A. M. Levinov
The learning mechanism of saccade formation
Keywords: saccade, microsaccade, fixation, learning, superior colliculus, saccade
speed.
Abstract. This work presents a learning mechanism of saccade and fixation formation.
The role of various parts of this mechanism is explained, and a construction that can
function in the presence of different types of muscular fibers, various parameters of motor neurons, and different numbers of their input and terminal synapses is described.
The described mechanism of learning does not require the existence of inborn computational programs in the nervous system or the ability to carry out computational procedures. The assumption about learning allows for the formulation of experiments to examine this supposition. In the first appendix, the requirement of the superior colliculus (SC)
and why the retinal mapping to the SC should be well organized are discussed. The possibility that the SC does not set the movement’s characteristics, but only initiates the
movement is presented. The second appendix describes the assumption that learning enables to receive the formula of saccade amplitude and its maximal speed connection.
Saccades and fixations: neurophysiology and models.
Neurophysiology of fixations and saccades. At present much attention is given to the
research of neurophysiological mechanisms involved in the movements of live organisms. Studying eye movements turned out to be convenient for investigating these mechanisms since these movements are rather easily accessible for experiments.
Eye movements are defined by three pairs of extraocular muscles: horizontal movements carried out by medial and lateral rectus muscles, vertical movements carried out by
the joint constriction of pairs of superior/inferior rectus and superior/inferior oblique
muscles, and torsional movements, rotation of the eye around its sight axis, carried out by
superior/inferior oblique muscles.
One of the most studied eye movements are saccades: the jumps that allow fast eye
movements from one viewed dot to another. These movements can shift the eye in any
direction, but they do not normally rotate the eyes around the sight axis. Since, here
mainly saccades are discussed, torsional eye movements will not be further taken into
consideration.
Extraocular muscles are guided by impulses from motor neurons. These neurons are
found in III (oculomotor), IV (trochlear), and VI (abducens) cranial nerve nuclei. The
amplitude, duration, and speed of saccades are defined by intensity, duration, and firing
rate of these neurons (Fuchs and Lushei, 1970; Schiller, 1970).
Different areas of the premotor brainstem participate in saccade performance (Fuchs et
al., 1993; Goldberg et al., 1998; Sylvestre and Cullen, 1999; Sparks, 2002; Brown et al.,
2003; Bullock and Grossberg, 2004; Van der Stigchel et al., 2006; Ludwig et al., 2007).
2
Signals to motor neurons come from exciting burst neurons (EBNs) located in the
paramedian pontine reticular formation (Moschovakis et al., 1996; Scudder et al., 2002;
Sparks and Hu, 2006). Horizontal movements are determined by premotor neurons in the
pons and medulla, while vertical movements are determined by premotor neurons in the
rostral midbrain (Kokkoroyannis et al., 1996; Dalezios et al., 1998; Sparks, 2002).
The viewing of objects is connected with small amplitude movements – tremor, drift,
and microsaccades – which accompany eye fixations (Alpern, 1962; Ярбус, 1965;
Леушина, 1971; Becker and Jürgens, 1979; Carpenter, 1988). At fixation, motor neurons
fire with constant frequency (step), the size of which linearly depends on the extent of
eye deviation from the position of balance (Fuchs and Lushei, 1970). Tonic activity of
many neurons in nucleus prepositus hypoglossi and the medial vestibular nucleus is proportional to the eye horizontal position: these cells generate tonic excitation, which is
necessary for motor neuron step activity.
EBNs are monosynaptically connected with motor neurons and are the basic source of
excitation for pulse activity, prompting the eye jump (Igusa et al., 1980; Strassman et al.,
1986). EBNs generate high frequency burst activity before ipsilateral saccades and send
signals to inhibitory burst neurons (IBNs), which inhibit the motor neurons of muscle antagonists.
Premotor neurons in the rostral midbrain generate pulse and step activity that
monosynaptically transferred to the motor neurons controlling vertical eye movements
(King et al., 1981; Kokkoroyannis et al., 1996). Duration, amplitude, and speed of vertical saccades are functions of burst duration, spike number, and burst frequency of corresponding neurons. Neurons in the interstitial nucleus of Cajal and in the vestibular nucleus fire tonically with a frequency that is linearly related to the eye position on the vertical
axis, and creates activity that generates a step signal (Dalezios et al., 1998).
EBNs receive excitation signals from long lead burst neurons (LLBNs) located in the
rostral pons, and from neurons of superior colliculus (SC). LLBNs are not as closely connected to the beginning of a saccade as EBNs. EBNs and IBNs are influenced by inhibitory signals from omnipause neurons (OPNs) that fire with rather constant frequency during fixation, but stop firing during saccades (Buttner-Ennever et al., 1988; Langer,
Kaneko, 1990; Moschovakis et al., 1996).
The governing of saccades is mainly transferred through the SC (Lee et al., 1988;
Quaia et al., 1998; Hanes and Wurtz, 2001), which is the brainstem area that generates
the basic input for the pulse–step activity of neurons. The SC receives signals from many
cortical and subcortical areas, and sends signals to those premotor areas that are connected with the control of eye movements (Robinson, 1972; Sparks, 1986, 2002; Bergeron et
al., 2003; Gandhi and Sparks, 2004; Krauzlis et al., 2004; Saito and Isa, 2005; Sparks and
Hu, 2006; Ramat et al., 2007).
It is suggested that SC codes the distance to the target (Bergeron et al., 2003), programs the optimal movement trajectory (Goossens and Van Opstal, 2006), and defines
the saccade beginning, speed, and shift size (Sparks and Hu, 2006). That SC activity corresponds to the initial position of the target, but not to saccade direction (Krauzlis et al.,
2004). It is assumed that SC defines the target of the movement (Krauzlis et al., 2004)
carries a signal describing the distance between the target and current sight direction
(Matsuo et al., 2004), and plays an important role in the control of saccade direction and
amplitude (Hanes and Wurtz, 2001). The problem is that the activity of SC neurons con-
3
nected with saccade performance does not correlate with saccade direction, amplitude,
and speed (Robinson, 1972; Schiller and Stryker, 1972; Sparks and Hartwich-Young,
1989; Stanford et al., 1996).
Another problem is that it is not clear how SC knows what signals should be sent to
the oculomotor muscles and when to send them: the mechanism responsible for the transformation of the choice of target to the movement launching is still unknown (Gandhi and
Sparks, 2004). It is not clear for what SC is necessary: why cannot the control of eye
movements proceed directly from the brain visual cortex?
About the saccadic system models. Modeling plays an essential role in studying various types of eye movements. It is assumed that models facilitate the understanding of
experimental results and indicate the direction of further studies (Glimcher, 2003/1999;
Sparks, 2002; Pola, 2002; Krauzlis et al., 2004; McPeek and Keller, 2004; Gandhi and
Sparks, 2004; Sparks and Hu, 2006; Ramat et al., 2007).
At the beginning of eye movement modeling, the mainly studied area was the reticular
formation area of the brainstem (Jürgens et al., 1981; van Gisbergen et al., 1985;
Grossberg and Kuperstein, 1986; Nichols and Sparks, 1995). Later the researchers began
to include SC (Ottes et al., 1986; Waitzman et al., 1991; Arai et al., 1994; Optican, 1995;
Grossberg et al., 1997; Trappenberg et al., 2001; Goossens and Van Opstal, 2006; Nakahara et al., 2006), the cerebellum (Dean, 1995; Gancarz and Grossberg, 1999; Quaia et
al., 1999; Optican and Quaia, 2002), the basal ganglia, and the cortex (Dominey and
Arbib, 1992; Gancarz and Grossberg, 1999; Mitchell and Zipser, 2003; Brown et al.,
2004; VanRulen, 2004).
