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Electrical recording. Single LGN neurons in the A laminae of the LGN were recorded with plastic-coated tungsten electrodes (AM Systems, Everett, WA). In some experiments, single units were recorded with electrodes of a multielectrode array (System Eckhorn Thomas Recording, Marburg, Germany). Recorded voltage signals were amplified, filtered, and passed to a personal computer running DataWave (Longmont, CO) Discovery software, and spike times were determined to 0.1 msec resolution. Preliminary spike discrimination was done during the experiment, but analysis is based on a more rigorous spike sorting from postprocessing of the recorded waveforms. The power spectra of the spike trains did not contain a peak at the stimulus frame rate (128 Hz). For the purposes of this analysis, it is crucial that any trial-to-trial variability in the recorded response be unequivocally attributable to neural noise rather than noise introduced at the level of data acquisition. Therefore, we report here only the results from extremely well isolated single units, with strict absolute refractory periods and spike amplitudes several SDs above the noise. Voltage traces surrounding each spike were examined for spike shape, as well as for precision of recorded spike times. Of the 27 cells recorded, 13 cells were excluded for imperfect spike isolation, and one cell was excluded because it responded poorly (<1 spike/sec) to our stimulus. The remaining recordings we believe to be entirely free of errors in the form of missed spikes, extraneous spikes, or spike time jitter greater than 0.1 msec. Furthermore, we used Monte Carlo simulations to simulate the effects of contamination with such errors, had they been present. We showed that if 2% of spikes were missed, 2% extraneous events were recorded as spikes, or 2% of spike times were jittered by the width of a spike, our information estimate would have decreased by 1.4, 2.8, or 0.1%, respectively. We define a "burst" as a group of action potentials, each of which is <4 msec apart, preceded by a period of >100 msec without spiking activity. This criterion was shown previously to reliably identify bursts that are caused by low-threshold calcium spikes in LGN relay cells of the cat (Lu et al., 1992
The Journal of Neuroscience, July 15, 2000, 20(14):5392-5400
Temporal Coding of Visual Information in the Thalamus
Pamela Reinagel and R. Clay Reid Department of Neurobiology, Harvard Medical School, Boston, Massachusetts 02115
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
The amount of information a sensory neuron carries about a stimulus is directly related to response reliability. We recorded from individual neurons in the cat lateral geniculate nucleus (LGN) while presenting randomly modulated visual stimuli. The responses to repeated stimuli were reproducible, whereas the responses evoked by nonrepeated stimuli drawn from the same ensemble were variable. Stimulus-dependent information was quantified directly from the difference in entropy of these neural responses. We show that a single LGN cell can encode much more visual information than had been demonstrated previously, ranging from 15 to 102 bits/sec across our sample of cells. Information rate was correlated with the firing rate of the cell, for a consistent rate of 3.6 ± 0.6 bits/spike (mean ± SD). This information can primarily be attributed to the high temporal precision with which firing probability is modulated; many individual spikes were timed with better than 1 msec precision. We introduce a way to estimate the amount of information encoded in temporal patterns of firing, as distinct from the information in the time varying firing rate at any temporal resolution. Using this method, we find that temporal patterns sometimes introduce redundancy but often encode visual information. The contribution of temporal patterns ranged from
3.4 to +25.5 bits/sec or from
Key words: LGN; neural coding; information theory; entropy; white noise; reliability; variability
INTRODUCTION
Cells in the lateral geniculate nucleus of the thalamus (LGN) respond to spatial and temporal changes in light intensity within their receptive fields. The collective responses of many such cells constitute the input to visual cortex. All stimulus discrimination at the perceptual level must ultimately be supported by reliable differences in the neural response at the level of the LGN cell population. We are therefore interested in measuring the statistical discriminability of LGN responses elicited by different visual stimuli.
