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 divided by the mean 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 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 and , where and are, respectively, the mean and standard deviation of the ISI distribution . Thus the Fano factor F is related to the coefficient of variation of the associated ISI distribution by ,
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 was held constant (as described in Sec. 2.2) in order to keep the mean postsynaptic current , and therefore the firing rate, constant. A large 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.