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Sequential Pattern Formation in the Cerebellar Granular Layer

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Abstract

Here, we introduce a novel mechanism for temporal recoding by the cerebellar granular layer based on three key properties: the granule cell-Golgi cell inhibitory feedback loop, bursting behaviour of granule cells and the large ratio of granule cells to Golgi cells. We propose that mutual inhibition of granule cells, mediated by Golgi cell feedback inhibition, prevents simultaneous activation. Granule cells are differentiated by firing threshold, resulting in sequential bursts of spikes. We demonstrate the plausibility of the mechanism through a computational simulation of a firing rate model, and further examine its robustness by developing a spiking model incorporating realistic postsynaptic potentials.

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References

  1. Marr D. A theory of cerebellar cortex. J Physiol 1969;202(2):437.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  2. Albus JS. A theory of cerebellar function. Math Biosci 1971;10(1):25–61.

    Article  Google Scholar 

  3. Fujita M. Adaptive filter model of the cerebellum. Biol Cybern 1982;45(3):195–206.

    Article  CAS  PubMed  Google Scholar 

  4. Porrill J, Dean P, Anderson SR. Adaptive filters and internal models: multilevel description of cerebellar function. Neural Netw 2013;47:134–49.

    Article  PubMed  Google Scholar 

  5. Bratby P, Sneyd J, Montgomery J. Computational architecture of the granular layer of cerebellum-like structures. Cerebellum 2016:1–11. doi:10.1007/s12311-016-0759-z.

  6. Medina JF, Garcia KS, Nores WL, Taylor NM, Mauk MD. Timing mechanisms in the cerebellum: testing predictions of a large-scale computer simulation. J Neurosci 2000;20(14):5516–25.

    CAS  PubMed  Google Scholar 

  7. Montgomery JC, Bodznick D. An adaptive filter that cancels self-induced noise in the electrosensory and lateral line mechanosensory systems of fish. Neurosci Lett 1994;174(2):145–8.

    Article  CAS  PubMed  Google Scholar 

  8. Bratby P, Montgomery J, Sneyd J. A biophysical model of adaptive noise filtering in the shark brain. Bull Math Biol 2014;76(2):455–75.

    Article  PubMed  Google Scholar 

  9. Kennedy A, Wayne G, Kaifosh P, Alviña K, Abbott LF, Sawtell N B. 2014. A temporal basis for predicting the sensory consequences of motor commands in an electric fish. Nat Neurosci. 2014;17(3):416–22.

  10. Lepora NF, Porrill J, Yeo CH, Dean P. Sensory prediction or motor control? Application of Marr–Albus type models of cerebellar function to classical conditioning. Front Comput Neurosci 2010:4.

  11. Bullock D, Fiala JC, Grossberg S. A neural model of timed response learning in the cerebellum. Neural Netw 1994;7(6):1101–14.

    Article  Google Scholar 

  12. Moore JW, Desmond JE, Berthier NE. Adaptively timed conditioned responses and the cerebellum: a neural network approach. Biol Cybern 1989;62(1):17–28.

    Article  CAS  PubMed  Google Scholar 

  13. Yamazaki T, Tanaka S. The cerebellum as a liquid state machine. Neural Netw 2007a;20(3):290–7.

    Article  PubMed  Google Scholar 

  14. Rössert C, Dean P, Porrill J. At the edge of chaos: how cerebellar granular layer network dynamics can provide the basis for temporal filters. PLoS Comput Biol 2015;11(10):e1004515.

    Article  PubMed  PubMed Central  Google Scholar 

  15. Yamazaki T, Tanaka S. A spiking network model for passage-of-time representation in the cerebellum. Eur J Neurosci 2007b;26(8):2279–92.

    Article  PubMed  PubMed Central  Google Scholar 

  16. D’Angelo E. Cerebellar granule cell. In: Handbook of the cerebellum and cerebellar disorders. Springer; 2013. p. 765–791.

