The saccadic control system#
Introduction#
Due to the highly inhomogeneous distribution of photoreceptors across the retina, as detailed in section subsection:topo, the highest visual resolution is achieved only when targets are projected onto the central part of the retina, specifically the fovea. In primates, the fovea’s effective diameter is less than one degree of the visual angle, making it a focal point for detailed vision. Consequently, when the primate visual system needs to examine the details of a peripheral visual target, it necessitates a precise redirection of the gaze line onto the object. This critical task is accomplished by the saccadic gaze control system.
The saccadic orienting response is not only extremely rapid, with peak velocities in some species like monkeys exceeding 1000 degrees per second (as illustrated in Fig. Typical position traces of saccadic eye movements as a function of time. Note that saccade duration increases with saccade amplitude (straight-line relation), and that the peak velocity of the saccade increases with amplitude in a saturating manner. The latter two relations are known as the ‘main sequence’ for saccades.), but it is also remarkably flexible. Goal-directed gaze shifts can be achieved through various means: solely by eye movements, or through a combination of movements involving the eyes, head, and even the body. This versatility allows for a dynamic and responsive visual system.
The saccadic control system’s straightforward function, coupled with its underlying complexity, has made it a subject of extensive study in numerous research laboratories, including the Nijmegen Lab of Biophysics. The system’s study has led to the development of a growing number of detailed quantitative models. These models seek to explain not just the mechanics of eye movements, but also their integration with cognitive processes and how they contribute to our overall visual experience.
In this module, we delve into the neurocomputational modeling of the saccadic eye movement system. We will explore how these models replicate the functionality of the saccadic system, how they account for the precision and speed of eye movements, and their role in efficiently navigating our complex visual environment. By understanding these models, we gain insights into one of the most fundamental aspects of visual perception and its neural underpinnings.
Fig. 51 Typical position traces of saccadic eye movements as a function of time. Note that saccade duration increases with saccade amplitude (straight-line relation), and that the peak velocity of the saccade increases with amplitude in a saturating manner. The latter two relations are known as the ‘main sequence’ for saccades.#
The results depicted in Fig. Typical position traces of saccadic eye movements as a function of time. Note that saccade duration increases with saccade amplitude (straight-line relation), and that the peak velocity of the saccade increases with amplitude in a saturating manner. The latter two relations are known as the ‘main sequence’ for saccades. underscore a pivotal characteristic of the saccadic system: its nonlinearity (for a deeper exploration, see Exercise exc:nonlinearms). This nonlinearity is fundamental to understanding the system’s dynamic response to visual stimuli and the intricate coordination required for rapid eye movements. Historically, research into this area, particularly during the seventies and eighties, was primarily focused on gaze shifts in a single dimension (1D; horizontal). However, a more comprehensive understanding began to unfold as studies expanded to incorporate two degrees of freedom (2D; horizontal and vertical), examining the coordination of both eye and head movements.
In recent years, the scope of research has broadened even further, embracing a three-dimensional perspective (3D; encompassing horizontal, vertical, and torsional aspects) of the gaze control system. This expansion acknowledges the complex, real-world scenarios where the eyes and head move freely and interdependently, navigating and responding to a dynamic visual environment. Such studies have significantly enhanced our understanding of the saccadic system, revealing its intricate multi-dimensional nature and its critical role in how we interact with and perceive the world around us.
Chapter Roadmap#
This chapter explores the intricacies of the saccadic control system, encompassing its physiological, neuroanatomical, and computational aspects. In the previous chapters chap:oculomotorsystem and chap:assignmentpulsestep, we saw how the pulse-step generator enables the brainstem to overcome the low-pass characteristics of the plant and generate a rapid and accurate saccade. In this chapter, we will look at how the pulse is generated, what role internal (non-visual) feedback plays, where the non-linearity in the main sequence of saccades may come from, and how coordinates are mapped from the visual field to the superior colliculus and from the superior colliculus to the eye movement. The roadmap for the chapter is as follows:
Introduction
Overview of the saccadic control system in visual processing.
The significance of the fovea and the necessity for precise gaze redirection.
The Evolution of Saccadic System Research
Understanding the nonlinearity in the saccadic system.
Historical progression from one-dimensional to multi-dimensional saccadic studies.
The Gaze Control System in One and Two Dimensions
Discussion of internal feedback mechanisms in saccade programming.
Analyzing experiments that demonstrate the role of internal feedback.
Internal Feedback in Saccade Programming and Generation
Exploration of the mechanisms for saccade accuracy and control.
The role of the brainstem in saccade generation.
One-Dimensional Model for Saccades
Introduction to a model for saccadic movements based on internal feedback.
Neurophysiological underpinnings of saccadic generation.
Two-Dimensional Eye Movements: The Common Source Model
Investigating spatial-temporal transformation in saccades.
Understanding linear and nonlinear processes in eye movement control.
Neural Programming of Saccades in the Central Nervous System
Overview of saccade control within the CNS.
Detailed examination of visual processing stages leading to saccades.
Neural Mapping Principles and Cortical Magnification Factor
Discussion on topographic and complex logarithmic mapping in visual processing.
Sensorimotor Mapping in the Superior Colliculus
Evaluation of afferent mapping in the superior colliculus.
Insights into the motor field and its role in saccade generation.
Population Coding of Saccades in the SC Motor Map
Investigating the hypothesis of movement field uniformity in the SC.
The gaussian distribution in SC population activity.
Connection Scheme of the Superior Colliculus with the Brainstem
How activated SC neurons project to the brainstem.