It has been shown that the LGN can respond to visual stimuli with remarkable temporal precision (Reich et al., 1997
). This implies that LGN neurons have the capability to signal information at high rates. Previous estimates of the information in LGN responses have used two general approaches. The first approach, stimulus reconstruction, relies on an explicit model of what the neuron is encoding, as well as an algorithm for decoding it (Bialek et al., 1991
The second approach, the "direct" method, relies instead on statistical properties of the responses to different stimuli (the entropy of the responses). Because this involves only comparisons of spike trains, without reference to stimulus parameters, we need not know what features of the stimulus the cell encodes. Analysis of this type can be simplified by using a small set of stimuli and describing neural responses in terms of a few parameters, as has been done in previous studies of the LGN (Eckhorn and Pöpel, 1975
Recently, a version of the direct method has been developed that can be applied to the detailed firing patterns of neurons in response to arbitrarily complex stimuli (Strong et al., 1998
Because the direct method does not constrain either the temporal resolution of the code or the role of temporal patterns, the result does not by itself tell us anything about how LGN cells encode stimuli. We therefore present two further analyses. First, we explore the temporal resolution of the neural code. Second, we introduce a measure of the contribution of temporal patterns.
We distinguish three broad possibilities: (1) temporal patterns do not exist or are irrelevant to the neural code; (2) temporal patterns exist and make the neural code more redundant; or (3) temporal patterns exist and encode useful information. In our data, we find a range of results. Some cells encode information redundantly, whereas others use temporal patterns to encode visual information. In the latter case, to extract all the information from the spike trains, it would be necessary to consider temporal firing patterns; the time-varying instantaneous probability of firing would not be sufficient at any temporal resolution.
MATERIALS AND METHODS
Experimental
Surgery and preparation. Cats were initially anesthetized with ketamine HCl (20 mg/kg, i.m.) followed by sodium thiopental (20 mg/kg, i.v., supplemented as needed and continued at 2-3 mg · kg1 · hr
i P(i) log2P(i), where P(i) is the probability of luminance i], for a rate of 924.4 bits/sec, which well exceeds the coding capacity (i.e., the measured entropy) of individual neural responses. The intensity distribution was dominated by low intensities, such that luminance was <33% of the maximum in 73% of frames, and changed by <33% of the total range in 77% of transitions. This produced moderate contrast compared with, for instance, a binary black-to-white flicker (variance, 0.04 vs 0.25). We constructed a single 20-min-long stimulus consisting of 128 repeats of a single 8 sec sample interleaved with 32 unique 8 sec samples. We presented this 20 min composite stimulus in a single continuous trial. Most cells responded well to this modulated full-field stimulus. Stimuli were presented on a computer monitor controlled by a personal computer with an AT-Vista graphics card, at 128 Hz and 8 bit gray scale, at a photopic mean luminance.
Numeric calculations
We measured the visual information in neural responses by measuring entropy rates from probability distributions of neural responses, as explained in Results (Fig. 2). First, neural responses were represented as binned spike trains in which the value of a time bin was equal to the number of spikes that occurred during that time interval. Then we analyzed short strings of bins, or words. This representation depends on two parameters, which we varied: the size of the time bins,, and the number of bins in the words, L. We measured two forms of response entropy, the noise entropy, Hnoise, which reflects the trial-to-trial variability of responses to a repeated stimulus, and the total entropy, Htotal, which reflects the variability of responses to all (nonrepeated) stimuli in the ensemble. Hnoise was calculated from the distribution of responses at a fixed time t relative to stimulus onset, in 128 repeats of the same sample of the white-noise stimulus. We performed a separate calculation of Hnoise for many different values of t (separated by one bin) within the 8 sec stimulus. We then averaged the result over all values of t to get the average noise entropy. Htotal was calculated in the identical manner. We estimated the entropy from the distribution of responses over responses to 256 unique stimuli at each time t and then averaged the result over t. Data adequacy for entropy calculations. In any analysis of this kind, misleading results could be obtained if the amount of data were insufficient for estimating the probabilities of each response. In pilot studies with 1024 repeats, we estimated that 128 repeats were adequate for measuring the entropy of repeated stimuli. We used twice as many samples for Htotal as for Hnoise to compensate for the approximately twofold difference in entropy. For every entropy we computed, we determined how our estimate of H converged as we used increasing fractions of the data and then corrected for finite data size according to the method of Strong et al. (1998)
RESULTS
Neural responses to a dynamic visual stimulus
We recorded from individual neurons in the LGN of anesthetized cats while presenting spatially uniform (full-field) visual stimuli with a random time-varying luminance. We repeated one 8 sec sample of this white-noise stimulus 128 times, interleaved among 256 sec of unique samples. We report results from 11 exceptionally well isolated cells. Results from one example cell are shown in Figures 1-4.