  17. Chabrol FP, Arenz A, Wiechert MT, Margrie TW, DiGregorio DA. Synaptic diversity enables temporal coding of coincident multisensory inputs in single neurons. Nat Neurosci 2015;18(5):718–27.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  18. Mapelli L, Solinas S, D’Angelo E. Integration and regulation of glomerular inhibition in the cerebellar granular layer circuit. Front Cell Neurosci 2014;8:55.

  19. Izhikevich EM, Moehlis J. Dynamical systems in neuroscience: the geometry of excitability and bursting. SIAM Rev 2008;50(2):397.

    Google Scholar 

  20. Jörntell H, Hansel C. Synaptic memories upside down: bidirectional plasticity at cerebellar parallel fiber-purkinje cell synapses. Neuron 2006;52(2):227–38.

    Article  PubMed  Google Scholar 

  21. Caporale N, Dan Y. Spike timing-dependent plasticity: a Hebbian learning rule. Annu Rev Neurosci 2008; 31:25–46.

    Article  CAS  PubMed  Google Scholar 

  22. Regehr WG. Short-term presynaptic plasticity. Cold Spring Harb Perspect Biol 2012;4(7):a005702.

    Article  PubMed  PubMed Central  Google Scholar 

  23. Stackman RW, Hammond RS, Linardatos E, Gerlach A, Maylie J, Adelman JP, Tzounopoulos T. Small conductance Ca 2+-activated K + channels modulate synaptic plasticity and memory encoding. J Neurosci 2002;22(23):10163–71.

    CAS  PubMed  Google Scholar 

  24. Nieus T, Sola E, Mapelli J, Saftenku E, Rossi P, D’angelo E. Ltp regulates burst initiation and frequency at mossy fiber–granule cell synapses of rat cerebellum: experimental observations and theoretical predictions. J Neurophysiol 2006;95(2):686–99.

    Article  PubMed  Google Scholar 

  25. Ermentrout GB, Terman DH. Mathematical foundations of neuroscience. vol. 35. Springer Science & Business Media; 2010.

  26. Cesana E, Pietrajtis K, Bidoret C, Isope P, D’Angelo E, Dieudonné S, Forti L. Granule cell ascending axon excitatory synapses onto golgi cells implement a potent feedback circuit in the cerebellar granular layer. J Neurosci 2013;33(30):12430–46.

    Article  CAS  PubMed  Google Scholar 

  27. Jörntell H, Ekerot C-F. Properties of somatosensory synaptic integration in cerebellar granule cells in vivo. J Neurosci 2006;26(45):11786–97.

    Article  PubMed  Google Scholar 

  28. Mapelli L, Rossi P, Nieus T, D’Angelo E. Tonic activation of gabab receptors reduces release probability at inhibitory connections in the cerebellar glomerulus. J Neurophysiol 2009;101(6):3089–99.

    Article  CAS  PubMed  Google Scholar 

  29. Crowley JJ, Fioravante D, Regehr WG. Dynamics of fast and slow inhibition from cerebellar golgi cells allow flexible control of synaptic integration. Neuron 2009;63(6):843–53.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  30. De Schutter E. Computational modeling methods for neuroscientists. Cambridge: MIT Press; 2009.

    Book  Google Scholar 

  31. van Beugen BJ, Gao Z, Boele H-J, Hoebeek F, De Zeeuw CI, et al. High frequency burst firing of granule cells ensures transmission at the parallel fiber to purkinje cell synapse at the cost of temporal coding. Front Neural Circuits 2013;7:95.

  32. Vergara C, Latorre R, Marrion NV, Adelman JP. Calcium-activated potassium channels. Curr Opin Neurobiol 1998;8(3):321–9.

    Article  CAS  PubMed  Google Scholar 

  33. Lee K, Duan W, Sneyd J, Herbison AE. Two slow calcium-activated afterhyperpolarization currents control burst firing dynamics in gonadotropin-releasing hormone neurons. J Neurosci 2010;30(18):6214–24.

    Article  CAS  PubMed  Google Scholar 

  34. Maex R, De Schutter E. Synchronization of golgi and granule cell firing in a detailed network model of the cerebellar granule cell layer. J Neurophysiol 1998;80(5):2521–37.

    CAS  PubMed  Google Scholar 

  35. D’Angelo E, Nieus T, Maffei A, Armano S, Rossi P, Taglietti V, Fontana A, Naldi G. Theta-frequency bursting and resonance in cerebellar granule cells: experimental evidence and modeling of a slow K +-dependent mechanism. J Neurosci 2001;21(3):759–70.

    PubMed  Google Scholar 

  36. Semyanov A, Walker MC, Kullmann DM, Silver RA. Tonically active GABA A a receptors: modulating gain and maintaining the tone. Trends Neurosci 2004;27(5):262–9.

    Article  CAS  PubMed  Google Scholar 

  37. Maass W, Natschläger T, Markram H. Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Comput 2002;14(11):2531–60.

    Article  PubMed  Google Scholar 

  38. Fang Y, Cohen MA, Kincaid TG. Dynamics of a winner-take-all neural network. Neural Netw 1996;9 (7):1141–54.

    Article  PubMed  Google Scholar 

  39. Ermentrout B. Complex dynamics in winner-take-all neural nets with slow inhibition. Neural Netw 1992;5(3): 415–31.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Peter Bratby & James Sneyd

  2. School of Biological Sciences, University of Auckland, Auckland, New Zealand

    John Montgomery

Authors
  1. Peter Bratby
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  2. James Sneyd
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  3. John Montgomery
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Correspondence to Peter Bratby.

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Appendix

Appendix

Spiking Model

Equation 3 represents the synaptic drive generated by a continuously varying firing rate r i (t). Here we derive an equivalent formulation based on a sequence of discrete spikes.

First, note that Eq. 3 can be restated in integral form:

$$ z_{i}(t) = {{\int}^{t}_{0}} \frac{1}{\tau_{i}}\exp\left( \frac{s-t}{\tau_{i}}\right)r_{i}(s-{\Delta}_{i})ds. $$
(16)

Now, the spike rate r i (t) represents the instantaneous average of a sequence of discrete spikes, which can be represented as follows:

$$ z_{i}(t) \approx {{\int}^{t}_{0}} \frac{1}{\tau_{i}}\exp\left( \frac{s-t}{\tau_{i}}\right)\left( \sum\limits_{t_{j}<t}T_{ISI}\delta(s-t_{j})\right)ds, $$
(17)

where δ is the delta function and the sequence of spikes t j is calculated as follows:

$$ t_{j+1} = t_{j} + \frac{T_{ISI}}{r(t_{j}-{\Delta}_{i})}, $$
(18)

where T ISI is a small arbitrary constant representing the interspike interval at r = 1. Rearranging, we arrive at the following equation for z i (t):

$$ z_{i}(t) = \sum\limits_{t_{j}<t}\frac{T_{ISI}}{\tau_{i}}\exp\left( \frac{-(t-t_{j})}{\tau_{i}}\right). $$
(19)

Setting τ i = τ d and for τ r≪τ d, we define a postsynaptic potential:

$$ I_{\text{PSP}}(t)=\frac{T_{\text{ISI}}}{\tau_{\mathrm{d}}-\tau_{\mathrm{r}}}\left( e^{-t/\tau_{\mathrm{d}}}-e^{-t/\tau_{\mathrm{r}}}\right), $$
(20)

and so:

$$ z_{i}(t) = \sum\limits_{t_{j}<t}I_{\text{PSP}}(t-t_{j}). $$
(21)

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Bratby, P., Sneyd, J. & Montgomery, J. Sequential Pattern Formation in the Cerebellar Granular Layer. Cerebellum 16, 438–449 (2017). https://doi.org/10.1007/s12311-016-0820-y

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  • DOI: https://doi.org/10.1007/s12311-016-0820-y

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