The population coding scheme for saccade vector encoding.
Exercises and Glossary
Exercises to reinforce understanding of key concepts.
A glossary of terms for reference.
The gaze control system in one and two dimensions.#
The two concepts that are important for the present discussion, are the notions of internal feedback and velocity integration. We will here provide the case for the need of internal feedback in the system.
Fig. 52 Evidence for internal feedback. Three lines of evidence for the use of internal, non-visual, feedback. Top: Experiments that indicate that feedback is used in saccade programming. Bottom: Experiment to show that feedback is also used during saccade execution. In all three experiments, saccades are generated in complete darkness, i.e. after the disappearance of the target(s). F = initial fixation position of the eyes.#
Internal feedback in saccade generation#
When a visual stimulus is presented onto the peripheral retina, the saccadic system could program an accurate eye movement to foveate the target, solely on the basis of the retinal error signal (defined as the difference between the current gaze direction and the desired gaze direction, ‘gaze error’). Based on the following arguments, however, the retinal error, although sufficient at first glance, is not the only source of information on which the saccadic programmer (i.e. the brain) can rely (see Fig. Evidence for internal feedback. Three lines of evidence for the use of internal, non-visual, feedback. Top: Experiments that indicate that feedback is used in saccade programming. Bottom: Experiment to show that feedback is also used during saccade execution. In all three experiments, saccades are generated in complete darkness, i.e. after the disappearance of the target(s). F = initial fixation position of the eyes., top panel):
Saccades accurately compensate for any intervening eye movement that may occur between target presentation and saccade initiation. For example, in a double-step paradigm in darkness, the system has to generate two saccades in series to briefly flashed targets, A and B, that are both extinguished well before the start of the first movement. Despite the absence of visual feedback, the two saccades, S1 and S2, accurately land on each target. The second saccade is even accurate when the second target was flashed during the first saccade (Fig. Evidence for internal feedback. Three lines of evidence for the use of internal, non-visual, feedback. Top: Experiments that indicate that feedback is used in saccade programming. Bottom: Experiment to show that feedback is also used during saccade execution. In all three experiments, saccades are generated in complete darkness, i.e. after the disappearance of the target(s). F = initial fixation position of the eyes., top left).
Monkeys can generate an accurate eye movement to an extinguished flashed target in darkness, even after the eyes had been driven away from their initial position, within the reaction-time period, by electrical stimulation of the midbrain superior colliculus (SC, STIM) (Fig. Evidence for internal feedback. Three lines of evidence for the use of internal, non-visual, feedback. Top: Experiments that indicate that feedback is used in saccade programming. Bottom: Experiment to show that feedback is also used during saccade execution. In all three experiments, saccades are generated in complete darkness, i.e. after the disappearance of the target(s). F = initial fixation position of the eyes., top right; for more details on the role of this neural structure, see below).
Accurate saccades can be generated to auditory targets in darkness, although the acoustic frame of reference is head-centered, rather than eye-centered. For this, the saccadic system needs to know the absolute position of the eyes in the head (Q: why?).
It was noted by Robinson and colleagues that visual feedback is far too slow to be of any use for the accurate control of rapid eye movements (visual delays are in the same order of magnitude as typical saccade durations, roughly 50-60 ms). Since normal saccades have quite stereotyped kinematics with relatively little scatter, the possibility exists that they are generated by a preprogrammed pattern generator in the brainstem. However, without going into the details, strong additional evidence supports the idea that also the brainstem saccade generator is controlled by internal feedback:
Saccades remain accurate despite a considerable, e.g. drug-induced or fatigue-related, variability in their kinematics.
Saccades that have been interrupted in midflight (by brief electrical microstimulation of the saccade gating system, embodied by the brainstem omnipause neurons (OPNs), see e.g. Fig. More detailed schematic of the 1-D Scudder model, which is also available in Matlab’s Simulink), accurately re-acquire the target in complete darkness (Fig. Evidence for internal feedback. Three lines of evidence for the use of internal, non-visual, feedback. Top: Experiments that indicate that feedback is used in saccade programming. Bottom: Experiment to show that feedback is also used during saccade execution. In all three experiments, saccades are generated in complete darkness, i.e. after the disappearance of the target(s). F = initial fixation position of the eyes., bottom panel).
Saccades that have been perturbed by brief air puffs that cause rapid closure of the eyelids (’blinks’) are highly variable, but their endpoints are as accurate as for normal, unperturbed saccades (see Chapter chap:assignmentnonlinearmodel, Fig. Effect of an intravenous injection of Diazepam (Valium) on human saccades toward flashed targets: although the Valium saccades (right) are still quite accurate, their kinematics are much more variable and slower than normal responses (left). Note also the differences in the reconstructed Pulse-Step signals (bottom). (After: Van Opstal et al., Vision Res., 1985)).
A one-dimensional model for saccades#
Since visual feedback plays no role in the experimental observations described above, it was hypothesized by Robinson that the saccade burst generator is in fact driven by an internal feedback circuit in which a desired eye movement (specified by the saccade programmer) is continuously compared with an efference copy (= a neural motor-command signal) of the actual movement program. The signal that activates the horizontal eye muscles is characterized by a pulse-step activity pattern (see the Powerpoint file of this Lecture for a real motoneuron recording in the Abducens Nucleus). The pulse is an intense short-duration burst of action potentials, that is directly derived from so-called short-lead burst neurons in the brainstem, and faithfully reflects the velocity profile of the eye movement. The output of the short-lead burst neurons, b(t), correlates well with the instantaneous eye velocity, e(t), and is needed to rapidly accelerate the slow eye against the low-pass filtering properties of the oculomotor plant (see also Lecture 3 on Linear systems). It is thought that the main-sequence nonlinearity (see above) of the saccadic system is due to a saturating input-output characteristic of these burst cells (in the second part of this Lecture, however, we will challenge that view).