The responses to different stimulus samples were highly variable (Fig. 1a, unique), whereas the responses to any repeated sample were highly reliable (Fig. 1a, repeat). From the peristimulus time histogram (PSTH) (Fig. 1a, PSTH), it is apparent that any given stimulus evoked very precise responses. When examined at a fine scale, many of the peaks in the PSTH have widths of ~1 msec (Fig. 1b) (
= 0.6 msec). This is somewhat higher than the temporal precision seen by Reich and colleagues in LGN responses to high-contrast drifting gratings (
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Figure 1. LGN responses to nonrepeated and repeated white-noise stimuli. Responses from an ON center X cell are shown (the cell in our sample that responded best to the full-field stimulus). a, Responses to 128 unique samples of the visual stimulus are shown at the top as rasters (unique) in which each row represents a different 1 sec segment of the neural response, and each dot represents the time of occurrence of a spike. Below the rasters is a PSTH from the 128 rasters, in which the value for each 1 msec bin is defined as the number of times a spike was observed in that bin in the 128 trials shown. The second set of rasters (repeat) and corresponding PSTH are from 128 repeats of one particular 1 sec sample of the white-noise visual stimulus. In the experiment, repeat and unique samples were interleaved but occurred in the order shown. The luminance time course of the stimulus that corresponds to the repeat rasters and PSTH is plotted at the bottom. The peak expanded in b is marked with an asterisk. The rasters expanded in c are marked by the bar. b, A narrow peak from PSTH in a, expanded and binned at 0.2 msec resolution. The peak defined by the shaded area (126 spikes in 128 repeats) was best fit by a Gaussian of
Information measured directly from spike train entropy
A neural response encodes information when there is a narrow distribution of responses to any given stimulus when it is repeated (Fig. 1a, repeat) compared with the distribution of possible responses to all stimuli (Fig. 1a, unique). The variability underlying these probability distributions can be quantified by their entropy (Shannon, 1948
Throughout this paper, we divide both entropy (H) and information (I) estimates by duration in time, such that all estimates are reported as entropy or information rates in units of bits per sec.
(1) To compute these quantities, we represented neural responses as binned spike trains in which the value of a time bin was equal to the number of spikes that occurred during that time interval. We then considered the probability distributions of short strings of bins, or words. The resulting estimate of entropy depends on two parameters: the temporal resolution of the bins in seconds,
where w is a specific word (spike pattern), W(L,
(2)
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Figure 2. Illustration of calculation of mutual information. Data are same as shown in Figure 1. a, Probability distribution of words of length L = 8, binned at 
), as described by Strong et al. (1998)
For any given word length L and resolution
According to this method, the responses shown in Figure 1 contain 102 bits/sec of visual information about this stimulus set, which is about an order of magnitude higher than had been demonstrated previously in the LGN (Eckhorn and Pöpel, 1975
Temporal resolution of visual information
First, we asked how our estimate of encoded information depended on the temporal resolution of analysis,
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Figure 3. Temporal resolution of visual information. For each bin size
Measuring the effect of temporal structure on coding
The previous analysis allowed for the possibility of any temporal structure within the spike train. There are many known physiological properties that would result in temporal patterns in the response, such as absolute and relative refractory periods and the presence of stereotyped bursts. Such patterns result in statistical dependence between time bins. However, it is not obvious what role these patterns play in coding visual information.
