The Cerebellum: Brain for an Implicit Self (19 page)

On the other hand, recent physiological analysis of
in vivo
single granule cell discharge has not always supported the preceding models. Chadderton et al. (
2004
) reported that sensory stimulation produced bursts of mossy fiber EPSCs that summated to trigger bursts of granule cell spikes. This discharge pattern was evoked by only a few quantal EPSCs. However, spontaneous mossy fiber inputs did not trigger spikes unless Golgi cell inhibition of granule cells (
Chapter 5
, “
Inhibitory Interneurons and Glial Cells in the Cerebellar Cortex
”) was reduced. These observations suggest that mossy fiber-granule cell synapses are characterized by a balance between the exquisite sensitivity of granule cells to mossy fiber input and a high signal-to-noise ratio. In the anesthetized mouse
in vivo
preparation, individual flocculus granule cell discharge was shown to represent faithfully the direction and velocity of whole-body motion via bidirectional modulation of EPSC frequency (
Arenz et al., 2008
). On the basis of differences in EPSC waveform being the signature of individual mossy fiber-granule cell synapses, Arenz et al. (
2008
) suggested that multiple vestibular, visual, and/or eye-movement-related signals converged onto single granule cells. They also emphasized that mossy fiber-granule synapses did not simply relay incoming mossy-fiber spike rates but rather integrated vestibular information with other stimulus features. In a whisker-related cerebellar area, however, Rancz et al. (
2007
) found in rat
in vivo
preparations that sensory stimulation produced spike bursting at very high instantaneous firing frequencies (>700 Hz) in single mossy fiber terminals. This, in turn, drove rapid bursts of EPSC discharge in granule cells. Even a single mossy fiber firing at
in vivo
rates could produce the burst discharge of granule cells in
in vitro
preparations. These various observations suggest that a single presynaptic mossy fiber afferent transmits a sensory message to a granule cell in a “detonator” manner.

Jörntell and Ekerot (
2006
) and Bengtsson and Jörntell (
2009b
) reported various features of mossy fiber-granule cell synapses in the C
₃
zone of decerebrate cats. In this zone, each granule cell receives three to four mossy fiber afferents arising from the same small cutaneous receptive field of a forelimb, with each afferent generating large EPSCs on stimulation. These authors suggested that in this way granule cells enhanced signal-to-noise ratio and, as such, they operated as a noise-reducing device. Bengtsson and Jörntell (
2009b
) also demonstrated that the tested granule cells maintained the firing pattern of mossy fibers. In this “similar” coding,
as they called it, granule cells processed mossy fiber signals, while not distorting the latter’s temporal pattern of activity.

The preceding four cases show that the pattern of mossy fiber-to-granule cell convergence varies between different areas of the cerebellum, as if to suggest that each subserves a unique functional requirement.

Purkinje cells receive mossy fiber signals via granule cells and a unique set of parallel-fiber axons, and their axons serve as the final common output path of the cerebellar cortex (
Chapter 4
, “
Input and Output Pathways in the Cerebellar Cortex
”). Whereas each parallel fiber supplies excitatory synapses to ~300 Purkinje cells, each Purkinje cell receives excitatory synapses from ~150,000 parallel fibers. However, only 3%–15% of the latter synapses are functional, the remainder being silent. Parallel fiber-Purkinje cell synapses are characterized by bidirectional synaptic plasticity, involving conjunctive LTD and nonconjunctive LTP (
Chapters 7
and
8
), these presumably being the major mechanisms of information storage in Purkinje cells. In recent investigations of network mechanisms in Purkinje cells, advanced simulation technology is utilized to reproduce Purkinje cell behavior from neuronal circuit models.

