81  Cerebellar learning: eyeblink conditioning and VOR adaptation

A guide to the two interactive simulations

Published

October 29, 2026

Launch: eyeblink conditioning ↗ Launch: VOR adaptation ↗

NoteBefore you start

Each simulation opens in a new tab, so you can switch between it and this guide. Each is a single file (cerebellum.html for eyeblink conditioning, cerebellum-vor.html for VOR adaptation) with no external dependencies and no internet connection required. Keep both in the same folder as this guide. They work best on a laptop or a projector at 1280 pixels wide or more. On a phone, the circuit and the trial traces scroll sideways. The two simulations share the same layout, colours and controls, so once students know one, they can read the other.

Quickest demonstration (2 minutes): in the eyeblink simulation, choose preset 1 Acquisition, press Play 1 trial and watch, then press Fast-forward for about 8 seconds and Pause. Compare the trial view before and after.

The first half of this guide covers eyeblink conditioning, which shows how the cerebellum learns a timed, anticipatory response. Chapter 86 covers the second simulation, adaptation of the vestibulo-ocular reflex (VOR), which shows learning of a reflex gain, an error that disappears because the movement corrects it, and a memory that moves from the cerebellar cortex to the brainstem. Chapter 87 connects the two, and both to experiments on predicting the sensory consequences of one’s own movements.

83 The science

83.2 The circuit

flowchart TB
  PN["Pontine nuclei<br/>(tone = CS)"] -->|mossy fibres| GC["Granule cells"]
  PN -->|mossy-fibre collateral| DCN["Interpositus<br/>projection neurons"]
  GC -->|"parallel fibres (plastic)"| PC["Purkinje cell"]
  GC -->|"parallel fibres (plastic)"| MLI["Basket / stellate<br/>interneurons"]
  MLI --o PC
  PC --o DCN
  PC --o NO["Nucleo-olivary<br/>neurons"]
  DCN --> RN["Red nucleus"]
  RN --> FN["Facial nucleus<br/>→ eyelid"]
  TG["Trigeminal nucleus<br/>(airpuff = US)"] --> IO["Inferior olive"]
  TG -.->|reflex blink| FN
  NO --o IO
  IO ==>|"climbing fibre<br/>(teaching signal)"| PC
Figure 83.1: The circuit in the simulation. Solid arrows are excitatory (glutamate); lines ending in a circle are inhibitory (GABA). The thick arrow is the climbing fibre, the teaching signal. The dotted arrow is the brainstem reflex, which does not involve the cerebellum.
Cell group Transmitter Role In the simulation
Granule cells → parallel fibres Glutamate Recode mossy-fibre input into many sparse channels 11 groups, each active at a different delay after tone onset
Basket and stellate cells (molecular-layer interneurons) GABA Feedforward inhibition of Purkinje cells Driven by the same parallel fibres; their input synapses are plastic
Purkinje cell GABA The only output of the cerebellar cortex; fires simple spikes spontaneously at roughly 50–100 Hz Spontaneous rate 60 Hz
Interpositus projection neurons Glutamate Cerebellar output; fire tonically at tens of Hz Baseline 40 Hz with Purkinje cells at 60 Hz
Nucleo-olivary neurons GABA Inhibit the inferior olive Baseline 10 Hz; disinhibited by a Purkinje pause
Inferior olive → climbing fibre Glutamate Fires at about 1 Hz; each spike causes a complex spike in its Purkinje cells Spontaneous 1 Hz; airpuff gives about an 85% chance of a spike

A few anatomical facts are worth stating precisely, because simplified diagrams often get them wrong:

  • Each adult Purkinje cell is typically innervated by one climbing fibre, from the contralateral olive, which forms a very large number of synapses on the proximal dendrites. The rule is not absolute: many mouse Purkinje cells and most human ones have more than one primary dendrite, and these can receive more than one climbing fibre (Busch & Hansel, 2023).
  • The complex spike recorded at the Purkinje cell soma is a large spike followed by a few smaller spikelets. The spikelets are fast Na⁺ spikes generated near the axon. The Ca²⁺ spikes happen in the dendrites, and that dendritic Ca²⁺ signal matters for plasticity (Davie et al., 2008). Simple spikes then pause briefly.
  • Simple spikes are partly intrinsic. Purkinje cells fire spontaneously even without synaptic input, helped by a resurgent Na⁺ current (Raman & Bean, 1997).
  • The cerebellar nuclei are not silent at rest. Their projection neurons fire tonically at tens of Hz even with Purkinje input intact. Purkinje inhibition moves that rate up and down.

83.3 The Marr–Albus–Ito theory

Marr (1969) and Albus (1971) independently proposed that the climbing fibre acts as a teaching signal for the Purkinje cell. When a climbing-fibre spike arrives, the parallel-fibre synapses that were active shortly beforehand should change. The two proposals differed in direction: Marr proposed that those synapses are strengthened, and Albus that they are weakened. Ito and colleagues then showed experimentally that pairing parallel-fibre and climbing-fibre activity produces long-term depression (LTD) of the parallel-fibre → Purkinje cell synapse (Ito, Sakurai & Tongroach, 1982; Ito & Kano, 1982). Albus’s direction was right.

Parallel-fibre activity without climbing-fibre activity produces the opposite change, long-term potentiation (LTP) (Lev-Ram et al., 2002; Coesmans et al., 2004). The climbing fibre therefore decides the direction of plasticity.