It is generally held that the input to the short-lead burst neurons is a dynamic motor error signal, \(me(t)\), which is defined as the instantaneous difference between the desired displacement of the eyes \(\Delta e_d\), and their current actual displacement \(\Delta e(t)\) :
The desired eye-displacement command is specified by the midbrain superior colliculus. This structure contains a map of saccadic eye movement vectors (described in detail below), in which the location of neural activation determines the desired eye displacement vector. The activated population of neurons in this so-called motor map sends a burst of action potentials to the brainstem to execute the saccade (see also the illustration on the cover of this syllabus). The saccadic system is driven by a continuous feedback that compares the desired and actual eye displacement commands. In a simple, quasi-realistic model of the saccadic system (proposed by Scudder in 1988; Fig. Simple scheme of the one-dimensional model of saccade control, as proposed by Scudder, 1988), the comparison in the feedback system is performed as neural integration by so-called long-lead burst cells (LLB) of the difference between the superior colliculus (SC) burst, and the burst from the short-lead burst cells that represents the actual eye velocity, \(\dot e(t)\):
Fig. 53 Simple scheme of the one-dimensional model of saccade control, as proposed by Scudder, 1988#
Fig. 54 More detailed schematic of the 1-D Scudder model, which is also available in Matlab’s Simulink#
Two-dimensional eye-movements: the Common Source Model#
The vertical saccadic pulse-step generator in the brainstem consists of burst cells in the medial longitudinal fasciculus (MLF), and the neural integrator resides in the nucleus of Cajal (NC). Neurophysiology has revealed that cells in this pulse-step generator are tuned to movements in the vertical/torsional plane. In this chapter, we will ignore the torsional dimension, and only consider eye movements with horizontal and/or vertical components. Both saccade burst generators receive a common input command from the SC, that represents a desired vectorial displacement of the eye (!ed) relative to the initial eye position (see also above). As will be described in great detail below, the SC signal is spatially encoded in a topographic motor map, in the sense that neighbouring recruited regions in the map encode similar saccade vectors by their location within the map, rather than by the intensity of the neural activity. The transformation of this vectorial eye-movement signal into the appropriate activation patterns of the eye muscles, is known in the literature as the spatial-temporal transformation stage.
The spatial-temporal transformation refers to the process by which the brain converts visual information about the location of an object in space (spatial information) into the appropriate timing and muscle contractions needed to move the eyes to focus on that object (temporal information). This transformation is a key aspect of the oculomotor system, which is responsible for controlling the movements of the eyes.
Here’s a more detailed explanation:
Spatial Information Processing: This involves understanding where an object is located in space relative to the observer. The visual system detects the position of an object in the visual field. This spatial information is essential for determining the direction and distance the eyes must move to fixate on the object.
Temporal Information Processing: Once the brain knows where the object is, it must calculate when and how to move the eyes. This involves timing the initiation of the eye movement and determining the speed and duration of the muscle contractions needed to move the eyeballs.
Coordination of Eye Muscles: The oculomotor system controls several muscles that move each eye. The spatial-temporal transformation helps in coordinating these muscles to ensure both eyes move accurately and simultaneously to focus on the same point.
Feedback and Adjustment: The oculomotor system continually receives feedback about the position of the eyes and adjusts the movements as needed. This is important for maintaining accurate eye alignment and focus, especially when tracking moving objects or when the head is moving.
Integration with Other Systems: The oculomotor system doesn’t work in isolation. It integrates information from the vestibular system (which helps with balance) and the proprioceptive system (which provides information about the position of the body) to help maintain gaze stabilization during head and body movements.
This process is essential for activities like reading, sports, and navigating through our environment. It involves complex neural pathways and is an area of significant interest in neuroscience and ophthalmology. For the purposes here, it involves two different processes:
Space: Vector decomposition (VD) into horizontal and vertical saccade displacement components .
Time: Pulse generation (PG) of motor error, \(m(t)\), into an eye velocity command, \(\dot e(t)\).
The vector decomposition stage is linear: \(\Delta H = R \cdot \cos{\Phi}, \Delta V = R \cdot \sin{\Phi}\), which scales linearly with the saccade amplitude. However, the PG stage is assumed to be nonlinear, since it is described by a saturating function that accounts for the observed saccade kinematics: \( \dot e = a (1-\exp (-b \cdot m))\), see Fig. Typical position traces of saccadic eye movements as a function of time. Note that saccade duration increases with saccade amplitude (straight-line relation), and that the peak velocity of the saccade increases with amplitude in a saturating manner. The latter two relations are known as the ‘main sequence’ for saccades.. As a result, the order in which these two processes are implemented in the system matters (it is a non-commutative operation): \(VD \circ PG \neq PG \circ VD \).
Study of oblique saccades has revealed that both saccade components are tightly synchronized and coupled, such that approximately straight saccades are elicited in all directions. In the so-called ‘common source scheme’, a vectorial burst generator issues a vectorially-encoded velocity pulse command (PG), that is subsequently decomposed (VD) into the respective horizontal and vertical velocity components (see Exercise exc:common_source).
