´ nueva la sobre Investigaciones asn ai 1949. Paris Masson, , aA.G.: ıa Perception direction (degrees) h aitllobes Parietal The oeo Neurocibern´ Modelos c rwHl,NwYr,1966 York, New Graw-Hill, Mc. , S ce H.: ecaen I i l n nycso ni-ectcrbu u oa to due cerebrum in-deﬁcit an on onlyacts and , nlf y.Setetext. the See eye. left on scooiceAaye inahlgshrF¨ hirnpathologischer Analysen Psychologische 15 mcsd aectblddnerviosa excitabilidad la de amicas E ,2826(1992). 218-226 6, , III i rod odn1953. London Arnold, , rmstimulus from eCre eerl td Neuro-psycho- Etude Cerebral. Cortex Le hp ,1312(1948) 163-182 8, Chap. , tcsd Din´ de eticos n ...Mdi 1978. Madrid U.P.M. on. mc eerl aatvddcerebral actividad La cerebral. amica αS S i S i mc Cerebral amica S j slna,bttecontri- the but linear, is ,where j rw,btnecessarily but grows, α ul C.S.I.C., Publ. , =0innormal leafGrund auf alle ,Ph.D.The- alle. Intersensorial Summation as a Nonlinear Contribution to Cerebral Excitation 101

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