83.4 Why weakening a synapse makes a movement

The circuit inverts the sign of the change twice:

weaker parallel-fibre → Purkinje synapses → less Purkinje firing during the tone → less inhibition of the nuclei → more nuclear firing → eyelid closes

There is a catch, and the simulation is built to show it. Weakening an excitatory synapse only removes the excitation that synapse supplied. On its own, it cannot push the Purkinje cell below the rate it would fire without that input. Yet trained Purkinje cells in the eyeblink region pause well below their spontaneous rate during the tone, with the deepest suppression just before the airpuff (Jirenhed et al., 2007). Something inhibitory must be involved.

In the model, that something is the molecular-layer interneurons. The same parallel fibres excite them, and they inhibit the Purkinje cell. Before learning, excitation and inhibition from the tone roughly cancel. As the parallel-fibre → Purkinje synapses weaken (and the parallel-fibre → interneuron synapses strengthen, the opposite plasticity reported by Jörntell & Ekerot, 2002), the tone’s net effect on the Purkinje cell becomes inhibitory, and the cell pauses. Preset 7 removes the interneurons to show the problem directly.

This is one answer, not the only one. Purkinje cells may also have a timing mechanism inside the cell itself (Johansson et al., 2014), and mice lacking interneuron inhibition of Purkinje cells still learn (Boele et al., 2018). See Section 83.8.

83.5 How learning is timed

The learned blink peaks near the moment the airpuff is due, and it moves if the interval is changed. Timing models in the Marr–Albus tradition explain this with a temporal basis: different granule cells (and hence different parallel fibres) are active at different delays after tone onset (Medina et al., 2000). LTD then depresses only the parallel fibres that were active shortly before each climbing-fibre spike, so the Purkinje pause is carved out at the right time.

The simulation uses 11 parallel-fibre groups that peak at 10–485 ms after tone onset; later groups have broader activity, so timing becomes less precise at longer intervals. A climbing-fibre spike tags each group by how much it was active in the preceding ~300 ms, weighted most strongly about 70 ms before the spike. Measured timing rules for LTD differ between cerebellar regions and appear matched to each region’s feedback delay (Suvrathan et al., 2016). The 70 ms value is a modelling choice, not a measurement for the eyeblink region.

How granule cells actually represent elapsed time is unresolved. The temporal basis is an assumption of the model.

83.6 The molecular mechanism of LTD

  1. Parallel-fibre glutamate activates AMPA receptors and the metabotropic receptor mGluR1 on Purkinje cell spines.
  2. mGluR1 → Gq → PLCβ produces IP₃ and DAG.
  3. IP₃ releases Ca²⁺ from internal stores. When this coincides with the large Ca²⁺ influx from a climbing-fibre spike, the Ca²⁺ signal in the spine is supralinear: a molecular coincidence detector (Wang et al., 2000).
  4. Ca²⁺ and DAG activate PKCα, which phosphorylates the AMPA receptor subunit GluA2. The receptors are then removed from the synapse by clathrin-mediated endocytosis (Chung et al., 2003).

The result is a smaller response to that parallel fibre: LTD.

83.7 Who teaches the teacher? The olivary feedback loop

If every airpuff evoked a climbing-fibre spike forever, LTD would continue without limit. Something must tell the olive that the airpuff has been predicted. That job falls to the nucleo-olivary neurons: small GABAergic neurons in the cerebellar nuclei, separate from the projection neurons, that inhibit the inferior olive. Their terminals end in the olivary glomeruli next to the gap junctions that couple olive neurons (de Zeeuw et al., 1990). There they both hyperpolarise the olive neurons and reduce their electrical coupling (Lefler et al., 2014). The inhibition is slow and long-lasting because GABA is released asynchronously (Best & Regehr, 2009).

After learning, the same Purkinje pause that drives the blink also disinhibits the nucleo-olivary neurons. Their inhibition reaches the olive in time to cancel much of the airpuff’s excitation. The olive therefore behaves like a comparator: its output is roughly airpuff excitation − predicted inhibition. Kim, Krupa & Thompson (1998) linked this to the Rescorla–Wagner learning rule. They found that olive responses to the airpuff are reduced once the learned blink is present, and that blocking GABA in the olive abolished the behavioural “blocking” effect that the rule predicts.

Two consequences follow, and the simulation makes both visible:

  • Learning settles where the climbing-fibre response is back near its spontaneous level, not at zero. Climbing-fibre activity above the spontaneous rate produces net LTD; below it, net LTP. The system stops changing when the two balance (Kenyon et al., 1998). In the simulation, the airpuff-evoked response falls from about 85% to about 20% per trial, compared with a spontaneous level of 8% in the same window.
  • Below-baseline climbing-fibre activity is itself a teaching signal. During extinction, the learned blink still drives nucleo-olivary inhibition, the olive fires less than its spontaneous rate, LTP wins and the blink fades. Medina, Nores & Mauk (2002) tested this directly. Blocking inhibition in the olive prevented extinction, and blocking the olive’s excitatory input caused extinction even while the airpuff was still delivered. Presets 3 and 4 reproduce these two experiments.

There is a second way an error signal can disappear: the movement itself can remove the error. In VOR adaptation, climbing fibres signal retinal image slip, and learning stops when the eye movement abolishes the slip. Preset 8 is a thought experiment that shows this route in isolation.

