A Case Study of Population Coding: Stimulus Localisation in the Barrel Cortex - Computational Neuroscience: A Comprehensive Approach - page 388

ence on the neural code. This is the essential difference between the series expan-
sion method and the D I method. In fact, we can characterise correlations as being
decoder-sensitive and decoder-insensitive . D I focuses on the decoder-sensitive type,
the series expansion considers both.

How do decoder-sensitive and -insensitive correlations relate to the spike pattern
components I ttb and I ttc of the series expansion? The answer is remarkably simple. It
turns out that D I , the decoder-sensitive contribution of cross-correlations to the pop-
ulation code, is precisely equal to the (cross-correlational) stimulus-dependent term
of the series expansion I ttc [20, 30]. This is true also of the general, exact decom-
position. In other words, one can think of the stimulus-dependent spike pattern term
as expressing a decodable spike pattern effect. In contrast, the stimulus-independent
term of the series expansion I ttb quantifies the effect of the intrinsic correlations in
the spike train, which cannot affect the performance of the decoder.

Finally, we note a second difference between D I and the series expansion; namely
that, as defined in [16], D I considers only correlations between different neurons,
whereas the series expansion assesses also the role of spike correlations within neu-
rons. However, the difference is minor, since the D I formalism could easily be ex-
tended to the case of either within-cell correlations alone, or within-cell and cross-
cell correlations together.

13.6 Conclusions

Although single trial discriminability (mutual information) is widely agreed to be the
right framework for addressing neural coding, information theory has been applied
mainly to single neuron coding, and little to the more general case of population cod-
ing. This is due to the problem of limited sampling. In this chapter, we have argued
that, for the class of sparsely responding neuronal ensembles, the series expansion
approach to information estimation allows population coding to be studied in a rigor-
ous, comprehensive manner. For rat barrel cortex, this method has revealed that there
is a temporal code of a simple kind: about 85% of the total information available in
the spike trains of neuron pairs concerning whisker location can be attributed to the
timing of individual spikes. Moreover, about 90% of this information is captured by
the first post-stimulus spike fired by each neuron.

Acknowledgments : Supported by European Community IST-2000-28127, Telethon
Foundation GGP02459, J.S. McDonnell Foundation 20002035, Wellcome Trust grant
066372 /Z /01 /Z, Italian MURST, Consiglio Nazionale delle Ricerche 02. 00536.
ST97 and Regione Friuli Venezia Giulia.

S.P. is supported by an MRC Research

Fellowship.