The Marr-Albus model of the cerebellum has been reformulated with linear system analysis. This adaptive linear filter model of the cerebellum performs a filtering action of a phase lead-lag compensator with learning capability, and will give an account for the phenomena which have been termed “cerebellar compensation”. It is postulated that a Golgi cell may act as a phase lag element; for example, as a leaky integrator with time constant about several seconds. Under this assumption, a mossy fiber-granule cell-Golgi cell input network functions as a phase lead-lag compensator. Output signals from Golgi-granule cell systems, namely, parallel fiber signals, are gathered together through variable synaptic connections to form a Purkinje cell output. From a general theory of adaptive linear filters, learning principles for these modifiable connections are derived. By these learning principles, a Purkinje cell output converges to the “desired response” to minimize the mean square error of the performance. In a more general sense, a Purkinje cell acquires a filtering function on the basis of multiple pairs of input signals and corresponding desired output signals. The mode of convergence of the output signal is described when the input signal is sin
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