83.8 Where the textbook model is challenged

The strongest evidence for the model:

  • In eyeblink conditioning, Purkinje cells in the relevant zone acquire a pause that is adaptively timed to precede the airpuff, extinguishes, and is reacquired faster (“savings”) (Jirenhed et al., 2007).
  • Olive responses to the airpuff shrink as the learned blink develops, and the nuclear inhibition responsible can be blocked to change learning as predicted (Kim et al., 1998; Medina et al., 2002).
  • The molecular coincidence detector for LTD is well characterised (Wang et al., 2000; Chung et al., 2003).

The strongest evidence against parallel-fibre LTD being the mechanism:

  • LTD may not be necessary. Three mouse lines in which PF–Purkinje LTD cannot be expressed learned VOR adaptation, eyeblink conditioning and a locomotion task normally (Schonewille et al., 2011). Conditioning was severely impaired only when LTD and interneuron inhibition of Purkinje cells were disrupted together (Boele et al., 2018). The two mechanisms can compensate for each other.
  • Timing may be intrinsic to the Purkinje cell. Purkinje cells learned correctly timed responses even when the conditioned stimulus was direct, unpatterned stimulation of parallel fibres, and with interneurons blocked pharmacologically. This points to a timing mechanism inside the cell rather than one that depends on timed input and changes in synaptic weight (Johansson et al., 2014).
  • Plasticity is distributed. Learning-related changes occur at several sites: parallel-fibre LTP, interneuron synapses, intrinsic excitability, and synapses in the cerebellar nuclei and vestibular nuclei. Different cerebellar modules may rely more on suppression or more on potentiation of Purkinje activity (De Zeeuw & ten Brinke, 2015).
  • Disinhibition vs rebound. The nuclei could also be driven by post-inhibitory rebound, a burst after strong, synchronous Purkinje inhibition ends. Rebound can drive movements under strong stimulation (Witter et al., 2013), but its role in normal behaviour is questioned (Alviña et al., 2008). This simulation uses disinhibition only.

The simulation lets students test two of these points against the model. Preset 6 blocks PF–Purkinje plasticity, and the model still learns through the interneuron synapses, as the mutant mice did. Preset 7 removes interneuron inhibition, and the model fails to learn even though real mice do. That gap shows what the model leaves out.

86 The VOR adaptation simulation

Launch: VOR adaptation ↗

86.1 The reflex and how it adapts

When the head turns, the vestibulo-ocular reflex (VOR) turns the eyes the opposite way at the same speed, so the image of the world stays still on the retina. Its gain (eye velocity ÷ head velocity) is close to 1 for distant targets. The fastest route is a three-neuron arc: vestibular afferent → vestibular-nucleus relay neuron → eye motor neuron, with a latency of only a few milliseconds. That is too fast for visual feedback to correct the movement as it happens, so the reflex has to be calibrated in advance.

The calibration is learned. An animal wearing magnifying spectacles sees the image move faster than the head turns, so a gain-1 VOR leaves the image slipping, and over hours to days the gain rises. Minifying spectacles make it fall. The error signal is retinal image slip. Direction-selective neurons in the pretectum and accessory optic system detect it, and the dorsal cap of the inferior olive relays it to the flocculus as direction-specific climbing-fibre signals (Simpson & Alley, 1974). Floccular Purkinje cells inhibit a subset of vestibular-nucleus neurons, the flocculus target neurons, which form a modifiable side-path of the reflex.

flowchart TB
  VA["Semicircular canals<br/>→ vestibular afferents"] -->|mossy fibres| GC["Granule cells"]
  VA -->|"plastic (brainstem site)"| FTN["Flocculus target neurons<br/>(vestibular nucleus)"]
  VA --> REL["Relay neurons<br/>(direct pathway)"]
  GC -->|"parallel fibres (plastic)"| PC["Purkinje cell<br/>(flocculus)"]
  GC -->|"parallel fibres (plastic)"| MLI["Basket / stellate<br/>interneurons"]
  MLI --o PC
  PC --o FTN
  FTN --> MN["Eye motor neurons"]
  REL --> MN
  MN -.->|"eye rotation,<br/>seen through lenses"| RET["Retina:<br/>image slip"]
  RET --> AOS["Pretectum / AOS"]
  AOS --> IO["Inferior olive<br/>(dorsal cap)"]
  IO ==>|"climbing fibre<br/>(retinal slip)"| PC
Figure 86.1: The circuit in the VOR simulation. Solid arrows are excitatory; lines ending in a circle are inhibitory. The thick arrow is the climbing fibre carrying retinal slip. The dotted arrow is not a synapse: it is the eye movement, which together with the lenses determines the slip.

86.2 Where is the memory? Two hypotheses

  • Ito’s flocculus hypothesis (Ito, 1982). Climbing fibres carrying retinal slip depress the vestibular parallel-fibre synapses on floccular Purkinje cells. This is the same Marr–Albus–Ito rule as in eyeblink conditioning. Less Purkinje inhibition of the target neurons during the head turn raises the gain.
  • Miles and Lisberger’s brainstem hypothesis (Miles & Lisberger, 1981). They argued, largely from recordings of floccular Purkinje cells in adapted monkeys, that the change in the flocculus could not by itself explain the new gain. They proposed that the lasting memory is stored at the vestibular inputs to the target neurons, with the Purkinje cells acting as the teacher.