Neural programming of saccades in the central nervous system#
In order to get familiarized with the nomenclature of the relevant structures that are involved in saccadic eye movement control in the central nervous system (CNS; including humans), Fig. Side view of the monkey brain with some of the gaze-control areas indicated: visual cortex, auditory cortex, motor cortex, cerebellum, and brainstem. FEF: frontal eye fields, Somato: somatosensory cortex (responsive to touch). The human brain has a very similar organization. gives a highly simplified schematic representation of the monkey brain.
Fig. 55 Side view of the monkey brain with some of the gaze-control areas indicated: visual cortex, auditory cortex, motor cortex, cerebellum, and brainstem. FEF: frontal eye fields, Somato: somatosensory cortex (responsive to touch). The human brain has a very similar organization.#
Fig. 56 Schematic overview of the different neural structures involved in the planning and execution of a goal-directed eye movement. While the foveae, F, looks straight ahead, a target, T, is presented to the left visual field and gets projected on the right side of both retinae. Code: L = left side of the brain, or left eye; l = left visual hemifield; R = right side of the brain, or right eye; r = right visual hemifield. E.g. ‘lL’ in LGN–R is a cell in the right LGN that receives input from the left eye’s left visual field; ‘LRr’ in V1-L denotes a cell that receives input from both eyes from the right visual field. In the SC it indicates neurons that drive both eyes into the right motor hemifield.#
The different processing stages in visuo-motor processing (Fig. Schematic overview of the different neural structures involved in the planning and execution of a goal-directed eye movement. While the foveae, F, looks straight ahead, a target, T, is presented to the left visual field and gets projected on the right side of both retinae. Code: L = left side of the brain, or left eye; l = left visual hemifield; R = right side of the brain, or right eye; r = right visual hemifield. E.g. ‘lL’ in LGN–R is a cell in the right LGN that receives input from the left eye’s left visual field; ‘LRr’ in V1-L denotes a cell that receives input from both eyes from the right visual field. In the SC it indicates neurons that drive both eyes into the right motor hemifield.) may be globally summarized as follows:
Light rays that enter the eye, are projected onto the retina through the eye lens, and yield visual signals that, after substantial pre-processing by the different cell types within the retina, are finally passed into the CNS by the output of the retinal ganglion cells, the so-called optic tract (the thick bundle of ganglion axons that leaves the front (!) of the retina via the blind spot) (see the Chapter on the Visual System for more details).
First, the visual signals related to the images on the left and right hemifield of each retina are split in the optic chiasm (place where the signals of both eyes first meet and than diverge again, like railroad tracks), so that after the chiasm, the left-visual input from both eyes is transmitted to the right side of the brain, and the right-visual input from the eyes goes to the left side of the brain. Nobody knows why this strange splitting process is needed but it occurs all over the animal kingdom and is therefore probably some remnant of the early evolution of the visual system.
The retinal output of each eye is then separately relayed to the Thalamus (to be more precise, into alternate neural layers of a thalamic nucleus called the Lateral Geniculate Nucleus, LGN). In the LGN, the colour- and shape sensitivity of the cells is further enhanced. Note, that individual cells in the LGN are still monocular (they receive input from either the ipsilateral, or the contralateral, eye), and that they all ‘look’ at the same contralateral visual hemifield.
The LGN directly projects its outputs to the Primary Visual Cortex, also known as the occipital cortex, area 17 (in cats), or area V1 (in primates). Here, for the first time the signals from the two eyes are combined (there are binocular cells). It is generally believed that the essential visual processing, needed for shape, colour, movement, recognition, visual awareness, first takes place at this level. The visual cortex is the most extensively studied object in the CNS, and despite the enormous amount of data and spectacular insights, the exact processes and functions are still rather obscure. But a localized lesion in this area causes functional blindness for stimuli at the associated visual coordinate.
An important aspect of V1 is the presence of a topographic map of visual space (see Lecture 4). This aspect is far from trivial, because the bundle of nerves within the optic tract does NOT contain any specific information with respect to the topography of visual space (since the nerves are randomly mixed as they leave the blind spot), and also the cells in the LGN do not appear to display a clear topography. Yet, in visual cortex cells are arranged such that spatially neighbouring cells respond to visual stimuli from spatially neighbouring locations (see below).
The result of the visual processing in the cortex is transmitted to many different sites in the brain. For example, it is combined with information from other sensory modalities, like auditory and somatosensory (tactile) information in the so-called Association Cortex. It is also used for the control of eye- and head movements, as well as for the guidance of the hand and body to visual targets. The former process involves a motor structure in the cortex known as the Frontal Eye Fields, the latter includes the recruitment of cells in the Motor Cortex.
When the cortex is ‘done’ with its processing (which involves target recognition/selection, and spatial orientation, but also memory), the signal is sent to the motor system to prepare the action/movement. For eye movement control, this process involves the Superior Colliculus, or SC in the midbrain, and which receives direct and simultaneous inputs from many cortical areas. This structure constitutes a crucial interface between the sensory and the motor worlds (and is thus a prime sensorimotor integration stage). Also the SC is organized topographically, the details of which will be dealt with below.
Once the SC ’knows’ what to do, a goal-directed eye movement may be executed in the way proposed by the Robinson or Jurgens models (see above). One further important detail: down from the SC a ‘recrossing’ of the projections takes place, such that the left SC is connected to the right brainstem, and the right SC to the left brainstem.