In addition, there are models related to other eye movement problems (Lebedev et al.,
1996; Moschovakis, 1997; Raphan, 1998; Findlay and Walker, 1999; Harwood et al.,
1999; Smith and Crawford, 2001; Mitchell and Zipser, 2003; Pola, 2004; Kuniharu and
Keller, 2004; Ludwig et al., 2007). An idea emerged suggesting the creation of a model
comprising all brain parts connected with saccade formation (Girard and Berthoz, 2005).
The opportunity to use modeling to explain various deviations from the norm and to
search for ways to eliminate them, is being investigated (Ramat et al., 2007). Therefore, it
is necessary to be able to estimate how close such models are to the facts found in experiments.
Early attempts of oculomotor system modeling have established that in order to overcome the viscous drag of orbital tissue and to ensure high speed movements, it is necessary for motor neurons to create pulse activity during a jump. After that, the pulsation
should gradually diminish, and a constant activity of neurons (step) should emerge and
support muscle exertion during fixation (Westheimer, 1954; Robinson, 1964).
In solving the problem of such activity formation, Robinson (1975) assumed that there
is a mechanism (comparator) that compares signals from the current eye position and
from the stimulus that defines the subsequent fixation. By the intensity of the signals that
emerge as a result of comparison, the pulse generator calculates the impulse size that
should be sent to muscle agonists to generate a jump.
Robinson (1973) assumed that the step signal intensity is proportional to the pulse intensity, and that after pulse signal determination, the step signal intensity can be mathematically calculated by the brain. Later on, this idea of pulse and step component formation was used in many models.
Most of the existing models of a saccadic system have modified the initial (positional)
4
model of Robinson (1975) so that now the input signals for the burst generator are determined as a desirable change in eye position (displacement model).
Early models of saccade generation were ballistic: the number of spikes in a motor
neuron burst was defined prior to the beginning of movement. In modern models, it is
assumed that saccades are under feedback control. Such a concept emerged when it was
found that patients with certain neurological diseases generate slow saccades, some of
which are changing during a jump (Zee et al., 1976). This fact stimulated the creation of
models with feedback (Zee et al., 1976). In such models, a copy of a motor command
(corollary discharge) was used as a source of a feedback for the saccade amplitude control. This assumption is applied in almost all modern models of saccadic systems.
Unresolved problems. Existing models allow the solving of various problems related
to different types of eye movements, but still many questions remain unanswered. For example, as Sparks (2002) points out, the type of feedback signal used for saccade control is
still unclear, and the location of neurons involved in the comparison of input and feedback signals has still not been found. The problem of the comparator anatomic location
remains unresolved.
It is known that the switching on of a luminous dot can cause an infant eye movement
both towards and from a stimulus (Aslin and Salapatek, 1975; Фонарев, 1977; Hanter
and Richards, 2003). The sure performance of saccades in the necessary direction occurs
only around the age of 3-4 years. Existing models do not explain this phenomenon.
At an early age, muscle agonists and antagonists are simultaneously strained in all
kinds of eye movements. After a while, at a saccade, a relaxation of the muscle antagonist
corresponds to the strain of the muscle agonist (Björk, 1955; Митрани, 1973; Goldberg
et al., 1991; Glimcher, 2003/1999): during a saccade, signals reach muscle agonists,
while the neurons controlling antagonists cease firing (Fuchs and Lushei, 1970; Schiller,
1970). This phenomenon is not considered in existing models (Sparks, 2002).
Eye movements in models are defined by signals going through a channel, imitating a
generalized motor neuron. The fact that each channel contains tens and/or hundreds of
different neurons and at the ends of their axons there could be a different number of synapses of different sizes and intensities of impact on subsequent neurons, is not considered
(Sparks, 2002, Ramat et al., 2005, 2007).
During a saccade, approximately 25-30% of SC neurons fire (Lee et al., 1988; Munoz
and Wurtz, 1995; Anderson et al., 1998). It is assumed that saccade accuracy results from
the average activity of these neurons (Baldi and Heiligenberg, 1988; Gandhi and Sparks,
2004), without consideration that averaging does not mean receiving signals of the necessary intensity and necessary distribution in time.
Extraocular muscles contain six types of muscular fibers with different exertion abilities (Porter et al., 1995; Goldberg et al., 1998; Goldberg and Shall, 1999). This is not taken into consideration in the majority of modern models (Sparks, 2002).
It is not clear how the comparator can innately define the necessary pulse frequency. It
is not explained how the nervous system establishes the signal intensity that defines muscular exertions at fixation. What is the meaning of "proportional" for step signals: "proportional" can be two or ten times more or less (Sparks, 2002; Glimcher, 2003/1999).
On the possibility of calculations in the nervous system. In his model, Robinson
(1975) assumed that the nervous system calculates the necessary eye movement, proceeding from its current position and the position of a new target that emerges in the visual
5
field. All the models of eye movements are based on the idea of calculations. It is assumed in models that the nervous system can innately define (calculate) the necessary intensity and duration of motor neuron excitation for saccade performance. Let us consider
the requirements such a system should satisfy.
Each fiber of an extraocular muscle reacts in its own way to the same sequence of motor neuron signals. For the necessary movement generation, the nervous system must
send signals onto an appropriate sequence of neurons and distribute these signals to certain muscular fibers. For this purpose, the nervous system should "remember" the reactions of all muscular fibers to every possible input impact.
In the nervous system, even neurons of the same type differ in their parameters. Neurons have different quantities of terminal synapses with varying amounts of input synapses of various sizes and intensity of impact terminating on them. In order to receive the
necessary signal, the nervous system has to consider the sequence of neurons and synapses that the signal will pass through. In order to accomplish this, the nervous system
should innately know all the interrelations of neurons on the signal's path from the retina
to a muscle, and should innately store in its memory the characteristics of all neurons
through which the signal from the retina can reach the extraocular muscles. The system
should also innately know the distribution and characteristics of these synapses, as well
as the structure of the interrelations of each neuron with both previous and subsequent
neurons and muscles. The nervous system should be able to innately define the set of signals that each neuron receives in order to generate the necessary movement
In addition, the nervous system should be able to innately coordinate signals from different parts of the brain, for example, for inclined saccade, execution should coordinate
the simultaneous participation of the pons, medulla, and rostral midbrain and their interaction.
It is hard to believe that the nervous system possesses sufficient capacity to store all
these data in its memory. It is also hard to believe that it possesses sufficient speed for the
performance of the necessary calculations during a saccade and that it can "remember"
the changes in the nervous system structure that occur during organism development.
The idea of the mechanism learning to shift the retina center to a
stimulus. Positing the problem. As an alternative to calculations, one can assume that
fixations and saccades are not innate and emerge as a result of learning. From the point of
view of the processes occurring in a live organism, the assumption of learning seems
more attractive than the idea of calculations performed by the nervous system. The possibility of the existence of mechanisms that provide such learning, the process of learning,
and some of its consequences, are considered below.
Structure of the further presentation. The system of all the mechanisms providing
different eye movements has turned out to be very cumbersome. Therefore, the formation
of saccades is described only for the simplest case, in which the retina center is initially
directed to a fixation dot, then a new stimulus emerges on the periphery and the eye shifts
to it. The process of the increasing of saccade amplitude, and saccade automation are not
considered; the mechanism that defines the termination of eye fixation and stimulates the
beginning of a saccade is not considered – all this lies beyond the framework of this
work.
Retina, motor neurons, synapses, and extraocular muscles. For simplification of
the presentation, instead of eye movements, only the movements of the retina will be
6
considered further. It will be assumed that these movements are governed by two pairs of
muscles: the horizontal muscles move the retina left and right and the vertical muscles
move the retina up and down. Figure 1 shows the retina, extraocular muscles, and the motor neurons governing them.
k
•K
C
f
A •
g
i
M
P
Q
N
Fig. 1
C – retina, A  retina center, K  dot on the retina periphery, f and g – horizontal muscles, i and k – vertical muscles; M, P, Q, N  motor neurons of oculomotor system, small
circles – excitatory synapses through which the signal reaches the motor neurons.