Later work suggests both are partly right. In cats trained to lower their VOR gain, blocking the flocculus with CNQX immediately after training returned the gain to its original value. After 3 days of training, the same block reduced the learned change only slightly (Kassardjian et al., 2005). A similar shift from the flocculus to the vestibular nuclei over several days has been reported for optokinetic adaptation in mice (Shutoh et al., 2006). The simulation implements both sites: a fast flocculus site and a slow brainstem site guided by Purkinje output, so the memory gradually moves.

86.3 What the VOR simulation adds

Eyeblink simulation VOR simulation
What is learned A timed response to a cue The gain of a reflex
Output Eyelid closes before the airpuff Eye counter-rotates with the head
Error signal Airpuff, via trigeminal nucleus → olive Retinal slip, via pretectum / AOS → dorsal cap of the olive
Why the error disappears Predicted: nucleo-olivary inhibition cancels it at the olive Removed: the corrected eye movement abolishes the slip
LTD timing rule in the model Peak at 70 ms (a modelling choice) Peak at 120 ms, the measured floccular rule (Suvrathan et al., 2016)
Learning sites Cerebellar cortex Cortex (fast) and vestibular nucleus (slow)

86.4 Evidence and caveats specific to the VOR model

  • Gain-up learning is the well-supported half. Pairing climbing-fibre stimulation with head rotation was enough to induce gain-up learning in mice, and a climbing-fibre spike on one trial predicted reduced Purkinje firing on the next (Kimpo et al., 2014).
  • Gain-down learning may not work this way. In the same study, climbing fibres were robustly activated during gain-down training, yet their activity did not predict trial-by-trial changes, and stimulating them did not produce gain-down learning. The authors concluded that the circuit gates whether climbing fibres can induce plasticity. In the simulation, gain-down learning comes from climbing-fibre firing below baseline, which lets LTP win. Treat that as the hypothesis this experiment tested and did not support.
  • Single turns instead of oscillations. Experiments usually rotate the head back and forth. With magnifying lenses, climbing fibres then fire more during one half of each cycle; with minifying lenses, during the other half. Here each trial is a single turn in one direction, so “eye too fast” appears as climbing-fibre firing below baseline rather than as extra firing during the opposite half-cycle.
  • No visual tracking. Real floccular Purkinje cells also carry eye-velocity and visual signals (“gaze-velocity” Purkinje cells; Lisberger & Fuchs, 1978), and visual tracking reduces slip during training. The model leaves these out.
  • Dark trials are pure tests. The model applies no plasticity in darkness. In real animals, rotation in darkness soon after training can disrupt a fresh memory (Kassardjian et al., 2005).

86.5 Using the VOR simulation

The layout, playback buttons, manipulations panel, Inspector and CSV export work as in the eyeblink simulation (Section 84.3 and the sections around it). The differences are:

Control or display What it does
Spectacles ×0.5 / ×1 / ×1.5 / ×2 Image magnification during training. ×2 requires a gain of 2; ×0.5 a gain of 0.5.
Vision: Light / Dark Light trials train: retinal slip is signalled to the olive. Dark trials are tests: the VOR runs without visual feedback and nothing is learned.
Manipulations PF → Purkinje plasticity; PF → interneuron plasticity; vestibular-nucleus (brainstem) plasticity; interneuron inhibition; flocculus blocked (Purkinje cells fire steadily at their resting rate and the flocculus cannot learn, as with CNQX).
Trial view: Head turn Each trial is one smooth 300 ms head turn peaking at 50°/s.
Trial view: Target neurons Flocculus target neurons in the vestibular nucleus (0–200 Hz).
Trial view: Inferior olive Climbing-fibre rate on a log axis. It follows retinal slip about 110 ms later.
Trial view: Eye vs head Thin grey: head velocity. Dashed: the eye velocity the lenses require. Solid: actual eye velocity. Shaded: retinal slip. The trial’s gain is shown at the right.
Learning curves Top: VOR gain, with the gain the lenses require as a dashed line. Middle: the chance of a climbing-fibre spike while slip from the turn reaches the olive; the spontaneous level in this 300 ms window is 26%. Bottom: the learned change in gain, split into the part stored in the flocculus (red) and in the vestibular nucleus (blue).
Weights chart Flocculus weights for 10 parallel-fibre groups, each representing a moment of the head turn. The brainstem weight v appears in the tiles and on the target-neuron box.

Fast-forward goes up to 300 trials per second, because VOR adaptation needs more trials than eyeblink conditioning in this model.

86.6 The VOR presets

Presets 3–6 start from an animal that has already been trained with lenses; the training trials appear to the left of a dashed line on the learning curves.

86.6.1 VOR preset 1: Gain up

Setup. Naive animal, ×2 magnifying lenses, light.

What to do. Press Play 1 trial: the eye turns at head speed, slip appears (shaded), about 110 ms later the olive fires, and a complex spike appears. Then fast-forward at 100 trials/s to about trial 1000, then at 300 trials/s to about trial 4000.

What to look for.

  • The gain rises from 1.0 to about 1.9 by trial 1000, and the slip shading shrinks.
  • The parallel-fibre groups covering the middle of the turn are depressed (green bars fall) and their interneuron weights rise. The Purkinje cell fires less during the turn (about 20 Hz at peak head velocity) and the target neurons fire more.
  • The climbing-fibre response falls from about 75% to near its 26% spontaneous level. Learning stops because the movement removed the error; there is no nucleo-olivary cancellation in this simulation.
  • In the bottom chart, the learned change is first stored almost entirely in the flocculus, then moves gradually to the vestibular nucleus. By trial 4000, about 90% is in the brainstem, and the floccular weights have drifted back towards their naive values.