Neural Mapping Principles#
Topographic map#
Most visual areas in the brain map the visual world in a topographic manner. i.e. neighbouring locations in the visual world evoke activity in neighbouring regions of the visual system. These locations can be described by coordinates (ordered pairs (or triplets) of points). The visual field is typically described by \href{https://openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates}{polar coordinates} (Fig. Mapping of visual space onto neural space. **AA,B; you can follow the link if you want to learn more about polar coordinates), rather than simple Cartesian coordinates (\(x\) and \(y\)). In the polar coordinate system, a point is represented by eccentricity \(r\) (the radial coordinate, the radiant) and direction \(\phi\) (the angular coordinate). The eccentricity on the retina is proportional to the distance from the fovea (central point with highest cell density) to the point on the retina stimulated by the visual stimulus.
The neural map is typically not a straightforward copy of the visual field. A transformation occurs from one coordinate system to another. As an example, we will first have a look at how the visual world is mapped into the neural space of of the primary visual cortex (retinotopic mapping). For that, we first have to consider the organization principles of the retina.
Cell density in the retina, \(d\), decreases with increasing eccentricity (distance) from the fovea, \(r\). In other words, cell density is a function of eccentricity:
\(d = d(r)\).
The sensitivity of a sensory neuron is described by its receptive field. In the retina, receptive fields of output ganglion cells are circular-symmetric. Experimental data have shown that the radius, \(\sigma\), is proportional to distance, r, of the cell from the fovea:
\(\sigma \propto r\).
A visual stimulus, matched to receptive field size, always influences the same number of retinal cells. The total number of stimulated retinal cells, \(N = \pi \sigma^2(r) \cdot d(r)\) is roughly constant (about N=35). Thus, cell density decreases as:
\(d(r) \sim \frac{1}{r^2}\)
Fig. 57 Mapping of visual space onto neural space. **A#
Given this organization of the retina, what does the retinotopic map* (the mapping of the retinal input onto the visual cortex) look like? Suppose that \(\Delta A\) is a small area in the visual field, that is represented by \(\Delta N\) retinal (ganglion) cells. We know that the receptive field area, \(A = \pi \sigma^2\), is represented by equal numbers of retinal ganglion cells, \(N\), irrespective of \(r\). Thus, a relative change in receptive field size results in an equal change in relative cell numbers. Therefore, the neural mapping should satisfy the following constraint:
:label: eqn:mapeqn1
\frac{\Delta A}{A}=\frac{\Delta N}{N}
From this equation eqn:mapeqn1, we can formulate the local magnification factor \(M(r,\phi)\): \(M(r,\phi)=\frac{\Delta N}{\Delta A}\), relating small changes in number of neurons across the retina to small changes in stimulated area of the visual field. Taking the limit as \(\Delta N\) and \(\Delta A\) go to zero, yields: \(M(r,\phi)=\frac{dN}{dA}\). To solve this equation, we can integrate, yielding the complex logarithmic map function (Fig. Mapping of visual space onto neural space. **AB; more on this later).
cortical magnification factor#
Similarly, rather than using cell density and receptive field size, let’s try to derive the transformation of the neural map in the primary visual cortex. In the Lecture, this map is described by horizontal and vertical coordinates \(X\) and \(Y\). These points on the cortical surface are a function of the eccentricity \(r\) and direction \(\phi\) of the corresponding points of the visual field: \(X(r,\phi)\) and \(Y(r,\phi)\). For a mathematical approximation, Both the X coordinate and the local magnification factor are assumed to be only a function of eccentricity:
For purely radial displacements, from experiments in the macaque monkey, results suggest that \(M(r) = \frac{\lambda}{r_0+r} \), with \(\lambda \approx 12\) mm, and \(r_0 \approx 1^\circ\). Integration (see \href{https://en.wikipedia.org/wiki/Lists_of_integrals}{rules for integrating rational functions}) yields (for \(r>>1^\circ\)):
Here we see the logarithm (of the complex logarithm map) popping up. What this function does, is transform (map) visual field coordinate \(r\) to neural coordinate \(X\) by taking the (natural) logarithm. This logarithmic mapping causes images of eccentric circles (iso-\(r\) curves in Fig. Mapping of visual space onto neural space. **AA) on the retina to be represented at straight vertical lines in the \(X\) direction (iso-\(u\) curves in Fig. Mapping of visual space onto neural space. **AC). From purely directional displacements, \(dY\), we could derive a similar equation (but we are not going to do this; see Lecture on the form of this function).
complex logarithmic mapping#
We have seen that transformation of retinal to cortical coordinates are achieved through logarithmic mapping and how complex number can represent two-dimensional vectors. Let’s combine these two concepts. We use the symbols \(u\) and \(v\) for the horizontal and vertical coordinates in the neural map (which we called \(X\) and \(Y\) before for the primary visual cortex), and \(x\) and \(y\) in the visual field. For the polar coordinates (eccentricity and direction/angle), we use \((r', \phi')\) and \((r, \phi)\) for the neural map and visual field, respectively. The relationship between the rectangular and polar coordinates is graphically represented in Fig. Complex coordinate system. From rectangular to polar coordinates.. The complex-logarithmic mapping function is given by:
where the complex numbers \(\mathbf{z}\) and \(\mathbf{w}\) are defined as \(\mathbf{z}=x+i \cdot y=r \cdot e^{i\phi}\) and \(\mathbf{w}=u+i \cdot v=r' \cdot e^{i\phi'}\) (see eqn. eqn:mapeqn3).
