Approximation examples of basic non-linear functions.

<p>(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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi-1002979-g002" target="_blank">Figure 2C</a>. In each row, a different non-linear interaction retrieved from the terms within the inverse dynamics of the planar double joint arm in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Hollerbach1" target="_blank">[79]</a> 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).</p

Piecewise-linear (PL) approximations in the cerebellar neuronal network.

<p>(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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979.e001" target="_blank">Eq. (1)</a>. 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.</p

Approximation error when the number of granule cells is increased.

<p>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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi-1002979-g004" target="_blank">Figure 4B</a>).</p

Comparison of network structures of ANNs and spinocerebellar systems.

<p>(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. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Dean1" target="_blank">[40]</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Jrntell5" target="_blank">[100]</a>).</p

The information provided by the spinocerebellar and spinoreticulocerebellar mossy fibers derives from the spinal interneuron circuitry.

<p>The vermis and pars intermedia of the cerebellum receives a substantial part of their mossy fiber inputs from the SCT/SRCT pathways <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Oscarsson1" target="_blank">[19]</a>. 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) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Oscarsson1" target="_blank">[19]</a>. 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002979#pcbi.1002979-Alstermark1" target="_blank">[99]</a>.</p

Simulated activity of SBCs receiving RST input and Ib inhibitory input.

<p>(A) Simulated intracellular SBC signal of RST excitation only during one step cycle. The amplitude of the intracellular signal in the model (7 mV) corresponds to that recorded in VSCT neurons under fictive locomotion in paralyzed cats (7.1 mV) <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107793#pone.0107793-Fedirchuk1" target="_blank">[9]</a>. (B) Corresponding simulated spike responses in SBCs, summarized in histograms with the instantaneous firing rate (5 ms bin width) for the full step cycle. The maximum firing rate of the model is indicated by the horizontal black line (264 Hz). (C) Same simulation as in (A), but in this case including inhibitory synaptic inputs from the Srt and VL Ib inhibitory interneuron input (black line, compare with the response without inhibition, illustrated by the grey line). (D) Spike responses for the SBC neuron with summation of RST excitation with the Ib afferent evoked inhibition.</p

Example of <i>in vivo</i> granule cell properties in the cerebellar target region of SBC inputs.

<p>(A) Example of an intracellular granule cell recording and its responses to rectangular current steps. (B) Relationship between current injected and average firing frequency (measured from the average interspike intervals between the first and the last spikes during the current step) for the same cell as in (A). The diagram illustrates the mean and standard deviation for firing frequencies obtained at different current values (N = 10 for each current value).</p

Simulated network scenario.

<p>Each granule cell was simulated to receive 4 mossy fiber inputs from the SBCs. Each simulated SBC received a massive, excitatory, monosynaptic reticulospinal input and a variable number of inhibitory interneuron inputs from the Ib interneurons driven by Ib afferents from two different hindlimb muscles, vastus lateralis (quadriceps) (VL(Q)) and Sartorius (Srt).</p

SBC model prediction compared to actual firing recorded in a VSCT neuron.

<p>The spike response of the SBC model to an input intracellular signal recorded during one step cycle (adapted from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107793#pone.0107793-Fedirchuk1" target="_blank">[9]</a>). The gray area indicates the 95% confidence bounds of the model response and the solid line the average measured instantaneous firing frequency from 12 cycles <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107793#pone.0107793-Fedirchuk1" target="_blank">[9]</a>. Note that while the model does not reproduce the measured response beyond 0.2 s, it captures the initial transient from 0 to approximately 300 Hz, and also the subsequent slow decay of the firing frequency. Since the input to the model and the firing frequency are naturally not from the same step cycle, perfect overlap cannot be expected.</p

Effect on granule cells spike responses of grading the synaptic weights at the level of the SBCs of the Ib inhibitory interneurons from different muscles.

<p>(A) Histogram with the instantaneous firing rate (5 ms bin width) of the model response, using the model that reproduced the response of granule cell 1 (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107793#pone-0107793-g003" target="_blank">Fig. 3</a>). In this case, all SBCs were simulated to receive Ib inhibition from the Srt muscle only. The weight of the inhibitory synapses between the Ib inhibitory interneurons were varied in five steps, 0.2–1.0 in arbitrary values. The effect of the different weights on the pattern and level of granule cell firing is shown in the histogram as bars in different shades of grey. (B) Similar display as in A, but in this case the synaptic weights of the Ib interneurons activated by the VL muscle were varied between 0.2–1.0, and the Srt weights were set to 0. Panels (C) and (D) are the same as (A) and (B), but using the model that reproduced the response of granule cell 2.</p