The probability of firing clearly varies with time (Fig. 1a, PSTH). We would neglect any additional temporal structure in the spike train if we simply considered the information in the average single time bin (Eqs. 1, 2, L = 1), instead of using long words. The information rate estimated from L = 1 represents an approximation on the assumption of independence between bins. If each bin were indeed independent of other bins (that is, if there were no further temporal interactions between bins beyond those reflected in the PSTH), then the information rate calculated at any word length L would be equal to that calculated with L = 1.
We introduce a new quantity Z for the correction in our information estimate when we take temporal patterns into account compared with the estimate we would have obtained if we had ignored them:
For a discussion of how Z relates to other quantities, see Appendix.
(3) If temporal patterns as such encode visual information, then the pattern correction term Z is positive, because one time bin is synergistic with other bins in encoding information. Conversely, if temporal patterns merely introduce redundancy into the neural code, the pattern correction Z is negative, because the stimulus information conveyed by one time bin is partly redundant with the information contained in other bins.
For the responses shown in Figures 1-3, the pattern correction Z was positive. The estimate obtained at 1 msec resolution but neglecting patterns, I(L = 1,
As we increased the word length L at
4), the fraction of nonempty words that contained multiple spikes increased linearly with word length, crossing 12.5% at L = 10.
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Figure 4. Effect of word length on estimated visual information. Information I between the neural response and the visual stimulus (Eq. 1) as a function of the reciprocal of word length 1/L, computed at temporal resolution
We suggest that the information in firing patterns is the result of temporal structure within the spike trains that occur on a short time scale. To test this, we performed a control in which words were composed not from L contiguous bins but from L bins well separated in time (chosen randomly from 8 sec samples). To compute Hnoise, a set of words was extracted using the same bins from each repeat. In these words of nonadjacent bins, our estimate of the information rate depended only very weakly on word length (Fig. 4, scrambled). This confirmed that the relevant temporal interactions are local in time.
Population results
Among the 11 cells we analyzed, we obtained a range of firing rates in response to this full-field white-noise stimulus. Absolute information rate was strongly correlated with firing rate, such that the information encoded per spike was relatively constant at 3-4 bits/spike (Fig. 5a). Coding efficiency can be defined as the fraction of response entropy at a given bin size
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Figure 5. Summary of results for all cells in this study. Each filled circle represents one cell. Results are shown for lim L
All cells showed a precision of spike timing of 2 msec or better. The estimate obtained at
Not all cells encoded information in spike patterns. Analyzed at 1 msec resolution, the pattern correction Z ranged from
One of our cells showed a substantial net redundancy in spike patterns, with a pattern correction of
Simulations
To gain insight into what kinds of neural mechanisms could produce either 0, negative, or positive Z, we analyzed three kinds of artificial spike trains. In condition A, spike trains were generated by a Poisson process according to a time-varying firing rate taken from that of a real neuron (Fig. 1a, PSTH). Thus firing probability was modulated with the same high temporal precision as the cell. However, the estimate of information did not vary with word length L, such that Z = 0 (Fig. 4, Poisson).