\begin{remark} Complex numbers: See chapter chap:complexnumbers on complex numbers and the difference between polar and rectangular coordinates. \end{remark}
We also need to account for the magnification factor in the actual transformation from retinal to cortical coordinates (see sections subsection:topo and section:cmf). Thus, the neural mapping is of the form:
and so, \(u=B \cdot \ln{\sqrt{x^2+y^2}}=B \cdot \ln{(r)} \) and \(v=B \cdot \arctan{\frac{y}{x}}=B \cdot \phi \) (\(r\) and \(\phi\) in radians), with B the neural magnification factor, which indicates the amount of neural space (in mm) per radian angular change in the retina. This type of function has been proposed above (see section section:cmf) to underlie the neural mapping from visual retinal space into visual cortical space. Note the singularity of this function at \(r = 0\), and note also the ‘forbidden regions’ given by \(|v| > B \cdot \frac{pi}{2}\). Thus, the entire 2D visual world (either in cartesian or in polar coordinates) is mapped onto a narrow, logarithmic ‘strip’ of neural tissue. You may verify that radiants (\(\phi\) = constant) are transformed into parallel horizontal lines, whereas circles (r = constant) transform into parallel vertical lines (see Fig. Mapping of visual space onto neural space. **A). For visual cortex, experiments have yielded a magnification factor of 1.4 mm/rad for both u and v coordinates (i.e. an isotropic mapping).
Sensorimotor mapping in the Superior Colliculus#
For the midbrain Superior Colliculus, the afferent (= input, sensory) mapping function appears to be only slightly different than the visual cortical complex-log map described above. First, the singularity at \(r = 0\) is resolved by introducing the shift vector \(A = (A, 0)\), which appears to be quantitatively different from the one proposed for the visual cortex (see also Exercise exc:com_log_map). Second, the mapping appears to be slightly anisotropic.
The experiment that led to a quantitative description of this mapping function is shown in Fig. Mapping of visual space onto neural space. **A (down and right). It is a remarkable fact that by applying a small electrical pulsatile current (threshold about \(10 \mu A\)) to a micro-electrode that is positioned within the SC, it is possible to elicit a perfectly normal saccadic eye movement. Neither the stimulation intensity, nor changes in the pulse-frequency have any influence on the ensuing saccade. Only by stimulating at different sites within the SC is it possible to yield different saccade vectors: the amplitude (\(R\)) and direction (\(\phi\)) of the vectors then appear to change smoothly as a function of the neural coordinates, \((u, v)\) (mm). Fig. Mapping of visual space onto neural space. **A shows the iso-amplitude (\( R = [2, 5, 10,... 50] \) deg) and isodirection (\(\phi = [-80, -40, 0, 40, 80]\) deg) lines of elicited saccades, superimposed on the cartesian \((u, v)\)-plane. Note that the resulting experimental map (the so-called motor map of the SC) has much in common with the mathematical complex-logarithmic mapping function. It was therefore proposed to describe the experimental data by the following function:
where \(\mathbf{B} \equiv (B_u,B_v)\) are slightly different magnification factors, for the horizontal and vertical domain in Cartesian neural space, respectively, and \(\mathbf{A} = (A, 0)\) (in deg) prevents the singularity at \(r = 0\). A nonlinear regression on the electrical stimulation data of Fig. Mapping of visual space onto neural space. **A (bottom right) has yielded as best-fit parameters for the monkey SC motor map: \(B_u=1.4\) mm, and \(B_v=1.8\) mm/rad, and \(A = 3\) deg. Thus, written in a slightly different way:
Properties of the map are as follows:
It is a conformal mapping, i.e. that angles (in receptor space) are invariant under the transformation and surface is preserved.
It is an inhomogeneous mapping (expansion for small \(R\) (small eye movements to targets near the fovea), and a compression for large \(R\) (large saccades)).
It is a slightly anisotropic mapping, i.e. no conservation of shape under the transformation (a square becomes an elongated rectangle) (see Exercises).
The Superior Colliculus Motor Field#
When the saccadic system prepares a saccadic eye movement, the correct site in the motor map of the SC is selected to activate the local neurons at that site. The stimulation experiment (Fig. Mapping of visual space onto neural space. **A) showed that, indeed, if a local population of such neurons is activated, a reliable saccade vector signal is sent to the brainstem. Now, what do recordings from these neurons tell us under normal conditions? For this, we have to introduce the concept of a movement field: if one records from a SC neuron, the cell is active only when the saccade vector belongs to a restricted range of amplitudes and directions. This range of saccade vectors is called the cell’s movement field (cf. receptive field for sensory neurons). The movement fields have a number of properties. First, the SC movement field is not rotationally-symmetric, but appears to be markedly skewed along the amplitude dimension (see e.g. Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5, bottom right), whereas it has roughly a gaussian shape along the direction coordinate. Second, movement fields of cells for small saccades have a small diameter, when compared to cells associated with large saccades. It turns out that the diameter of the movement field increases roughly linearly with saccade amplitude (cf. with the retinal receptive field, above!). Now, consider the following: since SC cells are not tuned to a delta-peaked function of saccade vectors, it must be concluded that whenever a saccade is made, say with amplitude \(R_o\) and direction \(\phi_o\), a large population of cells in the SC will be recruited (namely all cells for which \((R_o,\phi_o)\) belongs to the movement field). It may be expected, that the center of this population will correspond to the coordinates prescribed by the neural mapping function, Eq. eqn:mapeqn5. An interesting further question is, what shape the population has in the SC motor map, and how it explains the properties of the movement fields.
