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.

Fri Nov 28 10:17:14 PST 1997