The main ideas of E.N. Sokolov’s scientific career were described in his book “Восприятие и условный рефлекс. Новый взгляд” (“Perception and Conditioned Reflex. New Look”) in 2003. A chapter of the book translated by E.N. Sokolov in English is presented on this page.

Universal spherical model of cognitive and executive processes

• 1. Introduction
• 2. The spherical Model of Colour Vision: trichromatic vision
• 3. The spherical Model of Colour Vision: dichromatic vision
• 4. The spherical Model of Colour Vision: monochromatic color system
• 5. Perception and expression of emotions in the framework of spherical model
• 6. Vector encoding in space and time perception
• 7. Conclusion semantic
• 8. Conclusion. New horizons
• References

1. Introduction

Psychophysics and neuroscience

Psychophysics beyond sensation has at least two aspects:

1. extension of psychophysics to such complex processes as perception, memory and semantics;

2. appli­cation of psychophysical methods for evaluation of event-related potentials and spike discharges. Thus, psycho­physics beyond sensation is integrated into the framework of neuroscience where neuronal mechanisms of cognitive and executive processes are treated with neurophysiological and psychophysical methods.

A challenging task for the neurosciences is to explore the neuronal basis of subjective phenomena. In an attempt to integrate neuronal processes with their behavioral and subjective manifestations Fomin et. al. (1979) suggested a spherical model of cognitive and executive operations. Specific cognitive events and intentions to perform parti­cular acts are represented on the hypersphere in the four-dimensional Euclidean space.

A cognitive hypersphere

Four Cartesian coordinates of points representing cognitive events on the hypersphere correspond to exci¬tations of four neuronal channels constituting an input excitation vector of a constant length.

Spherical coordinates of cognitive events (three angles of the hypersphere) refer to subjective scales. The cognitive hypersphere is characterized by a multilayered isomorphic structure encoding percepts, memories and semantics. Respective cognitive events are represented on the spherical layers by specific neurons (detectors and gnostic units). Subjective differences between particular cognitive events correspond to distances between their locations on the hypersphere and are «neurocomputer» as the absolute values of differences between vectors encoding respective cognitive events. The input stimuli are transformed into excitation vectors and projected on the hypersphere constituting a cognitive neuronal map.

An executive hypersphere

An executive hypersphere is «embodied» in a cognitive one so that particular cognitive events can be associated with particular intentions of specific acts.

Spherical coordinates on the executive hypersphere correspond to subjective scales of intentions.

A particular act is initiated by an intention via a command neuron activating four modules (premotor neu¬rons) constituting an output excitation vector. The signals from the premotor neurons are distributed among the motor neurons generating a specific «gesture». The differences between particular acts correspond to distances between intentions on the hypersphere and can be computed as absolute values of differences between output vectors.

The cognitive units (detector neurons and gnostic units) are connected with intention neurons, which in turn are linked with command neurons. The input stimuli pro¬jecting on the cognitive hypersphere can activate specific intention neurons. The command neurons are excited only by particular activation levels leading to behavioral acts. The role of intention neurons is to create a delay between an intention to act and a real performance of behavior. Such a link between a cognitive unit and an intention neuron evolves from learned associations between cognitive units and intention neurons initiating intentions by a variety of cognitive events.

2. The spherical Model of Colour Vision: trichromatic vision

Colour space of human trichromats: from subjective colour differences to colour excitation vectors

The presented general cognitive model suggests that colours in particular are encoded by four-dimensional excitation vectors. Subjective colour differences correspond to absolute values of vectorial differences between respective colours. It assumes that coordinates of colour-coding vectors can be extracted from a matrix of subjective differences. Indeed, the multidimensional scaling of a matrix of subjective colour differences has shown that colours varying in hue, lightness and saturation are encoded by four-dimensional vectors of a constant length, so that specific colours are located on a hypersphere in the four-dimensional Euclidean space. The Cartesian coordinates of colour stimuli correspond to four colour-coding neurons found in the lateral geniculate body of primates. Distances between ends of colour coding vectors obtained from a matrix of subjective colour differences closely correlate with subjective colour differences of the experimental matrix supporting vectorial encoding of colours.

Spherical coordinates of colours (three angles of the colour hypersphere) refer to hue, lightness and saturation, respectively (Izmailov, Sokolov, 1991).

From colour naming to colour excitation vectors

Colour stimuli are selectively specified by colour names. Respective articulations are triggered from the executive hypersphere. Because of confusion of colour names for similar colours each colour stimulus is represented by a set of colour names generated with different probabilities — a vector of probabilities of colour names. A block of colour stimuli is represented by response probabilities of a confusion matrix.

Principal component analysis of such a confusion matrix reveals a four-dimensional colour space coinciding with the colour space obtained from subjective colour differences. Absolute values of vectorial differences computed from colour coding vectors in turn closely match subjective colour differences (Izmailov, Sokolov, 1991).

The presented data show that the colour system is composed of two subsystems: a specification subsystem and a difference computation subsystem. The specification subsystem projecting colour stimulus on a local area of the colour hypersphere transfers color information to intention neurons of colour semantics that generate colour names with participation of articulator command neurons. The difference computation is performed by subtracting neurons finding differences between the Cartesian coordinates of colour excitation vectors. The absolute values of vectorial differences are used to generate subjective estimates of colour differences. The operations performed by both subsystems are based on vector encoding of colours.