In condition B, spike trains were generated exactly as in A, but for every spike generated in A, a doublet of spikes several milliseconds apart was produced in B. In this case, separate 1 msec time bins carried partly redundant information, which would be "counted twice" if bins were assumed to be independent. Thus, we predicted that this kind of temporal structure would add redundancy to the neural code. Indeed, the estimate of information using L = 1 was an almost twofold overestimate of the true information (compared with lim L
Finally, in condition C, spike trains were generated as in A, but noise was added in the form of spike time jitter. Nearby spikes were jittered by a correlated amount, simulating a low-frequency noise source. The result was that the precision of individual spikes was degraded, but local temporal patterns were preserved. In this case, the noise (temporal jitter) would be counted twice if bins were assumed to be independent (L = 1). Thus, we predicted that responses would be more reproducible at the level of words than one would expect from the variability of each bin. Simulations confirmed that, in spike trains of this kind, L = 1 produced an underestimate of the true visual information. Therefore, the pattern correction was positive. For example, one such model with a 1/f noise spectrum and a spike jitter of
In our data, it is not uncommon to observe jitter from trial to trial in which the relationships between nearby spikes are preserved. Model C suggests that this would be one source of temporal coding in our data, but there are probably others. Another source of trial-to-trial noise is the variable presence of a spike as opposed to jitter in the timing of the spike. We sometimes observe positive correlations between two unreliable events; if one spike is present, both are. Conversely, we find cases of "either-or" firing in which events separated by several milliseconds in time occur in a mutually exclusive manner. We suspect that these types of temporally correlated noise also contribute to pattern coding. Figure 1c illustrates all three types of noise in our data.
DISCUSSION
We have demonstrated information rates in excess of 100 bits/sec and rates of 3-4 bits/spike in spike trains of single neurons in the cat LGN. Thus, LGN cells can carry high rates of information about dynamic stimuli, as a similar analysis has revealed of several other types of neurons (Berry et al., 1997
as et al., 1998
One of the fundamental questions of neuroscience is how sensory information is encoded by neural activity. The first evidence on this question was the discovery that the amount of pressure of a tactile stimulus correlated with the number of spikes elicited in a somatosensory neuron over the duration of the stimulus (Adrian and Zotterman, 1926
It has long been recognized that such a "rate code" may not exhaustively describe the neural code and that additional features of spike timing may also carry stimulus information (MacKay and McCulloch, 1952
Much of the discussion in the literature has confounded two different ideas. The first idea is that the firing rate (or probability of firing) may vary on a much finer time scale than was classically explored: milliseconds rather than hundreds of milliseconds. The second idea is that temporal patterns of firing must be considered to fully describe the neural code. To date, most of the evidence for information in spike timing in single neurons has been consistent with the first idea. We have proposed a way to uncouple these two phenomena and have demonstrated that both forms of coding are evident in our data.
Our results constitute an existence proof that, in the LGN, visual information can be encoded both by temporally precise modulation of firing rate, to ~1 msec resolution, and also in firing patterns that extend over time. The time scales and relative contributions of spike timing and spike pattern will presumably be different for different types of neurons. From our small sample of cells, we cannot yet say whether there will be systematic differences between different cell types in the LGN. Even for a single neuron, results may depend significantly on the particular set of stimuli used. Finally, we have not considered spike patterns across multiple cells, which may introduce redundancy or may encode additional visual information (Abeles et al., 1994
Temporal precision of firing
We began by analyzing the temporal precision with which spike timing is visually driven. Sharp peaks in the PSTH revealed that rate was modulated on time scales as fine as 1 msec (Fig. 1). As a result of this precision, our estimates of visual information depended strongly on the temporal resolution of the analysis, down to at least 1 msec (Fig. 3).
Similar studies have found high temporal precision in several other visual cell types (Berry et al., 1997
Previous discussions have proposed criteria for distinguishing a rate code from a temporal code on the basis of the time scale alone. For example, when long time scales are considered, firing rate can be defined by the number of events counted from an individual cell in a single trial. When very short time scales are used, firing probability would have to be estimated from many trials or cells. Thus, one relevant time scale is the mean interspike interval; time bins longer than this have on average >1 spike and shorter bins have on average <1 spike (Fig. 3, e). However, we note that the mode interval (Fig. 3, b) is typically far shorter than the mean interval. The absolute refractory period of the cell (Fig. 3, a) defines the absolute minimum interval and places a limit on the maximum firing rate the neuron could sustain.
