Nemenman et al.

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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