Question for students. Why does the flocculus give back its change as the brainstem learns? (Hint: as the brainstem change grows, the gain briefly overshoots 2. What does the olive then signal?)

86.6.2 VOR preset 2: Gain down

Setup. Naive animal, ×0.5 minifying lenses, light.

What to look for. The eye now turns too fast, the slip is reversed, and the climbing-fibre response drops below its spontaneous level (about 8% against 26%). LTP wins: the Purkinje cell fires more during the turn, the target neurons fire less, and the gain falls to about 0.45–0.5 by trial 1000. The memory then moves to the brainstem as in preset 1.

Caveat. Kimpo et al. (2014) found that climbing fibres did not drive gain-down learning in mice. This preset shows what the Marr–Albus–Ito rule predicts, which is the hypothesis they tested.

86.6.3 VOR preset 3: Remove the lenses

Setup. 600 trials with ×2 lenses (gain about 1.8), then normal vision (×1).

What to look for. The gain is now too high, the slip reverses, and climbing fibres fall below their spontaneous rate. The gain returns to about 1.0, but slowly (about 2000 trials; use 300 trials/s), because climbing fibres can signal “too little” far more strongly than “too much”. In the bottom chart, the vestibular nucleus keeps part of the old increase for a long time, masked by an opposite change in the flocculus. Behaviour is back to normal while the brainstem still holds a trace of the old memory.

86.6.4 VOR preset 4: Block the flocculus early

Setup. 300 trials with ×0.5 lenses (gain about 0.84), then the flocculus is blocked and the VOR is tested in darkness. This mirrors Kassardjian et al. (2005), who used CNQX in cats.

What to do. Press Play 1 trial or fast-forward briefly. Then untick Flocculus blocked under Manipulations (as when the drug washes out) and play again.

What to look for. With the flocculus blocked, the gain jumps back to about 0.98: almost all of the fresh memory was in the flocculus. Unblocked, it returns to about 0.84.

86.6.5 VOR preset 5: Block the flocculus late

Setup. 4000 trials with ×0.5 lenses (gain about 0.5), then the same block and test.

What to look for. The gain stays at about 0.53, so most of the memory survives, because the vestibular nucleus now holds it. Kassardjian et al. (2005) found the same pattern after 3 days of training.

86.6.6 VOR preset 6: Flocculus-only model

Setup. As preset 5, but brainstem plasticity is switched off throughout, leaving only Ito’s flocculus site.

What to look for. After 4000 trials the gain is about 0.5, as in preset 5, but blocking the flocculus returns it to 1.0: the whole memory is lost.

The point. Presets 5 and 6 make different predictions, and the experiment agrees with preset 5. That is evidence for a second learning site outside the flocculus. It does not make the flocculus unimportant: in the model, as in animals with floccular damage, nothing is learned without it.

88 Suggested lecture sequences

88.2 Adding VOR adaptation (8 minutes)

  1. Circuit comparison (2 min). Open the VOR simulation next to the eyeblink one. The cell types and colours are the same; the error source (retinal slip) is different, and there is a second learning site in the brainstem.
  2. Gain up (3 min). VOR preset 1: play one trial, then fast-forward to about trial 1000. Point out that learning stops because the movement removes the error. Keep going to trial 4000 and watch the memory move to the brainstem.
  3. Where is the memory? (3 min). VOR presets 4, 5 and 6 in turn. Ask students to predict each result before playing it.

To finish, Chapter 87 gives the bridge to forward models and the head-movement experiments, which works best as one or two slides.

89 The eyeblink model in detail

Time advances in 1 ms steps over a 600 ms trial. The tone starts at 100 ms; the airpuff starts at 100 ms + interval and lasts 30 ms; the tone ends with the airpuff.

Parallel fibres. Group k (k = 1…11) has activity

\[g_k(t) = G\,\exp\!\left(-\frac{(t - t_{\text{tone}} - c_k)^2}{2\sigma_k^2}\right)\cdot \text{tone}(t), \qquad \sigma_k = 10 + 0.15\,c_k \ \text{ms},\]

with peak delays \(c_k\) = 10, 35, 65, 100, 140, 185, 235, 290, 350, 415 and 485 ms. \(G\) normalises the summed activity to about 1 while the tone is on. After the tone ends, activity decays with a 15 ms time constant.

Purkinje cell (Hz): \[r_{PC}(t) = \big[\,60 + 45\sum_k w_k g_k(t) - 45\sum_k u_k g_k(t)\,\big]_+\] where \(w_k\) is the PF → Purkinje weight and \(u_k\) the PF → interneuron weight (both 1 when naive). The interneuron term is dropped when interneuron inhibition is switched off. \(\tilde r_{PC}\) is \(r_{PC}\) filtered with a 6 ms time constant, standing in for the synapse onto the nuclei.

Nuclei (Hz): projection neurons \(r_{DCN}(t) = 40 + 8\,\text{tone}(t) - 1.0\,(\tilde r_{PC} - 60)\), where the second term is the mossy-fibre collateral drive. Nucleo-olivary neurons: \(r_{NO}(t) = [\,10 - 0.5\,(\tilde r_{PC} - 60)\,]_+\).