Another intrinsic time scale in the experiment is the integration time of the neuron. For instance, this might be represented by the duration of the positive phase of the neural response to a brief stimulus pulse (Fig. 3, d). A convolution of the stimulus by the impulse response would be expected to temporally filter (blur) the stimulus at approximately this time scale.
Alternatively, one might compare the precision of the neural response with the time scale of the stimulus rather than any neural parameter. It has been argued that, when temporal structure in the response matches the temporal structure in the stimulus, the neuron simply represents time with time. According to this view, if the spike train encodes information using a higher temporal precision than the time scale on which the stimulus changes, the neuron would be using temporal patterns to encode something nontemporal (Theunissen and Miller, 1995
We believe that all of these arguments identify landmarks along a continuum and that information encoded in the firing rate of a cell at any time scale is a direct extension of the idea of a rate code. A qualitative distinction can be made between all of these forms of coding and any code that depends on relationships within the spike train. Such pattern information cannot be extracted from the rate (or probability) of firing alone, regardless of time scale.
Poisson models
A common model for a neuronal rate code is a homogeneous Poisson process whose mean rate is stimulus-dependent (Shadlen and Newsome, 1994
The information estimates from the cell and the Poisson model differ at L = 1 because the number of spikes in each time bin was more precise than expected from a Poisson process, as has been observed for several other types of neurons (Levine et al., 1988
Temporal patterns of firing
In our second analysis, we held the temporal resolution of the analysis constant and considered the role of temporal patterns in the neural code. We define a quantity, Z, that is positive when information is encoded in temporal patterns and negative when patterns constitute redundancy in the code. We found that as much as 25% of the visual information could be attributed explicitly to firing patterns. Using 1 msec bins, we had enough data to explore words of up to 10 msec, and information continued to increase with L up to this limit (Fig. 4). Using lower resolutions to explore longer time windows, we estimated that patterns contain information to at least 64 msec. Interestingly, a quite different analysis revealed a similar time scale for temporal patterns in cortical responses to low-entropy stimuli (periodic stimuli or transiently presented, static textures) (Victor and Purpura, 1997
Temporal patterns could be caused by a direct effect of one stimulus-evoked spike on the subsequent ability of the stimulus to evoke other spikes at nearby times in the same trial. Examples of this type that have been discussed in the context of information coding include refractoriness (Berry and Meister, 1998
Alternatively or in addition, the ability of the stimulus to evoke spikes could be modulated by an external noise source that has a long enough time scale to affect more than one spike. This requires merely that the noise have a lower frequency spectrum than the visual signal after temporal filtering by neural mechanisms, as in simulation C above. In either case, observing extended patterns of spikes would reveal that response patterns were more predictable than if the noise had been independent from bin to bin.
Although some cells encoded information in temporal patterns, we found other cells with a net temporal redundancy (Z < 0). The largest such effect, Z =
For all cells in this study, the majority of the encoded information was contained in the temporal precision of firing alone; whether negative or positive, the pattern correction Z was never greater than 25%. Nonetheless, short-range temporal interactions served to increase visual information substantially for many cells (Fig. 5). This indicates that, in those cells, any redundancy between time bins was more than compensated by synergistic effects. We have no reason to think that this result is particular to LGN neurons. Instead, it may be found that neurons in many parts of the brain can exploit both precise spike timing and temporal patterns of spikes to encode useful information. This suggests a possible function for the known sensitivity of neurons to the precise timing and patterns of spikes in their input (Usrey et al., 2000
FOOTNOTES Received Sept. 28, 1999; revised April 28, 2000; accepted May 1, 2000.
This work was supported by National Institutes of Health Grants R01-EY10115, P30-EY12196, and T32-NS07009. Christine Couture provided expert technical assistance. Sergey Yurgenson suggested the control of composing words from temporally distant bins. We thank our many colleagues who provided critical reading of this manuscript, particularly C. Brody, M. DeWeese, and M. Meister.