Fig. 58 Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. \((u, v)\) anatomical coordinates; \((R, \phi)\) polar coordinates of saccade vector corresponding to site \((u, v)\) through Eq. {ref}`eqn:mapeqn5#
Population coding of saccades in the SC motor map#
The following hypothesis has been proposed to account for the shape and size-relation of the movement fields: when the skewed movement fields of many SC cells are replotted in SC coordinates (by applying the mapping function, Eq. \ref{eqn:mapeqn5`) the following interesting property emerges:
Each remapped movement field is invariant of the SC site. In other words, the size of the remapped movement field is roughly equal for all sites, and also the asymmetry disappears so that the remapped fields are rotation-symmetric (Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5, bottom left).
Conceptually, the remapping of a movement field into SC coordinates leads to a reconstruction of the actual population activity profile in the motor map (provided the movement fields for cells at identical sites are also identical). Thus, the data indicated that the size and shape of the population of cells in the SC is identical for each saccade, only its location within the motor map changes for saccades of different amplitudes and directions (see Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5, center panel left). In summary, the population activity function in SC coordinates can be well described by a rotation-symmetric, shift-invariant, gaussian (Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5 bottom left):
with \(F_o\), and \(\phi_o\) fixed parameters (the same for every site in the SC, and \(\mathbf{w_o} = (u_o, v_o)\) corresponds to the center of the population activity which is given by the afferent mapping function Eq. eqn:mapeqn5). Recordings from a large number of SC cells have indicated that an optimal choice of parameters for the population activity profile is: \(F_o = 500\) Hz, and \(\phi_o = 0.5\) mm.
Connection scheme of the Superior Colliculus with the brainstem#
The next question then is, how such a population of activated SC neurons will have to project to the brainstem pulse-step generators in order to encode the correct, goal-directed saccade vector. In principle, there may be many ways in which the SC could encode the saccade vector, but the following simple model captures a biologically plausible way (because it is simple). According to this model, the following principles apply (see Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5, center panel):
All cells that are recruited within the gaussian population contribute to the saccade (i.e. not only the center (few) cells!). The model therefore constitutes a population coding scheme, or ‘coarse coding’ scheme.
Cells that have a high firing rate yield a larger contribution to the saccade vector than poorly active cells (like in a democracy!).
The resulting saccade vector is finally determined by a weighted sum of all cell contributions.
The weighting factors for each neuron of the population are determined by two factors: (1) each cell’s firing rate (which varies from saccade to saccade), and (2) by each cell’s synaptic connection strength with the brainstem (which is fixed!). The latter property of the SC map is called the efferent mapping (= output, or motor mapping) of the colliculus (see also Exercises) and depends exclusively on the location of each cell within the motor map (see Fig. Conceptual model of the mappings that take place between the early visual stage of the retina, and the final motor stage in the brainstem (top). Center: Complex-logarithmic motor map of the Superior Colliculus. (u, v) anatomical coordinates; (R, \phi) polar coordinates of saccade vector corresponding to site (u, v) through Eq. {ref}`eqn:mapeqn5, center right). Thus, to summarize,
with \(\mathbf{z_o}\) the saccadic movement vector, (the desired 2D displacement vector) \((\Delta x,\Delta y)\), \(F_i\), is the firing rate of cell \(i\) in the active guassian population \(F_i \in [0, F_o]\), and \(\mathbf{W}_i\) represents the synaptic weighting of cell \(i\) with the horizontal and vertical pulse-step generators in the brainstem. Conceptually, Eq. eqn:mapeqn7 entails that each cell in the population generates a small ‘mini-vector’ to the brainstem, the size of which is modulated by its firing rate. At the level of the brainstem all contributions are linearly summed, and this should exactly yield the required saccade vector. The question is, however, whether the gaussian weighting of the firing rates is adequate to fulfill this latter requirement. Although it is analytically very difficult to show that Eq. eqn:mapeqn7 indeed yields the correct saccade components to refixate the target, computer simulations have shown that this is indeed the case.
Exercises#
\begin{exercise}[Nonlinearity of main sequence] According to the results shown in Fig. Typical position traces of saccadic eye movements as a function of time. Note that saccade duration increases with saccade amplitude (straight-line relation), and that the peak velocity of the saccade increases with amplitude in a saturating manner. The latter two relations are known as the ‘main sequence’ for saccades. the saccadic system must be a nonlinear system.
[label=\alph*]
Why is this conclusion drawn?
Draw the saccadic ’main sequence’ for a linear system.
\end{exercise}
\begin{exercise}[Common Source] In the ‘Common Source’ model of the saccadic system (e.g. Fig. Simple scheme of the one-dimensional model of saccade control, as proposed by Scudder, 1988 and More detailed schematic of the 1-D Scudder model, which is also available in Matlab’s Simulink), the brainstem is driven by a nonlinear vectorial pulse generator that transforms a 2D motor error vector, \(\Delta e_{vec}\), into a vectorial eye-velocity command according to: \(\dot e_{vec}(t) = v_{max} \cdot [1 - \exp{-\frac{\lvert \Delta e_{vec}(t) \rvert }{m_o}}]\) with \(\lvert \cdot \rvert\) the magnitude of the vector. Give expressions for the velocity profiles of oblique saccades \([R,\phi]\), with identical horizontal saccade components (i.e. horizontal component amplitude is fixed, say at \(\Delta H\), but the vector rotates over angle \(\phi\) with respect to the horizontal direction).