ERPs evoked by colour change as measures of subjective colour differences

Close correspondence of results obtained from subjective colour differences and from confusion matrix of colour names strongly supports spherical model of cognitive and executive operations. The model assumes that subjective colour differences are «neurocomputed» from colour excitation vectors within specific neuronal networks with participation of «subtracting neurons» evaluating absolute differences of excitations in respective colour coding channels. A sum of such differences represents a vectorial difference of substituted colours stimuli. Amplitudes of ERPs evoked by colour changes should correlate with the absolute values of differences between colour-coding vectors and respective subjective colour differences. It was shown that the amplitudes of the N87 component evoked by colour change correlates with subjective estimates of colour differences supporting their neuronal basis (Izmailov et ai, 1998).

From a matrix of N87 amplitudes to a colour space

A positive correlation of N87 amplitudes evoked by colour change and respective subjective colour differences suggests that it is possible to construct an «objective» colour space using multidimensional scaling of a matrix of N87 amplitudes as an equivalent of a matrix of subjective dif­ferences. The results of the analysis have demonstrated a four-dimensional colour space coinciding with- colour space found from subjective colour differences and colour naming. Colour excitation vectors extracted from the matrix of N87 amplitudes being applied for computation of a vectorial differences reproduced amplitudes of N87 supporting the notion concerning a neuronal network extracting colour differences from colour excitation vectors (Izmailov et al., 1998).

From reaction time of colour detection to colour space

Coincidence of subjective differences with amplitudes of N87 to colour change emphasizes the concept of colour vector code. This conclusion follows from the results obtained by computing of colour differences from colour excitation vectors. It was found that absolute values of vectorial differences closely correlated both with subjective colour differences and amplitudes of N87. The colour differences are of importance for the detection of colour targets against a colour background. The greater is the colour difference the shorter the reaction time (RT) up to a non­reducible latency. Subjective colour differences and RTs are related according to an exponential law. This allows us to calculate of subjective differences from RT magnitudes. A matrix of subjective differences found from RTs and treated by the multidimensional scaling uncovers a four-dimensional space coinciding with colour space found directly from estimated subjective differences and N87 amplitudes. Thus colour differences «computed» in neuronal nets are used in the detection of coloured targets. When identification of colours is not required by instruction, detection RTs are shorter than those for colour identification. One can assume that a signal of colour specification follows a signal of colour difference. In this way, under the detection task a «response gesture» can be activated before colour identification occurred. Under an identification task the respective command neuron is activated by a subsequent vector code for colour specification. Thus RTs of colour detection are shorter than RTs of colour identification (Sokolov, 1998 a, b).

Working memory colour space

In psychophysical experiments subjects can correctly estimate colour differences when intervals between presented colour stimuli are increased in the range of minutes. This range of intervals suggests involvement of prefrontal working memory neurons preserving a stimulus trace. The multidi­mensional scaling of subjective colour differences obtained under 60s delay revealed a four-dimensional colour space coinciding with perceptual colour space. The correspondence of perceptual and working memory colour spaces is of great significance. It implies that the vector code found at the per­ceptual level is preserved at the working memory level. In the framework of a spherical model neurons of working memory constitute a spherical layer above the spherical layer of colour detectors (Vartanov et al., 1996).

Using an analogy between perceptual and working memory colour spaces one can assume that the memory trace is maintained as an excitation vector in prefrontal neurons. The following stimulus generating a new excitation vector is matched against the memory vector in «subtracting» neurons which generate a difference between the colour trace and actual colour.

Long term memory colour space

Spherical structure of four-dimensional colour space of the working memory suggests that colour vector codes are also implicated in long-term memory. A test of the long-term memory colour space requires a prolongation of intervals between standard and test colour stimuli up to several days. To construct a matrix of subjective differences for a long-term memory subjects were instructed to remember presented standard colour stimulus. After a week subjects were required to estimate subjective differences between the remembered standard colour and a new colour stimulus. The subjective differences constitute a row of a matrix of colour subjective differences. The procedure was repeated for other colour stimuli to generate a complete matrix. The multi­dimensional scaling revealed a four-dimensional long-term colour space. The four-dimensional structure of the long-term memory colour space coincides with perceptual and working memory colour spaces and assumes that long-term memory units constitute a spherical layer of the cognitive hypersphere (Sokolov,2000).

In working memory in long-term memory units and a trace of colour is preserved as an excitation vector. A test stimulus is matched against the long-memory trace «computing» in «subtracting» neurons a difference between long-term memory colour trace and an input colour excitation vector.

Semantic colour space

Units representing colour images in the long-term memory can be associated with visual, auditory or somatosensory signals that become «symbols» for respective colours. A set of such «colour symbols» constitutes a semantic layer of the cognitive hypersphere. Three basic operations based on «semantic space» can be distinguished.

1. Search of symbolised colour in the visual field. In this case the colour name activates long-term memory andtransfer a colour image into working memory for scanning i! against the screen.

2. Colour naming. In this procedure perceived colour reaches the respective long-term memory neuron and activates colour semantic unit which via the executive hypersphere stimulates a command neuron of the respective «articulation gesture».

