1464 Part G Human-Centered and Life-Like Robotics fibers (MFs)arise from the spinal cord and brainstem. Since the development of the Marr-Albus model sev- They synapse onto granule cells and deep cerebellar nu- eral cerebellar models have been introduced in which clei.Granule cells have axons which each project up cerebellar plasticity plays a key role.Limiting our to form a T,with the bars of the T forming the paral- overview to computational models,we will describe lel fibers(PFs).Each PF synapses on about 200 PCs. (1)the cerebellar model articulation controller(CMAC), The PCs,which are grouped into microzones,inhibit (2)the adjustable pattern generator (APG),(3)the the deep nuclei.PCs with their target cells in cerebel- Schweighofer-Arbib model,and(4)the multiple paired lar nuclei are grouped together in microcomplexes (see forward-inverse models (see van der Smagt [62.80,811). Ito [62.79]).Microcomplexes are defined by a variety of criteria to serve as the units of analysis of cerebellar The Cerebellar Model Articulation Controller influence on specific types of motor activity.The climb- (CMAC) ing fibers(CF)arise from the inferior olive.Each PC One of the first well-known computational models of the receives synapses from only one CF,but a CF makes cerebellum is the CMAC (Albus [62.70];see Fig.62.5b). about 300 excitatory synapses on each PC which it con- The algorithm was based on Albus'understanding of tacts.This powerful input alone is enough to fire the PC, the cerebellum,but it was not proposed as a biologically though most PC firing depends on subtle patterns of PF plausible model.The idea has its origins in the BOXES activity.The cerebellar cortex also contains a variety of approach,in which for n variables an n-dimensional hy- inhibitory interneurons.The basket cell is activated by percube stores function values in a lookup table.BOXES PF afferents and makes inhibitory synapses onto PCs. suffers from the curse of dimensionality:if each variable Golgi cells receive input from PFs,MFs,and CFs and can be discretized into D different steps,the hyper- inhibit granule cells. cube has to store D"function values in memory.Albus assumed that the mossy fibers provided discretized func- The Marr-Albus Model tion values.If the signal on a mossy fiber is in the In the Marr-Albus model (see Marr [62.68]and Al- receptive field of a particular granule cell,it fires onto bus [62.69])the cerebellum functions as a classifier of a parallel fiber.This mapping of inputs onto binary out- sensory and motor patterns received through the MFs. put variables is often considered to be the generalization Only a small fraction of the parallel fibers(PF)are ac- mechanism in CMAC.The learning signals are provided tive when a Purkinje cell(PC)fires and thus influence by the climbing fibers. the motor neurons.Both Marr and Albus hypothesized Albus'CMAC can be described in terms of a large set that the error signals for improving PC firing in re- of overlapping,multidimensional receptive fields with sponse to PF,and thus MF input,were provided by the finite boundaries.Every input vector falls within the climbing fibers(CF),since only one CF affects a given range of some local receptive fields.The response of PC.However,Marr hypothesized that CF activity would CMAC to a given input is determined by the average of strengthen the active PF/PC synapses using a Widrow- the responses of the receptive fields excited by that input. Hoff learning rule,whereas Albus hypothesized they Similarly,the training for a given input vector affects would weaken them.This is an important example of only the parameters of the excited receptive fields. a case where computational modeling inspired impor- The organization of the receptive fields of a typical tant experimentation.Eventually,Masao Ito was able Albus CMAC with a two-dimensional input space can Part to demonstrate that Albus was correct-the weakening be described as follows.The set of overlapping receptive of active synapses is now known to involve a process fields is divided into C subsets,commonly referred to as 0 called long-term depression (Ito [62.79]).However,the layers.Any input vector excites one receptive field from each layer,for a total of Cexcited receptive fields for any 62.3 rule with weakening of synapses still known as the Marr-Albus model,and remains the reference model input.The overlap of the receptive fields produces input for studies of synaptic plasticity of cerebellar cortex.generalization,while the offset of the adjacent layers of However,both Marr and Albus viewed each PC as func-receptive fields produces input quantization.The ratio tioning as a perceptron whose job it was to control an of the width of each receptive field (input generaliza- elemental movement,contrasting with more plausible tion)to the offset between adjacent layers of receptive models in which PCs serve to modulate the involvement fields (input quantization)must be equal to C for all di- of microcomplexes (which include cells of the deep nu- mensions of the input space.This organization of the clei)in motor pattern generators (e.g.,the APG model receptive fields guarantees that only a fixed number,C, described below). of receptive fields is excited by any input.1464 Part G Human-Centered and