To calculate the information storage capacity of Purkinje cells, Brunel et al. (
2004
) used the classical perceptron model, a prototypical single-layer feedforward network with excitatory weights (
Rosenblatt, 1962
;
Minsky and Papert, 1988
) (Recall that the initial simple perceptron model was introduced in
Chapters 1
and
3
). Brunel et al. (
2004
) assumed that the operation of parallel fiber-Purkinje cell synapses was binary (zero or one action potential) despite the fact that parallel-fiber-evoked EPSPs have a slow temporal profile, which is quite unlike binary pulses. They assumed that these EPSPs were trimmed by the IPSPs evoked in Purkinje cells via basket/stellate cells with a brief delay after the EPSPs (
Chapter 5
). They confirmed that these IPSPs were recruited concomitantly with EPSPs over a relatively wide range of stimulus intensities such that even the weakest of parallel fiber inputs could generate brief depolarizing pulses. Brunel et al. (
2004
) reasoned that because of the biphasic EPSP-IPSP responses to granule cell stimulation, the depolarizing pulses effectively summed in Purkinje cells only when they occurred coincidently. In this way, in the presence of basket/stellate cell inhibition, only coincident granule cell inputs could sum to excite effectively a Purkinje cell. They found that coincident inputs summed reasonably linearly, which matched the linearity approximation in the perceptron model.

Brunel et al. (
2004
) assigned to the perceptron model a task that consisted of learning the largest possible number of random input/output associations given a particular reliability level. They calculated the distribution of synaptic weights associated with the maximal storage capacity in their model and found that it contained >50% of the silent synapses. This fraction increased with storage reliability. Silent synapses (
Chapter 8
, “
Multiplicity and Persistency of Synaptic Plasticity
”) therefore appeared to be a necessary byproduct of optimizing learning and reliability. The distribution of synaptic weight theoretically derived by Brunel et al. (
2004
) resembled that obtained experimentally for granule cells-Purkinje cell synapses (
Isope and Barbour, 2002
). It was estimated that a Purkinje cell could learn up to 5 KB of information in the form of 40,000 input-output associations.

To estimate the pattern recognition capacity of Purkinje cells, a multicompartmental model was developed by De Schutter and Bower (
1994a
,
b
). This model of a Purkinje cell received parallel fiber synapses on 150,000 dendritic spines (
Harvey and Napper, 1988
, 1991). All the spines were activated independently by a random sequence of parallel fiber inputs, firing at an average rate of 0.3 Hz. This background excitation was balanced by tonic background inhibition, and the model Purkinje cell fired simple spikes at an average frequency of 48 Hz, just as in real Purkinje cells.

For a similar Purkinje cell model, Walter and Khodakhah (
2006
,
2009
) added experimentally determined response variability and pattern size based on a linear algorithm. Each pattern was generated by randomly selecting 650 different inputs from the entire pool of 150,000 inputs. This model’s response was a linear function of the strength of its input, with the 650 inputs increasing the model’s firing rate by 200 spikes/s. Learning occurred in this Purkinje cell model by a process that mimicked LTD of the parallel fiber-Purkinje cell synapses. The result was a 50% decrease in the strength of all the inputs that comprised the learned pattern. The model was also used for evaluating the capacity of a Purkinje cell to participate in pattern recognition. This was accomplished by altering the number of patterns the model had to learn and quantifying its ability to distinguish between learned and novel patterns. To quantify the latter, the authors calculated the resulting signal-to-noise (s/n) ratio of the maximum firing rate of the model in response to learned versus novel patterns. From the results of this calculation, Walter and Khodakhah (
2009
) concluded that their linear algorithm provided efficient pattern recognition.

To create yet another unique Purkinje cell model, Steuber et al. (
2007
) postulated that optimal pattern recognition capacity was obtained if Purkinje cells encoded information using pauses in their discharge rather than acceleration. Such a pause was observed in computer-simulated or experimentally recorded Purkinje cell discharges after applying a strong shock to granule cells. The applicability of this model to real cerebellar tissues under natural conditions has been questioned, however, because a relatively milder and temporally dispersed (i.e., more natural) granule cell activation failed to cause such a pause (
Walter and Khodakhah, 2009
). To compare with the pause coding, linear encoding of information would enable cerebellar nuclear neurons to use a simple averaging mechanism. In contrast to encoding patterns with pauses, Purkinje cells using a linear algorithm could recognize a large number of both synchronous and asynchronous input patterns in the presence or absence of inhibitory synaptic transmission. Walter and Khodakhah (
2009
) emphasized that under all conditions, the number of patterns recognized by Purkinje cells using a linear algorithm would be greater than that achieved by encoding information in pauses.

