One striking feature of a lot of brain structures – from the retina to the cerebral cortex, from the optic tectum to the cerebellum – is that they are layered. One example can be found in this figure, which was taken from Squire et al.’s, Fundamental Neuroscience:
This image represents the cellular organization of the sensory cortex into six layers (labeled I-VI). Notice that the input and output is highly organized: inputs from the thalamus terminate mainly in layers III and IV, and output to that same structure is from pyramidal neurons in layer VI. An anthropocentric view tends to see this remarkable feat of organization as the pinnacle of evolution in terms of laminated structures; some textbooks refer to the cerebral cortex as neocortex because it has six laminae, as opposed to the three laminae found in “archicortex” (hippocampus). Since hippocampus is usually regarded as an “ancient” structure (part of the “paleomammallian brain” proposed by Paul MacLean), it should follow that lamination is a sophisticated and recent development in evolution (which, one always assume, involves only the neocortex).
A closer examination reveals that this is not the case. Even in the human brain, lots of other structures are laminated: we already mentioned the retina, the optic tectum (called “superior colliculus” in humans), the hippocampus and the cerebellum. The number of layers in each of those structures varies considerably: the retina has four nuclear and two plexiform layers (a nuclear layer contains axons and cell bodies [e.g., layers II-III and V-VI in the neocortex], whilst a plexiform layer contains dendrites and their synaptic contacts [such as layers I and IV in the neocortex]; not every laminated structure is as easily organisable as this, though), whilst, as mentioned, the hippocampus has three layers. Sure, intra-specifically – that is, considering the variation of layer numbers among structures in humans – the neocortex is highly laminated. What do we say, though, about laminated structures in other species? The goldfish has a structure, called gustatory lobe, which has more than 20 layers . The input to those layers comes from taste receptors in his mouth and in other regions around it (some fish species have taste receptors everywhere, it’s amazing), and is organized in a way that is similar to the organization of the human neocortex. The output of this structure goes to the vagal lobe, which is also highly layered and constitutes the motor output for this circuit. Whenever the fish swallows something that does not taste good, this information is first processed in the gustatory lobe and then passed on to the vagal lobe, which orders cranial muscles to spit that something .
This organization - a sensory laminated structure which is connected, or in register, with a motor laminated structure - is reminiscent of another pair of structures in the human cerebral cortex: the primary somatosensory area (S1) and the primary motor area (M1). In this case, the primary somatosensory area acquires information about, for example, the relative position of the limbs (proprioception) and relays this information to the appropriate region of the primary motor area, which would execute a motor program involved in this.
Already two principles are emerging from a preliminary analysis: (A) there are many laminated structures in the brains of vertebrates, and this lamination helps to organize inputs and outputs; and (B) sometimes those structures are connected to other laminated structures. In order to understand the evolution of those structures, we will make a short detour to the input-output integration problem.
Sensorimotor integration: How to integrate inputs and outpus?
One of the most important challenges any nervous system can face is the integration of inputs and outputs. The philosophers Patricia Churchland  and Paul Churchland  devised a cartoon story which can be used to assess the difficulty of the problem. They imagined the world of Roger, a “crab-like critter” equipped with a pair of eye-like devices which detect the position of his favorite food item (an apple) in a two-dimensional external world:
The position of the apple in this external space can be given by drawing a pair of coordinate axes and specifying the position in terms of the coordinates. Its position can also be represented in visual phase space – that is, its position in the natural coordinate system of Roger’s sensory equipment. (Ref. , p. 420).
The main problem that poor Roger faces is that of sensorimotor coordination: how can he reach the apple?
Just as Roger has a 2-D visual space in which the position of the object is represented, so he has a 2-D motor space in which its arm position can be represented. But, and this is crucial, these two phase spaces are very different (Ref. , p. 421).
Roger’s motor phase space (state space would be more appropriate) is characterized as the two angles by which his arm deviates from its standard position. If the zero position is aligned with the horizontal axis, a position of 45 on the horizontal arm axis will represent a horizontal-arm position of 45 degrees off the horizontal. By the same token, a position of 78 on the vertical axis represents a vertical-arm position of 78 degrees off the vertical.
The eyes, having detected the apple, announce its position: “Apple at (55, 85)”. Notice that if the arm were to take as its command, “Go to (55, 85)”, then it would execute that command by putting the upper arm at 55 and the forearm at 85. This would be disastrous, because it would put the arm nowhere near the apple. In other words, the position of the apple in visual space is not the position of the apple in motor space (…). What Roger needs, therefore, is something to tell him what coordinates in motor space correspond to a given set of coordinates in visual space. (Ref. , pp. 423-424.
Sensorimotor coordination – the coordination of a sensory input and a motor output – is a complex problem, as Roger showed us. The main problem here (assuming that the position of an object in the environment can be thought of as coordinates in a phase space) is that the coordinates from one space and the other are not equal; in order to reach the apple, Roger’s nervous system must make a transformation of coordinates.
