BRIEF MATHEMATICAL BACKGROUND
Log In Sign Up. Pietro Morasso. Vittorio Sanguineti. Substantial advances have been achieved since the pioneering work in the 50's and 60's by Mountcastle, Hubel, Wiesel, and Evarts, among others, thus gaining an understanding of the cortex as a continuously adapting system, shaped by competitive and co-operative interactions.
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However, the greatest part of the effort has been devoted to the investigation of the receptive-field properties of cortical maps, whereas relatively little attention has been devoted to the role of lateral connections and the cortical dynamic processes that are determined by the patterns of recurrent excitation Amari , Kohonen , Grajski and Merzenich , Reggia et al , Martinetz and Schulten , Sirosh and Miikkulainen , Sanguineti et al a, Levitan and Reggia , In this chapter, we explore the hypothesis that lateral connections may actually be used to build topological internal representations and propose that the latter are particularly well suited for the processing of high-dimensional "spatial" variables and solving complex problems of motor control that involve sensorimotor information.
In particular, we apply the methods to the case of speech motor control in which acoustic and articulatory variables are typically high-dimensional, and describe an approach to articulatory speech synthesis that is based on the dynamic interaction of two computational maps. Lateral connections in cortical maps It has been observed that cortical areas can be seen as a massively interconnected set of elementary processing elements the so-called cortical "columns" , which constitute what is called a "computational map" Knudsen et al The "somatotopic" or "ecotopic" layout of many cortical areas has long suggested a kind of topologic organisation, which has been often associated with a dimensionality reduction of the representational space Durbin and Mitchison This also served as inspiration for a large family of neural network models.
From its beginning, however, the effort has been affected by a number of misconceptions, partly due to the over-emphasis in the neurophysiological community on the receptive field properties of cortical neurons. Only recently, a new understanding of the cortex has emerged as a dynamical system, which has focused the attention on the competitive and cooperative effects of lateral connections Sirosh and Miikkulainen Moreover, it has been shown that cortico-cortical organisation is not static but changes with ontogenetic development together with patterns of thalamo-cortical connections Katz and Callaway The "flatness" assumption that characterises the classic map models Amari , Kohonen is contradicted by the fact that the structure of lateral connections is not genetically determined, but depends mostly on electrical activity during development.
More precisely, the connections have been observed to grow exuberantly after birth and reach their full extent within a short period; during the subsequent development, a pruning process takes place so that the mature cortex is characterised by a well defined pattern of connectivity, which includes a large amount of non-local connections. In particular, the superficial connections to non-adjacent columns are organised into characteristic patterns: a collateral of a pyramidal axon typically travels a characteristic lateral distance without giving off terminal branches and then it produces tightly terminal clusters possibly repeating the process several times over a total distance of several millimetres.
Such characteristic distance is not an universal cortical parameter and is not distributed in a purely random fashion, but is different in different cortical areas Gilbert and Wiesel , Schwark and Jones , Calvin , Das and Gilbert As the development of lateral connections depends on the cortical activity caused by the external inflow, they may be used to capture and represent the hidden correlation in the input channels. Each individual lateral connection is "weak" enough to go virtually unnoticed while mapping the receptive fields of cortical neurons but the total effect on the overall dynamics of cortical maps can be substantial, as is revealed by cross-correlation studies Singer Recurrent excitation is likely to be the underlying mechanism that produces the synchronised firing observed in distant columns.
The existence and preponderance of massive recurrent excitation in the cortex is in contrast with what could be expected, at least in primary sensory areas, considering the ubiquitous presence of peristimulus competition or "Mexican-hat pattern" which has been observed time ago in many pathways, like the primary somatosensory cortex, and has been confirmed by direct excitation of cortical areas as well as correlation studies; in other words, in the cortex there is a significantly larger amount of long-range inhibition than expected from the density of inhibitory synapses.
In general, "recurrent competition" has been assumed to be the same as "recurrent inhibition", with the goal of providing an antagonistic organisation that sharpens responsiveness to an area smaller than would be predicted from the anatomical funneling of inputs. Thus, an intriguing question is in which manner long-range competition can arise without long-range inhibition and a possible solution is a mechanism of gating inhibition based on a competitive distribution of activation Reggia et al.
A computational map is a set F of processing elements PE or filters, which model cortical columns. This means that lateral connections are reciprocal and excitatory. This is a simple distributed model that allows to manipulate input patterns in different ways, according to the intrinsic dynamics induced by lateral connections and the structure of the mixing function. In any case, the symmetry of recurrent connections provides asymptotic stability of the whole state. In the classical model proposed y by Amari f. The inhibitory connections required by the Mexican hat model are not consistent with the preponderance of excitatory lateral connections in the cerebral cortex, but a similar sharpening effect can be obtained by different non linear mechanisms that only use excitatory connection weights.
It has been noted Sanger , that such a code has the important property of being independent of the coordinate system in which x is measured, i. The properties of the implicit and bi-directional mapping between X and F are entirely determined by the thalamo-cortical transformation eq. As for the map geometry, it has often been suggested that cortical maps are "topologically continuous", i.
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A more stringent concept has been formulated by Martinetz and Schulten by requiring a bidirectional topological continuity: PE's vs. This leads to the so called TRN Topology Representing Network model, in which the matrix of lateral connections, C, reflects the topology of X: if X is a n-dimensional manifold, the map F should be organised as a n-th order lattice.