Inferior olive. The inhibition reaching the olive, \(I(t)\), is the excess nucleo-olivary rate \([r_{NO} - 10]_+\), delayed by 10 ms and filtered with a slow 30 ms time constant. The climbing-fibre rate is \[\lambda(t) = \big[\,1 + A(t) - 3\,I(t)\,\big]_+ \ \text{Hz},\] where \(A(t)\) = 63 Hz from 15 to 45 ms after airpuff onset (otherwise 0). This gives about an 85% chance of a spike in a naive animal. In preset 8, \(A\) is multiplied by (1 − eyelid closure at airpuff time). Climbing-fibre spikes are drawn at random from \(\lambda(t)\) with a 100 ms refractory period. Only increases in nucleo-olivary activity above baseline affect the olive, a simplification.

Eyelid. The learned blink follows \(0.022\,[\,r_{DCN}(t - 20) - 40 - 10\,]_+\) (capped at 1) with a 25 ms time constant. The 20 ms delay stands for the red nucleus and facial nucleus; the 10 Hz threshold means small changes in the nuclei do not move the eyelid. The reflex blink closes the eye fully from 8 to 70 ms after airpuff onset.

Plasticity. Each group’s eligibility \(E_k(t)\) is its recent activity, weighted by a kernel that peaks when parallel-fibre activity precedes the climbing-fibre spike by 70 ms: \[E_k(t) = \frac{1}{Z}\sum_{\Delta=0}^{299} g_k(t-\Delta)\,K(\Delta), \qquad K(\Delta) = \frac{\Delta}{70}\,e^{\,1-\Delta/70}.\] After each trial, \[\Delta w_k = \eta_w\Big[\bar\lambda\!\int\! E_k\,dt \;-\; \sum_{\text{CF spikes } j} E_k(t_j)\Big], \qquad \Delta u_k = -\,\eta_u\,\big[\text{same bracket}\big],\] with \(\eta_w = 0.025\), \(\eta_u = 0.014\), and \(\bar\lambda\) = 0.91 Hz (the 1 Hz spontaneous rate corrected for the refractory period). Weights are bounded: \(0.1 \le w_k \le 1.5\) and \(0.5 \le u_k \le 2.2\). Each climbing-fibre spike causes LTD of \(w\) (and potentiation of \(u\)). Parallel-fibre activity causes a steady LTP term, sized so that on average the two cancel when the olive fires at its spontaneous rate. The expected change is therefore

\[\mathbb{E}[\Delta w_k] = -\,\eta_w \int \big(\lambda(t) - \bar\lambda\big)\,E_k(t)\,dt,\]

net LTD where the olive fires above its spontaneous rate and net LTP where it fires below. This is the equilibrium idea of Kenyon et al. (1998).

Why these numbers. Rates are in the ranges described in the circuit table. Learning rates were chosen so that acquisition takes a few hundred trials, the same order as typical experiments that run over several daily sessions. The 1 Hz spontaneous olive rate and the slow nucleo-olivary inhibition follow the physiology cited above.

90 The VOR model in detail

Time advances in 1 ms steps over a 600 ms trial. Each trial contains one head turn from 100 to 400 ms: \[H(t) = 50\cdot\tfrac12\big(1-\cos(2\pi (t-100)/300)\big)\ \text{°/s}, \qquad \hat H = H/50.\]

Parallel fibres. Ten groups represent successive moments of the turn: \(g_k(t) = \hat H(t)\, b_k(t) / \sum_j b_j(t)\), where \(b_k\) is a Gaussian (σ = 24 ms) centred 0, 33, 67, …, 300 ms after turn onset. Their activity sums to \(\hat H(t)\), so changing a group’s weight changes the Purkinje cell’s response to head velocity at that moment of the turn.

Purkinje cell (Hz): \(r_{PC} = [\,60 + 45\sum_k w_k g_k - 45\sum_k u_k g_k\,]_+\), or a constant 60 Hz when the flocculus is blocked. \(\tilde r_{PC}\) is this rate filtered with a 6 ms time constant.

Flocculus target neurons (Hz): \(r_{FTN}(t) = 50 + 40\,v\,\hat H(t-5) - 1.5\,(\tilde r_{PC}(t) - 60)\), where \(v\) is the brainstem (vestibular → target neuron) weight, 1 when naive.

Eye velocity (°/s, time in ms): \[8\,\frac{dE}{dt} = -E + 0.5\,H(t-6) + 0.625\,\big(r_{FTN}(t-3) - 50\big).\] The first input is the direct pathway (fixed gain 0.5); the second is the target-neuron pathway (gain 0.5 when naive). VOR gain is \(\int E\,dt \big/ \int H\,dt\) over 100–460 ms.

Retinal slip and climbing fibres. In the light, slip is \(s(t) = M\,H(t) - E(t)\), where \(M\) is the lens magnification; in darkness there is no slip signal. The climbing-fibre rate is \(\lambda(t) = [\,1 + 0.15\,s(t - 110)\,]_+\) Hz, with spikes drawn at random and a 100 ms refractory period.

Flocculus plasticity. The same rule as the eyeblink model (Chapter 90), but the eligibility kernel peaks at 120 ms. \(\eta_w = 0.03\), \(\eta_u = 0.017\), \(0.1 \le w_k \le 1.6\) and \(0.4 \le u_k \le 2.2\).

