Synaptic Plasticity and Pattern Recognition in Cerebellar Purkinje Cells

Abstract

Many theories of cerebellar learning assume that long-term depression (LTD) of synapses between parallel fibres (PFs) and Purkinje cells is the basis for pattern recognition in the cerebellum. Here we describe a series of computer simulations that use a morphologically realistic conductance-based model of a cerebellar Purkinje cell to study pattern recognition based on PF LTD. Our simulation results, which are supported by electrophysiological recordings in vitro and in vivo, suggest that Purkinje cells can use a novel neural code that is based on the duration of silent periods in their activity. The simulations of the biologically detailed Purkinje cell model are compared with simulations of a corresponding artificial neural network (ANN) model. We find that the predictions of the two models differ to a large extent. The Purkinje cell model is very sensitive to the amount of LTD induced, whereas the ANN is not. Moreover, the pattern recognition performance of the ANN increases as the patterns become sparser, while the Purkinje cell model is unable to recognise very sparse patterns. These results highlight that it is important to choose a model at a level of biological detail that fits the research question that is being addressed.

Appendix: Calculation of the Signal-to-Noise Ratio for the ANN

We here derive a simple approximation for the signal-to-noise ratio of the ANN responses to stored and novel patterns. We assume, as in Fig. 26.4, that the synapses take binary values: they are either depressed or non-depressed, and without loss of generality (see below) we assign them weights zero and one. In this case, the responses to stored patterns are always zero, as is their variance. Hence we need only to calculate the mean and variance of the responses to novel patterns. For this we proceed in two steps: (a) find the number D of depressed synapses; (b) draw A synapses (A being the number of active synapses per pattern) from a pool of n = S synapses of which D are depressed. Let us start with the second problem, so D is supposed to be known.

Calculating the response to a novel pattern is now equivalent to drawing, without replacement, A balls from an urn containing S balls of which D are white (depressed) and S − D balls are red (non-depressed). The distribution of the number of red balls drawn on each trial is known to be hypergeometric. However, if S be very large compared to A, the numbers approximate a binomial distribution with the probability p of drawing a non-depressed synapse equal to $(1-D/S)$. A binomial distribution has mean n p and variance n p (1 − p), hence the average output of the ANN for novel patterns will be $A(1-D/S)$ with variance $A(1-D/S)(D/S)$. The signal-to-noise ratio becomes:

$$ \begin{array}{l}\text{s}/\text{n}=\frac{{\left({\mu}_{s}-{\mu}_{n}\right)}^{2}}{0.5\left({\sigma}_{s}^{2}+{\sigma}_{n}^{2}\right)}\\ =2\frac{{(A(1-D/S))}^{2}}{A(1-(D/S)(D/S)}\\ =2\frac{A(S-D)}{D}\end{array}$$

Note that the signal-to-noise ratio does not depend on the exact values of the weights assigned to the binary synapses. If the depressed and non-depressed synapses were to be assigned weights a and b, respectively, instead of 0 and 1, we would obtain from the binomial distributions μ _s = a n, ${m}_{n}=an(1-p)+bnp$, σ _s ² = 0, and ${s}_{n}^{2}={(a-b)}^{2}np(1-p)$, which after substitution yields the same value for s∕n.

We now turn to the problem of assessing the number D of depressed synapses. If the number of learned patterns L is very small and/or the patterns are very sparse (A small), then the degree of overlap between learned patterns will also be small, so that D becomes approximately equal to L A (each depressed synapse is unique).

Then the signal-to-noise ratio reduces to:

$$ \begin{array}{cccc}& \begin{array}{l}\text{s}/\text{n}=2\frac{A(S-LA)}{LA}\\ =2\frac{(S-LA)}{L}\\ \approx 2\frac{S}{L}\end{array}& & \end{array}$$

where it is assumed that L ∗ A ≪ S. Hence the “s/n” signal-to-noise ratio is proportional to the number S of synapses and inversely proportional to the number L of learned patterns.

If L A is large, we cannot assume L A to be a good approximation of D. But let us assume all synapses are independently depressed with probability A∕S for each pattern, then each synapse has probability ${(1-A/S)}^{L}$ of not having been depressed by any of the L patterns, and consequently the mean number D of depressed synapses will approximate $S-S{(1-A/S)}^{L}$.