Neural Evidence of the Cerebellum as a State Predictor

Abstract

We here provide neural evidence that the cerebellar circuit can predict future inputs from present outputs, a hallmark of an internal forward model. Recent computational studies hypothesize that the cerebellum performs state prediction known as a forward model. To test the forward-model hypothesis, we analyzed activities of 94 mossy fibers (inputs to the cerebellar cortex), 83 Purkinje cells (output from the cerebellar cortex to dentate nucleus), and 73 dentate nucleus cells (cerebellar output) in the cerebro-cerebellum, all recorded from a monkey performing step-tracking movements of the right wrist. We found that the firing rates of one population could be reconstructed as a weighted linear sum of those of preceding populations. We then went on to investigate if the current outputs of the cerebellum (dentate cells) could predict the future inputs of the cerebellum (mossy fibers). The firing rates of mossy fibers at time t + t₁ could be well reconstructed from as a weighted sum of firing rates of dentate cells at time t, thereby proving that the dentate activities contained predictive information about the future inputs. The average goodness-of-fit (R²) decreased moderately from 0.89 to 0.86 when t₁ was increased from 20 to 100 ms, hence indicating that the prediction is able to compensate the latency of sensory feedback. The linear equations derived from the firing rates resembled those of a predictor known as Kalman filter composed of prediction and filtering steps. In summary, our analysis of cerebellar activities supports the forward-model hypothesis of the cerebellum.

Keywords: Dentate cell; Internal forward model; Kalman filter; Mossy fiber; Motor control; Purkinje cell.

Fig. 1

a Distribution of the spatiotemporal separability index (STSI) computed for mossy fibers (MFs) (red), Purkinje cells (PCs) (blue), and dentate cells (DCs) (green). Three colored vertical lines depict the median values of the three cell populations. b Probability densities of firing rates of MFs (left), PCs (middle), and DCs (right). The densities are binned into 50 bins. Dashed lines overlapped with the histograms represent best-fit Gamma distributions

Fig. 2

Two representative examples of linear reconstructions of PCs. a Original and reconstructed firing rates of a representative PC (#52). Eight upper panels compare time series of the original (black) and the reconstructed (red) firing rates for each movement direction. R² is 0.961 for this PC. Two lower panels provide the same firing rates of the original (left) and the reconstructed (right) firing rates in contour plots. b Original and reconstructed firing rates of another representative PC (#56), presented in the same format of a. R² is 0.964 for this PC

Fig. 3

Two representative examples of linear reconstructions of DCs: a DC cell #48 (R² value 0.95) and b DC cell #55 (R² value 0.96), in the same format of Fig. 2

Fig. 4

Comparison of Akaike information criterion (AIC) between the linear, threshold, quadratic, and FIR models for PC cell fitting. A typical example of model fitting to the firing rates of a PC (#52) shown in a time series and b contour plots. In a, the firing rates of the original, linear, threshold, quadratic, and FIR models are shown with black, red, blue, green, and cyan lines, respectively. c Box plots of the goodness-of-fit for the four models at the pronated (left) and supinated (right) postures. On each box, the central mark is the median, and the edges of the box are the 25th and 75th percentiles, respectively

Fig. 5

Comparison of AIC between the linear, threshold, and quadratic models for DC cell fitting. A typical example of model fitting to the firing rates of a DC (#33) shown in time series (a) and contour plots (b). In a, the firing rates of the original, linear, threshold, and quadratic models are shown with black, red, blue, and green lines, respectively. c Box plots of the goodness-of-fit for the three models at the pronated (left) and supinated (right) postures

Fig. 6

Generalization from fitting at one posture to another posture of multiple models. a Box plots of goodness-of-fit (left) trained in supinated posture and tested in pronated posture and (right) trained in pronated posture and tested in supinated posture for PCs. b Box plots of goodness-of-fit (left) trained in supinated posture and tested in pronated posture and (right) trained in pronated posture and tested in supinated posture for DCs

Fig. 7

Two representative examples of linear predictions of MFs (#16 in a (R² value 0.91) and #37 in b (R² value 0.93)) at time t + t₁ from DCs at time t. Here t₁ was set to 40 ms. In each panel, the original activities and the predicted activities were compared in terms of time series and contour plots

Fig. 8

Two representative examples of linear predictions of MFs at time t + t₁ from DCs at time t. These are the same MFs (#16 in a (R² value 0.85) and #37 in b (R² value 0.89)) presented in Fig. 7. Here t₁ was set to 80 ms

Fig. 9

Goodness-of-fit of linear predictions with an increasing time advance t₁ ranging from 0 to 200 ms with an interval of 20 ms. Error bars indicate standard deviations at each time advance. The goodness-of-fit was computed separately for the two postures: pronated (black solid line) and supinated (red dashed line)

Fig. 10

Distributions of weights of linear reconstruction models: a MF → PC connectivity, b PC → DC, and c MF → DC connectivity. Note that the PC → DC weights are nonpositive, and the MF → DC weights are nonnegative. The signs of PC → DC weights were flipped for a visual presentation. d DC → MF weights of linear prediction model trained with the time-advance parameter t₁ = 40 ms. These distributions were normalized as probability density functions and were plotted in linear (left) and logarithmic (right) scales. Dashed lines indicate exponential distributions best fitted to the experimental distributions

Fig. 11

Summary schematic of our findings overlaid on the cerebellar circuit. MF, mossy fiber (red); PC, Purkinje cell (green); DC, dentate cell (light blue). Granule cells (orange) and inhibitory interneurons (blue) that are not analyzed in this work are included to show the basic structure of the cerebellar neuron circuitry. Three stages of linear computation obtained in our analysis are accompanied with the three types of computation of Kalman filter explained in the main text