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I Nemenman, W Bialek, and R. de Ruyter van Steveninck. Submillisecond spike timing precision in adaptive encoding of natural motion stimuli. In preparation.

'''Extended Abstract'''

Information theoretic methods provide a general framework for
understanding the structure of the neural code. They have been used
extensively, particularly for analyzing responses to complex,
dynamical stimuli. For white noise stimuli the neural code is
efficient (utilizing over 50% of its total entropy), a spike contains
over 1 bit of information, and coarsening spike timing beyond a few
milliseconds leads to significant information loss (Strong et
al. 1998, Reinagel and Reid 2000). One may ask, however, whether these
results generalize to natural settings. Lewen et al. (2001) recorded
from a blowfly motion sensitive neuron, with the fly outdoors in its
natural environment, subjected to angular motion representative of
natural flight. In these conditions spike timing precision improved
with increasing ambient light intensity. At midday brightness,
stimulus zero crossings generated initial spikes with 0.7 ms
precision, and interspike intervals with a precision of 0.2 ms (de
Ruyter van Steveninck and Bialek, 2001). But it remains to be seen
whether such accurate intervals with slightly more variable absolute
timing are used to transmit absolute temporal information with
sub-millisecond precision. And even if that were the case, these
accurate features might be too rare to contribute significantly to the
overall information capacity.

Systematic information theoretic analysis of such experiments could
answer these questions, but faces serious conceptual problems since
straightforward estimation of information quantities relies on a
thorough sampling of the stimulus-response joint distribution. High
neuronal timing precision in natural settings implies an enormous
space of distinguishable responses, requiring an extreme number of
instances for proper sampling. On the other hand, natural flight
trajectories are correlated over relatively long time scales of order
100ms, so that only a relatively small number of independent stimuli
can be presented during a neurophysiological experiment. So far, this
undersampling problem has precluded successful information theoretic
analysis of experiments with natural stimuli.

Recently we proposed a novel estimator for information quantities with
comparatively modest data requirements (Nemenman et al., 2002;
Nemenman, 2002). We have shown that it gives robust results even in
data-starved neurophysiological experiments (Nemenman et al.,
2004). Here we use the technique to analyze the neural code in the
outdoors experiments described in Lewen et al. (2001). We calculate
the information rate in the spike train as a function of two
parameters: the time resolution (the discretization of spike times)
and the observation time (length of code words). The upper limit on
the observation time is set by the flies' 30 ms behavioral decision
time (Land and Collett 1974). With this observation time we can
reliably estimate information quantities with resolutions between 0.2
and 30ms. We find coding efficiencies greater than 50 per cent for all
resolutions above 1 ms, approaching 80 per cent at 30 ms. This indicates that
the code is optimized for the natural distribution of
stimuli. Further, compared to counting spikes over 30ms observation
times, taking account of spike timing at high resolution is almost
three times more informative; there also is strong evidence that the
spike train contains information about the stimulus even at a
resolution as high as 0.2 - 0.3 ms. This is one of the highest spike
timing precisions relevant for transmitting bits in a single neuron
ever observed.

The measured information rate is ~150 bits/s, or about 1
bits pike. Notably, the latter number is close to what is found in
other experiments (Strong et al., 1998; Reinagel and Reid 2000), even
though (a) the average spike rate is very high, and (b) long
correlations in the stimulus imply lower stimulus entropy and,
potentially, smaller transmitted information. This hints at an
intriguing design principle underlying the neural code, where
competition between the metabolic cost of producing a spike and
delayed information arrival due to rare spikes results in a spike
being emitted when, on average, 1 bit of information is to be
transmitted.

At fixed time resolution, the information rate has a clear maximum at
a code word length of 3 ms (see also de Ruyter van Steveninck and
Bialek, 2001). This means that, just like in a human language,
quantifiably more information is carried by words (structured
combinations of letters/spikes), than by individual symbols. At yet
larger observation times the information rate drops again because of
smoothness, thus redundancy, in the natural stimulus itself.

We note a few more observations about the code: First, the rank
ordered distribution of the words used by the fly obeys Zipf's law---a
paradigmatic characterization of complex natural signals, such as
human languages. Second, neural firing rate and information rate
covary with the ambient light intensity, indicating that the fly is so
good at extracting information from photon arrival times that even a
small drop in intensity, for example about 0.3 log units due to a
cloud covering the sun, results in measurably decreased
performance. Finally, analysis of observation times of less than 1ms
suggests that neural refractoriness leads to synergy in the neural
code.

Bibliography
*MF Land and TS Collett. J Comp Physiol 89, 331-357 (1974)
*SP Strong, R Koberle, RR de Ruyter van Steveninck, and W Bialek, Phys. Rev. Lett. 80, 197 (1998)
*P Reinagel, and RC Reid, J. Neurosci. 20:5392-5400 (2000)
*GD Lewen, W Bialek, and RR de Ruyter van Steveninck, Network: Comput. Neural Syst. 12, 312 (2001)
*RR de Ruyter van Steveninck and W Bialek, in Methods in Neural Networks IV, J van Hemmen, JD Cowan, and E Domany, eds. (Springer-Verlag, Heidelberg, New York, 2001), pp. 313-371
*I Nemenman, F Shafee, and W Bialek, in Advances in Neural Information Processing Systems 14, TG Dietterich, S Becker, and Z Ghahramani, eds. (MIT Press, Cambridge, MA, 2002)
*I Nemenman, physics/0207009
*I Nemenman, W Bialek, and RR de Ruyter van Steveninck, Phys. Rev. E, 69:056111, 2004

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nemenman>Ilya at 01:49, 13 November 2006