Finally, another Purkinje cell model has been proposed for fish. In certain areas of the cerebellum-like structures of electric fish, climbing fibers are lacking, and parallel fiber stimulation alone induces homosynaptic LTD in Purkinje-like cells. When a fish moves voluntarily, it causes water perturbations that stimulate lateral line organs to generate sensory signals, which eventually cause the fish to move. Homosynaptic LTD has been shown to play a role in forming negative images of such sensory signals. Adding the negative images to actual sensory signals minimized the neural response to predictable sensory features (
Bell et al., 1997
).

Golgi cells have characteristically large divergence and convergence numbers. Each Golgi cell receives ~4,788 excitatory inputs to its dendrites in the molecular layer from parallel fibers and also ~228 mossy fiber terminals onto its descending dendrites (
Pelionisz and Szentágothai, 1973
). Each Golgi cell, in turn, extends a broadly branching axon to up to ~5,700 granule cells (cat;
Palkovits et al., 1971
). These mutual connections form a loop involving granule cells and Golgi cells. Marr (
1969
) emphasized the action of Golgi cells in regulating codon size. More recent researchers, however, have considered the contribution of Golgi cells to be more
important for the generation of signals with temporal profiles, which are essential for determining the timing of movements. Unlike earlier models based on conduction delays in parallel fibers, arrays of elements with different time constants, or populations of elements oscillating at different frequencies, recent models assume that temporal coding emerges from the dynamics of those cerebellar circuits that include Golgi cells.

Fujita (
1982a
) proposed an adaptive filter model of the cerebellum. It incorporated a phase-converter that consisted of the mossy fiber-granule cell-Golgi cell circuit (
Figure 16
). This phase converter generated a set of multiphase versions of the mossy fiber input. This set was, in turn, conveyed by a set of parallel fibers and converged through modifiable synaptic connections to Purkinje cells, which provide the output signals of the cerebellar cortex. The learning principle used by these modifiable connections was that in each Purkinje cell the weight of the parallel fiber synapses was reduced when they were activated conjunctively with signals from climbing fibers. This learning principle insured that a Purkinje cell output converged to the “desired response” in order to minimize the mean square error of the performance.

Since then, attempts have been made to substantiate the possible clock function of the granule cell-Golgi cell-granule cell pathway. In Buonomano and Mauk’s (
1994
) model for eye-blink conditioning (
Chapter 11
, “
Somatic and Autonomic Reflexes
”), the population vector of granule cell activity encodes both the particular mossy fiber input pattern and the time since its onset. For example, assume in this model that a particular periodic mossy fiber input pattern activates a subset of granule cells. This would then activate a subset of Golgi cells, which, in turn, would inhibit another, partially overlapping subset of granule cells. When this granule cell-Golgi cell-granule cell negative feedback loop retains realistic divergence/convergence ratios between granule cells and Golgi cells, it would create a dynamic, nonperiodic population vector of granule cell activity. Thus, the population vector of granule cell activity would encode not just the constellation of stimuli impinging on the organism, but also the time since the onset of the stimuli, and thereby serve as a clock.

Yamazaki and Tanaka (
2005
) proposed another unique clock model of the granule cell-Golgi cell loop (
Figure 21
). It featured a set of random connections that generated ensemble patterns of the discharge of activated granule cells. As shown in
Figure 22
, this assembly of discharge patterns continued to change unless it was reset by a large mossy fiber input to the pathway. The time-dependent changes of the discharge pattern ensemble represented the passage of time as in a clock. When this Golgi cell clock was incorporated into a neuronal circuit model for the eye-blink reflex, it reproduced appropriately timed-conditioned responses of the reflex.

Figure 21. The granule cell-Golgi cell loop as an internal clock.

Schematic of the granular layer network wherein granule cells (small white circles) receive mossy fiber inputs and excite a set of Golgi cells (large gray circles), which are aligned in parallel with the granule cells’ axons; that is, the parallel fibers. Note how a single Golgi cell inhibits, in turn, a set of granule cells within its axonal arborization (dashed rectangle). If random connections between granule cells and Golgi cells occur, this recurrent inhibitory network then acts as an internal clock (an idea of
Yamazaki and Tanaka, 2005
). (Courtesy of Tadashi Yamazaki.)