The computational neurobiologists Andras Pellionisz and Rodolfo Llinás (in their tensor network theory [5, 6]) proposed that this problem can be modeled by vector-matrix transformations. From observations derived from the cytoarchitectural and hodological organization of the cerebellum, they proposed that the connectivity of arrays of neurons is crucial to explaining how a given input yields a given output. The problem, then, becomes very similar to that found by Roger – that is, to find what is the relation between input arrays and output arrays.
What Pellionisz and Llinás proposed is that this relationship can be modeled by means of a tensor – a generalized mathematical function for vector transformation, no matter what frames of reference are involved. In geometrical terms, vectors are represented as line segments in a specified coordinate system (frame of reference), such as the visual and motor phase spaces available to Roger; the components of the vector are given in terms of their values as specified in relation to the relevant coordinate axes.
If each neuron in a network of input neurons specifies an axis of a coordinate system, then the input of an individual neuron – its spiking frequency – defines a point on the axis, and the input of the whole array of neurons can then be very neatly given as a vector in that space. Similarly, the output of an array can be specified as a vector in the space defined by the set of output neurons. (Ref. , p. 417).
The model reasons that the connectivity relations between input and output neurons serve as a matrix; any input vector is transformed into an output vector by matrix multiplication. The model seems simple enough: by means of a tensor, an input vector is transformed into an appropriate output vector; representations from input and output are positions in phase spaces (vector representations), and computation is coordinate transformation via tensors. Paul Churchland  complicated matters a little more proposing that tensor network theory can be best applied when the relevant phase spaces are two-dimensional:
If we suppose that our crab contains internal representations of both its sensory phase space and its motor phase space, then the following arrangement will effect the desired transformation. Let its sensory phase space be represented by a physical grid of signal-carrying fibers, a grid that is metrically deformed in real space in just the way displayed in [the following figure]. Let its motor phase space be represented by a second grid of fibres, in undeformed orthogonal array. Place the first grid over the second, and let them be connected by a large number of short vertical fibres, extending downwards from each vertex in the sensory grid to the nearest vertex in the underlying motor grid, as in [this] figure.
The system functions as follows. Depending on where each eye is positioned, a signal is sent down a particular fibre in the appropriate bundle, a fibre which extends into the sensory grid from the appropriate radial point. Joint eye position is thus reliably registered by a joint stimulation at a unique coordinate intersection.
In the lower grid, a stimulation at any coordinate intersection is conveyed outwards along the relevant pair of motor fibres, each of which induces its target limb joint to assume the appropriate angle. We need now suppose only that the vertical connections between the upper and lower grids all function as "AND-gates" or "threshold switches", so that a stimulatory signal is sent down a vertical connection to the motor grid exactly if and exactly where the relevant sensory intersection point is simultaneously stimulated by both of its intersecting fibers. Such a system will "compute" the desired coordinate transformation to a degree of accuracy limited only by the grain of the two grids, and be the density of vertical connections. Because the upper map is metrically deformed in the mannter described, the points in sensory space are brought into appropriate register with the points in motor space. As the eyes fixate, so fixates the arm: it reaches out to precisely the point triangulated by the crab's eyes. (Ref. ).
This description is suspiciously similar to the relations between topographic maps and lamination in real brains. Sensory regions in the brain are differentially sensitive to stimuli presented in different regions of the external space; thus, a “map”, codified in action potentials, exists in those structures. This spatial organization involves the juxtaposition of neural fibers and cell bodies side by side, according to the spatial position of receptors. The spatial organization of a structure (or part of a structure) in maps, layered in suitable register, generates a simple way in which neurons can execute 2-D to 2-D transformation. Another benefit of this organizational principle is that it tends to minimize connection lengths between neurons:
If multiple sensory and/or motor maps are distributed across different layers but in register with one another, then corresponding points in different maps can be connected by dendrites that extend across the laminae and by short, radially coursing axons. If the maps were located in different brain regions, much longer connections would be needed to connect the corresponding points (…). This is important because space is limited within the skull and because neural tissue is metabolically expensive to build and operate (…). In addition, minimizing connection lengths should minimize conduction times and, therefore, increase processing speed. (Ref. , p. 188)Thus, the organization of topographic maps is advantageous because it allows for a mechanism of vector-to-vector transformation between two frames of reference in a way that is economic to the brain [3-6, 8]. As the evolutionary neuroscientist Georg Striedter  pointed, “another likely reason for lamination to evolve is that it may be ‘easy’ to develop” (p. 188).
