Tensor network theory is a theory of brain function (particularly that of the cerebellum) that provides a mathematical model of the transformation of sensory space-time coordinates into motor coordinates and vice versa by cerebellar neuronal networks. The theory was developed by Andras Pellionisz and Rodolfo Llinas in the 1980s as a geometrization of brain function (especially of the central nervous system) using tensors. [1] [2]
The mid-20th century saw a concerted movement to quantify and provide geometric models for various fields of science, including biology and physics. [3] [4] [5] The geometrization of biology began in the 1950s in an effort to reduce concepts and principles of biology down into concepts of geometry similar to what was done in physics in the decades before. [3] In fact, much of the geometrization that took place in the field of biology took its cues from the geometrization of contemporary physics. [6] One major achievement in general relativity was the geometrization of gravitation. [6] This allowed the trajectories of objects to be modeled as geodesic curves (or optimal paths) in a Riemannian space manifold. [6] During the 1980s, the field of theoretical physics also witnessed an outburst of geometrization activity in parallel with the development of the Unified Field Theory, the Theory of Everything, and the similar Grand Unified Theory, all of which attempted to explain connections between known physical phenomena. [7]
The geometrization of biology in parallel with the geometrization of physics covered a multitude of fields, including populations, disease outbreaks, and evolution, and continues to be an active field of research even today. [8] [9] By developing geometric models of populations and disease outbreaks, it is possible to predict the extent of the epidemic and allow public health officials and medical professionals to control disease outbreaks and better prepare for future epidemics. [8] Likewise, there is work being done to develop geometric models for the evolutionary process of species in order to study the process of evolution, the space of morphological properties, the diversity of forms and spontaneous changes and mutations. [9]
Around the same time as all of the developments in the geometrization of biology and physics, some headway was made in the geometrization of neuroscience. At the time, it became more and more necessary for brain functions to be quantified in order to study them more rigorously. Much of the progress can be attributed to the work of Pellionisz and Llinas and their associates who developed the tensor network theory in order to give researchers a means to quantify and model central nervous system activities. [1] [2]
In 1980, Pellionisz and Llinas introduced their tensor network theory to describe the behavior of the cerebellum in transforming afferent sensory inputs into efferent motor outputs. [1] They proposed that intrinsic multidimensional central nervous system space could be described and modeled by an extrinsic network of tensors that together describe the behavior of the central nervous system. [1] By treating the brain as a "geometrical object" and assuming that (1) neuronal network activity is vectorial and (2) that the networks themselves are organized tensorially, brain function could be quantified and described simply as a network of tensors. [1] [2]
In 1986, Pellionisz described the geometrization of the "three-neuron vestibulo-ocular reflex arc" in a cat using tensor network theory. [10] The "three-neuron vestibulo-ocular reflex arc" is named for the three neuron circuit the arc comprises. Sensory input into the vestibular system (angular acceleration of the head) is first received by the primary vestibular neurons which subsequently synapse onto secondary vestibular neurons. [10] These secondary neurons carry out much of the signal processing and produce the efferent signal heading for the oculomotor neurons. [10] Prior to the publishing of this paper, there had been no quantitative model to describe this "classic example of a basic sensorimotor transformation in the central nervous system" which is precisely what tensor network theory had been developed to model. [10]
Here, Pellionisz described the analysis of the sensory input into the vestibular canals as the covariant vector component of tensor network theory. Likewise, the synthesized motor response (reflexive eye movement) is described as the contravariant vector component of the theory. By calculating the neuronal network transformations between the sensory input into the vestibular system and the subsequent motor response, a metric tensor representing the neuronal network was calculated. [10]
The resulting metric tensor allowed for accurate predictions of the neuronal connections between the three intrinsically orthogonal vestibular canals and the six extraocular muscles that control the movement of the eye. [10]
Neural networks modeled after the activities of the central nervous system have allowed researchers to solve problems impossible to solve by other means. Artificial neural networks are now being applied in various applications to further research in other fields. One notable non-biological application of the tensor network theory was the simulated automated landing of a damaged F-15 fighter jet on one wing using a "Transputer parallel computer neural network". [11] The fighter jet's sensors fed information into the flight computer which in turn transformed that information into commands to control the plane's wing-flaps and ailerons to achieve a stable touchdown. This was synonymous to sensory inputs from the body being transformed into motor outputs by the cerebellum. The flight computer's calculations and behavior was modeled as a metric tensor taking the covariant sensor readings and transforming it into contravariant commands to control aircraft hardware. [11]
Ataxia is a neurological sign consisting of lack of voluntary coordination of muscle movements that can include gait abnormality, speech changes, and abnormalities in eye movements, that indicates dysfunction of parts of the nervous system that coordinate movement, such as the cerebellum.
