Brain modeling by tensor network theory and computer simulation. The cerebellum: Distributed processor for predictive coordination

Abstract A fundamental problem regarding the functional interpretation of neural networks in the central nervous system is that of establishing the principles of their parallel, distributed organization. Available morphological and physiological knowledge concerning the cerebellum suggests that the central nervous system may use organizational principles other than the traditionally assumed ‘random connectivity’, ‘reflex loops’ or ‘redundancy’. We propose formally, and demonstrate by computer modeling, that the firing frequencies of individual cells over a cerebellar cortical area may be interpreted as a spatially distributed, finite, series expansion of a time function, which is reconstructed, by summation, in the nucleus where the cortical cells project. Thus, for example, the firing of Purkinje cells, when considered as representing a Taylor expansion, yield a prediction in the cerebellar nuclei of the frequency-time function of the input arriving at the cortex. This ‘lookahead’ (Δ) is an emergent property of the inherently parallel, distributed network. In order to analyze how a Taylor expansion-like process is used by the cerebellum on a system level, the linear algebraic matrix- and vector-representation of a distributed network was generalized in such a way as to regard the neuronal networks as tensors. Thus, the brain is envisioned as a set of tensorial systems which communicate with each other through vectorial channels (the pathways). These pathways carry multidimensional frequency vectors which are transformed by the tensors. In these terms the function of a particular subsystem of the central nervous system, in the present case the cerebellum, is represented in a multidimensional space. The frequency-hyperspace is characterized by the matrix of the cerebellar tensor which specifies a curved set of trajectories: a cerebellar vector field. Dynamic posture and balance are interpreted as displacement or stabilization of the functional status vector of the motor system along these curved trajectories. Cerebellar coordination of ballistic movements can be described as guiding the movement onto the ‘wired in’ trajectories of the vector field, by virtue of the coordination and inhibition vectors provided by the cerebellum. A proposal is also introduced that the climbing fiber system operates by momentary perturbations of the vector field, leading to deformations of the trajectories of the cerebellar frequency hyperspace. To bridge the gap between an attempt to treat parallel, distributed, neuronal networks, such as the cerebellum, as geometrical objects (leading, at the first approximation, to a linear mathematical formulation) and the simultaneous task of incorporating and interpreting experimental data, computer simulation methods are required. This combined approach of mathematical treatment (which provides an abstract language) and computer simulation (which, by accommodating data into the model, explores the significance of deviations from linear character) appears at present to be the most adequate technique for dealing with central nervous system function.