The development of laminated structures
In 1963, the neurobiologist Roger Sperry  proposed that topographical projections develop mainly because axon outgrowth and termination are largely “chemotactic” functions . In a nutshell, the embryonic nervous system is filled with molecules that are expressed in (largely overlapping) concentration gradients, which are “detected” and “interpreted” by growing axons. These developing fibers grow to specific locations within a concentration gradient and then terminate there. In fact, axons are guided by simultaneous and coordinate actions of four types of guidance mechanisms (figure from Squire et al.’s Fundamental Neuroscience):
Sperry reasoned that, if neighboring neurons have similar “instructions” about where to terminate (a likely assumption, since neighboring cells are exposed to similar molecular environments when they begin their development), then topographical projection maps result. Those maps may not be terribly accurate at first, but activity-dependent mechanisms can reduce the overlap between adjacent terminals .
Let’s study an example here, retinotectal projections in the mice. Representation of the retina on the tectum is typically simplified to the mapping of two sets of orthogonally oriented axes: the temporal-nasal axis of the retina is mapped on the anterior-posterior axis of the tectum, whilst the ventral-dorsal axis of the retina is mapped on the medial-lateral axis of the tectum. Based on Sperry’s affinity hypothesis, each point in the tectum should have a unique “address” determined by the gradient of topographic guidance molecules, and this would result in a position-dependent response to them by retinal ganglion cell axons (the output from the retina). Well, over the second half of the 20th century, the specificity of the projections of retinal ganglion cell axons to the tectum was investigated using a plethora of techniques; one could, for example, trace axonal projections after sectioning the optic nerve and after waiting for it to regenerate (it is not only Wolverine which is capable of such deeds); or one could accompany, step by step, the development of retinal ganglion cell axons directed to the optic tectum. The main conclusion from those experiments is that, even when the axons from retinal ganglion cells are experimentally deflected within the tectum or forced to enter the axon in abnormal positions, they still reach their targets, and a topographic map is formed.The dynamics of topographical map development are very complicated. To signal with yet another example, let’s accompany the story of chicks. Ephrins are real cool proteins which have been shown to control the topographic map of (yet again) retinal ganglion cells onto the optic tectum. In chick (a common developmental model for visual systems, especially the retina and its close connections), the protein ephrin-A2 is expressed in an increasing anterior-posterior gradient in the entire tectum, whilst ephrin-A5 is expressed in a steeper gradient which is limited to the posterior regions of the tectum. Taken together, both proteins form and increasing gradient across the anterior-posterior tectal axis:
In this image (again, Squire et al’s Fundamental Neuroscience), we see a scheme of the approximate gradient profiles for EphA receptors and ephrin-A ligands in the chick retina and the optic tectum. Retinal ganglion cells show a temporal(T)-nasal(N) gradient of receptor expression. Ephrin-A2 and ephrin-A5 combine to form a anterior(A)-posterior(P) gradient of expression across the tectum. Because temporal retina with high levels of EphA3 maps to anterior tectum with low levels of eprhin-A (and vice-versa), ephrin-A2 and ephrin-A5 probably act as “repellants” that affect the temporal axons more strongly than nasal ones [12, 13]. The graded expression of eprhin-A2 and eprhin-A5 and their differential repulsion of axons which sprung from temporal versus nasal ganglion cells implicate them as guidance molecules, as would be predicted by the Sperry model.So far, so good: chemical cues guide axons to proper locations in the optic tectum, a structure that is known to be both laminated and to possess a topographical map of the retina. However, only a single map is usually of no use; in fact, most sensory areas are characterized by the existence of topographic maps at every level of processing. In the visual system of primates, for example, there are maps in the lateral geniculate nucleus of the thalamus and in the V1 area of the occipital cortex.
The basic mechanism here would be the same; axons must also reach other structures, leaving the most rostral ones, and they probably do so by the same “chemotactic” cues which were laid before. Striedter  extended Sperry’s model and proposed that those basic mechanisms may be also used to build laminar structures with multiple maps:
Imagine, for example, that two different sets of neurons project topographically to the same target, using the same molecular gradient to find their appropriate termination sites. In that case, two overlapping topographical maps would form [as in this figure; yeah, I know the quality sucks, but my scanner is tweaked].
If the target structure contains an additional molecular gradient that is oriented orthogonally to the topography-determining gradient, and if the two sets of axons preferentially grow to different locations within that additional gradient, then the two topographical maps would become at least partially segregated along the radial dimension. That is, the presence of an additional “biasing” gradient would cause one of the two maps to lie above the other one, at least on average. If this partial segregation were subsequently enhanced by activity-dependent terminal segregation, a distinctly laminar termination pattern would result and the two maps would “automatically” be in register with one another (since they were based on the same topography-determining molecular gradient). (Ref. , pp. 189-190.It seems, then, that evolution used the same principle for the developmental construction of laminated structures with on-register maps. All you need is some molecular gradients which would guide axons in their development, a few experience-based adjustments, and you have a good solution for the input-output problem!
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