Skip to main content
Log in

A computational study of synaptic mechanisms of partial memory transfer in cerebellar vestibulo-ocular-reflex learning

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

There is a debate regarding whether motor memory is stored in the cerebellar cortex, or the cerebellar nuclei, or both. Memory may be acquired in the cortex and then be transferred to the cerebellar nuclei. Based on a dynamical system modeling with a minimal set of variables, we theoretically investigated possible mechanisms of memory transfer and consolidation in the context of vestibulo-ocular reflex learning. We tested different plasticity rules for synapses in the cerebellar nuclei and took robustness of behavior against parameter variation as the criterion of plausibility of a model variant. In the most plausible scenarios, mossy-fiber nucleus-neuron synapses or Purkinje-cell nucleus-neuron synapses are plastic on a slow time scale and store permanent memory, whose content is passed from the cerebellar cortex storing transient memory. In these scenarios, synaptic strengths are potentiated when the mossy-fiber afferents to the nuclei are active during a pause in Purkinje-cell activities. Furthermore, assuming that mossy fibers create a limited variety of signals compared to parallel fibers, our model shows partial memory transfer from the cortex to the nuclei.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Aizenman, C. D., & Linden, D. J. (1999). Regulation of the rebound depolarization and spontaneous firing patterns of deep nuclear neurons in slices of rat cerebellum. Journal of Neurophysiology, 82, 1697–1709.

    PubMed  CAS  Google Scholar 

  • Aizenman, C. D., & Linden, D. J. (2000). Rapid, synaptically driven increases in the intrinsic excitability of cerebellar deep nuclear neurons. Nature Neuroscience, 3, 109–111.

    Article  PubMed  CAS  Google Scholar 

  • Aizenman, C. D., Manis, P. B., & Linden, D. J. (1998). Polarity of long-term synaptic gain change is related to postsynaptic spike firing at a cerebellar inhibitory synapse. Neuron, 21, 827–835.

    Article  PubMed  CAS  Google Scholar 

  • Akemann, W., & Knöpfel, T. (2006). Interaction of Kv3 potassium channels and resurgent sodium current influences the rate of spontaneous firing of Purkinje neurons. Journal of Neuroscience, 26, 4602–4612.

    Article  PubMed  CAS  Google Scholar 

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

    Article  Google Scholar 

  • Armstrong, D. M., & Rawson, J. A. (1979). Activity patterns of cerebellar cortical neurons and climbing fibre afferents in the awake cat. Journal of Physiology, 289, 425–448.

    PubMed  CAS  Google Scholar 

  • Boyden, E. S., Katoh, A., & Raymond, J. L. (2004). Cerebellum-dependent learning: The role of multiple plasticity mechanisms. Annual Review of Neuroscience, 27, 581–609.

    Article  PubMed  CAS  Google Scholar 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Dayan, P., & Abbott, L. F. (2001). Theoretical neuroscience—computational and mathematical modeling of neural systems. MIT.

  • Davies, P., & Melvill Jones, G. (1976). An adaptive neural model compatible with plastic changes induced in the human vestibulo-ocular reflex by prolonged optical reversal of vision. Brain Research, 103, 546–550.

    Article  PubMed  CAS  Google Scholar 

  • De Schutter, E., & Bjaalie, J. G. (2001). Coding in the granular layer of the cerebellum. Progress in Brain Research, 130, 279–296.

    Article  PubMed  Google Scholar 

  • du Lac, S., Raymond, J. L., Sejnowski, T. J., & Lisberger, S. G. (1995). Learning and memory in the vestibulo-ocular reflex. Annual Review of Neuroscience, 18, 409–441.

    Article  PubMed  Google Scholar 

  • Fujita, M. (1982). Adaptive filter model of the cerebellum. Biological Cybernetics, 45, 195–206.

    Article  PubMed  CAS  Google Scholar 

  • Goldberg, J. M., & Fernandez, C. (1971). Physiology of peripheral neurons innervating semicircular canals of the squirrel monkey. I. Resting discharge and response to constant angular accelerations. Journal of Neurophysiology, 34, 635–660.

    PubMed  CAS  Google Scholar 

  • Hansel, C., Linden, D. J., & D’Angelo, E. (2001). Beyond parallel fiber LTD: the diversity of synaptic and nonsynaptic plasticity in the cerebellum. Nature Neuroscience, 4, 467–475.

    PubMed  CAS  Google Scholar 

  • Ito, M. (2001). Cerebellar long-term depression: characterization, signal transduction, and functional roles. Physiological Reviews, 81, 1143–1195.

    PubMed  CAS  Google Scholar 

  • Ito, M., Jastreboff, P. J., & Miyashita, Y. (1982a). Specific effects of unilateral lesions in the flocculus upon eye movements in albino rabbits. Experimental Brain Research, 45, 233–242.

    Article  CAS  Google Scholar 

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

    CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  • Kleim, J. A., Freeman Jr., J. H., Bruneau, R., Nolan, B. C., Cooper, N. R., Zook, A. et al. (2002). Synapse formation is associated with memory storage in the cerebellum. Proceedings of the National Academy of Sciences of the United States of America, 99, 13228–13231.

    Article  PubMed  CAS  Google Scholar 

  • Kramer, P. D., Shelhamer, M., & Zee, D. S. (1995). Short-term adaptation of the phase of the vestibulo-ocular reflex (VOR) in normal human subjects. Experimental Brain Research, 106, 318–326.

    Article  CAS  Google Scholar 

  • LeDoux, M. S., Hurst, D. C., & Lorden, J. F. (1998). Single-unit activity of cerebellar nuclear cells in the awake genetically dystonic rat. Neuroscience, 86, 533–545.

    Article  PubMed  CAS  Google Scholar 

  • Lisberger, S. G. (1988). The neural basis for learning of simple motor skills. Science, 242, 728–735.

    Article  PubMed  CAS  Google Scholar 

  • Lisberger, S. F., & Sejnowski, T. J. (1992). Motor learning in a recurrent network model based on the vestibulo-ocular reflex. Nature, 360, 159–161.

    Article  PubMed  CAS  Google Scholar 

  • Luebke, A. E., & Robinson, D. A. (1994). Gain changes of the cat's vestibulo-ocular reflex after flocculus deactivation. Experimental Brain Research, 98, 379–390.

    Article  CAS  Google Scholar 

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

    PubMed  CAS  Google Scholar 

  • Masuda, N., & Amari, S. (2006). Modeling memory transfer and savings in cerebellar motor learning. Advances in Neural Information Processing Systems, 18, 859–866 (Y. Weiss, B. Scholkopf, J. Platt Eds.).

  • Mauk, M. D. (1997). Roles of cerebellar cortex and nuclei in motor learning: Contradictions or clues? Neuron, 18, 343–346.

    Article  PubMed  CAS  Google Scholar 

  • Mauk, M. D., & Donegan, N. H. (1997). A model of Pavlovian eyelid conditioning based on the synaptic organization of the cerebellum. Learning & Memory, 3, 130–158.

    Article  Google Scholar 

  • McClelland, J. L., McNaughton, B. L., & O'Reilly, R. C. (1995). Why there are complementary learning systems in the hippocampus and neocortex: Insights from the successes and failures of connectionist models of learning and memory. Psychological Review, 102, 419–457.

    Article  PubMed  CAS  Google Scholar 

  • Medina, J. F., Garcia, K. S., & Mauk, M. D. (2001). A mechanism for savings in the cerebellum. Journal of Neuroscience, 21, 4081–4089.

    PubMed  CAS  Google Scholar 

  • 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.

    PubMed  CAS  Google Scholar 

  • Medina, J. F., & Mauk, M. D. (1999). Simulations of cerebellar motor learning: Computational analysis of plasticity at the mossy fiber to deep nucleus synapse. Journal of Neuroscience, 19, 7140–7151.

    PubMed  CAS  Google Scholar 

  • Medina, J. F., Repa, J. C., Mauk, M. D., & LeDoux, J. E. (2002). Parallels between cerebellum- and amygdala-dependent conditioning. Nature Review Neuroscience, 3, 122–131.

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  • Miyachi, S., Hikosaka, O., & Lu, X. (2002). Differential activation of monkey striatal neurons in the early and late stages of procedural learning. Experimental Brain Research, 146, 122–126.

    Article  Google Scholar 

  • Miyachi, S., Hikosaka, O., Miyashita, K., Kárádi, Z., & Rand, M. K. (1997). Differential roles of monkey striatum in learning of sequential hand movement. Experimental Brain Research, 115, 1–5.

    Article  CAS  Google Scholar 

  • Morishita, W., & Sastry, B. R. (1996). Postsynaptic mechanisms underlying long-term depression of GABAergic transmission in neurons of the deep cerebellar nuclei. Journal of Neurophysiology, 76, 59–68.

    PubMed  CAS  Google Scholar 

  • Muellbacher, W., Ziemann, U., Wissel, J., Dang, N., Kofler, M., Facchini, S. et al. (2002). Early consolidation in human primary motor cortex. Nature, 415, 640–644.

    Article  PubMed  CAS  Google Scholar 

  • Nagao, S., Kitamura, T., Nakamura, N., Hiramatsu, T., & Yamada, J. (1997). Differences of the primate flocculus and ventral paraflocculus in the mossy and climbing fiber input organization. Journal of Comparative Neurology, 382, 480–498.

    Article  PubMed  CAS  Google Scholar 

  • Nagao, S., & Kitazawa, H. (2003). Effects of reversible shutdown of the monkey flocculus on the retention of adaptation of the horizontal vestibulo-ocular reflex. Neuroscience, 118, 563–570.

    Article  PubMed  CAS  Google Scholar 

  • Osanai, R., Nagao, S., Kitamura, T., Kawabata, I., & Yamada, J. (1999). Differences in mossy and climbing afferent sources between flocculus and ventral and dorsal paraflocculus in the rat. Experimental Brain Research, 124, 248–264.

    Article  CAS  Google Scholar 

  • Ouardouz, M., & Sastry, B. R. (2000). Mechanisms underlying LTP of inhibitory synaptic transmission in the deep cerebellar nuclei. Journal of Neurophysiology, 84, 1414–1421.

    PubMed  CAS  Google Scholar 

  • Palkovits, M., Mezey, É., Hámori, J., & Szentágothai, J. (1977). Quantitative histological analysis of the cerebellar nuclei in the cat. I. numerical data on cells and on synapses. Experimental Brain Research, 28, 189–209.

    Article  CAS  Google Scholar 

  • Pasupathy, A., & Miller, E. K. (2005). Different time courses of learning-related activity in the prefrontal cortex and striatum. Nature, 433, 873–876.

    Article  PubMed  CAS  Google Scholar 

  • Penhune, V. P., & Doyon, J. (2002). Dynamic cortical and subcortical networks in learning and delayed recall of timed motor sequences. Journal of Neuroscience, 22, 1397–1406.

    PubMed  CAS  Google Scholar 

  • Penhune, V. B., & Doyon, J. (2005). Cerebellum and M1 interaction during early learning of timed motor sequences. Neuroimage, 26, 801–812.

    Article  PubMed  CAS  Google Scholar 

  • Perrett, S. P., Luis, B. P., & Mauk, M. D. (1993). Cerebellar cortex lesions disrupt learning-dependent timing of conditioned eyelid responses. Journal of Neuroscience, 13, 1708–1718.

    PubMed  CAS  Google Scholar 

  • Perrett, S. P., & Mauk, M. D. (1995). Extinction of conditioned eyelid responses requires the anterior lobe of cerebellar cortex. Journal of Neuroscience, 15, 2074–2080.

    PubMed  CAS  Google Scholar 

  • Peterson, B. W., Baker, J. F., & Houk, J. C. (1991). A model of adaptive control of vestibuloocular reflex based on properties of cross-axis adaptation. Annual New York Academy Science, 627, 319–337.

    Article  CAS  Google Scholar 

  • Pugh, J. R., & Raman, I. M. (2006). Potentiation of mossy fiber EPSCs in the cerebellar nuclei by NMDA receptor activation followed by postinhibitory rebound current. Neuron, 51, 113–123.

    Article  PubMed  CAS  Google Scholar 

  • Racine, R. J., Wilson, D. A., Gingell, R., & Sunderland, D. (1986). Long-term potentiation in the interpositus and vestibular nuclei in the rat. Experimental Brain Research, 63, 158–162.

    Article  CAS  Google Scholar 

  • Raymond, J. L., Lisberger, S. G., & Mauk, M. D. (1996). The cerebellum: A neuronal learning machine? Science, 272, 1126–1131.

    Article  PubMed  CAS  Google Scholar 

  • Repa, J. C., Muller, J., Apergis, J., Desrochers, T. M., Zhou, Y., & LeDoux, J. E. (2001). Two different lateral amygdala cell populations contribute to the initiation and storage of memory. Nature Neuroscience, 4, 724–731.

    Article  PubMed  CAS  Google Scholar 

  • Roland, N. C., & Jaeger, D. (2005). Coding of tactile response properties in the rat deep cerebellar nuclei. Journal of Neurophysiology, 94, 1236–1251.

    Article  Google Scholar 

  • Raymond, J. L., & Lisberger, S. G. (1998). Neural learning rules for the vestibulo-ocular reflex. Journal of Neuroscience, 18, 9112–9129.

    PubMed  CAS  Google Scholar 

  • Sakurai, M. (1987). Synaptic modification of parallel fibre-Purkinje cell transmission in in vitro guinea-pig cerebellar slices. Journal of Physiology, 394, 463–480.

    PubMed  CAS  Google Scholar 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Stickgold, R., Hobson, J. A., Fosse, R., & Fosse, M. (2001). Sleep, learning, and dreams: Off-line memory reprocessing. Science, 294, 1052–1057.

    Article  PubMed  CAS  Google Scholar 

  • Sugihara, I., Ebata, S., & Shinoda, Y. (2004). Functional compartmentalization in the flocculus and the ventral dentate and dorsal group y nuclei: an analysis of single olivocerebellar axonal morphology. Journal of Compensation and Benefits, 470, 113.

    Google Scholar 

  • Thach, W. T. (1968). Discharge of Purkinje and cerebellar nuclear neurons during rapidly alternating arm movements in the monkey. Journal of Neurophysiology, 31, 785–797.

    PubMed  CAS  Google Scholar 

  • Walker, M. P., Brakefield, T., Hobson, J. A., & Stickgold, R. (2003). Dissociable stages of human memory consolidation and reconsolidation. Nature, 425, 616–620.

    Article  PubMed  CAS  Google Scholar 

  • Yamazaki, T., & Tanaka, S. (2005). Neural modeling of an internal clock. Neural Computation, 17, 1032–1058.

    Article  PubMed  Google Scholar 

  • Zhang, W., & Linden, D. J. (2006). Long-term depression at the mossy fiber-deep cerebellar nucleus synapse. Journal of Neuroscience, 26, 6935–6944.

    Article  PubMed  CAS  Google Scholar 

Download references

Acknowledgments

We thank S. Nagao, T. Knöpfel, and C. Rockland for helpful discussions and carefully reading the manuscript. This work is supported by Special Postdoctoral Researchers Program of RIKEN.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Naoki Masuda.

Additional information

Action Editor: Erik De Schutter

Appendices

Appendix A: Fast–slow analysis with static MF and PF firing and MF–VN plasticity

When the vestibular signal is static, we set ω=0, θ= π/2, and hence set the vestibular signal sin(ωt + θ)=1. Given that the MF and PF firing rates are static with f(a)=1, we have <sin(ωt + θ)u> i =<sin(ωt + θ)x> i =<u> i =<x> i  = <ux t> i,j =<xu t> j,i =<xx t> i,j =1. Here, <u> i , for example, is time average of the i-th MF signal. Because the PC–VN synapse is assumed to be static under MF–VN plasticity, we set b = 1. Let us assume m = n = 1 (and quit bold notations) and perform fast–slow analysis. Similar analysis works for general m and n.

In an early stage of learning, the PF–PC synapse w evolves much faster than the MF–VN synapse v does. The fast nullcline defined by setting dw/dt = 0 in Eq. (8) is given by

$$ v = v_{0} + R - R_{0} + \frac{{\eta _{1} + \eta _{3} }} {{\eta _{1} }}{\left( {w - w_{0} } \right)}. $$
(27)

On a short timescale, w and v converge onto this line.

1.1 A.1 CF-driven MF–VN plasticity

For the CF-driven learning, the slow nullcline to which w and v converge in a long run is given by setting dv/dt = 0 in Eq. (11):

$$ v = v_{0} + \frac{{\eta _{4} }} {{\eta _{4} + \eta _{6} }}{\left( {R - R_{0} } \right)} + \frac{{\eta _{4} }} {{\eta _{4} + \eta _{6} }}{\left( {w - w_{0} } \right)}. $$
(28)

A trajectory of the synaptic weights in the w–v plane approaches somewhere on the fast nullcline (Eq. (27)) and then slides along it toward the equilibrium obtained as the crossing of the two nullclines. The crossing is given by

$$ w^{*} = w_{0} - \frac{{\eta _{1} \eta _{6} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{6} + \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}, $$
(29)
$$ v^{*} = v_{0} + \frac{{\eta _{3} \eta _{4} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{6} + \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}. $$
(30)

The error at the equilibrium is

$$\begin{aligned} & e^{*} = R - v^{*} + w^{*} + y_{0} - z_{0} \\ & = \frac{{\eta _{3} \eta _{6} {\left( {R - R_{0} } \right)}}}{{\eta _{1} \eta _{6} + \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}, \\ \end{aligned} $$
(31)

which is small given η 1 ≫ η 3.

1.2 A.2 Hebbian MF–VN plasticity

For the Hebbian learning, the w-nullcline is given by Eq. (27), and the v-nullcline derived from Eq. (13) becomes

$$ v = v_{0} + \frac{{\eta _{4} }} {{\eta _{4} - \eta _{6} }}{\left( {w - w_{0} } \right)}. $$
(32)

The equilibrium is given by

$$ w^{*} = w_{0} + \frac{{\eta _{1} {\left( {\eta _{4} - \eta _{6} } \right)}{\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{6} - \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}, $$
(33)
$$ v^{*} = v_{0} + \frac{{\eta _{1} \eta _{4} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{6} - \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}, $$
(34)
$$ e^{*} = \frac{{\eta _{3} {\left( { - \eta _{4} + \eta _{6} } \right)}{\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{6} - \eta _{3} \eta _{4} + \eta _{3} \eta _{6} }}. $$
(35)

Because the relative magnitudes of η 1 η 6 and η 3 η 4 are indecisive, the sign of η 1 η 6 − η 3 η 4 + η 3 η 6 is indefinite.

1.3 A.3 PC-driven MF–VN plasticity

For the PC-driven learning, the slow nullcline derived from Eq. (15) becomes

$$ v = v_{0} - \frac{{\eta _{4} }} {{\eta _{6} }}{\left( {w - w_{0} } \right)}, $$
(36)

and the equilibrium is given by

$$ w^{*} = w_{0} - \frac{{\eta _{1} \eta _{6} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{4} + \eta _{1} \eta _{6} + \eta _{3} \eta _{6} }}, $$
(37)
$$ v^{*} = v_{0} + \frac{{\eta _{1} \eta _{4} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{4} + \eta _{1} \eta _{6} + \eta _{3} \eta _{6} }}, $$
(38)
$$ e^{*} = \frac{{\eta _{3} \eta _{6} {\left( {R - R_{0} } \right)}}} {{\eta _{1} \eta _{4} + \eta _{1} \eta _{6} + \eta _{3} \eta _{6} }}. $$
(39)

Appendix B: Fast–slow analysis for PC-driven PC–VN plasticity

For PC-driven PC–VN plasticity, we can analytically obtain the equilibrium solutions. The initial synaptic weights are w = w 0, v = v 0, and b = b 0, which are chosen to yield the initial gain R 0. This condition together with Eqs. (16) and (18) provides the relations:

$$ R_{0} - v_{0} + b_{0} {\left( {w_{0} + y_{0} } \right)} - z_{0} = 0, $$
(40)
$$\kern6pt \eta _{4} {\left( {w_{0} + y_{0} - y_{{ref}} } \right)} - \eta _{6} b_{0} = 0. $$
(41)

Because we assumed η 1 ≫ η 3 and η 4 ≫ η 6, there are two equilibria (w*, b*) given by

$$ {\left( {w_{0} - \frac{{\eta _{6} {\left( {R - v_{0} } \right)}}} {{\eta _{4} {\left( {w_{0} + y_{0} } \right)}}},\frac{{v_{0} - R}} {{w_{0} + y_{0} }}} \right)} $$
(42)

and

$${\left( { - y_{0} + \frac{{\eta _{6} {\left( {R - v_{0} - z_{0} } \right)}}}{{\eta _{4} {\left( {w_{0} + y_{0} } \right)}}},b_{0} - \frac{{\eta _{4} {\left( {w_{0} + y_{0} } \right)}}}{{\eta _{6} }} + \frac{{R - v_{0} - z_{0} }}{{w_{0} + y_{0} }}} \right)}.$$
(43)

In both solutions, the final error is

$$ e^{*} = \frac{{\eta _{6} {\left( {R - v_{0} - z_{0} } \right)}{\left( {R - R_{0} } \right)}}} {{\eta _{4} {\left( {w_{0} + y_{0} } \right)}^{2} }}, $$
(44)

which is small. We discard the second solution because it is unstable.

Appendix C: Gain and phase learning with dynamic neural responses and MF–VN plasticity

With f(a) = sin(a), we obtain <x> = <u> = 0, <sin(ωt + θ)x> i  = cos(θ − φ i )/2, <xu t> i,j =<ux t> j,i =cos(ψ j  − φ i )/2, and <xx t> i,j =cos(φ j  − φ i )/2. The signals with closer phase leads are more correlated. Then, Eq. (8) reads

$$\frac{{{\text{d}}w_{i} }}{{{\text{d}}t}} = - \frac{{\eta _{1} }}{2}{\left[ {{\left( {R - R_{0} } \right)}\cos {\left( {\theta - \phi _{i} } \right)} - {\sum\limits_{j = 1}^m {\cos {\left( {\psi _{j} - \phi _{i} } \right)}{\left( {v_{j} - v_{{0j}} } \right)} + {\sum\limits_{j = 1}^n {\cos {\left( {\phi _{j} - \phi _{i} } \right)}{\left( {w_{j} - w_{{0j}} } \right)}} }} }} \right]} - \eta _{3} n{\left( {w_{i} - w_{{0i}} } \right)}.$$
(45)

3.1 C.1 CF-driven MF–VN plasticity

With the CF-driven learning rule, Eq. (11) reads

$$\frac{{{\text{d}}v_{i} }}{{{\text{d}}t}} = \frac{{\eta _{4} }}{2}{\left[ {{\left( {R - R_{0} } \right)}\cos {\left( {\theta - \psi _{i} } \right)} - {\sum\limits_{j = 1}^m {\cos {\left( {\psi _{j} - \psi _{i} } \right)}{\left( {v_{j} - v_{{0j}} } \right)} + {\sum\limits_{j = 1}^n {\cos {\left( {\phi _{j} - \psi _{i} } \right)}{\left( {w_{j} - w_{{0j}} } \right)}} }} }} \right]} - \eta _{6} m{\left( {v_{i} - v_{{0i}} } \right)}.$$
(46)

In terms of the order parameters defined by

$$\begin{aligned} & W_{c} = {\sum\limits_{i = 1}^n {\cos \phi _{i} {\left( {w_{i} - w_{{0i}} } \right)}} },{\text{ }}W_{s} = {\sum\limits_{i = 1}^n {\sin \phi _{i} {\left( {w_{i} - w_{{0i}} } \right)}} }, \\ & V_{c} = {\sum\limits_{i = 1}^m {\cos \psi _{i} {\left( {v_{i} - v_{{0i}} } \right)}} },{\text{ V}}_{s} = {\sum\limits_{i = 1}^m {\sin \psi _{i} {\left( {v_{i} - v_{{0i}} } \right)}} }, \\ \end{aligned} $$

the equilibrium for the desired output R sin (ωt + θ) is obtained by solving

$$\begin{aligned} & \begin{array}{*{20}c} {{W^{*}_{c} }} & { = } & {{ - \frac{{\eta _{1} }}{{2\eta _{3} n}}\left[ {{\left( {R - R_{0} } \right)}\cos \theta {\sum\limits_{i = 1}^n {\cos ^{2} \phi _{i} + {\left( {R - R_{0} } \right)}\sin } }\theta {\sum\limits_{i = 1}^n {\cos \phi _{i} \sin \phi _{i} } }} \right.}} & {{}} \\ {{}} & {{}} & {{ + \left. {(W^{*}_{c} - V^{*}_{c} ){\sum\limits_{i = 1}^n {\cos ^{2} \phi _{i} + } }(W^{*}_{s} - V^{*}_{s} ){\sum\limits_{i = 1}^n {\cos \phi _{i} \sin \phi _{i} } }} \right],}} & {{}} \\ {{W^{*}_{s} }} & { = } & {{ - \frac{{\eta _{1} }}{{2\eta _{3} n}}\left[ {{\left( {R - R_{0} } \right)}\cos \theta {\sum\limits_{i = 1}^n {\cos \phi _{i} \sin \phi _{i} + {\left( {R - R_{0} } \right)}\sin } }\theta {\sum\limits_{i = 1}^n {\sin ^{2} \phi _{i} } }} \right.}} & {{}} \\ {{}} & {{}} & {{\left. { + {\left( {W^{*}_{c} - V^{*}_{c} } \right)}{\sum\limits_{i = 1}^n {\cos \phi _{i} \sin \phi _{i} + {\left( {W^{*}_{s} - V^{*}_{s} } \right)}} }{\sum\limits_{i = 1}^n {\sin ^{2} \phi _{i} } }} \right],}} & {{}} \\ \end{array} \\ & \begin{array}{*{20}c} {{V^{*}_{c} }} & { = } & {{\frac{{\eta _{4} }}{{2\eta _{6} m}}\left[ {{\left( {R - R_{0} } \right)}\cos \theta {\sum\limits_{i = 1}^m {\cos ^{2} \psi _{i} + {\left( {R - R_{0} } \right)}\sin } }\theta {\sum\limits_{i = 1}^m {\cos \psi _{i} \sin \psi _{i} } }} \right.}} & {{}} \\ {{}} & {{}} & {{\left. { + {\left( {W^{*}_{c} - V^{*}_{c} } \right)}{\sum\limits_{i = 1}^m {\cos ^{2} \psi _{i} + {\left( {W^{*}_{s} - V^{*}_{s} } \right)}} }{\sum\limits_{i = 1}^m {\cos \psi _{i} \sin \psi _{i} } }} \right],}} & {{}} \\ {{V^{*}_{s} }} & { = } & {{\frac{{\eta _{4} }}{{2\eta _{6} m}}\left[ {{\left( {R - R_{0} } \right)}\cos \theta {\sum\limits_{i = 1}^m {\cos \psi _{i} \sin \psi _{i} + {\left( {R - R_{0} } \right)}\sin } }\theta {\sum\limits_{i = 1}^m {\sin ^{2} \psi _{i} } }} \right.}} & {{}} \\ {{}} & {{}} & {{\left. { + {\left( {W^{*}_{c} - V^{*}_{c} } \right)}{\sum\limits_{i = 1}^m {\cos \psi _{i} \sin \psi _{i} + } }{\left( {W^{*}_{s} - V^{*}_{s} } \right)}{\sum\limits_{i = 1}^m {\sin ^{2} \psi _{i} } }} \right].}} & {{}} \\ \end{array} \\ \end{aligned} $$

The phase leads φ i and ψ i are assumed to be distributed uniformly on [0,2π] and [−Δ ψ , Δ ψ ], respectively. Assuming that n and m are large, we have, for example,

$$ {\sum\limits_{i = 1}^n {\cos ^{2} } }\phi _{i} = \frac{n} {{2\pi }}{\int_0^{2\pi } {\frac{{1 + \cos 2\phi }} {{2\phi }}} }d\phi = \frac{n} {2}. $$
(47)

For notational convenience, we write A c (Δ ψ ) = 1 + sin 2Δ ψ /2Δ ψ and A s (Δ ψ ) = 1 − sin 2Δ ψ /2Δ ψ . We note that 1 ≤ A c  ≤ 2, and A s decreases in Δ ψ . Particularly, A s  ≅ 2Δ ψ 2/3 becomes small as Δ ψ  → 0. Then, the equilibrium is given by

$$ V^{*}_{\alpha } = \frac{{\eta _{3} \eta _{4} {\left( {R - R_{0} } \right)}\cos \theta A_{\alpha } {\left( {\Delta _{\psi } } \right)}}} {{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} + \eta _{3} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)}}}, $$
(48)
$$ W^{*}_{\alpha } = - \frac{{\eta _{1} \eta _{6} {\left( {R - R_{0} } \right)}\cos \theta }} {{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} + \eta _{3} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)}}}, $$
(49)

where α = c or s. The amount of the memory stored in the MF–VN synapses is given by

$${\sum\limits_{i = 1}^m {\sin {\left( {\omega t + \psi _{i} } \right)}{\left( {v^{*}_{i} - v^{*}_{{0i}} } \right)} = V^{*}_{c} \sin \omega t + V^{*}_{s} \cos \omega t = r_{D} {\left( {R - R_{0} } \right)}\sin {\left( {\omega t + \theta _{D} } \right)},} }$$
(50)

where

$$ r_{D} = {\sqrt {\frac{{\cos ^{2} \theta }} {{{\left[ {1 + \frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{3} \eta _{4} A_{c} {\left( {\Delta _{\psi } } \right)}}}} \right]}^{2} }} + \frac{{\sin ^{2} \theta }} {{{\left[ {1 + \frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{3} \eta _{4} A_{s} {\left( {\Delta _{\psi } } \right)}}}} \right]}^{2} }}} }, $$
(51)
$$ \theta _{D} = \tan ^{{ - 1}} {\left[ {\frac{{1 + \frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{3} \eta _{4} A_{c} {\left( {\Delta _{\psi } } \right)}}}}} {{1 + \frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{3} \eta _{4} A_{s} {\left( {\Delta _{\psi } } \right)}}}}}\tan \theta } \right]}. $$
(52)

If MFs create any delay elements (Δ ψ  = π, A c (Δ ψ ) = A s (Δ ψ ) = 1), Eq. (52) results in θ D  = θ, that is, the perfect phase learning by the MF–VN synapses. However, r D  = η 3 η 4/(η 1 η 6 + η 3 η 4 + 4η 3 η 6) can be considerably smaller than the ideal value (= 1) because η 1 η 6 may be as large as η 3 η 4 in general. The MF–VN synapses learn the desired gain only for a restricted parameter range, and gain transfer is not robust against parameter variation. This is in line with the result of the gain-only theory.

Furthermore, the discrepancy between θ D and θ cannot be ignored when Δ ψ  = O((η 1 η 6/η 3 η 4)1/2) i.e. when Δ ψ is of the order of (η 1 η 6/η 3 η 4)1/2. Because η 1 ≫ η 3 and η 4 ≫ η 6, one cannot tell without additional information whether η 1 η 6 ≫ η 3 η 4, η 1 η 6 ≅ η 3 η 4, or η 1 η 6 ≪ η 3 η 4. Because η 1 η 6/η 3 η 4 is not necessarily small, transfer can degrade even for a large Δ ψ .

3.2 C.2 Hebbian MF–VN plasticity

With the Hebbian rule, Eq. (13) reads

$$\frac{{{\text{d}}v_{i} }}{{{\text{d}}t}} = \frac{{\eta _{4} }}{2}{\left[ {{\sum\limits_{j = 1}^m {\cos {\left( {\psi _{j} - \psi _{i} } \right)}{\left( {v_{j} - v_{{0j}} } \right)} - {\sum\limits_{j = 1}^n {\cos {\left( {\phi _{j} - \psi _{i} } \right)}{\left( {w_{j} - w_{{0j}} } \right)}} }} }} \right]} - \eta _{6} m{\left( {v_{i} - v_{{0i}} } \right)}.$$
(53)

By solving the equilibrium in combination with Eq. (45), we have

$$V^{*}_{\alpha } = \frac{{\eta _{1} \eta _{4} {\left( {R - R_{0} } \right)}\cos \theta A_{\alpha } {\left( {\Delta _{\psi } } \right)}}}{{ - 4\eta _{3} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)} + 4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }},$$
(54)
$$ W^{*}_{\alpha } = \frac{{\eta _{1} {\left( {R - R_{0} } \right)}\cos \theta {\left[ {\eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)} - 4\eta _{6} } \right]}}} {{ - 4\eta _{3} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)} + 4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }}, $$
(55)

which leads to

$$ r_{D} = {\sqrt {\frac{{\cos ^{2} \theta }} {{{\left[ {\frac{{4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }} {{\eta _{1} \eta _{4} A_{c} {\left( {\Delta _{\psi } } \right)}}} - \frac{{4\eta _{3} }} {{\eta _{1} }}} \right]}^{2} }} + \frac{{\sin ^{2} \theta }} {{{\left[ {\frac{{4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }} {{\eta _{1} \eta _{4} A_{s} {\left( {\Delta _{\psi } } \right)}}} - \frac{{4\eta _{3} }} {{\eta _{1} }}} \right]}^{2} }}} }, $$
(56)
$$ \theta _{D} = \tan ^{{ - 1}} {\left[ {\frac{{\frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{1} \eta _{4} A_{c} {\left( {\Delta _{\psi } } \right)}}} - \frac{{\eta _{3} }} {{\eta _{1} }}}} {{\frac{{\eta _{1} \eta _{6} + 4\eta _{3} \eta _{6} }} {{\eta _{1} \eta _{4} A_{s} {\left( {\Delta _{\psi } } \right)}}} - \frac{{\eta _{3} }} {{\eta _{1} }}}}\tan \theta } \right]}. $$
(57)

When MFs create any phase leads (Δ ψ  = π), Eq. (57) implies perfect transfer of the target phase (θ D  = θ). However, r D  = η 1 η 4/(4η 1 η 6 + 16η 3 η 6 − 4η 3 η 4), derived from Eq. (56), is indefinite because 4η 1 η 6 + 16η 3 η 6 − 4η 3 η 4 can take an arbitrary value. Consequently, unrealistic phenomena such as overlearning (r D  > 1) can arise in the model. Regarding phase learning, the error is prominent when Δ ψ is as small as Δ ψ  = O((η 6/η 4)1/2) (recall η 4 ≫ η 6), which is suitable.

3.3 C.3 PC-driven MF–VN plasticity

With the PC-driven learning, Eq. (15) reads

$$\frac{{dv_{i} }}{{dt}} = - \frac{{\eta _{4} }}{2}{\sum\limits_{j = 1}^n {\cos {\left( {\phi _{j} - \psi _{i} } \right)}{\left( {w_{j} - w_{{0j}} } \right)}} } - \eta _{6} m{\left( {v_{i} - v_{{0i}} } \right)}.$$
(58)

We derive

$$ V^{*}_{\alpha } = \frac{{\eta _{1} \eta _{4} {\left( {R - R_{0} } \right)}\cos \theta A_{\alpha } {\left( {\Delta _{\psi } } \right)}}} {{\eta _{1} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)} + 4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }}, $$
(59)
$$ W^{*}_{\alpha } = - \frac{{4\eta _{1} \eta _{6} {\left( {R - R_{0} } \right)}\cos \theta }} {{\eta _{1} \eta _{4} A_{\alpha } {\left( {\Delta _{\psi } } \right)} + 4\eta _{1} \eta _{6} + 16\eta _{3} \eta _{6} }}, $$
(60)

which in combination with Eq. (50) yields r D (Eq. (23)) and θ D (Eq. (24)). The amount of the memory stored in the PF–PC synapses is represented by

$$ - W^{*}_{c} \sin \omega t - W^{*}_{s} \cos \omega t = r_{I} {\left( {R - R_{0} } \right)}\sin {\left( {\omega t + \theta _{I} } \right)}, $$
(61)

which yields r I (Eq. (25)) and θ I (Eq. (26)).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Masuda, N., Amari, Si. A computational study of synaptic mechanisms of partial memory transfer in cerebellar vestibulo-ocular-reflex learning. J Comput Neurosci 24, 137–156 (2008). https://doi.org/10.1007/s10827-007-0045-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-007-0045-7

Keywords

Navigation