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Reliability of mean rate coding

It may seem obvious that because multiple functional contacts increase the fidelity with which a presynaptic signal is propagated, it can overcome the ``noise'' induced by synaptic failures and quantal fluctuations and thereby increase the fidelity of neuronal signaling. In the previous section we quantified this intuition under the hypothesis that the precise timing of spikes carries information. To what extent does this conclusion depend on the particular assumptions we are making about the neural code?

According to the ``mean-rate'' hypothesis for the neural code, the ``signal'' is carried not by the times at which spikes occur, but instead by the number of output spikes generated in some relatively long window. Under this hypothesis, multiple functional contacts can actually have the seemingly paradoxical effect of decreasing the transmitted information.

We use the Fano factor [Fano, 1947] to assess the reliability of coding under the mean rate hypothesis. The Fano factor is defined as the variance tex2html_wrap_inline1473 divided by the mean tex2html_wrap_inline1475 of the spike count N in some time window W. The Fano factor can be viewed as a kind of ``noise-to-signal'' ratio; it is a measure of the reliability with which the spike count could be estimated from a time window that on average contains several spikes. In fact, for a renewal process like the neuronal spike generator considered here, the distribution tex2html_wrap_inline1481 of spike counts can be shown (by the central limit theorem; [Feller, 1971]) to be normally distributed (asymptotically, as the number of trials becomes large), with tex2html_wrap_inline1483 and tex2html_wrap_inline1485, where tex2html_wrap_inline1487 and are, respectively, the mean and standard deviation of the ISI distribution tex2html_wrap_inline1333. Thus the Fano factor F is related to the coefficient of variation tex2html_wrap_inline1493 of the associated ISI distribution by tex2html_wrap_inline1495,

Fig. 5 shows the Fano factor as a function of the number of functional contacts. The spike trains are the same as those analyzed in Fig. 4B. The Fano factor increases monotonically with the firing rate. Since the reliability with which the spike count can be estimated is inversely related to its variability, an increase in the number of functional contacts results in a decrease in the effective signal-to-noise ratio. This suggests that if a ``mean-rate'' coding scheme is used, an increase in the number of functional contacts could actually decrease the coding fidelity. This behavior stands in marked contrast to that observed in the previous section, where the increase in functional contacts produced the expected increase in information rate.

How can we account for this seemingly paradoxical decrease in signal-to-noise with increased redundancy? The resolution rests in the normalization used to increase the connection redundancy. In these simulations, the net input Poisson rate tex2html_wrap_inline1195 was held constant (as described in Sec. 2.2) in order to keep the mean postsynaptic current tex2html_wrap_inline1109, and therefore the firing rate, constant. A large tex2html_wrap_inline1185 leads to a ``redistribution'' of the presynaptic spikes into a small number of highly synchronous events, surrounded by longer periods during which no spike occurred. Normalizations that do not increase the effective synchrony might have given a different result.

These synchronous events have two effects. First, they tend to trigger postsynaptic action potentials at precise times. This increased timing precision decreases the conditional entropy--and thereby increases the total information (by eq. 8)--under the coding assumptions analysed in Sections 3.2-3.4, but has no effect on the available information under the mean rate hypothesis. Second, the increased synchrony increased the variance of the postsynaptic input current, which in turn leads to an increase in the output variance (as assessed by the Fano factor). This increases the total entropy and hence the total information under the coding assumptions analysed in Sections 3.2-3.4, but actually decreases the effective signal-to-noise ratio under the mean rate hypothesis. Thus increased the connection redundancy has diametrically opposed effects on the available information, depending on how the spike trains are decoded.


next up previous
Next: Discussion Up: Results Previous: Information rate depends on

Tony Zador
Fri Nov 28 10:17:14 PST 1997