Brainstem plasticity. After each light trial, \[\Delta v = 0.012 \int \hat H(t-5)\,\frac{60 - \tilde r_{PC}(t)}{60}\,dt \quad (t \text{ in seconds}),\qquad 0 \le v \le 3.5.\] The vestibular synapse strengthens when vestibular input coincides with less Purkinje inhibition than usual, and weakens when it coincides with more. If the flocculus is blocked, so that the Purkinje cell stays at its resting rate, it does not change.

No learning in the dark. Dark trials apply no plasticity at either site.

Where the change is stored. The bottom learning curve splits the change in gain into a brainstem part, \(0.5\,(v - 1)\), and a flocculus part, the contribution of Purkinje modulation to \(\int E\,dt / \int H\,dt\).

Why these numbers. The naive gain is 1.0, half from each pathway. The 120 ms kernel matches the measured floccular LTD timing rule. It also matches the model’s own loop: the ~110 ms slip delay plus the motor delays. So the synapses tagged are the ones that drove the eye movement that caused the slip. Learning rates were chosen so that adaptation takes about a thousand trials and the shift to the brainstem a few thousand. The absolute timescale is not meant to match the hours or days of real training.

91 Simplifications and limitations

The VOR model’s specific simplifications are listed in the VOR evidence-and-caveats section above. For the eyeblink model:

  • One Purkinje cell population, one nuclear population and one olive population stand in for a whole cerebellar microzone. Real olive neurons are electrically coupled and oscillate, which can synchronise climbing-fibre spikes; the model ignores this.
  • The temporal basis of parallel-fibre activity is assumed, not derived from granule-cell physiology.
  • Plasticity is limited to two sites. Mossy-fibre → nucleus plasticity, intrinsic plasticity in Purkinje cells, and any timing mechanism inside the Purkinje cell are left out.
  • The nuclei respond to Purkinje cells by disinhibition only; there is no rebound firing.
  • Each trial starts fresh: there are no carry-over effects between trials, no savings mechanism beyond what the weights retain, and no sensory adaptation.

92 References

Albus, J. S. (1971). A theory of cerebellar function. Mathematical Biosciences, 10, 25–61.

Alviña, K., Walter, J. T., Kohn, A., Ellis-Davies, G., & Khodakhah, K. (2008). Questioning the role of rebound firing in the cerebellum. Nature Neuroscience, 11, 1256–1258.

Best, A. R., & Regehr, W. G. (2009). Inhibitory regulation of electrically coupled neurons in the inferior olive is mediated by asynchronous release of GABA. Neuron, 62, 555–565.

Boele, H.-J., Peter, S., ten Brinke, M. M., Verdonschot, L., IJpelaar, A. C. H. G., Rizopoulos, D., Gao, Z., Koekkoek, S. K. E., & De Zeeuw, C. I. (2018). Impact of parallel fiber to Purkinje cell long-term depression is unmasked in absence of inhibitory input. Science Advances, 4, eaas9426.

Brooks, J. X., Carriot, J., & Cullen, K. E. (2015). Learning to expect the unexpected: rapid updating in primate cerebellum during voluntary self-motion. Nature Neuroscience, 18, 1310–1317.

Busch, S. E., & Hansel, C. (2023). Climbing fiber multi-innervation of mouse Purkinje dendrites with arborization common to human. Science, 381, 420–427.

Chung, H. J., Steinberg, J. P., Huganir, R. L., & Linden, D. J. (2003). Requirement of AMPA receptor GluR2 phosphorylation for cerebellar long-term depression. Science, 300, 1751–1755.

Coesmans, M., Weber, J. T., De Zeeuw, C. I., & Hansel, C. (2004). Bidirectional parallel fiber plasticity in the cerebellum under climbing fiber control. Neuron, 44, 691–700.

Cullen, K. E., & Roy, J. E. (2004). Signal processing in the vestibular system during active versus passive head movements. Journal of Neurophysiology, 91, 1919–1933.

Davie, J. T., Clark, B. A., & Häusser, M. (2008). The origin of the complex spike in cerebellar Purkinje cells. Journal of Neuroscience, 28, 7599–7609.

de Zeeuw, C. I., Ruigrok, T. J. H., Holstege, J. C., Jansen, H. G., & Voogd, J. (1990). Intracellular labeling of neurons in the medial accessory olive of the cat: II. Ultrastructure of dendritic spines and their GABAergic innervation. Journal of Comparative Neurology, 300, 478–494.

De Zeeuw, C. I., & ten Brinke, M. M. (2015). Motor learning and the cerebellum. Cold Spring Harbor Perspectives in Biology, 7, a021683.

Ito, M. (1982). Cerebellar control of the vestibulo-ocular reflex: around the flocculus hypothesis. Annual Review of Neuroscience, 5, 275–296.

Ito, M., & Kano, M. (1982). Long-lasting depression of parallel fiber–Purkinje cell transmission induced by conjunctive stimulation of parallel fibers and climbing fibers in the cerebellar cortex. Neuroscience Letters, 33, 253–258.

Ito, M., Sakurai, M., & Tongroach, P. (1982). Climbing fibre induced depression of both mossy fibre responsiveness and glutamate sensitivity of cerebellar Purkinje cells. Journal of Physiology, 324, 113–134.

Jirenhed, D.-A., Bengtsson, F., & Hesslow, G. (2007). Acquisition, extinction, and reacquisition of a cerebellar cortical memory trace. Journal of Neuroscience, 27, 2493–2502.

Johansson, F., Jirenhed, D.-A., Rasmussen, A., Zucca, R., & Hesslow, G. (2014). Memory trace and timing mechanism localized to cerebellar Purkinje cells. Proceedings of the National Academy of Sciences, 111, 14930–14934.

Jörntell, H., & Ekerot, C.-F. (2002). Reciprocal bidirectional plasticity of parallel fiber receptive fields in cerebellar Purkinje cells and their afferent interneurons. Neuron, 34, 797–806.

Kassardjian, C. D., Tan, Y.-F., Chung, J.-Y. J., Heskin, R., Peterson, M. J., & Broussard, D. M. (2005). The site of a motor memory shifts with consolidation. Journal of Neuroscience, 25, 7979–7985.

Kenyon, G. T., Medina, J. F., & Mauk, M. D. (1998). A mathematical model of the cerebellar-olivary system I: Self-regulating equilibrium of climbing fiber activity. Journal of Computational Neuroscience, 5, 17–33.

Kim, J. J., Krupa, D. J., & Thompson, R. F. (1998). Inhibitory cerebello-olivary projections and blocking effect in classical conditioning. Science, 279, 570–573.

Kimpo, R. R., Rinaldi, J. M., Kim, C. K., Payne, H. L., & Raymond, J. L. (2014). Gating of neural error signals during motor learning. eLife, 3, e02076.

Lefler, Y., Yarom, Y., & Uusisaari, M. Y. (2014). Cerebellar inhibitory input to the inferior olive decreases electrical coupling and blocks subthreshold oscillations. Neuron, 81, 1389–1400.

Lev-Ram, V., Wong, S. T., Storm, D. R., & Tsien, R. Y. (2002). A new form of cerebellar long-term potentiation is postsynaptic and depends on nitric oxide but not cAMP. Proceedings of the National Academy of Sciences, 99, 8389–8393.

Lisberger, S. G., & Fuchs, A. F. (1978). Role of primate flocculus during rapid behavioral modification of vestibuloocular reflex. I. Purkinje cell activity during visually guided horizontal smooth-pursuit eye movements and passive head rotation. Journal of Neurophysiology, 41, 733–763.

Marr, D. (1969). A theory of cerebellar cortex. Journal of Physiology, 202, 437–470.

McCormick, D. A., & Thompson, R. F. (1984). Cerebellum: essential involvement in the classically conditioned eyelid response. Science, 223, 296–299.

Medina, J. F., Garcia, K. S., Nores, W. L., Taylor, N. M., & Mauk, M. D. (2000). Timing mechanisms in the cerebellum: testing predictions of a large-scale computer simulation. Journal of Neuroscience, 20, 5516–5525.

Medina, J. F., & Mauk, M. D. (2000). Computer simulation of cerebellar information processing. Nature Neuroscience, 3, 1205–1211.

Medina, J. F., Nores, W. L., & Mauk, M. D. (2002). Inhibition of climbing fibres is a signal for the extinction of conditioned eyelid responses. Nature, 416, 330–333.

Miles, F. A., & Lisberger, S. G. (1981). Plasticity in the vestibulo-ocular reflex: a new hypothesis. Annual Review of Neuroscience, 4, 273–299.

Raman, I. M., & Bean, B. P. (1997). Resurgent sodium current and action potential formation in dissociated cerebellar Purkinje neurons. Journal of Neuroscience, 17, 4517–4526.

Roy, J. E., & Cullen, K. E. (2004). Dissociating self-generated from passively applied head motion: neural mechanisms in the vestibular nuclei. Journal of Neuroscience, 24, 2102–2111.

Schonewille, M., Gao, Z., Boele, H.-J., et al. (2011). Reevaluating the role of LTD in cerebellar motor learning. Neuron, 70, 43–50.

Shutoh, F., Ohki, M., Kitazawa, H., Itohara, S., & Nagao, S. (2006). Memory trace of motor learning shifts transsynaptically from cerebellar cortex to nuclei for consolidation. Neuroscience, 139, 767–777.

Simpson, J. I., & Alley, K. E. (1974). Visual climbing fiber input to rabbit vestibulo-cerebellum: a source of direction-specific information. Brain Research, 82, 302–308.

Suvrathan, A., Payne, H. L., & Raymond, J. L. (2016). Timing rules for synaptic plasticity matched to behavioral function. Neuron, 92, 959–967.

Thompson, R. F. (1986). The neurobiology of learning and memory. Science, 233, 941–947.

Wang, S. S.-H., Denk, W., & Häusser, M. (2000). Coincidence detection in single dendritic spines mediated by calcium release. Nature Neuroscience, 3, 1266–1273.

Witter, L., Canto, C. B., Hoogland, T. M., de Gruijl, J. R., & De Zeeuw, C. I. (2013). Strength and timing of motor responses mediated by rebound firing in the cerebellar nuclei after Purkinje cell activation. Frontiers in Neural Circuits, 7, 133.

Zobeiri, O. A., & Cullen, K. E. (2024). Cerebellar Purkinje cells combine sensory and motor information to predict the sensory consequences of active self-motion in macaques. Nature Communications, 15. https://doi.org/10.1038/s41467-024-48376-0