3. Evaluation of semantic differences. In this case the subject is presented with two colour names and required to estimate their «semantic differences».

During this operation each colour name activates neurons of long-term memory producing two excitation vectors. An absolute value of vectorial difference represents semantic differences between presented colour names. The procedure of getting semantic differences between colour names is very similar to estimation of differences between real colours. But real colours are addressed to colour detectors while colour names refer to long-term memory. Multidimensional scaling of a matrix of semantic differences between colour names demonstrates a four-dimensional colour semantic space. Colour names positioned themselves on a hypersphere close to the colours specified by the colour names. These results are in accordance with data concerning four-dimensional long-term memory colour space, which in turn is a «replica» of four-dimensional perceptual colour space (Vartanov, Sokolov, 1995).

Colour space of arbitrary colour names

Colour names in the «mother language» are acquired in the childhood. Parallel systems of colour names are formed during learning of foreign languages. To understand the process of formation of the four-dimensional semantic colour space arbitrary symbols of colours are used. At the beginning of experiments application of such arbitrary symbols to identify colours is based on random guesses. In the process of learning a confusion matrix of responses probabilities becomes more and more regular. After learning the multidimensional scaling of semantic differences between arbitrary colour names demonstrates the four-dimensional spherical structure. Thus, perceptual, memory and semantic colour spaces constitute isomorphic spherical structures being represented by different types of neurons which are overlapping each other (Izmailov, Sokolov, 1991).

From instrumental colour learning to colour space in monkey

Colour naming in humans has demonstrated that specific colours are represented by probability vectors of colour names. A matrix of such probability vectors contains information about perceptual colour space. This approach can be applied for animals, but instead of specific colour names differential conditioned reflexes to colour stimuli have to be elaborated. To produce a matrix of probability vectors a sequential re-learning has to be introduced using as conditional stimuli different colours. Thus in a set of colour stimuli a particular colour is selected as a conditional stimulus (CS) being reinforced with food. The other colour stimuli presented randomly constitute a subset of differential stimuli. In the process of learning the response probability to CS increases, reaching a plateau at a level greater than 80%. The response probabilities to differential stimuli are stabilized at different level in accordance with their deviations from the CS: the more a colour deviates from the CS the lower was its response probability level.

A sequential change of conditional stimuli enables one to construct a confusion matrix of response probabilities. The principal component analysis of the confusion matrix revealed a hypersphere in the four-dimensional space coinciding with the colour space found in humans. Such a result implies that in the process of colour learning synaptic weight vectors are modified and become equal to respective presynaptic inputs of the colour excitation vector generated by the conditional stimulus. Obtaining excitation vectors for all colour stimuli used in the experiments allows one to compute the inner products of conditional and differential stimuli. The computed inner products closely correlate with their respective response probabilities. This result suggests that colour learning is based on the process when synaptic weights are acquiring magnitudes equal to excitation vector component of the CS. Then responses to differential stimuli are equal to the inner products of their excitation vectors and synaptic weight vector established by the conditional stimulus in a command neuron of a respective response pattern (Latanov et al,2000).

From instrumental colour learning in carp to its colour space

Carp has a trichromatic colour vision as demonstrated in instrumental learning experiments. Using a particular colour as a conditional stimulus interposed randomly with differential colour stimuli one can obtain a vector of response probabilities. Repeating experiments with different colours used as conditional stimuli allows for the development of a confusion matrix. The cells of the matrix correspond to response probabilities. Principal component analysis of the carp’s confusion matrix has demonstrated a hypersphere in the four-dimensional colour space. Colour excitation vectors found from the response probability matrix enable the computation of inner products for a conditional stimulus and differential stimuli. The inner products computed from conditional stimulus excitation vector and excitation vectors of differential stimuli used in parallel closely correlate with respective response proba­bilities. Such a correspondence implies that the vector code for colour stimuli has modified according ly synaptic contacts that have to be organised also as weight vectors. These weight vectors of plastic synapses refer to a command neuron responsible for instrumental conditioned reflexes (Latanov et al., 2000).

Four-dimensional colour space at bipolar cells of carp retina

To get information concerning the neuronal basis of the four-dimensional colour space found from instrumental conditional reflexes in carp intracellular recordings from bipolar cells of an isolated carp retina were perfomed using monochromatic light stimuli. It was found six types of bipolar cells with receptive fields having an antagonistic organisation of «centre-periphery». «Red+ green-» and «red—green+» were depolarised on «either — or» basis. The same refers to «yellow+ blue-» and «yellow-blue+» cells. In a certain degree were depolarised «brightness» and «darkness» neurons. Thus a maximum offour bipolar cells were depolarised by any colour stimulus. This suggests that colour stimuli are represented at the level of bipolar cells with excitation vectors having four coordinates: two coor­dinates refer to colour opponent cells and two ones - to cells having no colour opponency. The sum of squared amplitudes off their depolarisation for any wavelength of the input stimulus was equal to a constant value demonstrating constant lengths of colour excitation vectors.