The central nervous system (CNS) is the part of the nervous system consisting primarily of the brain and spinal cord. The CNS is so named because the brain integrates the received information and coordinates and influences the activity of all parts of the bodies of bilaterally symmetric and triploblastic animals—that is, all multicellular animals except sponges and diploblasts. It is a structure composed of nervous tissue positioned along the rostral to caudal axis of the body and may have an enlarged section at the rostral end which is a brain. Only arthropods, cephalopods and vertebrates have a true brain, though precursor structures exist in onychophorans, gastropods and lancelets.
A neuron, neurone, or nerve cell is an excitable cell that fires electric signals called action potentials across a neural network in the nervous system. They are located in the brain and spinal cord and help to receive and conduct impulses. Neurons communicate with other cells via synapses, which are specialized connections that commonly use minute amounts of chemical neurotransmitters to pass the electric signal from the presynaptic neuron to the target cell through the synaptic gap.
The cerebellum is a major feature of the hindbrain of all vertebrates. Although usually smaller than the cerebrum, in some animals such as the mormyrid fishes it may be as large as it or even larger. In humans, the cerebellum plays an important role in motor control and cognitive functions such as attention and language as well as emotional control such as regulating fear and pleasure responses, but its movement-related functions are the most solidly established. The human cerebellum does not initiate movement, but contributes to coordination, precision, and accurate timing: it receives input from sensory systems of the spinal cord and from other parts of the brain, and integrates these inputs to fine-tune motor activity. Cerebellar damage produces disorders in fine movement, equilibrium, posture, and motor learning in humans.
The sense of balance or equilibrioception is the perception of balance and spatial orientation. It helps prevent humans and nonhuman animals from falling over when standing or moving. Equilibrioception is the result of a number of sensory systems working together; the eyes, the inner ears, and the body's sense of where it is in space (proprioception) ideally need to be intact.
The brainstem is the posterior stalk-like part of the brain that connects the cerebrum with the spinal cord. In the human brain the brainstem is composed of the midbrain, the pons, and the medulla oblongata. The midbrain is continuous with the thalamus of the diencephalon through the tentorial notch, and sometimes the diencephalon is included in the brainstem.
The vestibulocochlear nerve or auditory vestibular nerve, also known as the eighth cranial nerve, cranial nerve VIII, or simply CN VIII, is a cranial nerve that transmits sound and equilibrium (balance) information from the inner ear to the brain. Through olivocochlear fibers, it also transmits motor and modulatory information from the superior olivary complex in the brainstem to the cochlea.
Computational neuroscience is a branch of neuroscience which employs mathematics, computer science, theoretical analysis and abstractions of the brain to understand the principles that govern the development, structure, physiology and cognitive abilities of the nervous system.
In neuroanatomy, a neural pathway is the connection formed by axons that project from neurons to make synapses onto neurons in another location, to enable neurotransmission. Neurons are connected by a single axon, or by a bundle of axons known as a nerve tract, or fasciculus. Shorter neural pathways are found within grey matter in the brain, whereas longer projections, made up of myelinated axons, constitute white matter.
Rodolfo Llinás Riascos is a Colombian neuroscientist. He is currently the Thomas and Suzanne Murphy Professor of Neuroscience and Chairman Emeritus of the Department of Physiology & Neuroscience at the NYU School of Medicine. Llinás has published over 800 scientific articles.
Climbing fibers are the name given to a series of neuronal projections from the inferior olivary nucleus located in the medulla oblongata.
