Morphodynamical Enaction: the Case of Color

With Francisco Varela's death, one of the most beautiful minds I ever met has gone to heaven. The inspired way he succeeded in synthesizing cognitive neurosciences, dynamical models of global brain activity and the phenomenological dimension of consciousness was truly amazing and wonderful. His scientific production was constitutive of a spiritual vocation. In science, each generation has to tackle some critical problems whose correct s c i e n t i f i c f o r m u l a t i o n i s a l r e a d y a challenge. For our generation, one of the primary challenges will have been to naturalize the mind and the intentionality of consciousness using: 1. the empirical results of integrative, c o g n i t i v e , a n d c o m p u t a t i o n a l neurosciences, 2. the mathematical tools for modeling self-organized complex structures. Today, with the fantastic development of n e u r o m i m e t i c m o d e l s , d y n a m i c a l perspectives have become widely accepted – and even dominant – in many fields of neuroscience. We have quite forgotten how difficult it was to develop them thirty years ago in the context of the functionalist symbolic paradigm. One thing that is particularly striking in Francisco's scientific trajectory is the astonishing and marvelous permanence of his fundamental choices. He saw the dynamical perspective as essential from the beginning. In his final years, it culminated in very interesting works concerning large scale synchronization phenomena and global binding in the brain. In some cortical areas, such as V1 in the primary visual cortex, the synchronization coding the spatial coherence of the percept is well established (e.g. by the experiments of Gray, Singer, König, etc.). But when we look at the mathematical models of synchronization of elementary oscillators (Kuramoto, Daido, Ermentrout, Kopell, etc.), we find that the synchronization process is too slow. In conjunction with Francisco showed that oscillators of Hodgkin-Huxley type or, more simply, of Fitzhugh-Nagumo type, which present bursts of activation, can be synchronized quickly. Francisco also worked hard, especially with Evan Thompson, on the problems concerning filling-in processes. He strongly emphasized the integration of the different modular processings in the brain through large scale binding. For him, the …

phenomena and global binding in the brain.In some cortical areas, such as V1 in the primary visual cortex, the synchronization coding the spatial coherence of the percept is well established (e.g. by the experiments of Gray, Singer, König, etc.).But when we look at the mathematical models of synchronization of elementary oscillators (Kuramoto, Daido, Ermentrout, Kopell, etc.), we find that the synchronization process is too slow.In conjunction with Heinz Schuster and Michel Le Van Quyen, Francisco showed that oscillators of Hodgkin-Huxley type or, more simply, of Fitzhugh-Nagumo type, which present bursts of activation, can be synchronized quickly.
Francisco also worked hard, especially with Evan Thompson, on the problems concerning filling-in processes.He strongly emphasized the integration of the different modular processings in the brain through large scale binding.For him, the holistic character of brain dynamics was essential.
At the espistemological level, Francisco was one of the major defenders of the dynamical and enactive approaches as alternatives to the classical functionalist symbolic one.He rejected the thesis that only syntactic structures could be realized at the causal physical level and that semantic structures were therefore epiphenomenal.According to him, these conceptions were not plausible at the neural level and, as he said, were separated from their «biological roots.»I think that Francisco believed that classical cognitivism only could describe competences (roughly in a Chomskyan-Fodorian sense) in a syntactic way, using e x p l i c i t f o r m a l r u l e s , a n d t h a t i t s philosophical mistake was to try to endow such descriptions with an explicative power while explanations must be causal and can concern only performances.
Another of Francisco's strong criticism against classical cognitivism concerned its strong representationalist theses.The most critical point was the dogmatic assumption that only a predefined world can be represented.On the contrary, according to the enactive perspective, the properties that are relevant for a cognitive system do not preexist, but rather are produced by the interaction between the system and its environment.It is the coupling which enacts t h e e x t e r n a l w o r l d , t h u s m a k i n g i t meaningful for the system.
I would now like to reflect upon a technical paper that Francisco wrote in collaboration with Evan Thompson and Adrian Palacios (Thomson et al, 1992) on color: «Ways of coloring: Comparative color vision as a case study in cognitive science.»ON COLOR By using the term 'enactive perception,' the authors seek to move beyond the traditional conflicts expressed by the following antinomies: (i) c o m p u t a t i o n a l o b j e c t i v i s m v s .neurophysiological subjectivism, (ii) external objectivity vs. internal cognitive processing, (iii) recovering objective properties vs.
constructing enacted properties, (iv) heteronomous input-output systems v s .a u t o n o m o u s s e l f -o r g a n i z i n g systems.