This, bipolar cells constitute a basis of four-dimensional colour space in carp. An additional test for correspondence of directly recorded responses of bipolar cells and components of excitation vectors extracted from behavioral experiments can be done by a comparison of «neuronal» and «behavioral» colour spaces. The coincidence of colour points found by intracellular recording and by behavioral experiment on the hypersphere in the four-dimensional space strongly support the contribution of vector codes in perception of colours and colour conditioning.

From three-dimensional space at the receptor level to four-dimensional space at the bipolar cells level

At the receptor level input colours are encoded by vectors with components equal to responses of L, M and S cones. The overlapping response characteristics of cones suggest a non-orthogonal basis. The receptor excitation vectors lengths depend on light intensity so that stimuli of different wavelengths and intensity occupy a colour body. At the level of photopic horizontal cells due to specific synaptic contacts between cones and horizontal cells an orthogo-nalization takes place: response characteristics of horizontal cells become independent. The lengths of vectors still depend on light intensity so that colour stimuli build up a colour body. At the bipolar level a radical transformation takes place. An additional «darkness» neuron is supplemented which is characterised by background activity under darkness conditions. Opponent colour neurons also are modified. A «red—green» horizontal cell was replaced by «red+green-» and «red-green+» arise. A «yellow-blue» horizontal cell now is «divided» into «yellow+blue—» and «yellow— blue+» bipolar cells. The brightness and darkness neurons remain monophasic. That reorganisation enables normalisation of the bipolar excitation vectors. The normalisation is performed in «city-bоск» metric while the sum of excitations of the set of cells remains constant under all variations of wavelengths and light intensity. Such normalization is achieved by sub­traction of excitations of opponent cells from excitations of brightness and darkness neurons and additional subtraction from darkness neuron the brightness neuron excitation. A transition from «city-bоск» to a Euclidean metric one is realised by neuronal adaptation reducing extreme values of excitation so that Euclidean metric emerges and colour stimuli become projected on a hypersphere in the four-dimensional space. Now colour stimuli instead of colour body evident at receptor level and horizontal cell levels are extended along a surface occupied with colour selective detectors. That colour map composed of colour detectors encodes colour stimuli by specific locations — by a colour topic projection similar to the well-known retinotopic, somatotopic and tonotopic projections (Chernorizov, 1999).

Colour space extracted from colour-selective orienting reflex habituation

Repeated presentations of colour stimuli result in habituation of event-related desynchronisation of alpha-waves. This habituation is colour-selective. A novel colour stimulus reinstates the desynchronisation again. The duration of alpha-blocking increases as the difference between repeated and novel colour stimuli increases. Using pair-wise tests composed from repeated and novel colours one gets a matrix of magnitudes of desynchronisations representing differences between colour traces and actual cojour stimuli. Multidimensional scaling reveals a four-dimensional memory space contributing to novelty generation. It suggests that orienting reflex habituation is due to vector representation of a stimulus trace.

Colour constancy

In the process of evolution colour vision progressed from perception of local sources of illumination towards perception of specific reflexive object properties under different illumination conditions. To perform the “neural computation” of reflexive characteristics of surfaces in addition to local illumination encoding a specific neuronal channel to evaluate total illumination of a scene has to be present. The local colours are encoded by bipolar cells, but total illumination is represented by a set of horizontal cells.

Excitations of horizontal cells encoding total illumi­nation are subtracted from the respective excitations of bipolar cells, so that the results of subtraction correspond to a coefficient of reflection on the local area. Lightness constancy under achromatic illumination is due to the difference between excitation of a local brightness bipolar cell and excitation of L-cells (luminosity horizontal cells). By equal excitation of bipolar cells representing local light emission and horizontal cell representing background illumination response of local brightness neuron is equal to zero and only darkness bipolar cell is active resulting in a perception of grey local colour.

If background illumination becomes greater than local illumination then a local «bаск» bipolar cell is activated due to an input from the background. With an increase of tin-background illumination excitation of the black neuron increases and the local color becomes progressively black. Excitation of the black neuron is subtracted from the local darkness neuron ensuring vector normalisation. This model corresponds to the results of experiments with disc-ring configuration showing that lightness is encoded by an angle on the plane composed from bright-black axis and an orthogonal axis represented by a darkness neuron. Now Id us regard a local surface illuminated by background light. Any change of the background illumination would respectively modify light reflected from the local surface. The difference between local brightness neurons and L-cells would remain constant under all levels of the background illumination ensuring lightness constancy of the surface Similar mechanisms refer to chromatic constancy under соnstant brightness. Local red illumination excites R+G— bipolar cell and red background illumination activates R(> horizontal cell. If red background illumination becomes equal to the local one the response of the R+G— cells is equal to zero and only the local darkness neuron is active resulting in a grey colour in the local area.

With an increase of red background illumination an R G+ bipolar cell is activated and the local area appears greenish. If a local area is occupied by a surface illuminated by red background any change in the background illumination would change the light emitted from the local surface but the difference between locally emitted light and background illumination would remain constant. Thus white paper reflecting equally well all rays illuminated by red background would look white because of subtraction of RG horizontal cell excitation from R+G— bipolar cell. The same mechanism of colour constancy refers to Y+B— and Y—B+ bipolar cells. In general a four — dimensional colour code has to be taken into account.

