Figure 1.
The information provided by the spinocerebellar and spinoreticulocerebellar mossy fibers derives from the spinal interneuron circuitry.
The vermis and pars intermedia of the cerebellum receives a substantial part of their mossy fiber inputs from the SCT/SRCT pathways [19]. The SCT/SRCT pathways consist of spinal interneuron projections either directly as mossy fibers (rostral spinocerebellar tract, RSCT), via a relay in the lateral reticular nucleus of the brainstem (spino-reticulo cerebellar path, SRCT), or via relay cells located in the spinal cord (ventral and dorsal spinocerebellar tracts, VSCT and DSCT) [19]. These spinal interneurons can project directly to alpha-motorneurons and likely form an integral part of the spinal motor circuitry, by conveying sensory feedback and motor commands to the motor nuclei of the spinal cord [99].
Figure 2.
Piecewise-linear (PL) approximations in the cerebellar neuronal network.
(A) Using the excitatory input directly from PF and the inhibitory pathway through molecular layer interneurons, the PC can construct a PL approximation of arbitrary non-linear functions of the input reaching the GrCs. (top) Two PFs innervate the PC directly (3 & 4), while the other two innervate a stellate interneuron (1 & 2). (middle) The four GrCs have slightly different thresholds and varying MF efficacy leading to varying activity slopes. (bottom) The PC modulates its output using the input coming from the GrCs according to Eq. (1). The path through the inhibitory molecular layer interneurons allows the weight and thus the slope of the curve to be negative. Each GrC threshold corresponds to one knot in the PL PC output. (B) The distribution of GrC thresholds over the input range determines how well the PC can approximate the non-linear regions of the approximated function. (top) Several receptive fields can contribute to measure a single intrinsic dimension. In this case, the skin stretch can be used to deduce the joint angle. (middle) The different receptive fields allow the GrC thresholds to be spread over a larger input range than that using only a varying degree of Golgi cell tonic inhibition. (bottom) Sum of activity of all GrCs activated from the three receptive fields. As the population GrC activity rises over the entire input range, their output could be used to approximate non-linearities over the entire input range. (C) A naïve approach to enable the PC output to approximate functions of two-dimensions. In this example, afferent information from skin stretch and Ib afferents are added separately in the PC, generating an approximated surface. At each point in the input space, the PC output is calculated by adding the contribution from GrCs innervated by the two separate afferent types.
Figure 3.
Comparison of network structures of ANNs and spinocerebellar systems.
(A) A standard feed-forward ANNs with one hidden layer (GrCs), where every input is available to all units in the hidden layer. (B) In contrast, in the spinocerebellar system, MF inputs to GrCs have a focal termination, where different functional types of MFs are connected to different sets of GrCs. In this arrangement it is possible for recombination of the sensorimotor inputs to take place already at the level of the SCT/SRCT units, while the recombination at the granule layer is restricted to the approximately four functionally similar MFs that innervate every GrC. In the biological system, the GrCs have only excitatory synapses upon the PCs, i.e. only positive weights. It is however possible to obtain inhibitory GrC to PC efficacies by mediating the GrC signal via the inhibitory interneurons of the molecular layer (Int) (cf. [40], [100]).
Figure 4.
Approximation examples of basic non-linear functions.
(A) Approximated surfaces using a single or two projections (left and middle columns, respectively) compared to the approximated or target function surface (right-hand column) (i.e RMSE = 0). The colors represent the height of the surface ranging from negative values (blue) to positive values (red), comparable to the surface in Figure 2C. In each row, a different non-linear interaction retrieved from the terms within the inverse dynamics of the planar double joint arm in [79] is used. The illustrated projections had the lowest RMSE of 100 tested projections, each tested projection having a random direction. The actual RMSE values can be found in (B). The approximated surfaces also display the actual projections used as dashed lines above the surfaces. The value of the elbow angle variable, range between
and
, to capture an entire period of the sin function that is approximated. (B) RMSE of approximations of three two-dimensional non-linear terms in A. The approximations where constructed using random projection directions and a total of 60 GrCs. 100 approximations where constructed for each box. The mean RMSE is shown by the center line of the box, the boxes themselves extend to the 25th and 75th quartile and the whiskers extends to the most extreme RMSE not considered to be outliers, which are instead shown as black crosses. The red markers with an arrow from “raw signal” show the RMSE of approximations using the raw signals as projection directions, i.e. without recombination of inputs and those with an arrow from “in A” show the RMSE of the approximations shown in (A).
Figure 5.
Three-dimensional approximations with random projections.
Similar to the approximations in Figure 4B, but the approximated function is instead the three-dimensional term from Eq. (5), approximated over a three-dimensional grid. As in Figure 4, 60 GrCs were used and 100 approximations with random projections were constructed for each of the boxes. The mean RMSE is shown by the center line of the box, the boxes themselves extend to the 25th and 75th quartile and the whiskers extends to the most extreme RMSE.
Figure 6.
Approximation error when the number of granule cells is increased.
The figure shows how the RMSE is reduced when the number of GrCs is increased as the functions in the figure legend were approximated using the specified number of projections. To search for the optimal approximations, the approximation directions were also optimized along with the GrC to PC weights. The first equation , is three-dimensional and was approximated using both 4 and 8 projections, while the others are two-dimensional. Note that the last equation
was approximated both using 2 and 3 projections to investigate the relatively large differences found using random projections (see Figure 4B).