Correspondence should be addressed to Clay Reid, Department of Neurobiology, Harvard Medical School, 220 Longwood Avenue, Boston, MA 02115. E-mail: clay_reid{at}hms.harvard.edu.
APPENDIX
We define a pattern correction term Z as the difference between the extrapolated information rate for infinitely long words, I(lim L
The term I(L = 1,
(3) For the interested reader, we derive below an alternative, equivalent expression for Z directly in terms of the statistical structure within spike trains.
The need for Z arises because of the statistical dependence between time bins within the spike train. This property of a spike train can be measured by the average mutual information between a single bin and all other time bins. We will call this Iint, for "internal" information. Iint measures the average reduction in uncertainty about one bin if you know the others, which is to say, the entropy of a single bin minus the entropy of a bin given all other bins. The average entropy of one bin given all others is equal to the average entropy per bin in the entire sequence (Shannon, 1948
We stress that Iint is not information about the stimulus. Rather, it measures the predictability of one time bin of the response from the rest of the response. Note that Iint takes into account statistical dependencies of all orders, not just those revealed by an autocorrelation function. In the following discussion, the dependence of all values on the choice of
(4) If time bins are statistically independent of one another, then H(L = 1) = H(lim L
A simple rearrangement of Equation 4 produces
By substituting the right side of Equation 4' for both Htotal and Hnoise in Equation 1, we can rewrite the mutual information between the response and the stimulus as follows:
(4`)
![=[H<SUB><UP>total</UP></SUB>(L=1)−I<SUB><UP>int,total</UP></SUB>]−](/content/vol20/issue14/fulltext/5392/img008.gif)
![[H<SUB><UP>noise</UP></SUB>(L=1)−I<SUB><UP>int,noise</UP></SUB>]](/content/vol20/issue14/fulltext/5392/img009.gif)
![=[H<SUB><UP>total</UP></SUB>(L=1)−H<SUB><UP>noise</UP></SUB>(L=1)]+](/content/vol20/issue14/fulltext/5392/img010.gif)
![[I<SUB><UP>int,noise</UP></SUB>− I<SUB><UP>int,total</UP></SUB>]](/content/vol20/issue14/fulltext/5392/img011.gif)
![=I(L=1)+[I<SUB><UP>int,noise</UP></SUB>− I<SUB><UP>int,total</UP></SUB>]](/content/vol20/issue14/fulltext/5392/img012.gif)
Finally, substituting Z from Equation 3, we arrive at an alternative expression for the pattern correction Z in terms of two measurements of Iint:
![<LIM><OP><UP>lim</UP></OP><LL>L → ∞</LL></LIM>I(L)− I(L=1)=[I<SUB><UP>int,noise</UP></SUB>− I<SUB><UP>int,total</UP></SUB>]](/content/vol20/issue14/fulltext/5392/img013.gif)
This formalizes the intuitive notion that the pattern correction Z is entirely attributable to the statistical dependence between bins within spike trains. Recall that all values in Equation 5 depend on the temporal resolution
(5) Although both Iint,noise and Iint,total must be positive, the difference between them, Z, may be positive or negative. This is why I(L = 1) is neither an upper bound nor a lower bound on the true information rate. To the extent that the dependence between bins renders the visual information in different bins redundant, L = 1 will produce an overestimate, our estimate will decrease with increasing L, and the pattern correction will be negative, Z < 0. On the other hand, to the extent that relationships between bins are used to encode stimulus information, L = 1 will produce an underestimate, and our estimate will increase with increasing L, reflecting a positive pattern correction, Z > 0.
In the special case of a spike train in which the time bins are statistically independent, Iint,noise = 0 and Iint,total = 0, leading trivially to Z = 0 (Eq. 5) and thus I(lim L
Note added in proof
Two different variants of the direct-information method have been used recently to support similar conclusions for visual neurons in the fly (Brenner et al., 1999
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