3. The spherical Model of Colour Vision: dichromatic vision

Four-dimensional subjective colour space of human dichromats

Four-dimensional spherical space demonstrated for trichromatic vision in humans and animals suggests four independent modules located above the receptor level. It might be assumed that the same colour-coding channels are operating also in dichromatic vision. The two receptors are linked with four neuronal channels activating them in different degree, so that four-dimensional colour space is preserved. The following normalisation of excitation vectors would result in a spherical structure of the colour space. The outlined assumptions suggest that dichromatic colour space is also spherical and four-dimensional. At the same time one has to expect a reduction of a colour area on the hypersphere. The predictions were studied in dichromats using the same set of monochromatic colours that were used in trichromatic subjects. Subjective differences between pairwise presented colours were treated by multidimensional scaling. The results of experiments conducted in protonopic subjects demonstrate a four-dimensional colour space. As in trichromats colour stimuli were projected on the hypersphere. At the same time total area occupied by colours on the hypersphere decreased with respect to reddish colours. It was evident by projecting colour points on the hue plane: red colour shifted to the centre of the plane where in trichromats achromatic colours are located.

This, multidimensional scaling revealed in dichromats a four-dimensional colour space. This paradox can be explained by the existence of R+G- and R-G+ channels. The protanopic vision is characterised by an absence of a long wave length absorbing pigments. It produces a reduction of R+G channel while R-G+ one is still active. The four-dimensional colour space remains intact due to of R-G+ channel contributing to inputs coming from Y+B-, Y-B+, brightness and darkness neurons (Paramey, Izmailov, Sokolov, 1991; Paramey, Bimler, 2002).

Evoked potentials reveal four-dimentional colour space in dichromats

The psychophysical data demonstrating four-dimen sional colour space in dichromats suggest that colour coding N87 components as equivalents of colour differences will also demonstrate a four-dimensional structure. The matrix of amplitudes of N87 analysed by multidimensional scaling has shown that «objective» colour space of protanopic subjects coincides with their subjective colour spaces. This suggests that woth the elimination of «red pigment» four neuronal modules are still active, however, reduced with respect to R+G- channel.

Rabbits behavioral colour space

Rabbits lacking «red pigment» represent a natural model of dichromatic vision. The above consideration and experiments in Human dichromats strongly suggest that rabbits colour space would also be four-dimensional. Using instrumental conditional reflexes to colour stimuli a confusion matrix of response probabilities was constructed. The principal component analysis of the matrix has demonstrated a hypersphere in the four-dimensional Eucli dean space. The area on the hypersphere occupied by colours was reduced due to a shift of red colours to position represented in trichromats by achromatic colours (white, grey) very similar to human protanopl (Polyanski et al., 2000).

4. The spherical Model of Colour Vision: monochromatic color system

Rod and cone monochromats

Achromatic stimulation (white, grey) of the trichroma­tic eye results in a switching-off of opponent hue coding channels. The remaining excitations of two achromatic channels (brightness and darkness neurons) constitute an excitation vector, which is normalised. Thus different intensities of achromatic stimuli are localised along a semi­circle being represented by intensity detectors. In the case of a rod-monochromatic vision cones are not operating and all light stimuli excite only highly sensitive rods making rod-monochromatic vision equivalent to normal trichromat vision under achromatic stimulation. Thus rod-monochromat has a two dimensional spherical colour space with reduction of colours to the achromatic domain. A particular characteristic of rod-monochromat is low threshold resulting in feelings of discomfort which is due to the absence of cone mechanism normally inhibiting the rod system under high light intensity. A cone monochromat is characterised by the fact that only one pigment is found in all cones. This is equi­valent to switching off colour opponent channels occurring under achromatic illumination in normal trichromats when only brightness and darkness channels remain acting. Thus colour vision of the cone-monochromat is reduced up to a two-dimensional one being extended along a semicircle due to normalisation procedure.

Monochromatic vision in snail

A natural model of monochromatic vision is the visual system of snails. Thus Helix has a camera-like eye with a crystal lens. Photoreceptors constituting a retina are not inverted as in vertebrates but directed towards the light source. Rhodopsin is a single pigment expressed in photo­receptors of Helix. Two types of photoreceptors with depo­larising and hyperpolarising responses are distinguished in snails however. In insects photoreceptors are depolarised by light due to opening of sodium ionic channels. In vertebrates photoreceptors are hyperpolarised due to the closing of sodium ionic channels by light. In snail one sees a combination of both types of photoreceptors with domination of depolarising responses. The depolarisation and hyper polarisation are intrinsic features of photoreceptors. An intracellular recording from single photoreceptors isolated from retina demonstrates either a depolarisation or hyperpolarisation type excluding the possible contribution of lateral inhibition in hyperpolarizing responses. Thus light sti­muli evoke in the retina a combination of depolarising and hyperpolarizing responses in two subsets of photoreceptors assuming two-dimensional vector encoding similar to encoding of achromatic colours in normal human trichromats or light encoding of rod and cone human monochromats. The two-dimensional colour space in snails has a spherical structure constituting a semicircle along which different light intensities are located.

