## 2 Overview of Cerebellar Functions  
From classical literature, the cerebellum is well known to govern movement coordination and motor learning [3; 4; 5] (for more recent reviews, see [6; 7]), but it has been also recognized as involved in cognitive and emotional processing [8; 9]. Taking into account the peculiar anatomy of the cerebellum with a parallel repetition of microcircuits [8], its dense connectivity with cerebral cortex, basal ganglia, brainstem, and spinal cord, and that it contains about 50% of the brain neurons [10], it is not inappropriate to think that the cerebellum is involved in so many different processes. One of the most important cerebellar functions is correlated to ensure that the anticipated actions are correctly in tune with changes in the environment [1; 2]. The cerebellum is thought to contain neural representations reproducing the dynamic properties of the body and to exploit them to create sensorimotor predictions; this allows performing accurate motor forecasts linked to environmental stimuli and to body kinematics [11]. This subcortical structure is essential not only in the prediction of incoming information, but also in controlling the temporal relationship between task-relevant events [12]. To grant a correct output in motor coordination, it is necessary to define spatiotemporal sequences of body segment movements. In this regard, the cerebellum is able to operate as an internal \"timing machine\", providing a precise temporal representation for motor and nonmotor tasks [13]. Another characteristic function of the cerebellum is sequence learning; in fact, in a work of [PERSON] and [PERSON] (2003) [14], patients with cerebellar damage did not show learning the spatial and temporal sequences simultaneously presented. The wide neural network in which the cerebellum communicates with the cerebral cortex, basal ganglia, and limbic system allows not only the control of timing, as well as the predictive and learning function, but also some high-level cognitive and emotional processing, like attention, language, memory, and reasoning [15; 16; 17]. Consequently, patients with cerebellar lesions can also show cognitive-affective alterations [18]. The most common motor outcome of cerebellar lesions is ataxia (from the Greek word \(\alpha\tau\alpha\xi\iota\sigma\), i.e., lack of order). Ataxia is a neuropathological state consisting of lack of movement coordination. This condition is characterized by hypotonia, dysmetria, asynergy, dyschronometria, and dysdiadochokinesia [11; 19]. In addition, the cognitive-affective alterations linked to this pathology lead to a lack in the organization of thought, called \"dysmetria of thought\", comprising impairment of executive functions (abstract reasoning, working memory, planning) and difficulties with spatial cognition [20]. Nevertheless, the \"lack of order\" mainly affects the control of balance and gait. [PERSON] [5] reported that a cerebellar lesion disrupted the coordination between voluntary movement and equilibrium stabilization, demonstrating the important involvement of the cerebellum in postural organization [21]. Cerebellar patients typically show enhanced postural sway [22], abnormal response to perturbations, reduced control of equilibrium during movements of other body parts, and irregular oscillations of the trunk. Moreover, these patients often suffer from dysfunctional co-contractions, that is, co-activation of muscle pairs with opposing actions of major limb joints, during postural control [23]. Gait ataxia is often described as a \"drunken gait\" characterized by walking incoordination, variable foot placement, abnormal foot trajectories, a widened base of support, a deviating path of movement, and irregular interjoint coordination patterns [24; 25; 26].