They develop two main arguments from external irreducibility and from perceiver relativity.More precisely, their purpose is «to offer a new empirical and philosophical perspective on color vision» (1992:1) using a concept that would simultaneously be e x p e r i e n t i a l i s t ( n o t o b j e c t i v i s t ) a n d ecological (not subjectivist).After having presented the classical theories of color with their 3D color space (hue, saturation, brightness) and their phenomenological hue-opponency (Red/Green, Yellow/Blue), they analyze the neurophysiological and p s y c h o p h y s i c a l c o r r e l a t e s a n d t h e covariance between phenomenal and biological properties.
Let S be a colored surface with current point s.We have the map R:→S∆, s→R(s) of S to the space ∆ of color channels (the 3 types of cones).But, due to the post-receptor color channels, particularly the opponent chromatic channels R/G, Y/B, and the nonopponent achromatic channel White/Black, we have in fact a composed color map C(s) = MR(s), where M is the post-receptor transformation.
To generate color constancy, one has to disentangle in the irradiance of the visible s u r f a c e s t h e i l l u m i n a t i o n a n d t h e reflectance.It is a difficult and technically ill-posed problem of inverse optics whose solution requires the introduction of a priori constraints concerning models of light and reflectance, as well as segmentation processes, etc.
The core of Francisco's paper, however, is epistemological and looks for a way to break with the objectivist/subjectivist antinomy.Owing to the species variability of color channels (some species are tetraand even penta-chromats), it is impossible to identify perceived color with spectral reflectance.In fact, as was well established by neurophysiological experiments and phenomenological eidetic description, the segmentation of visual scenes into different patches and the construing of perceived objects themselves are perceiver-relative.It is a constitution in the strongest phenomenological sense (see Petitot, 1999).M o r e o v e r , a n a r g u m e n t o f e x t e r n a l irreducibility adds to this argument of perceiver-relativity.The hue-opponency belongs to the experience of color but lacks a n y o b j e c t i v e p h y s i c a l c o u n t e r p a r t .Nevertheless, the alternative of physicalist objectivism cannot be for all that a radical subjectivism which would amount to a sort of neurophysiological solipsism.
The antinomy is formulated by the authors in the following way (p.21).
Thesis: «The distal world can be specified independently of the animal; it casts images on the perceptual system whose task is to recover the world appropriately from them.»Antithesis: «The perceptual system projects its own world and the apparent reality of this world is merely a reflection of internal laws of the system.» The solution, of course, must be sought out in theories of adaptation, evolution and co-evolution.But this is not sufficient.We are also committed to adopt an active conception of perception as an actiondriven process.This enactive conception has a Gibsonian orientation, but is immune from the direct realism so prejudicial to Gibson's own ecological approach.
An organism and its environment are evolutionarily co-determined, and the problem is «to specify the sensory-motor patterns that underlie the visual guidance of animal activity in its local situation» (Thomspon et al 1992:22).In the case of color, it is to understand the relational, rather than the purely objective or purely subjective essence of this phenomenal quality.

MORPHODYNAMICAL MODELS
We will now explain how morphodynamical models can formulate this relational essence.For that, we must pay enough attention to the mathematical problems of modeling.We will show that the simplest mathematical models that can explain perceptive situations naturally include a solution to the antinomy.
Let us adopt first an «objectivist» computational point of view according to which the cognitive task of the visual system is to recover 3D visual scenes from highly ambiguous 2D projections.Let V be a visual scene consisting of visible surfaces moving in 3-space R 3 .It is a highly difficult problem t o u n d e r s t a n d h o w t h e g e o m e t r i c a l information about V is encoded in the optical signal.Indeed, this information is essentially embedded in the qualitative discontinuities (QDs) as the apparent contours of shapes, and we must therefore understand how discontinuities can be encoded in and transported by solutions of the wave equation.
At the retinal level, V is projected to a bidimensional pattern I(x,y), and the problem of inverse optics is to recover V from I(x,y).In order to perform this difficult computational task, the visual system must solve two different inverse problems: one geometrical and the other qualitative.
The geometrical inverse problem consists of recovering the shapes of the objects from their apparent contours.For this, the visual system must (at least) be able to: (i) detect the QDs embedded in I(x,y); (ii) detect among them those who share an objective meaning; (iii) interpret some of these objective QDs as apparent contours of shapes; (iv) reconstruct the shapes from these contours.