Corneal recording of ERG in snail for substitution of colour stimuli of different wave lenglhs enables one to generate a matrix of ERG amplitudes measuring stimulus differences. Multidimensional scaling of the ERG matrix revealed two axes equivalent to brightness and darkness channel found in vertebrates under achromatic stimulation. In snail the brightness neuron refers to depolarising photoreceptor and darkness neuron to hyperpolarizing one. Thus two channels encoding light intensity in snails are present already at the receptor level while in vertebrates separation of brightness and darkness channels starts only at the bipolar cell level. The presence of two encoding channels in colour vision in snail at the receptor level is followed by two types of axons emitted by receptors and extended to cerebral ganglia. Such neuronal organisation of photorecep­tors in snails opens the possibility of recording directly from axons spikes evoked by light stimulation.

Two types of axon activity were found: on and off-ones. In on axons spike discharges were evoked by light stimula­tion, while in off axons spiking starts by termination of light stimulation. Using different light intensity the numbers of spikes in on- and off- axons can be estimated. It was found that the same of numbers of spikes in on- and off- channels for any light intensity are equal to a constant value sug­gesting that spikes in on- and off- axons represent excitation vectors of a constant length. Taking into account two types of receptors with depolarising and hyperpolarizing responses one can conclude that on axons belong to depolarising photo­receptors while off axons are stimulated by hyperpolarizing ones.

It might be assumed that two-dimensional encoding of light intensity by excitation vectors of a constant length is used in snails for selective activation of intensity detectors and elaboration of selective conditioned reflexes to light intensity.

Comparison of colour vision at different level of evolution demonstrates universal principles of vector encoding in the framework of a spherical model.

5. Perception and expression of emotions in the framework of spherical model

Block diagram of perception and expression of emotions

Universal vector encoding can be demonstrated also in emotion processes.

The father is smiling. The child is smiling in response. This simple act of imitating on emotional expression has a very complicated neurophysiological mechanism. In its most general form it can be represented by a scheme including a «father-blоск» of generation of emotional expression, «child-blоск» of perception of emotional expression, «child-blоск» of generation of facial expression and «father-blоск» for perception of child emotional expression supporting the emotional contact.

The fact that the child smiles in response to smiling faces of adults indicates the contribution of inborn mechanisms in linking together a neuron identifying emotional expression and a neuron generating respective emotional expression. The problem of the neuronal code realising the concordance between perception of emotion and generation of its respective expression is still unsolved. In accordance with universal spherical model of cognitive and executive processes it is proposed that perception of emotional expression is achieved by a hypersphere in the four-dimensional space similar to colour space. The generation of emotional expressions is performed with participation of command neurons linked with four executive modules building up a four-dimensional output space isomorphic with perceptual space of emotional expressions.

Spherical structure of the space for perceiving emotional expressions

To test the predictions concerning perception of emotional expression pairs of computer generated schematic faces generated by a computer were pairwise successively presented to the subject. Emotional expressions of such schematic faces were produced by different inclinations of lines simulating brow and mouth. The task of the subject was to estimate the difference between presented faces in numbers from «0» to «9» and to input them into a computer. A matrix of subjective differences between faces was ana­lysed by multidimensional scaling. It was found that faces having different emotional expressions were located on a hypersphere in the four-dimensional space very similar to the perceptual colour space.

Three angles of the emotion hypersphere match respectively emotional tone, emotional intensity and emotional saturation (Sokolov, Boucsein, 2000). The Cartesian coordinates of points representing different emotional expressions are assumed to involve excitations of four predetector of specific emotions. Particular emotions are encoded by gestalt-detectors representing particular loci on the hypersphere. The Euclidean distances between tops of vectors encoding emotions closely correlate with subjective differences found during presentation of pairs of faces. It implies that subjective differences between emotions are «computed» from excitation vectors that encode respective emotion (Izmailov et al., 1999).

From N180 component of evoked potential to spherical emotion space

The presented data concerning reconstruction of emo­tion space from subjective differences between emotional expressions suggest that neuronal nets are «computing» the differences from respective excitation vectors. It can be assumed that evoked potentials produced by the substitution of faces will be equal to subjective indices of perceived differences between emotions. Experiments with substitution of schematic faces has shown that amplitudes of a negative component arising with 180 ms latency (N180) indeed closely correlate with subjective differences between respective faces. A matrix of N180 amplitudes implicitly contains information concerning the basis of emotional vector space. It was found from N180 amplitudes that schematic faces were located on a hypersphere in the four-dimensional space identical to emotion space obtained from subjective estimates of emotional differences.

Motor space of facial expressions of emotions

To evaluate the output space of emotional expressions one has to use multichannel EMG recording from facial muscles. A set of local face EMGs at a particular time interval after presentation of a face constitutes a EMG-vector. Using different schematic faces one can collect a matrix of EMGs. Computing absolute values of EMG­vectors one can construct an EMG-matrix of differences between emotional expressions. Multidimensional scaling of the EMG-matrix should reveal the motor space of emotional expression. Preliminary analysis suggests that emotion motor space is a spherical and four-dimensional one. It suggests that the space dealing with the perception of emotions and the space for expressing emotions are isomorphic enabling a concordance between perception and expression of emotions.

How are perception and expression of emotions are linked together?

The gestalt-units selectively activated by perception of particular emotional expressions are apparently connected with «mirror neurons» of the frontal lobe selectively activated before generation of a particular emotional expressions. Such «mirror neurons» belong to a class of «intention units». The «mirror neurons» are linked with respective command neurons. Particular command neurons are activated at the subsequent step — the generation of an “emotion gesture”. Command neurons are connected with four modules (premotor neurons). Each of the modules is linked with a pool of motor neurons producing a specific combination of constrictions of facial musculs. A combi­nation of four pools of motor neurons defines a specific emotional expression.

Emotion semantics

Presentation a subject with a schematic face expressing a particular emotion under instruction to name it results in the generation of an emotion label. Using a set of schematic faces one gets a matrix of probabilities of emotion labels, so that each row of the matrix corresponds to a particular emotional expression. Principal component analysis of the matrix demonstrates the four-dimensional space in which particular facial expressions are located on the hypersphere. Thus, schematic faces reach semantic units which determine emotion names. Presenting subjects with pairs of emotion names and asking them to estimate a semantic difference between respective emotion names enables one to construct a matrix of subjective differences. Multidimensional scaling reveals four dimensions of the spherical emotion semantic space.

It is assumed that emotion names activate neurons of emotion in long-term memory. Excitation vectors generated in declarative memory by emotion names are matched against each other in subtracting neurons which contribute to the generation of a semantic difference. Thus two emotion names evoking identical populations of long-term memory units are synonyms.

To match perception of emotional expression against emotion names a combined matrix of subjective differences between schematic faces and emotion names has to be obtained. The joint four-dimensional space has a spherical structure. On the hypersphere schematic faces and respective emotion names are located close to each other demonstrating a common vector code.

Instruction to simulate emotional expression according to presented emotion names produces EMGs coinciding with EMGs elicited by presentation of emotionally loaded faces. Principlal component analysis of the EMG-matrix produced by facial muscle activity evoked by emotion names reveals a four-dimensional spherical output space of facial expression. The coincidence of output spaces obtained by presentation of faces and by application of emotion names implies that emotion semantic units are linked with command neurons responsible for generation of specific emotional expressions. By acquiring a foreign language neurons encoding new verbal symbols establish synaptic links with long-term memory units on the one hand and with command neurons for generation of verbal responses on the other hand. It implies that the semantic spaces of emotion names are isomorphic for different languages.

6. Vector encoding in space and time perception

Vector encoding of the physical space perception

Spherical structure of color and emotion spaces suggests similar structure of subjective perception of the physical space. A matrices of subjective differences between points presented on the monitor screen and being analyzed by multidimensional scaling demonstrate a hypersphere in the four-dimensional functional space of neuronal channels both by monocular and binocular observation. Four neurons mea­sure horizontal and vertical positions, depth and approach to the central point. Three angles of the hypersphere de monstrate:

1. orientation of points against the central fixa­tion point;
2. perceived depth;
3. deviation from the center.

The spherical structure of the perceived physical space is characterized by constancy. The perceived points don’t change their positions by eye movements when their projection are displaced against retina. The constant perception is due to correcting signals from neurons cont­rolling eye positions. The control excitation vectors determi­ning eye position have four components and constant lengths so that control eye movement space constitute a hypersphere isomorphic to perceptual space. The subtraction of eye position coordinates from the coordinates of a point on the retinotopic screen is accompanied by vector normalization. It results in a projection of perceived point on the constant screen of the hypersphere where perceived positions of points remain constant by all kinds of eye movements.

Four-dimensional subjective representation of the physical space has a spherical structure and the number of independent variables remains equal to three. But now three Cartesian coordinates of the physical space are replaced by three angles of the subjective hypersphere having a constant radius.

Vector code in time perception

The principle of vector encoding can be used to explain perception of time. A start of stimulation trigger in one time — predetector gradually increasing depolarization, which is subtracted from a spontaneous excitation in a twin neuron. In this way the excitations of two predetectors create excitation vector normalized in city-block metric which later on is trans formed into Evclidean one. The outputs of the predetectors converge on a set of time-selective detectors. Specific time intervals are represented by specific excitation vectors which sequentially activate time-selective detectors transforming time -unter into excitations of specific neurons.

Vector code in the time domain contributes to the process of anticipation and time-selective conditioning. In an indirect way time-selective detectors take part in evaluation of distances measuring a delay between the onset and return of the ultrasound in bats.

Vector code and brain oscillations

Selective tuning of feature detectors and command neurons with respect to an input stimulus depends on the coincidence of the excitation vector with respective synaptic weight vectors of the detector or command neuron. Selective tuning of the neurons can be improved by introduction of rhythmic processes at the presynaptic and postsynaptic levels. In this case maximal responses of detectors and command neurons will be achieved only under coincidence of frequency and phase lag of oscillations at presynaptic and postsynaptic mechanisms. To such resonance effects contribute pacemaker potentials.

The endogenous pacemaker potentials in pacemaker neurons contribute to prolongation of the synaptic response, frequency- selective filtering and potentiation of synaptic connections.

The prolongation of the postsynaptic response leading to prolongation of behavioral activity is evident in command neurons in which a short postsynaptic excitation can trigger a long-lasting pacemaker activity. The pacemaker dependent spiking is accompanied by gradually developing behavioral response. Pacemaker potentials present in presynaptic neurons generate bursts of spikes which are enhancing responses in a postsynaptical cell.