It is known that it is possible to develop almost any eye movement in response to almost any stimulus. In order for this to happen, it is necessary that a signal be able to reach
any motor neuron of the oculomotor system from each dot of the retina. Therefore, one
may assume that for each dot of the retina (through intermediate neurons), there exists a
synapse with each motor neuron. For example, figure 2 shows the connection of several
dots of retina C with two motor neurons P and Q (intermediate neurons are not shown).
C •
•
•
P
Q
Fig. 2
Designations as in figure 1.
Generally speaking, "a dot" stimulus usually excites not a dot, but a small area of the
retina. In figure 3, such an area is designated by Z. Further, "a dot" will mean such a
small area of the retina. The signals reaching motor neurons P and Q (fig. 2) include
summary excitation of such an area. A set of channels through which the signal reaches
the input synapses of motor neurons corresponds to each dot of the retina: each dot has its
7
own set of synapses.
Figure 3 shows that each retina dot can have many synapses with the same motor neuron. In order not to encumber the figures, the set of all synapses of a retina dot with a motor neuron will be represented by one input synapse of this neuron (see fig. 1).
C
Z
P
Q
Fig. 3
Z – a small area ("dot") of the retina; other designations as in figure 1.
Assumptions connected to the process of learning. The eyes of an infant can move
in response to the presentation of a stimulus, therefore: a) extraocular muscles innately
react to signals from motor neurons; and b) motor neurons innately react to signals coming from various parts of the retina. The problem is not in the ability of neurons and muscles to react in response to a stimulus, but that they do not "know" how to react in order
to turn the retina center toward the stimulus.
C
f
A•
1
P
K•
g
2
Q
Fig. 4
1 and 2 – excitatory synapses, K – dot stimulus on the horizontal axis of the retina;
other designations as in figure 1.
The absence of a rigid connection between the stimulus position and the eye movement is observed in children up to age 3-5 years (Ярбус, 1965; Запорожец, Венгер,
Зинченко, Рузская, 1967; Aslin and Salapatek, 1975; Фонарев, 1977; Hanter and Richards, 2003). It can be assumed that the establishment of such a connection is defined by
learning (Левинов, 1970, 1974). Therefore, it is necessary to show that with the help of
learning, it is possible to form connections between the retina and the extraocular system
8
in such a way that stimulations of any retinal dot will cause the retina's center to shift toward the source of excitation.
In order to define the mechanisms necessary for such learning, let us first address the
case of horizontal movements: when an excitatory stimulus is projected onto dot K located on the horizontal straight line passing through the retina center A (fig. 4).
As can be seen from figure 4, at the excitation of dot K, the signal will pass through
synapses 1 and 2 and can prompt the activity of motor neurons P and Q. The system is
not innately attuned, therefore, the signal from these neurons, reaching extraocular muscles, can leave the retina motionless or shift it to the right or left (stimulus can cause constriction of other extraocular muscles, but here only the horizontal muscles are considered).
At the final stage of learning, in the stimulus position shown in figure 4, muscle g
should contract and shift the retina to the right, and muscle f should cease to contract in
order to create an opportunity for a jump. To provide a possibility for such a change, it
can be assumed, for example, that at a repeated stimulation of dot K, the volume of synapse 2 gradually increases up to a certain limit, and the volume of synapse 1 gradually
decreases and, as a result, ceases to excite the motor neuron.
Thus, the process of learning can be reduced to the change of the size of motor neuron
input synapses: the increase of synapses whose action assists the retina center to shift towards the stimulus, and the decrease of synapses whose action counteracts such a shift.
Let us consider mechanisms that have to participate in the process of learning.
The condition of expectation. From what was said above, it follows that as soon as a
signal comes through synapses 1 and 2 to neurons P and Q (fig. 4) and these neurons fire,
both synapses should be ready for changes. They have to be only ready: changes can
begin only when there will be muscle constriction and a displacement of the retina defined by this constriction.
Hence, after neurons P and Q fire, synapses 1 and 2 should be in a condition under
which further retinal shift will define their change. Such a synapse condition will be referred to as the condition of expectation. The time during which this condition lasts will
be referred to as the time of expectation. This time lasts from the moment when synapses
1 and 2 start to act until the arrival of the signals that indicate the direction of the shift of
the retina.
In the experiments, the synapse condition, which is referred to as tagging, is indeed
observed: the transition of synapses to the analogue of the condition of expectation and
their further strengthening or weakening are dependent on the subsequent activation (Frey
and Morris, 1997; Martin and Kosik, 2002; Sajikumar et al., 2005). Some authors (Nader
et al., 2000; Martin and Kosik, 2002; Dudai, 2004) point out the possible importance of
tagging in memorization. Several studies consider the possible models of such memorization (Yuste and Urban, 2004; Govindarajan et al., 2006). The mechanism described below differs from the offered models in that during repetitions, the size of synapses can
change repeatedly.
Differences between signals from the center and from the periphery of the retina.
The learning of motor neuron input synapses consists of their changes at the retina shifts
to or from the stimulus. This means that the nervous system should distinguish whether
the stimulus after retina displacement is more closely projected to its center or further
away from it.
9
Let us consider a mechanism capable of distinguishing between signals from the same
stimulus at its projection closer to or further from the retina center. Assuming that learning occurs at a bright enough illumination, it is possible to limit oneself to the presence of
only cones in the retina. It is known that the density of cones increases at transition from
the periphery of the retina to its center, reaching its maximum in the central part of the
retina (Wässle, Boycott, 1991; Reid, 2003/1999) (fig. 5). Proceeding from this (not taking
into consideration the possible influence of retina cell interrelations  see, for example,
Calkins, 2001/1999; Dacey and Lee, 2001/1999), it can be assumed that the closer the
same luminous dot (a small area of the retina) is projected to the retina center, the more
photoreceptors will be activated, and the signal that will come onto the extraocular muscles will be more intensive.
у
х
Fig. 5
Distribution of cones density depending on the distance to the retina center: on axis х
 distance from the retina center; on axis y  quantity of cones in the area unit.
Hence, if a dot is projected onto the retina periphery and then the retina is shifted, the
dot will be projected closer to the center and the signal from it will be amplified. If the
retina is shifted so that the dot appears further from the center, the signal will weaken.
The mechanism of comparison of retina excitation intensity at the projection of a
stimulus to the retina center and its periphery. Let us consider the mechanism that
finds out the difference in signal strength at a projection of the same stimulus to the center and to the periphery of the retina.
C
f
K •
L•
4
3
5
m
6
n
R
g
S
U
p
Fig. 6
C – the retina, K and L – two consecutively excited dots of the retina, R, S neurons of
10
area E, U, an intermediate neuron, 3 and 4 – excitatory synapses, 5 and 6 – inhibitory
synapses, m, n, p – axons, f and g – horizontal muscles.
It is assumed that a comparison of intensities ensues by means of neurons of some
brain area E that is outside the retina. It can be assumed that each dot of the retina has an
excitatory synapse with a certain neuron in this area.
Let R and S be two area E neurons (fig. 6). The signal from dot L of the retina goes
through synapse 3 onto neuron S, and the signal from dot K of the retina goes through
synapse 4 onto neuron R. Due to the distribution of the retina cones, as shown in figure 5,
the signal transferred by synapse 4 will be more intensive than that transferred by synapse
3.
Comparison of the excitation intensities of neurons R and S is carried out by means of
mutual inhibition of these neurons by synapses 5 and 6 (fig. 6). It is assumed that the
higher the level of excitation of a neuron in area E, the higher the level of inhibition
prompted by it. This, in particular, means that at a projection of the same stimulus on dots
K and L, the level of inhibition prompted by synapse 6 should be higher than the level of
inhibition by synapse 5.