To this regard, it has been pointed out Braitenberg that lattices are effective ways to represent multiple dimensional spaces with their associated topology on a 2-dimensional sheet like the cortical surface. The observed phenomena have suggested Munoz et al a continuous movement of the peak of activation in the corresponding maps and in fact attempts have been made Droulez and Berthoz , Lukashin and Georgopoulos to model such a moving-hill mechanism of continuous remapping in terms of recurrent neural networks.
In the case of a sensory topology-representing map, the dynamic behaviour described by equations 1 and 2 can be interpreted as keeping the internal state "in register" with the incoming sensory information, while carrying out a function of edge-sharpening. Dynamic remapping in a 1-dimensional map. Evolution of the internal status of the map at different time steps top and time course of the population vector bottom. The equilibrium state of the map is again a sharp pattern, but it has a single peak, centered on a local maximum of the input pattern, and the position of this peak depends on the initial state of the map.
In other words, the effect of the shunting action on the input pattern is to select a feature of the input a "target" and cluster the map activity around it. It should be noted that the "moving-hill" becomes broader as the speed of the trajectory becomes higher and this means that there is a trade-off with the static and dynamic precision of this collective computational mechanism.
Moreover, if we consider higher-order lattices it is possible to show that the trajectory of the population vector will reflect the topology of F. For instance, if the momo- dimensional map of figure 1 represents a circle embedded in a plane, then the time course of the population vector in the plane will also be circular. Let us also consider two maps, Fx and Fy, which represent X and Y, respectively, and can be obtained from a suitable learning process, of competitive type Martinetz and Schulten These connections can be learned in a very similar way to the intra-connections, i. A subsequent combined training phase builds the inter-connections: filled circles in the figure indicate the "virtual" prototypes, implicitly defined by the inter-connections that are visualised as horizontal and vertical segments, respectively.
Inter-connectivity matrix and "virtual" prototypes. Open circles: neurons; filled circles: virtual prototypes. In the next section we show in which way the intrinsic map dynamics carries out the actual inversion, in fact selecting a specific inverse solution. In summary, the proposed approximation scheme has a number of distinctive features: i it operates directly on population codes, different from standard artificial neural network models like RBF Radial Basis Function or NGBF Normalised Gaussian Basis Function , and this feature has been observed in biological sensorimotor networks Zipser and Andersen , Salinas and Abbott ; ii the architecture is symmetric or bi-directional, thus performing both forward and inverse transformations; iii the scheme is efficient and modular, because a given map or representation layer might be shared by different computational modules, each implementing a specific transformation or association, in a complex network of maps.
The horizontal axis of the same panel shows five possible solutions: the peaks of the population code projected from Fy to Fx. How can the network choose one of them and which is the criterion? Figure 4 illustrates the process. The dashed curve on the horizontal axis is the arbitrary initial state of the map Fx. Thus the criterion of selection embedded in the network model is a criterion of minimum distance in the map.
The solution is reached in a continuous fashion, as a moving hill from the dashed peak to the continuous peak. In other words, the neural architecture operates at the same time as a regularisation mechanism of the inversion process and as a mechanism of "trajectory formation". Figure 4. Dynamic inversion of the y x mapping. This generates a multi-bump pattern of input stimulation in the x-map via the cross-connections between the two maps.
Application of cortical maps to articulatory speech synthesis Like all motor activities, the process of speech synthesis can be investigated in a computational framework: the plant is the whole oral cavity jaw, lips, tongue, and larynx plus the vocal cords and the lungs and it must be carefully controlled by the brain in order to produce speech sequences Sanguineti et al b, The problem is also relevant from an engineering point of view: with respect to state-of-art techniques, articulatory speech synthesis is expected to provide a much greater degree of adaptivity to different speech rates, stress conditions, geometry of speaker's oral cavity, etc.
In both cases, the generation of articulatory trajectories implies a kind of coordinate transformation, which is not uniquely determined because of redundancy, in the sense that there are several configurations corresponding to the same audible sound. On the other hand, the human articulatory system is able to fully exploit redundancy, as demonstrated by the phenomenon of compensation in the classical bite-block and the lip-tube experiments: if the jaw or the lips are mechanically constrained, a subject is still able to generate acoustically intelligible phonemes i.
Compensation suggests that the inverse acoustic-to-articulatory transformation is not defined a-priori, i. Moreover, redundancy is critically important in the process of chaining a sequence of phonemes: it has been observed that the realisation of a particular phoneme is heavily influenced by its "context", thus suggesting that each phoneme is prepared in advance by exploiting the excess degrees of freedom of the oral cavity in order to generate trajectories that are more "economic" in terms of smoothness and energy expenditure ; this phenomenon is known as coarticulation Ohman However, such a geometric or static model does require a separate dynamic mechanism to account for trajectory formation with an high degree of adaptativity: in fact, it should be able to account for the effect of different speech rates and different stress conditions the phenomenon of vowel reduction on a given sequence of phonemes.
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The issue of sequence generation also puts into evidence that targets have both a spatial and a temporal aspect. In fact, it has been observed that vowels and consonants have a somehow dual nature: vowels are well specified spatially in the acoustic space , whereas consonants are more accurately specified in time in the sense that the time interval in which they are "active", or context, is smaller than in vowels. Thus, potential fields acting on spatial maps are natural mechanisms for capturing the generality of motor control problems outlined above.
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