Frequency-selective filtering occurring in pacemaker cells is due to resonance between a synaptic train of spikes and phase-locked pacemaker frequency of the postsynaptic cell.

The frequency-selective tuning is related to resetting of the pacemaker wave which makes pacemaker phase-locked with respect to the input.

Pacemaker gamma-oscillations, evoked by high fre­quency spiking result in potentiation of synaptic efficency known as a homosynaptic potentiation. Such potentiation can be heterosynaptic when high frequency spiking refers to a set of neurons. A potentiation of synaptic contacts on selective detectors results in an extension of excitation area on the detector map.

The potentiation of synapses connecting detectors with a command neuron is augmenting respective conditioned reflexes.’ In general high frequency oscillations increase performance, while low frequency ones results in its reduction.

7. Conclusion semantic

Extracellular and intracellular recording, evoked potential study, behavioral and psychophysical experiments in humans and animals demonstrate that trichromats and dichromats utilize a universal four-dimensional colour space in which colours are represented by points located on the hypersphere constituting a colourtopic map. Each colour is encoded by an excitation vector that is of a constant length. Subjective colour differences are extracted from excitation vectors in specific neuronal nets as absolute values of vectorial differences.

Conditional reflexes to colour stimuli are elaborated due to reinforcement which modifies plastic synaptic contacts on command neurons so that the plastic synaptic vector coincides with the excitation vector generated by the conditional stimulus.

It was predicted that similar spherical models are re­levant for representing other modalities (Sokolov,2000). It was found it to be true for emotional space.

Perception of facial emotional expressions is performed by an «emotion analyser» similar to the «colour analyser». The input signals to the «emotion analyser» originate from orientation detectors. Four modules representing emotion predetectors determine a four-dimensional emotion space. Specific emotions are encoded by gestalt-detectors located on a hypersphere in the four-dimensional space. Motor space for expression of emotions is composed of «mirror neurons» (a kind of intention units) and command neurons linked with four output modules (premotor neurons) that determine a four-dimensional output space of emotional expressions. The connections of four premotor neurons with their respective pools of motor neurons defines a specific «emotion gestures*. The connections between gestalt-detectors of emotion via intention neurons with command neurons identifies a «one-to-one» relationship between perceived and expressed emotions. Isomorphism of perceptual and output spaces enables such concordant encoding of perception and expression of emotions.

The universal character of the vector model of encoding is supported in psychophysical and electrophysiological experiments related to perception of the physical space. Subjective differences between point on the monitor screen and evoked potentials to change of their position reveal a hypersphere in the four-dimensional space similar to color space and emotion space. Four axes of the perceptual space correspond to signals received from two neurons measuring orientation, and two neurons evaluating depth and approach to the central point, respectively. Three angels of the hyper­sphere represent orientation, depth and deviation from the central line.

Perceptual space of physical space is evidently a cons­tant one. It constancy is achieved by signals coming from neurons controlling positions of eyes.

The space of eye control and perceptual space of the physical space are isomorphic so that activation of a selective space-detector can generate a saccade addressed to the point of the stimulus location.

The vector encoding is operating in time-selectiv feature detectors. Pacemaker activity contributes to frequency se­lective filtering and prolongation of spiking in neuronal net­works.

Synaptic potentiation is elicited by high frequency spi­king dependent on pacemaker activity in the range of gamma-rhythm.

8. Conclusion. New horizons

Two basic operations can be distinguished in neuronal networks: specification of a stimulus at the detector level and neurocomputation of perceptual differences. These two ope­rations can be integrated in the framework of vector encoding. Input stimulus at the predetectors level is trans­formed into multidimensional excitation vectors of a cons­tant length acting in parallel on a set of selective detectors characterized by non-plastic synaptic weight vectors waving constant length but differing in orientation. Each selective detector multiplies presynaptic excitations on the postsy­naptic weights and sums up then products computing in this way a scalar product of the input excitation and its synaptic weight vectors. The selective detector with synaptic weight vector coinciding with the input excitation vector will be excited in maximum degree executing stimulus specification.

By an abrupt stimulus change one excitation vector is replaced by another one. The differences between coordinates are computed in phasic «on» and «off» neurons converging on «on-off» neurons summating these coordinates’ differences. The output of the «on-off» neuron results in an absolute of vectorial difference. The spike discharges of a phasic «on-off» neuron determine the amplitude of the evoked potential. It means that the amplitude of the evoked potential to stimulus change is a measure of the stimuli difference. It suggest that a matrix of evoked potential amplitudes implicitly contains information concerning vector space representing input sti­muli. The concept of stimulus specification and stimulus chan­ge can be extended on episodic memory.

In this case gnostic units of complex stimuli represent an identification process equivalent to the specification at the detector level. An abrupt change of object names results in an evoked potential with amplitude, which parallels seman­tic difference between respective words. A matrix of evoked potentials to abrupt change of different word implicitly con­tains information about semantic space of respective words. Responses of «on-off» neurons measuring differences between stimuli are characterized by high frequency spike discharges rising up with increase of the differences. At the evoked po­tential level these spike discharges parallel gamma rhythm frequency.

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