The mechanism of equating the levels of excitation and inhibition. In the considered mechanism, it is necessary that inhibition by neuron R through synapse 6 will block
the output channel n of neuron S, while inhibition by neuron S through synapse 5 under
the same conditions will not block output channel m of neuron R.
The existence of such an opportunity does not follow from what is said above. For example, the input synapses of neuron S can be innately larger than those of neuron R.
Then, at a lower level of retinal dot L excitation, the signal from neuron S can be more
intensive than neuron R excitation by a signal from dot K. As a result, inhibition by synapse 5 will block neuron R, and the mechanism will work as if excitation of dot L is
higher than that of dot K. A mechanism is needed that will prevent such a situation.
As a matter of fact, it is necessary that at equal excitations of dots K and L (fig. 6), the
level of excitation of neuron R will be equal by absolute value to the level of its inhibition
by synapse 5. Thus, at a higher level of dot K excitation in comparison with dot L, a signal will pass through axon m, while at a lower level no signal will come.
m
4
R
5
h
3
S
Fig. 7
h – an excitatory synapse from axon m to inhibitory synapse 5 of neuron S; other designations as in figure 6.
We need to achieve equality (by absolute value) of excitation of neuron R by synapse
4 and its inhibitions by synapse 5 (fig. 6) at dots K and L levels of excitation equality. For
this purpose, the excitatory synapse h is inserted from axon m of neuron R to inhibitory
synapse 5 in the scheme shown in figure 7. It is assumed that if a signal comes through
axon m, synapse h will be excited, and this will lead to the increase of synapse 5. If synapse 5 inhibits neuron R and a signal to synapse h does not come through axon m, there
11
will be a decrease of synapse 5.
After a number of repetitions, this scheme will lead to the approximate equality of
neuron R level of excitation by synapse 4 and inhibition by synapse 5 under the condition
of equality of levels of excitation of dots K and L. It can be assumed that after that, synapse h will cease to work, for example, it will degenerate.
It can also be assumed that such a form of learning occurs within the first days or
weeks of an organism’s development. As already mentioned, at this time the eye usually
moves slowly and without jumps. The intensity of signals from consecutively excited
dots is approximately identical, that allows reaching an approximate equality of neuron R
excitation level and its inhibition by synapse 5 at equality of levels of excitation of the
retina corresponding dots.
The indicator of the difference between retinal dot excitation levels. Let us address
the problem of signals that will be the indicators of the excitation difference of retina dots
at the movement to the stimulus that emerged on the retina periphery.
One can assume that at some moment, a signal from a stimulus comes onto dot K at
the retina periphery. As the system is not trained, the retina can shift to any side. Let us
initially assume that the retina shifted to the left. As a result, the signal will come to some
dot L (fig. 6), which is more weakly excited than dot K.
The signal from K will first pass through neuron R and axon m, then reach neuron U
and prompt its firing. As a result, a signal will emerge on axon p. Simultaneously, the
signal from neuron R will prompt the action of inhibitory synapse 6. As the signal from
neuron R is more intensive than from neuron S, inhibition by synapse 6 will block the
output signal of neuron S. Therefore, after the shift from dot K to dot L, the signal will
not pass through axon n, and neuron U will not fire.
Let us now assume that dot L is excited first, and then dot K. The signal from dot L
will pass through neuron S, will prompt neuron U to action and prompt the emergence of
a signal on axon p. The signal from neuron S will prompt the action of inhibitory synapse
5. At the shift from dot L to dot K, inhibition by synapse 5 will be weaker than the excitation of neuron R, therefore, inhibition by synapse 5 will not block neuron R. The signal
from it will go through axon m, prompt the firing of neuron U and the emergence of a
signal on axon p.
Thus, at the shift from a more excited to a less excited dot, the signal from neuron U
will pass through axon p only once, before the shift. If the shift occurs from a less excited
to a more excited dot, the signal through axon p will pass twice: before and after the shift
(about the signals corresponding to the termination of a saccade see, for example,
Ohtsuka and Noda (1991) and Fuchs et al. (1993); the presence of such signals, Ramat et
al. (2007) name the primary secret of the cerebellum).
A mechanism for comparing excitation of neurons defining muscle exertions. In
order for the system to function, one more mechanism is necessary. To show this, let us
suppose that at the beginning of learning, at the stimulus position shown in figure 4, the
retina shifts to the right. After the signal from dot K comes onto synapses 1 and 2, they
will initially pass to a condition of expectation, and then both will increase, since the shift
to the right is "correct". But in order for learning to occur at such a retina shift and at such
a position of stimulus, it is necessary that as a result, synapse 2 will increase and synapse
1 will decrease.
In other words, if the muscle pulls to the "correct" side, the learning synapse of a mo-
12
tor neuron that stimulates movement to the "correct" side should increase, and if the muscle pulls to the "wrong" side − it should decrease. To provide this, a mechanism that determines whether the muscle pulls to the "correct" or to the "wrong" side is necessary. Or,
what is equivalent, estimates the difference of motor neuron excitation that defines the
levels of muscles f and g exertion (fig. 4) at a stimulation of some dot of the retina. For
example, compare the intensity of output signals of neurons P and Q at dot K stimulation.
C
f
A •
K•
g
1
P
2
Q
m
V
8 10
7
n
W
9
Fig. 8
V and W – neurons with axons m and n; 7 and 9 – excitatory synapses, 8 and 10 – inhibitory synapses. Other designations as in figure 4.
It is assumed that comparison takes place on neurons V and W (fig. 8). Excitatory signals come onto them from motor neurons P and Q through synapses 7 and 9, and inhibitory signals from the same neurons come through synapses 8 and 10.
In addition, it is assumed that the degree of neuron V excitation by synapse 7 is equal
to the degree of neuron P excitation; the degree of neuron V inhibition by synapse 8 is
equal in absolute volume to the degree of neuron Q excitation. Generally speaking, such
equality does not follow from what was said above: signals onto motor neurons come
through many intermediate neurons with essential dispersion of parameters. (There is no
guarantee that signals on synapses 7, 9 and 8, 10 will correspond to the intensity of excitation of neurons P and Q. It is necessary to assume that a mechanism similar to that
shown in figure 7 forms the corresponding equality as a result of learning.)
Let us suppose that stimulus K is located as shown in figure 8, and excitation of neurons P and Q prompts the eye to shift to the right. This means that neuron Q has been excited more intensively than neuron P. (Generally speaking, here, too, a mechanism that
forms more intensive excitation of extraocular muscles for a more intensive excitation of
the retina dot is necessary. This can be done by analogy to the mechanism shown in figure 7. I will not do it here so as not to complicate the presentation).
Then the excitatory signal from motor neuron P will come onto neuron V through synapse 7, and the inhibitory signal from motor neuron Q will come through synapse 8. The
excitatory signal from motor neuron Q will come through synapse 9 onto neuron W, and
the inhibitory signal from neuron P will come through synapse 10. Since stimulation has
prompted a shift to the right, the strength of the signal from Q is higher than that from P.
This means that neuron V inhibition will be higher than excitation, and no signal will be
transferred through axon m. At the same time, neuron W excitation will be higher than
inhibition, and the signal will go further through axon n. Such a signal – by the level of
13
its intensity – shows the difference in levels of excitation of neurons P and Q.
The process of a motor neuron input synapse learning. To consider the process of a
motor neuron learning, let us turn to figure 9, which represents the mechanisms shown in
figures 6 and 8 together. (In figure 9, the synapse numeration changed compared with
figures 6 and 8.)
Let us assume that a stimulus has excited the retina dot K. The emerging signal will
pass through neurons P and Q, and will prompt muscles f and g constriction and a shift of
the retina. As the system is at its initial stage of learning, a shift can occur both to the
right and to the left (a case in which the retina remains practically motionless will be considered in connection with small amplitude movements).