Regarding (i), it is now quite certain that the retinal ganglion cells perform a wavelet analysis of the signal.Wavelet analysis is a local and multi-scale Fourier analysis that is able to extract the QDs (and therefore geometrical information) from the signal.As David Marr (Marr, 1982) has pointed o u t , t h i s a n a l y s i s i s p e r f o r m e d b y convolution with ganglion cells whose receptive profiles are Laplacians of Gaussians.«Marr's conjecture» was solved by Stéphane Mallat (Mallat-Zhong, 1989): it is possible to reconstruct I(x,y) from the QDs provided by wavelet analysis.Regarding (ii), the criterion for a QD to be objective is that it can be detected at an entire range of different scales.
Regarding (iii) and (iv), it is a very deep mathematical problem that can be solved only using singularity theory (Whitney, Thom, Arnold, see Petitot, 1990).
The solution of the geometrical inverse problem therefore leads to a geometrical configuration (W,K), where W is the spatial extension of the visual scene V, and K is a set of QDs.
And what about the qualities that fill in the different domains of W delimited by K? Let's suppose that we still accept the computational objectivist perspective.If we restrict ourselves to a single quality, n a m e l y c o l o r , w e w o u l d m a k e t h e hypothesis: (i) that at each point w of W there is a well-defined (objective) value R(w) of the reflectance of the surfaces in V; (ii) that R(w) varies continuously (and even differentiably) as a function of w, except along K; (iii) that the perceptual quality of color C(w) encodes R(w).
But the key point is that perception, from retinal transduction to post-receptors color channels, converts the reflectance R(w) into the perceptual quality of color C(w) in a more complex way than a mere encoding.
In fact, the situation is the following: (i) We can interpret (W,K) as either objective or subjective data because retinotopy is well preserved up to the primary visual cortex and there are isomorphisms between the retina and cortical layers via neural maps.(i) for the objectivist, C(w) is a mere encoding of R(w), and σ can be neglected because it reduces to an isomorphism between R and Q; (ii) for the subjectivist, C(w) is irreducible to R(w), and σ can't be a simple map grounding objectivism.
The solution to this version of the antinomy is that, as a color space that can be interpreted in sensorial (color channels), phenomenal (hue, saturation, brightness) or neurological (perceptual contents = neural states) terms, Q is an internal space of the perceptual system S while the values R(w) are external stimuli.
The first key remark is that there is no direct correspondence R(w)→C(w) but only a triggering of an internal state C(w) by the stimulus R(w): C(w) is a dynamical response of the system S to the stimulus R(w).The hue-opponency property is a characteristic aspect to this fact.
The second key remark is that such internal perceptual states are attractors of some internal dynamics.All the material the authors bring to bear concerning the global computations, the structure of hues, the ago-antagonism of color-channels and the self-organization of the system of colors is mathematically expressible by the fact that color qualities (hues for instance) are attractors of an internal self-organizing dynamics X defined on the internal space Q.Q and X are of course highly speciesdependent while R is objective.
The discontinuous sections C(w) are then the result of a complex dynamical process.In fact, we get a field X w of internal dynamics controlled by W, and C(w) is the attractor of X w which is selected and actualized by R(w).Then, R(w) is a control parameter for X w and WxR is a control space.In such a morphodynamical model, the qualitative discontinuities of K are r e c o v e r e d a s d y n a m i c a l e v e n t s o f bifurcation: at the crossing of K, the attractor selected by R(w) bifurcates toward another attractor and, consequently, C(w) presents a QD.
We see that the relationship s between R(w) and C(w) has nothing to do with a mere map between simple spaces.It is in fact the composition of (at least) four processes: (iii) an objective map W→R, w→R(w); (iv) a field of dynamics σ: W→X, w→ X w , embedding the extension W of the visual scene V in the functional (infinite dimensional) space X of internal dynamics X; (v) a s e l e c t i o n p r o c e s s s e l e c t i n g , according to the objective input R(w) PETITOT Biol Res 36, 2003, 107-112 (ii) R(w) and C(w) covary but belong to different spaces, respectively R (reflectance) and Q (quality).(iii) If we consider the fiber bundles WxR→W and WxQ→W (projections of Cartesian products on their first factor) we get sections: R: w∈→W(w,R(w))∈WxR of WxR→W and C: w∈W→(w,C(w))∈WxQ of WxQ→W which are discontinuous along K.The fundamental problem is to understand the relationship σ between R(w)∈R and C ( w ) ∈ Q.I n d e e d i t i s t h e d r a m a t i c misunderstanding of σ that is at the origin of the antinomy: