Elsevier

Biosystems

Volume 94, Issues 1–2, October–November 2008, Pages 10-17
Biosystems

Event-driven simulation of cerebellar granule cells

https://doi.org/10.1016/j.biosystems.2008.05.007Get rights and content

Abstract

Around half of the neurons of a human brain are granule cells (approximately 1011granule neurons) [Kandel, E.R., Schwartz, J.H., Jessell, T.M., 2000. Principles of Neural Science. McGraw-Hill Professional Publishing, New York]. In order to study in detail the functional role of the intrinsic features of this cell we have developed a pre-compiled behavioural model based on the simplified granule-cell model of Bezzi et al. [Bezzi, M., Nieus, T., Arleo, A., D’Angelo, E., Coenen, O.J.-M.D., 2004. Information transfer at the mossy fiber—granule cell synapse of the cerebellum. 34th Annual Meeting. Society for Neuroscience, San Diego, CA, USA]. We can use an efficient event-driven simulation scheme based on lookup tables (EDLUT) [Ros, E., Carrillo, R.R., Ortigosa, E.M., Barbour, B., Ags, R., 2006. Event-driven simulation scheme for spiking neural networks using lookup tables to characterize neuronal dynamics. Neural Computation 18 (12), 2959–2993]. For this purpose it is necessary to compile into tables the data obtained through a massive numerical calculation of the simplified cell model. This allows network simulations requiring minimal numerical calculation. There are three major features that are considered functionally relevant in the simplified granule cell model: bursting, subthreshold oscillations and resonance. In this work we describe how the cell model is compiled into tables keeping these key properties of the neuron model.

Introduction

The cerebellum is a well structured neural system conformed by three layers: granular, molecular and Purkinje layer. The granular layer contains approximately 1011 granule cells that represent in number of neurons half of the cells of the whole human brain (Kandel et al., 2000). The granule cells receive their inputs through the mossy fibers. The axons of the granule cells are called parallel fibers that connect with different Purkinje cells. The granular layer represents a highly divergent structure (there are approximately 103 granule cells per mossy fiber). Therefore they seem to be responsible for building a sparse representation of the mossy fibers inputs Marr, 1969, Albus, 1971, Coenen et al., 2001, D’Angelo et al., 2005. But the dynamical properties of the cell are still under study Magistretti et al., 2006, Armano et al., 2000, D’Angelo et al., 2005, Nieus et al., 2006, Mapelli and D’Angelo, 2007, Rossi et al., 2006 and detailed cell models are being built to evaluate the functional role (D’Angelo et al., 2001) of these dynamics. The neuron models can be simulated with different simulators (NEURON (Hines and Carnevale, 1997), Genesis (Bower and Beeman, 1998), EDLUT (Ros et al., 2006)) at different levels of detail. Recently an efficient event-driven lookup-table-based simulator (EDLUT) (Ros et al., 2006) has been developed to allow large-scale network simulations based on pre-compiled models and therefore avoiding intense numerical calculation during the neural-network simulation. Using EDLUT requires compiling previously the single cell behaviour into tables. This is done by means of massive calculation to characterize how the cell state changes in response to an input spike (depending on its initial status). For this purpose, lookup tables (LUTs) are built compiling the characteristic cell status traces in response to input spikes. Once these tables are built we can run event-driven large-scale network simulations without redoing any numerical calculation. The neuron state can be retrieved from these cell-characterizing LUTs at any instant in response to any input spike.

After building up cell models based on characterizing LUTs we need to validate the model in two ways:

  • 1.

    Accuracy validation. The number of samples in each dimension of the table can be critical to the accuracy of the table-based cell approach. Therefore we simulate the cell model with a classical numerical calculation method (for instance, Euler method with a very short time step) and we compare the output spike train obtained in response to different input spike trains with the results obtained using EDLUT simulator. The comparison of the output spike trains obtained by the two methods is done using the van Rossum distance (van Rossum, 2001).

  • 2.

    Functional validation. Key cell features must be kept. If we want to abstract a cell model that includes certain cell features that are considered relevant we also need to validate that the table-based model is able to reproduce the cell features under study.

Section snippets

Integrate-and-fire cerebellar granule-cell model

A detailed Hodgkin–Huxley model (Hodgkin and Huxley, 1952), of a granule cell defined in NEURON (with more than 15 differential equations describing its dynamics) was built to reproduce in detail the cell dynamics and evaluate the significant variables of the model (D’Angelo et al., 2001). Based on that model, Bezzi et al. (2004) presented a simplified integrated-and-fire cell model with threshold mechanism which kept important dynamical properties of the granule cell, such as subthreshold

Table-based approach

The neuron behaviour has been compiled into six tables. In order to use the event-driven simulator (EDLUT) the neuron state (membrane potential, synaptic conductances and other variables such as the gating variable n) need to be defined as functions of the neuron state at the instant in which it was updated the last time. Since it is an event-driven scheme the neuron state is updated each time that an event is produced (output spikes) or an input event is received (input spikes).

The model has

Experimental results

Here we show some illustrative simulations in which the behaviour of the cell model described in NEURON is compared with the behaviour of the model compiled into tables and simulated with EDLUT (Ros et al., 2006). The presented model can reproduce synaptic activation of a granule cell. Activation of 1 and 2 synapses makes subthreshold EPSPs which, in the immediately subthreshold region, become slower due to activation of persistent Na current. Activation of 3 synapses elicits a spike, which

Accuracy validation

In this section we evaluate the accuracy of the model captured on lookup tables that are used in the EDLUT approach. For this purpose we run some reference simulations using intensive numerical calculation (Euler method with a very short integration time constant (0.5 μs)) with the original differential equations of the simplified model Bezzi et al. (2004). After this, we perform the same simulations in EDLUT. Finally we compare the output spike trains obtained by the two approaches calculating

Computation requirements and performance

EDLUT simulator allows efficient simulation of large-scale spiking neural networks, since its performance (computation speed) does not depend on the network size but on the network activity. This simulator is especially suitable for neural structures with sparse coding. This is the case of the granular layer (Smith et al., 2000). We have simulated a medium-scale granular layer with 2000 granule cells (and 4 Golgi cells) which produces sparse coding in the parallel fibers (i.e. at the granule

Discussion

We have implemented using an event-driven lookup-table simulator, an integrate-and-fire neuron model with extended dynamical properties to allow oscillatory, bursting and resonance behaviours. This allows the use of efficient event-driven simulation engines such as EDLUT (Ros et al., 2006) to address large-scale network simulations. We have validated the model reproducing oscillatory, bursting and resonance behaviours easily in different experiments. This validates the model and illustrates how

Acknowledgements

This work has been supported by the EU project SENSOPAC (IST-028056) and the National Spanish Grant DEPROVI (DPI 2004-07032).

References (27)

  • J.S. Albus

    A theory of cerebellar function

    Math. Biosci.

    (1971)
  • M. Bezzi et al.

    An integrate-and-fire model of a cerebellar granule cell

    Neurocomputing

    (2004)
  • E.M. Izhikevich

    Resonate-and-fire neurons

    Neural Netw.

    (2001)
  • D. Philipona et al.

    Model of granular layer encoding of the cerebellum

    Neurocomputing

    (2004)
  • S. Armano et al.

    Long-term potentiation of intrinsic excitability at the mossy fiber - granule cell synapse of rat cerebellum

    J. Neurosci.

    (2000)
  • M. Bezzi et al.

    Information transfer at the mossy fiber - granule cell synapse of the cerebellum

  • M. Bezzi et al.

    Quantitative characterization of information transmission in a single neuron

  • J.M. Bower et al.

    The Book of GENESIS

    (1998)
  • J.H.E. Cartwright et al.

    The dynamics of Runge–Kutta methods

    Int. J. Bifurcation Chaos

    (1992)
  • O.J.-M.D. Coenen et al.

    Parallel fiber coding in the cerebellum for life-long learning

    Autonomous Robots

    (2001)
  • O. Coenen et al.

    Information theoretic quantification of neural transmission following changes in release probability

    (2007)
  • E. D’Angelo et al.

    Modeling Synaptic Transmission and Quantifying Information Transfer in the Granular Layer of the Cerebellum

    Lecture Notes Comput. Sci.

    (2005)
  • E. D’Angelo et al.

    Theta-frequency bursting and resonance in cerebellar granule cells: experimental evidence and modeling of a slow K+-dependent mechanism

    J. Neurosci.

    (2001)
  • Cited by (0)

    View full text