C
A•
f
5
1
P
•K
3
•L
g
2
4
Q
5
6
7
9
10
R
11
h
8
S
12
m
n
13
k
14
U
15
V
16
W
17
18
Fig. 9
C is the retina with center A and dots K and L on the retinal horizontal axis, P and Q
are motor neurons, 1-10 and 13-18 are excitatory synapses, 11, 12, 17, and 18 are inhibitory synapses, R, S, U, V, and W are neurons, f and g are horizontal muscles, and h, k, m,
and n are axons.
The change of the synapses during the retinal shift to the "wrong" side. Let us initially
address the case of the shift to the left, in which the projection of the stimulus will appear
further away from the retina center, for example, at dot L. Let us consider the processes
that will occur at such a shift.
Even before the retina shifts, the signal from dot K through synapses 1 and 2 will
come onto motor neurons P and Q and prompt their firing. Synapses 1 and 2 will pass to
the condition of expectation.
14
Signals from neurons P and Q will come onto neurons V and W through excitatory
synapses 15 and 16 and through inhibitory synapses 17 and 18. The retina's shift to the
left means that the signal from neuron P is more intensive than the signal from neuron Q.
Therefore, the inhibition of neuron W prompted by synapse 18 will be more intensive
than the excitation by synapse 16. Consequently, neuron W outcome signal will not pass
through its axon k and will not reach neuron Q.
Using similar reasoning, one shall receive that the signal will come onto motor neuron
P through neuron V axon h. Thus, after a shift to the left, the signal through synapse 7
will reach neuron P, but the signal through synapse 8 will not reach neuron Q.
Let us address the impact of the change of level of excitation of the retinal dots as a result of the shift. Before the shift, the signal from dot K will pass through neurons R and
U, come onto synapses 5 and 6, and force these synapses to act.
As dot L is further away from the retina center than dot K, the level of its influence
will be weaker than that of dot K. Therefore, the signal from neuron R will be more intensive than one from neuron S, inhibition through synapse 12 will block neuron S, and
the signal from it will not go further, that is, after the shift, through axons m and n no signal will reach neurons P and Q.
Thus, before the retina shift:
 Input synapses 1 and 2 will pass to the condition of expectation.
 Neurons P and Q, through synapses 5 and 6, will receive signals of the intensity of
excitation of the retina dot.
 Neuron P, through synapse 7, will receive a signal of the comparative signal intensity
from P and Q.
After the shift:
 Neuron P, through synapse 5, will not receive a signal of the retina intensity of stimulation change.
 No signals will reach neuron Q through synapses 6 and 8.
Proceeding from this, the rule of its size change can be formulated for synapse 1 as
follows. Let us assume that before the retina shift, the motor neuron's P learning input
synapse has already changed to the condition of expectation; the signal of the excitation
level of a viewed dot reached this neuron, and the signal has come about the difference
between the excitation levels of motor neurons defining the shift to the left or right. If after the retina shift no signal reaches neuron P, then synapse 1, which is in the condition of
expectation, will decrease.
The rule for synapse 2 can be formulated as follows: let us assume that before the retina shift, the motor neuron Q input synapse already transferred to the condition of expectation, the signal about the viewed dot degree of excitation reached this neuron, and the
signal about the difference of excitation intensity of motor neurons P and Q did not come.
If no signal will come onto neuron Q after the retinal shift, synapse 2, which was in the
condition of expectation, will increase.
The change of the synapse during the retina shift to the "correct" side. Let us now consider a shift to the "correct" side: from dot L to dot K.
Even before this shift, the signal from dot L will pass onto both motor neurons, and
synapses 3 and 4 will transfer to the condition of expectation. The shift to the right means
that neuron Q has been excited more strongly than neuron P. Hence, neuron W will be
more strongly excited than neuron V, and an inhibition through synapse 17 will block
15
neuron V. Therefore, the signal onto neuron P through synapse 7 will not arrive before
the shift. At the same time, synapse 18 will not block neuron W, and the signal from it
through synapse 8 will come onto neuron Q.
Let us address the change of excitation intensity of retina dots. Before the shift, the
signal will pass through synapse 10 onto neuron S and reach motor neurons through neuron U. The same signal through synapse 11 will block neuron R. The excitation of retinal
dot K is more intensive than that of dot L, therefore, inhibition by synapse 11 will not
block neuron R. After the shift, the signal from R through synapses 5 and 6 will reach
neurons P and Q.
In order for the mechanism to operate, it is necessary that synapse 3, which stimulated
the retina's shift to the "wrong" direction, will decrease, and synapse 4, which stimulated
the necessary shift, will increase. For synapse 3, the rule can be formulated as follows:
before the retinal shift, motor neuron P input synapse 3 transferred to the condition of expectation, the signal of the excitation degree of the viewed dot came through synapse 5 to
this neuron, and the signal of the difference between excitation intensities of motor neurons P and Q did not come through synapse 7.
If no signal will come to this neuron after the retina shift, synapse 3, which is in the
condition of expectation, will decrease.
For synapse 4, the rule can be formulated as follows: before the retinal shift, motor
neuron Q input synapse 4 transferred to the condition of expectation, the signal of the excitation degree of a viewed dot came through synapse 6 to this neuron, and the signal of
the difference of excitation intensities of motor neurons P and Q came through synapse 8.
If, after the retina shift, the signal of the difference between excitation intensities of the
retinal dots before and after the shift will come onto neuron Q through synapse 6, then
synapse 4, being in the condition of expectation, will increase.
Summary. It has turned out that a signal from a stimulus that prompts retina shift
should pass through motor neuron input synapses and, therefore, these synapses will
transfer to the condition of expectation. In addition, even before the shift, the signal of initial stimulus brightness comes onto these motor neurons. One can think that such a signal
simply supports the condition of expectation of the synapse, and is not related to future
changes.
If, after that, the motor neuron: a) will be excited by a signal that has prompted more
intensive muscle tension than other neurons, and, simultaneously, that the level of the retina stimulation has increased; or b) will not receive any of these signals, then the input
synapse of this motor neuron, being in the condition of expectation, will increase.
If, after a motor neuron's learning input synapse transition to the condition of expectation and after support of this condition by a signal from neuron U, only one signal will
come onto this motor neuron: either about comparative muscle tension or about greater
brightness of the retina final dot of stimulation, then the synapse, being in the condition
of expectation, will decrease.
Thus, at the position of excitatory stimulus on the retina's horizontal axis to the right
of its center A, the learning input synapses (2, 4) of motor neuron Q increase at the retina
shift to the left or right. At the same time, the motor neuron's P input synapses (1, 3) at
the same position of stimulus decrease at the retina shift to the left or right. Therefore, in
the mechanism being studied, those learning input synapses of motor neurons that stimulate movement to the "correct" side always increase, and those that stimulate movement
16
to the "wrong" side, decrease.
About stabilization of synapse growth. It is necessary to introduce one more mechanism. It is obvious that a learning synapse that stimulates movement to the "wrong" side
can decrease until it ceases to stimulate constriction of the muscles (for example, till synapse size becomes equal to zero).
At the same time, those learning synapses that stimulate the "correct" muscle tension
during the retina shift to the "wrong" side should increase. This is provided by the fact
that after the retina shift, no signals come onto such a neuron. It is assumed that in the absence of signals after the retina shift, the learning synapse – being in the condition of expectation – increases by some constant size.
At the retina shift to the "correct" side, learning synapses cannot grow endlessly: their
growth rate should gradually decrease and after a while, stop altogether. This decrease is
defined by the fact that two signals come onto a motor neuron after the shift to the "correct" side: that the tension of muscle agonists was more intensive than that of muscleantagonists, and that the level of retinal stimulation has increased. Upon the arrival of two
signals after the retina shift, the growth rate of the learning synapse – being in the condition of expectation – will decrease. At consecutive repetitions of these two signals, the
synapse size will stabilize.
As the synapse that stimulates movement to the necessary side can grow to a large
enough size, and the synapse connected to the opposite movement will not, as a result,
stimulate the muscle, the movement to a peripheral stimulus will be made by a jump.