Purkinje cells or Purkinje neurons, named for Czech physiologist Jan Evangelista Purkyně who identified them in 1837, are a unique type of prominent large neurons located in the cerebellar cortex of the brain. With their flask-shaped cell bodies, many branching dendrites, and a single long axon, these cells are essential for controlling motor activity. Purkinje cells mainly release GABA neurotransmitter, which inhibits some neurons to reduce nerve impulse transmission. Purkinje cells efficiently control and coordinate the body's motor motions through these inhibitory actions.
The dentate nucleus refer to a pair of deep cerebellar nuclei deep within the white matter of the cerebellum of the brain with a dentate – tooth-like or serrated – edge. The dentate forms the largest pathway between the cerebellum and the remainder of the brain. It is the largest and most lateral of the four pairs of deep cerebellar nuclei, the others being the globose and emboliform nuclei, which together are referred to as the interposed nucleus, and the fastigial nucleus.
In neuroscience, Golgi cells are the most abundant inhibitory interneurons found within the granular layer of the cerebellum. Golgi cells can be found in the granular layer at various layers. The Golgi cell is essential for controlling the activity of the granular layer. They were first identified as inhibitory in 1964. It was also the first example of an inhibitory feedback network in which the inhibitory interneuron was identified anatomically. Golgi cells produce a wide lateral inhibition that reaches beyond the afferent synaptic field and inhibit granule cells via feedforward and feedback inhibitory loops. These cells synapse onto the dendrite of granule cells and unipolar brush cells. They receive excitatory input from mossy fibres, also synapsing on granule cells, and parallel fibers, which are long granule cell axons. Thereby this circuitry allows for feed-forward and feed-back inhibition of granule cells.
Neural oscillations, or brainwaves, are rhythmic or repetitive patterns of neural activity in the central nervous system. Neural tissue can generate oscillatory activity in many ways, driven either by mechanisms within individual neurons or by interactions between neurons. In individual neurons, oscillations can appear either as oscillations in membrane potential or as rhythmic patterns of action potentials, which then produce oscillatory activation of post-synaptic neurons. At the level of neural ensembles, synchronized activity of large numbers of neurons can give rise to macroscopic oscillations, which can be observed in an electroencephalogram. Oscillatory activity in groups of neurons generally arises from feedback connections between the neurons that result in the synchronization of their firing patterns. The interaction between neurons can give rise to oscillations at a different frequency than the firing frequency of individual neurons. A well-known example of macroscopic neural oscillations is alpha activity.
The flocculus is a small lobe of the cerebellum at the posterior border of the middle cerebellar peduncle anterior to the biventer lobule. Like other parts of the cerebellum, the flocculus is involved in motor control. It is an essential part of the vestibulo-ocular reflex, and aids in the learning of basic motor skills in the brain.
Models of neural computation are attempts to elucidate, in an abstract and mathematical fashion, the core principles that underlie information processing in biological nervous systems, or functional components thereof. This article aims to provide an overview of the most definitive models of neuro-biological computation as well as the tools commonly used to construct and analyze them.
The Anatomy of the Cerebellum can be viewed at three levels. At the level of gross anatomy, the cerebellum consists of a tightly folded and crumpled layer of cortex, with white matter underneath, several deep nuclei embedded in the white matter, and a fluid-filled ventricle in the middle. At the intermediate level, the cerebellum and its auxiliary structures can be broken down into several hundred or thousand independently functioning modules or compartments known as microzones. At the microscopic level, each module consists of the same small set of neuronal elements, laid out with a highly stereotyped geometry.
Cerebellar granule cells form the thick granular layer of the cerebellar cortex and are among the smallest neurons in the brain. Cerebellar granule cells are also the most numerous neurons in the brain: in humans, estimates of their total number average around 50 billion, which means that they constitute about 3/4 of the brain's neurons.
The name granule cell has been used for a number of different types of neurons whose only common feature is that they all have very small cell bodies. Granule cells are found within the granular layer of the cerebellum, the dentate gyrus of the hippocampus, the superficial layer of the dorsal cochlear nucleus, the olfactory bulb, and the cerebral cortex.
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