A Case Study of Population Coding: Stimulus Localisation in the Barrel Cortex - Computational Neuroscience: A Comprehensive Approach - page 375

of individual spikes is the sum of an independent spike timing term ( I t ) and a PSTH
similarity term ( I tta ). The former term expresses the information that would be con-
veyed were the spikes to carry independent information; the latter term corrects this
for any redundancy arising from similarity of PSTHs across stimuli. The remaining
two terms ( I ttb and I ttc ) express any further effect that correlated spike patterns might
have beyond that of individual spikes. By evaluating these terms separately, the se-
ries expansion yields direct insight into a question of great current interest in neural
coding - the role of individual spikes compared to correlated spike patterns.

13.2.3.1 Independent spike timing

If within-trial spike patterns do not convey information, then all information must
be in the timing of individual spikes. Under these circumstances, the time-varying
firing rate (PSTH) is a complete description of the neuronal response, and is the only
statistic required in order to estimate the information. The greater the diversity in
PSTH structure across stimuli, the greater is the information. If each spike provides
independent information about the stimulus set, Equation (13.4) (the independent
spike timing term) gives the total information available in the response [4, 7, 21].

¯ s

n ais

Â
a

I t = n ais log 2 (13.4)

n ais s

, i

n ais is the number of spikes in time bin i of cell a elicited by stimulus s on a particular
trial. The bar ¯ · means an average over trials, thus n ais is simply the corresponding ¯

PSTH. The angle brackets ··· s denote an average over stimuli, weighted by the

stimulus probabilities P ( s ) .

13.2.3.2 PSTH Similarity

If there is any redundancy between spikes, Equation (13.4) can overestimate the
information. Redundancy is present if the PSTH value at a given time bin correlates
across the stimulus set with the PSTH value at a different time bin for the same cell,
or correlates with the PSTH value at any time bin for a different cell. This type
of correlation has been termed signal correlation and Equation (13.5) quantifies the
amount of redundancy that it introduces. The PSTH similarity term is:

CS aib j 1

ECS aib j

CS aib j
MS ai MS bj

1
2log e 2

Â

= − −

I tta log e (13.5)

a , b , i , j

Here MS ai = n ais s is the average of the PSTH over stimuli for time bin i of cell

a; CS aib j = n ais n bjs s is the signal correlation between time bin i of cell a and bin

j of cell b; ECS aib j = MS ai MS bj is the expected value of CS aib j for PSTHs that are

uncorrelated across stimuli. I tta is always negative or zero. I tta and I t together express
any information that the population conveys purely by the timing of individual spikes
(time-varying firing rate).