Similar reasoning can be applied to the case in which the stimulus is projected onto the
horizontal axis on the left of the retina center.
Jump amplitude adjustment. Let us now address the opportunity to achieve the necessary jump amplitude.
H• A • •K •L •G
Fig. 10
A − retina center, G, L, K, H – dots on the horizontal axis.
In the considered mechanism, learning is gradual, and jumps to the same dot can vary
in amplitude. It can be assumed that the higher the intensity of the retina excitation after a
jump, the more intensively will synapses 1-4 be tuned (fig. 9). Thus, the shift from dots G
to A (fig. 10) will be better remembered than the shift from dots G to K. The problem is
that a jump is possible directly from G to H: as H is closer to the retina center than G, the
jump will be remembered as a correct one. But then, saccades with retina center crossing
will be possible, and this is not desirable.
To prevent such jumps formation, it can be assumed that at the initial stages of learning, only small jumps can be formed. For example, from G to L, then from L to K, and
only after that, from K to A. In the process of learning, these small shifts will unite in
larger movements, corresponding to the scheme of motor learning: from small compo-
17
nents to larger ones.
Vertical movements. Let us look at motor neurons M and N (fig. 1), which are related
to vertical movements. At a stimulus position on a horizontal axis, neurons M and N
should prevent the retina fluctuations in a vertical direction. Such learning is reached
simply enough: at the action of any one of these neurons, not compensated by the action
of another, the projection of a luminous dot will appear further from the center. Such
movements will be perceived as "wrong", and synapse changes will restore the equality
of the influence of these neurons on muscles. As a result, the muscles will keep the retina
from fluctuating vertically.
In the case of a stimulus on the vertical axis passing through the retina center, learning
occurs in a way similar to that described for stimulus on the horizontal retina axis.
Inclined saccades. Let us consider the case of inclined saccades: when the stimulated
dot is not on the vertical or horizontal axis of the retina (dot K in figure 11). Mathematically, the inclined saccade can be presented in the form of its vector sum of horizontal
and vertical components. If these components in a living organism were independent and
had different sizes, they would have various durations, and the inclined saccades would
be bent. But experiments show that inclined saccades are bent insignificantly (Guitton
and Mandl, 1980; King et al., 1986; Becker and Jürgens, 1990; Smit et al., 1990). This
does not follow directly from models of horizontal and vertical saccades, and it was necessary to create special models simulating such behavior of inclined saccades (Van
Gisbergen et al., 1985; Grossman and Robinson, 1988; Becker and Jürgens, 1990).
• L
•K
A•
•G
Fig. 11.
A is the retina center, K and L are dots that are not on the horizontal or vertical axes,
and G is a dot on the horizontal axis.
Let us consider inclined saccade formation in the process of learning. For dot K in figure 11, it is possible that some muscles will shift the retina in the necessary direction during learning while others will prompt an incorrect movement. For example, horizontal
muscles will displace the retina in the necessary direction while vertical muscles will shift
it incorrectly, and as a result, the dot projection will appear closer to the retina center: in
dot L. Since the stimulation closer to the center is more intensive, the neurons N, M, P,
and Q (fig. 1) will "remember" the movement as a correct one, and this should not be.
In order to prevent such a situation, it can be assumed that during learning, the retina is
shifted in such a way that the stimulus will be projected with equal probability either further from the retina horizontal axis (dot L, figure 11) or closer to it (dot G). Assuming
that the memorization of a stronger excitatory stimulus is better than that of the weaker
one, the "correct" reactions of neurons will be better remembered than the "wrong" ones
and, finally, learning will be achieved.
18
Thus, for any stimuli appearing on the retina periphery, an adjustment of synapses for
the retina center shift to the source of excitation occurs. As a result of the learning, a certain saccade will correspond to each retina dot, and its speed will be defined by the motor
neuron input synapse size formed as the result of learning.
Drift, tremor, and microsaccades. During eye fixation, drift, tremor, and
microsaccades are observed. There are different assumptions about their formation and
role (Ditchburn, 1973; Carpenter, 1988; Hafed and Clark, 2002; Horwitz and Albright,
2003; Martinez-Conde et al., 2004; Engbert and Mergenthaler, 2006; Martinez-Conde et
al., 2006; Rolfs et al., 2008; Collewijn and Kowler, 2008). Let us consider an occurrence
of such movements within the limits of the considered mechanism.
If a small stimulus occurs in the area of the retina center, then muscles agonists and
antagonists will be simultaneously strained at the beginning of learning. Therefore, the
retina movements will be slow, with small amplitude (as in a newborn organism). Excitation of cones will change slightly, and the signals sent by motor neurons will be "remembered" as the correct ones.
There are many muscular fibers and they are innervated by different motor neurons.
Therefore, exertions of agonist and antagonist muscles during fixation might differ. A
"noise" of the firing neurons will occur (Gandhi and Sparks, 2004), and a drift and tremor
will take place, accompanying the eye fixation.
Under the influence of the noise, the stimulus projection can shift from the central part
of the retina. Then, the periphery excitation will become more intensive, and the system
will shift the retina so that the viewed dot will be projected onto its center again. After
this is repeated several times, synapses of muscle agonist motor neurons will amplify, antagonists will be weakened, and a microjump will be generated.
The idea of the formation of microjumps for eye movement correction at fixations is
known (Engbert and Mergenthaler, 2006). The assumption of similarity of usual saccade
and microsaccade formation mechanisms was also suggested (Zuber and Stark, 1965;
Troncoso et al., 2009).
Consequences. The independence of learning from parameter dispersion of
neurons and synapses. It has already been mentioned that in a real nervous system, signals travel through channels that contain tens and hundreds of neurons. These neurons
differ among themselves; there can be a different number of synapses at the end of their
axons with different intensity of impact on the subsequent neuron. Extraocular muscles
contain fibers with various abilities to constrict.
In the above mentioned mechanism, such a variety does not hamper learning: the
learning synapses that prompt the necessary movement will increase and the synapses
that were excited but did not prompt the necessary movement will decrease. For the process of learning, it is insignificant whether one synapse or many are trained, whether
muscular fibers, neurons, and the synaptic connections between them differ among themselves, from which retina sites the signals come, and through how many channels these
signals pass onto the extraocular muscles.
In this mechanism, the muscle agonists and antagonists are strained at the beginning of
learning. As a result of learning, signals from neurons only come onto the muscle agonists, the neurons controlling the antagonists cease to fire, and saccades emerge.
The number of exercises during learning. The above mentioned mechanism can
demand a considerable number of exercises during learning. In a real nervous system, this
19
number is also large and can be approximately calculated. It is known that in children,
about two saccades per second occur, i.e., 7000 per hour. Eye movements are formed in
full by about age three years (actually by 3-5), that is approximately 1000 days after birth.
Presuming that a child is awake on the average of about 10 hours a day, we can multiply
all these numbers. It turns out that by age three, there are approximately 70,000,000 saccades. In other words, in order for the nervous system to learn to carry out correct eye
movements, about 70 million exercises are necessary.
The opportunity for relearning. In the considered mechanism, the motor learning of
the oculomotor system can be carried out at any moment during an organism’s development. The opportunity for such learning is known from experiments. For example, for
studying the adaptation of eye movements, the so called two step paradigm is used: a luminous dot emerges on the screen periphery and when the eye begins to jump to it, it disappears, and a new dot emerges at another place on the screen. The eye starts to move
toward it but since the first movement has begun, the amplitude of the eye movement
turns out to be too small or too big in relation to the new target.
Researches show that in such a situation, the mechanism of learning switches on and
after a while, the movements reach the target again. The process of learning demands
several hundred exercises for humans and about 1000-3000 exercises for monkeys. The
results improve with the increase in learning time (Frens and Van Opstal, 1994; Straube
et al., 1997; Watanabe et al., 2000; Robinson et al., 2003; Alahyane and Pélisson, 2005;
Robinson et al., 2006). Adaptation to saccades in one direction does not influence saccades in the other direction (Frens and Van Opstal, 1994; Albano, 1996; Straube et al.,
1997), which follows from the described mechanism.
As eye movements are formed by learning, the final movements will not necessarily
be precise. Even with saccades from the same starting point to the same final point, there
will be dispersion in amplitude, duration, and speed of movement (Bernstein, 1947;
Ярбус, 1965; Van Opstal and Van Gisbergen, 1989; Smit and Van Gisbergen, 1990;
Smeets, Hooge, 2003; Van der Stigchel et al., 2006).
In the offered mechanism, learning to execute inclined saccades ensues under the same
scheme as for other saccades, and they are not equal to the vector sum of their vertical
and horizontal components (Guitton and Mandl, 1980; King et al., 1986; Becker and
Jürgens, 1990; Smit et al., 1990).
Signals from different parts of the brain participate in the performance of eye movements. For example, inclined saccades are determined by the neuron activity of the pons,
medulla, and rostral midbrain. Coordination of the interaction of brain parts represents an
essential complexity for computational models of eye movements. For the mechanism
described here, such a problem does not exist, as there is no necessity for calculations. In
addition, the problem of the search for a comparator is eliminated since, in the scheme of
learning, a comparator is not necessary.
The considered approach offers one and the same mechanism that explains the occurrence and performance of the eye movements connected with both eye fixations and saccades.
It is assumed that saccade accuracy results from averaging of the activity of these neurons (Baldi and Heiligenberg, 1988; Dean, 1996; Gandhi and Sparks, 2004). But averaging does not guarantee that signals of the necessary intensity and distribution in time will
be obtained. The offered mechanism removes this problem as it uses tuning, not averag-
20
ing, to reach the necessary result.
How saccade parameters are defined. It is known that for a saccade designation, it is
necessary to set its amplitude, speed (or, what is equivalent, duration), and direction
(Sparks, 2002; Glimcher, 2003/1999).
It can be assumed that the further from the retina center the stimulus is projected, the
longer the signal from the corresponding retina neurons lasts. The duration of the signal
will define the saccade amplitude. This corresponds to the results of Goossens and Van
Opstal (2006), who showed that the number of spikes of SC neurons corresponding to an
excited retina dot is proportional to the duration of a saccade to this dot.
Eye movement speed can depend on the intensity of the firing of neurons, that is, from
these neurons' input synapse size. This means that speed can be the result of the learning
of these synapses. Lastly, the direction of the jump is defined by premotor neurons that
are switched on by the excited SC area that is also defined by learning. Thus, the idea of
learning explains how the nervous system assigns the intensity and distribution of signals
necessary for saccade execution coming onto motor neurons and how it defines the signal
intensity necessary to support fixation.
The existence of various types of excitatory and inhibitory synapses and changes in
their interrelations allows the designing of various memorization and learning schemes
(Kandel, 1991; Fregnac and Bienenstock, 1998; Hasselmo and McClelland 1999; Rolls
and Milward, 2000; Bi and Poo, 2001; Kepecs et al., 2002; Kistler, 2002; Todorov, 2003;
Byrne, 2003/1999; Lamprecht and LeDoux, 2004). In Hebb’s scheme (1949, p. 62), it is
assumed that the consecutive activation of two synaptically connected neurons leads to
the strengthening of the connection between them. The learning described above is also
related to synapse size change during the process of neuron functioning. But in this learning, the change in the connection level is defined by an estimation of the gained result:
the things that will be remembered are the things that can be useful for the organism in
the future.
If the result is not necessary, it will not be remembered. If it ceases to be necessary, it
will be forgotten. In such learning, not only is the activity of cells important, but also the
estimation of the results of activity what is close to the idea of operant learning.
Both types of connection formation, with and without an estimation of the usefulness
of what is remembered, can be important for an organism. For recognition, it could be
necessary to memorize all perceived pictures without the estimation of their usefulness,
as in Hebb’s scheme. At the learning of motor reactions, an estimation of resulting usefulness can be important. The possibility of the existence of different neural mechanisms
of learning requires further study.
The offered mechanism does not explain the necessity of SC to the nervous system,
and what for are the intermediate systems of neurons (OPN, LLBH, EBN, IBN) that
transfer signals from SC to motor neurons. It is not clear what is the necessity of the intermediate burst and step neurons and why learning does not take place directly at the
motor neurons? Why does the retina not directly control the eye movements? The problem of SC necessity and the importance that neurons in SC are distributed in a definite
manner, are considered in Appendix 1, and the possible role of learning in the formation
of a connection between saccade peak speed and amplitude is considered in Appendix 2.
The possibility of experimental testing of the existence of learning in the formation of saccades. Let us consider the opportunity to experimentally test some of the
21
ideas described above.
Experiment 1. Whether the amplitude, speed, and direction of a saccade are defined
innately or formed as a result of learning, it is possible to test by an experiment that is
easy to formulate, thought I do not know whether it can be executed. It is necessary to
transpose the extraocular muscles. For example, for the muscles moving the eye horizontally, to transpose the left muscle to the right, and the right muscle – to the left. If eye
movements are formed as a result of learning, then at some time after an operation the
nervous system will learn to move the eye towards the stimulus. If movements are calculated innately, learning will not occur.
Experiment 2. If synapses grow during learning, and the maximal speed of a jump is
defined by their size, it is desirable to test the dependence of the given amplitude maximal jump speed by age: in the assumption of learning, speed should grow with age till it
reaches its maximum.
Experiment 3. It is known that horizontal movements are set by premotor neurons in
pons and medulla. If learning is formed on these neurons, their output signals will increase for muscle-agonists and decrease for muscle-antagonists with the years. Similar
changes will occur for the other directions of the eye movements.
Conclusion. The offered mechanism replaces the idea of calculations that are carried out by the nervous system by notion of learning. Saccades are an example of the
formation of stimuli and reaction interrelations during learning, and their changes depend
on the quantity of exercises.
The assumption of learning explains why it is not essential that signals be transferred
by sets of neurons that possess different characteristics and excite various types of
extraocular muscle fibers. It also explains the connection of mechanisms of saccade,
microsaccade and fixation formation.
The offered approach raises questions on how the learning of eye movements ensues,
how the speed of learning depends on the number of repetitions, what are the mechanisms
of learning, and where are they.
The suggested assumption raises the question about the role of unconscious learning,
especially important for a newborn organism. It shows that changes at the synapse level
occur according to an estimation (conscious or not) of the received result; and that feedback during saccade performance is not necessary.
For the mechanism described above, the problem of learning process convergence has
remained unexamined. But, first of all, practice shows that if there are many parameters,
they can be picked up so that the investigated processes will converge. Second, the problem of this work is not in proving the convergence of some processes, but in the attempt
to find an alternative to the notion of the nervous system as a structure with a lot of innate
knowledge and the ability to make calculations. Third, I wanted to show only an idea, as
it seems improbable that exactly the mechanisms described above are realized in a nervous system: they are shown as an example of what can be carried out from elements similar to those existing in the nervous system.
From my point of view, the main thing is to show that learning can play an essential
role in eye movement formation. The assumption that such learning exists can suggest
approaches to the solution of some problems that do not yet have a satisfactory explanation. The assumption of learning raises the question about the existence of a unified
mechanism for eye movement formation: from the emergence of the very first move-
22
ments up to automatism formation.
Appendix 1. For the mechanism considered above, the necessity of the SC remains
unclear: why can the control of the eye movements not proceed directly from the retina?
To understand the role of SC, let us note that at the time of fixation, saccades are undesirable, therefore, the neurons prompting a saccade should be blocked during fixation.
For example, the blockade can be carried out if the neurons of the retina center inhibit its
periphery during fixation: such an inhibition will block the opportunity of perception of
the stimuli that can generate a jump.
Such an assumption looks simple enough and can be easily realized. The point is only
that inhibition at the time of fixation will hinder peripheral vision, which is important for
organism survival. To avoid this, it is necessary to assume that such inhibition exists, but
is realized in another part of the nervous system.
For example, it can be assumed that an intermediate area F is located between the retina and motor neurons onto which visual signals come from the retina and that send the
signals defining eye movements. In area F, there should be fixational neurons that set the
activity similar to the neuron step activity, defining fixations. In order for the fixations
not to be interrupted by saccades, it is necessary to assume that fixational neurons of area
F prompt inhibition of the other neurons of this area in the intervals of fixation. Assuming
that the greater the distance from a fixational neuron the more intensive such an inhibition, it will impose certain requirements for the arrangement of neurons in area F.
Indeed, if fixational neurons will be distributed evenly in area F, they will – depending
on the distance between them – inhibit each other to a different degree, thus interfering
with fixation. In order for inhibition to be well regulated, fixation neurons should be
grouped together. The retina center should be projected onto those neurons of area F that
are responsible for fixation, and the periphery – onto those that are connected with saccade performance.
Similarly, in order for the burst and buildup neurons not to hamper each other, it can
be assumed that the neurons defining a saccade should be located close to each other.
Signals from area F can come to a set G of the neurons defining eye movements. The
ordered mapping from F to G is not necessary since – as it was shown above – the motor
neurons learn irrespective of their mutual location.
Thus, during stimulation of the central retina part, the neurons of area F that prompt
fixation and inhibit other neurons of area F, will be excited. The inhibition stops if there
is an absence of the center of area F excitation. If, at this time, some part of the retina periphery will be excited, then the corresponding neurons of area F will respond with activity that will prompt the jump, shifting the retina center to the new dot of fixation.
From this notion, it follows that the signals governing eye motor activity should not
come directly from the retina as occurs in the above mentioned mechanism, but from an
intermediate area F. Area F only maps the retina excitation, but does not bear data about
direction, amplitude, or jump speed. F can be compared to the keyboard of a piano: pressing a key generates a sound, but the key does not define sounds. The activity of a small
site in area F switches on a certain set of neurons that prompts a corresponding movement. The location of such a "key" in the F area defines which subsequent neurons will
be excited or inhibited, assigning corresponding eye movement.
Let us consider how much the functioning of area F is analogous to SC functioning. It
is known that fixational neurons existing in SC are located in the SC part that corresponds
23
to the retina center and defines eye fixations. Both burst and buildup neurons responsible
for saccades exist on the SC periphery. Experimental data show the existence of distant
inhibition in SC of burst and buildup neurons by fixation neurons (Dorris et al., 1997;
Munoz and Istvan, 1998; Meredith and Ramoa, 1998; Zhu and Lo, 2000). The possible
role of such inhibition in SC was already considered in the works of Van Opstal and Van
Gisbergen (1989), Arai et al. (1994), Kopecz and Schöner (1995), and Das et al. (1996).
The neurons that prompt saccades are located in SC in an orderly way. Neurons in rostral SC fire before movements of small amplitude, and neurons in caudal SC fire before
larger movements. Upward saccades are presented medially, and downward saccades are
presented laterally (Robinson, 1972; Schiller and Stryker, 1972).
The activity of SC neurons is closely connected with the beginning of a saccade, but
the firing itself does not correspond with movement to a certain position. In other words,
the activity of SC neurons at a saccade performance does not correlate with the direction
and speed of the saccade (Robinson, 1972; Schiller and Stryker, 1972; Sparks and
Hartwich-Young, 1989; Stanford et al., 1996). The information on saccade direction and
speed is presented as a place code: a corresponding cell position on the SC map (Robinson, 1972; Schiller and Stryker, 1972; Sparks and Hartwich-Young, 1989; Stanford et al.,
1996).
Such SC functioning is similar to that of area F described above. Therefore, it can be
assumed that the signal from SC only defines a channel through which a signal will go
further. Being at the end of such a channel, motor neurons respond to the incoming signal
with a certain intensity, distribution, and duration of the outcoming firing, and with this,
they assign the direction, speed, and size of the saccade.
Thus, starting with the assumption of the importance of peripheral visual stimuli perception during fixation, it is possible to explain the necessity of SC. Such an assumption
explains the arrangement of the neurons in SC corresponding to the retina center and periphery. But it still remains unclear why the systems transferring signals from SC to motor neurons (OPN, LLBH, EBN, IBN) are necessary, what are the intermediate burst and
step neurons necessary for, and why learning does not occur directly on motor neurons.
Appendix 2. A possible role of learning in the formation of an interrelation between saccade peak speed and amplitude. Let us now address the known fact of an interrelation between saccade peak speed and amplitude (Bahill et al., 1975; Baloh et al.,
1975; Bollen et al., 1993; Chen et al., 2002; Smeets and Hooge, 2003). The graph for not
very big amplitudes is represented in figure 12. Graph width shows that distribution of
the dots found in the experiment has comparatively wide dispersion, therefore, by coefficient selection, it is possible to describe the dependence in the form of hyperbolic, exponential, or some other function (Inchingolo and Spanio, 1985; Van Opstal and Van
Gisbergen, 1987; Lebedev et al., 1996; Chen et al., 2002; Ramat et al., 2007).
Actually, the problem is not in selecting a function for the dependence between saccade peak speed and amplitudes found in the experiments, but in establishing the cause of
this dependence. In other words, it is desirable to receive the formula that describes dependence not from the graph, but from the cause of such an interrelation occurrence. The
above considered assumption of synapse growth during the process of learning to perform saccades, allows deduction of the corresponding formula.
24
Vp
600
400
200
A
0
10
20
30
40
50
Fig. 12
Graph of peak speed of a saccade from its amplitude dependence: A – saccade amplitude in degrees, Vp – its peak speed in degrees in second.
As mentioned above, during saccade formation, it is necessary to prevent an opportunity of unlimited growth of motor neuron learning synapses. For example, assuming
that the more time is already spent for learning, the more slowly the volume of these synapses grows. In the simplest case, it is possible to assume that this dependence is inversely proportional to the time of learning. If the changes of synapse size during exercises to
designate through q, time spent for exercises to designate by T, the equation of interrelation can be written down as:
dq b

dT T
(1)
where b is a constant.
Let us assume that the initial stage of learning passes without a delay of synapse
growth, and a delay begins after some time h. Then equation (1) will become:
dq
b

dT T  h
Solving it, we shall receive:
q = b ln c (T − h)
(2)
where c is a constant.
As it was said, the bigger the total size of the motor neuron input synapses, the more
intense the outcome signal will be. Accordingly, it is possible to assume that the maximal
intensity of muscle constriction and the maximal speed v of the eye movement are proportional to this size. Designating the total size of such neuron's input learning synapses
by p, we receive:
25
v=dp
(3)
where d is a constant.
It can be assumed that for learning saccade performance, the bigger the saccade amplitude, the more time is necessary. The more the amplitude, the greater should be the size
of input learning synapses. Thus, more time will be necessary for the learning. That is:
T=eA
(4)
where e is a coefficient of proportionality. Substituting (2) and (3) in (4), we receive:
v = d b ln c (e A − h)
or
v = g ln h (A − k)
(5)
where g = d b, h = c e, and k=h:e. It turns out that the saccade maximal speed depends
logarithmically on its amplitude. The formula obtained in a certain measure corresponds
to known experimental data shown in figure 12. It is possible to receive other expressions
of interrelation, for example, assuming that in the denominator at the right part of equation (1) the power of T differs from one, then the dependence will be expressed in the
form of a power function (Ярбус, 1965; Inchingolo and Spanio, 1985; Lebedev et al.,
1996). The obtaining of expression (5), and the opportunity of other types of interrelations posit a question about the research of the saccade peak speed and its amplitude dependence in connection with learning time of saccade performance.
26
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