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 Politécnica 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.

Load more