Life-Like Robotics fibers (MFs) arise from the spinal cord and brainstem. They synapse onto granule cells and deep cerebellar nu￾clei. Granule cells have axons which each project up to form a T, with the bars of the T forming the paral￾lel fibers (PFs). Each PF synapses on about 200 PCs. The PCs, which are grouped into microzones, inhibit the deep nuclei. PCs with their target cells in cerebel￾lar nuclei are grouped together in microcomplexes (see Ito [62.79]). Microcomplexes are defined by a variety of criteria to serve as the units of analysis of cerebellar influence on specific types of motor activity. The climb￾ing fibers (CF) arise from the inferior olive. Each PC receives synapses from only one CF, but a CF makes about 300 excitatory synapses on each PC which it con￾tacts. This powerful input alone is enough to fire the PC, though most PC firing depends on subtle patterns of PF activity. The cerebellar cortex also contains a variety of inhibitory interneurons. The basket cell is activated by PF afferents and makes inhibitory synapses onto PCs. Golgi cells receive input from PFs, MFs, and CFs and inhibit granule cells. The Marr–Albus Model In the Marr–Albus model (see Marr [62.68] and Al￾bus [62.69]) the cerebellum functions as a classifier of sensory and motor patterns received through the MFs. Only a small fraction of the parallel fibers (PF) are ac￾tive when a Purkinje cell (PC) fires and thus influence the motor neurons. Both Marr and Albus hypothesized that the error signals for improving PC firing in re￾sponse to PF, and thus MF input, were provided by the climbing fibers (CF), since only one CF affects a given PC. However, Marr hypothesized that CF activity would strengthen the active PF/PC synapses using a Widrow– Hoff learning rule, whereas Albus hypothesized they would weaken them. This is an important example of a case where computational modeling inspired impor￾tant experimentation. Eventually, Masao Ito was able to demonstrate that Albus was correct – the weakening of active synapses is now known to involve a process called long-term depression (Ito [62.79]). However, the rule with weakening of synapses still known as the Marr–Albus model, and remains the reference model for studies of synaptic plasticity of cerebellar cortex. However, both Marr and Albus viewed each PC as func￾tioning as a perceptron whose job it was to control an elemental movement, contrasting with more plausible models in which PCs serve to modulate the involvement of microcomplexes (which include cells of the deep nu￾clei) in motor pattern generators (e.g., the APG model described below). Since the development of the Marr–Albus model sev￾eral cerebellar models have been introduced in which cerebellar plasticity plays a key role. Limiting our overview to computational models, we will describe (1) the cerebellar model articulation controller (CMAC), (2) the adjustable pattern generator (APG), (3) the Schweighofer–Arbib model, and (4) the multiple paired forward-inverse models (see van der Smagt [62.80,81]). The Cerebellar Model Articulation Controller (CMAC) One of the first well-known computational models of the cerebellum is the CMAC (Albus[62.70]; see Fig. 62.5b). The algorithm was based on Albus’ understanding of the cerebellum, but it was not proposed as a biologically plausible model. The idea has its origins in the BOXES approach, in which for n variables an n-dimensional hy￾percube stores function values in a lookup table. BOXES suffers from the curse of dimensionality: if each variable can be discretized into D different steps, the hyper￾cube has to store Dn function values in memory. Albus assumed that the mossy fibers provided discretized func￾tion values. If the signal on a mossy fiber is in the receptive field of a particular granule cell, it fires onto a parallel fiber. This mapping of inputs onto binary out￾put variables is often considered to be the generalization mechanism in CMAC. The learning signals are provided by the climbing fibers. Albus’CMACcan be described in terms of a large set of overlapping, multidimensional receptive fields with finite boundaries. Every input vector falls within the range of some local receptive fields. The response of CMAC to a given input is determined by the average of the responses of the receptive fields excited by that input. Similarly, the training for a given input vector affects only the parameters of the excited receptive fields. The organization of the receptive fields of a typical Albus CMAC with a two-dimensional input space can be described as follows. The set of overlapping receptive fields is divided into C subsets, commonly referred to as layers. Any input vector excites one receptive field from each layer, for a total of C excited receptive fields for any input. The overlap of the receptive fields produces input generalization, while the offset of the adjacent layers of receptive fields produces input quantization. The ratio of the width of each receptive field (input generaliza￾tion) to the offset between adjacent layers of receptive fields (input quantization) must be equal to C for all di￾mensions of the input space. This organization of the receptive fields guarantees that only a fixed number, C, of receptive fields is excited by any input. Part G 62.3