\end{exercise}
\begin{exercise}[Complex logarithmic mapping] The complex-log map has a singularity at \(z = 0\). In order to provide a more realistic model for the neural mapping onto the visual cortex, the singularity has to disappear. Therefore, a simple extension of the map is given by the following function: \(\mathbf{w}=\ln{\mathbf{z}+\mathbf{a}}\), where \( \mathbf{z} \equiv (x, y)\), and \( \mathbf{a} \equiv (1, 0)\).
[label=\alph*]
Make a 2D graph of this new complex-log mapping function (as in Fig. Mapping of visual space onto neural space. **A).
Give a mathematical description for the inverse mapping of this function: \(\mathbf{z}=M^{-1}[\mathbf{w}]\).
\end{exercise}
\begin{exercise}[Efferent mapping] The complex-log mapping of the superior colliculus is given by Eq. eqn:mapeqn5. Give an expression for the inverse mapping function (the efferent map) that relates the horizontal (\(\Delta x\)), and vertical (\(\Delta y\)) saccade components (\(\mathbf{z_o} = \Delta x, \Delta y)\)) to the collicular neural coordinates, \(\mathbf{w_o}=(u_o,v_o)\): \(\mathbf{z_o}=EFF.MAP(\mathbf{w_o})\).
\end{exercise}
\begin{exercise}[Anisotropy and end-point variability] Suppose that the brain somehow selects the center of the active gaussian collicular cell population and that the center determines the size and direction of the saccade vector, \(\Delta e\). Assume that this selection process is noisy, so that the center of the population (in response to identical target presentations) varies from saccade to saccade. Suppose that the scatter is described by a small circular region within the collicular complex-log map. Derive an expression for the distribution of saccade vector endpoints that results from this scatter, and show that under these assumptions the anisotropy of the motor map (the finding that \(B_u \neq B_v\)) is directly reflected in the saccade endpoint distribution.
\end{exercise}
Glossary#
\begin{description} \item[Saccadic Control System] A mechanism in the brain that rapidly redirects the line of sight, enabling the high-resolution central part of the visual field (the fovea) to focus on objects of interest. \item[Fovea] A small depression in the retina where visual acuity is highest. It is densely packed with cones and is critical for sharp central vision. \item[Retinal Error Signal] The difference between the current gaze direction and the desired gaze direction, which is used to program eye movements. \item[Superior Colliculus (SC)] A layered structure in the midbrain that plays an important role in initiating and guiding eye movements. \item[Neurocomputational Modeling] The use of computational methods and mathematical models to understand the neural mechanisms of cognitive functions, such as saccadic eye movements. \item[Magnification Factor] A measure used in neuroscience to describe how much a part of the visual field is magnified in its representation in the visual cortex. \item[Motor Map] A representation in the superior colliculus where the location of neural activation determines the desired eye displacement vector. \item[Pulse-Step Activity Pattern] A characteristic pattern of neural activity in eye muscle control, where a pulse of intense activity is followed by a sustained step, correlating with eye movement velocity and position. \item[Efference Copy] A copy of a motor command sent to the muscles, used internally in the brain to predict the expected sensory feedback and to adjust movements accordingly. \item[Main Sequence for Saccades] A relationship in saccadic movements where the duration and peak velocity of a saccade increase with its amplitude. \item[Gaze Shifts] Movements of the eyes, head, and sometimes the body to redirect the line of sight. \item[Nonlinear System] A system in which the change of the output is not proportional to the change of the input. In the context of the saccadic system, it refers to the complex relationship between saccade amplitude, duration, and velocity. \item[Burst Neurons] Neurons in the brainstem that produce a rapid series of action potentials to initiate saccadic eye movements. \item[Brainstem] The part of the brain that connects the cerebrum with the spinal cord and plays a key role in regulating vital body functions, including eye movements. \item[Omnipause Neurons (OPNs)] Neurons in the brainstem that pause during saccades, helping to regulate the timing and speed of eye movements. \item[Visual Cortex] The part of the cerebral cortex responsible for processing visual information. \item[Frontal Eye Fields (FEF)] Areas of the frontal cortex involved in controlling voluntary eye movements. \item[Iso-Amplitude Circles] Circles in a coordinate system representing locations with equal saccade amplitudes. \item[Conformal Mapping] A type of mathematical transformation that preserves angles but not necessarily lengths. \item[Population Coding] A neural coding scheme in which a population of neurons represents a particular stimulus or action, rather than relying on individual neurons.
\item[Afferent Mapping] The process by which sensory information is transmitted to the central nervous system. In the context of the superior colliculus, it refers to the mapping of visual or other sensory inputs onto the neural structure.
\item[Efferent Mapping] The transmission of neural signals from the central nervous system to the peripheral effectors, such as muscles. In the superior colliculus, this involves the mapping of neural activation patterns to motor commands for eye movement.
\item[Complex Logarithmic Mapping] A mathematical transformation used to represent visual information in the brain. It maps the visual field onto a neural space using logarithmic scaling, which helps in efficiently encoding wide ranges of visual information.
\item[Anisotropy] The property of being directionally dependent, as opposed to isotropy, which suggests identical properties in all directions. In neuroscience, anisotropic mapping can refer to how different neural pathways may process information differently based on direction.
\item[Common Source] In the context of the saccadic control system, this refers to a shared neural origin for the signals that control eye movements. It implies that both horizontal and vertical components of a saccade originate from a common neural source before being separated into individual pathways.
\item[Rotation-Symmetric] A property of a system or function where it remains invariant under rotation. In the context of neural mapping, it suggests that the mapping function preserves the relative orientation of elements in the visual field.
\end{description}