1Central Research and Development, E. I. Du Pont De Nemours and Co., Inc., Wilmington, Delaware 19880-0328; and 2Department of Neuroscience, University of Pennsylvania School of Medicine, Philadelphia, Pennsylvania 19104-6074
![]() |
ABSTRACT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Rogers, Robert F., Jacob D. Runyan, A. Ganesh Vaidyanathan, and James S. Schwaber. Information Theoretic Analysis of Pulmonary Stretch Receptor Spike Trains. J. Neurophysiol. 85: 448-461, 2001. Primary afferent neurons transduce physical, continuous stimuli into discrete spike trains. Investigators have long been interested in interpreting the meaning of the number or pattern of action potentials in attempts to decode the spike train back into stimulus parameters. Pulmonary stretch receptors (PSRs) are visceral mechanoreceptors that respond to deformation of the lungs and pulmonary tree. They provide the brain stem with feedback that is used by cardiorespiratory control circuits. In anesthetized, paralyzed, artificially ventilated rabbits, we recorded the action potential trains of individual PSRs while continuously manipulating ventilator rate and volume. We describe an information theoretic-based analytical method for evaluating continuous stimulus and spike train data that is of general applicability to any continuous, dynamic system. After adjusting spike times for conduction velocity, we used a sliding window to discretize the stimulus (average tracheal pressure) and response (number of spikes), and constructed co-occurrence matrices. We systematically varied the number of categories into which the stimulus and response were evenly divided at 26 different sliding window widths (5, 10, 20, 30, ... , 230, 240, 250 ms). Using the probability distributions defined by the co-occurrence matrices, we estimated associated stimulus, response, joint, and conditional entropies, from which we calculated information transmitted as a fraction of the maximum possible, as well as encoding and decoding efficiencies. We found that, in general, information increases rapidly as the sliding window width increases from 5 to ~50 ms and then saturates as observation time increases. In addition, the information measures suggest that individual PSRs transmit more "when" than "what" type of information about the stimulus, based on the finding that the maximum information at a given window width was obtained when the stimulus was divided into just a few (usually <6) categories. Our results indicate that PSRs provide quite reliable information about tracheal pressure, with each PSR conveying about 31% of the maximum possible information about the dynamic stimulus, given our analytical parameters. When the stimulus and response are divided into more categories, slightly less information is transmitted, and this quantity also saturates as a function of observation time. We consider and discuss the importance of information contained in window widths on the time scales of an excitatory postsynaptic potential and Hering-Breuer reflex central delay.
![]() |
INTRODUCTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
In efforts to understand
neuronal responses to physiologically relevant stimuli, the most
widely used and accepted experimental paradigm may be described as
follows: the animal/system is presented with a (usually static)
stimulus set many times, and a peristimulus time histogram is generated
in which the accumulated response to many stimulus presentations is, in
effect, averaged. The result is called the cell's "response to the
stimuli," given the experimental conditions, etc. However, as others
have pointed out (Reike et al. 1997), this is rather
unrelated to the problem that the nervous system must solve. The
nervous system must, often in short order, perform accurate,
appropriate computations and generate behavioral responses and/or motor
commands given just one stimulus presentation, or, more commonly, given
a continuously changing stimulus. In this biological scheme, the
currency of information content and exchange among and between neurons
is the ongoing action potential trains of cells, be they sensory
(primary afferent), motor, or any other. The central problem for the
neurophysiologist lies in decoding these spike trains.
The application of information theory has been particularly useful in
the spike train decoding problem (Reike et al. 1997; Richmond and Optican 1990
; Warland et al.
1997
; Werner and Mountcastle 1965
), especially
in sensory systems. Two basic experimental models have been utilized.
The first one involves repeated presentations of a set of temporally
discrete stimuli, as described above. One may then ask the question,
"given a single spike train response, can we predict (and with what
certainty) which stimulus was presented?" This method has been used
successfully by several investigators with impressive results (e.g.,
Richmond and Optican 1990
; Theunissen and Miller
1991
; Werner and Mountcastle 1965
), and the
approach provides a measure of performance of a task that is related to an actual problem the nervous system must solve, stated above. Recently, another approach has been developed (Bialek and Zee 1990
; Bialek et al. 1991
) and applied
(Reike et al. 1997
; Theunissen et al.
1996
; Warland et al. 1997
) by investigators to
determine how well a spike train can describe a continuous, dynamic
stimulus in real time. This is done by determining the quality of the
stimulus reconstruction (derived directly from convolving the spike
train with an optimal filter), as compared to the actual stimulus.
The present study examines the continuous spike trains of a specific
type of vagal primary afferent, the slowly adapting pulmonary stretch
receptor (PSR). These visceral mechanoreceptors respond to deformations
of the pulmonary tree/lungs and normally produce a burst of action
potentials during each breath. They provide the CNS with feedback
regarding rate and depth of respiration. These neurons project
centrally into the nucleus of the solitary tract and have been
anatomically and physiologically demonstrated to form synaptic contacts
with two types of neurons of the dorsal respiratory group
(Anders et al. 1993; Averill et al. 1984
;
Backman et al. 1984
; Berger and Dick
1987
). Ultimately, the signal from these afferents is used to
continuously control respiratory effort and cardiovascular function
(see Kubin and Davies 1995
for review).
The PSR spike trains analyzed in the present study were generated in
response to a dynamic tracheal pressure stimulus, analyzed in a
temporally discrete manner. We are interested in decoding the spikes of
these transducers in response to a continuous, physiologically relevant, dynamic stimulus. Using this approach, we hope to gain insight into the central problem of decoding the spike response in the
context of a signal that the system actually processes in real time.
This is done by defining the information content in the spike trains
using metrics that are ultimately based on information theory
(Shannon 1948). Our methods combine aspects of
both information theoretic approaches mentioned above. Instead of
attempting to optimize a reconstruction of the stimulus from the spike
train, which requires a Gaussian-distributed stimulus to confidently
place a lower limit on the rate of information transmission (see
Reike et al. 1997
), we take the direct approach of
defining information by a reduction in entropy or uncertainty of the
joint probability distribution that assumes mutual independence between
the stimulus and response (Shannon 1948
). The parameters of our analysis are chosen to examine a continuum between two specific
perspectives: that of an "ideal observer" and that of the actual
biological system.
A preliminary report of this work was published in abstract form
(Rogers et al. 1999).
![]() |
METHODS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Nineteen adult male New Zealand White (3.15-4.45 kg) rabbits were used in this study. All protocols and procedures were approved by the DuPont Neural Computation Group's Institutional Animal Care and Use Committee. Our group's program is fully accredited by the Association for Assessment and Accreditation of Laboratory Animal Care (AAALAC) International.
Surgical preparation
The animals were initially anesthetized intravenously with saffan (8-12 mg in a marginal ear vein) and then urethan (1.6 g/kg) was slowly infused over the course of ~1/2 h. One femoral artery and vein were cannulated for measurement of arterial pressure and intravenous infusion, respectively. The level of anesthesia was assessed by limb withdrawal reflex and the stability of arterial pressure and cardiac rate following a noxious stimulus (hindlimb pinch). It was never necessary to supplement the original dose of urethan.
The trachea was intubated below the larynx, and animals respired spontaneously. A lateral incision was made between the anterior tip of the scapula and the base of the pinna. Approximately 3-4 cm of the cervical vagus nerve (VN), as close to the clavicle as possible, was dissected free from the surrounding tissue and placed intact on bipolar hook electrodes for stimulation and recording (see Stimulus and recording techniques). The nodose ganglia were dissected free from surrounding tissue and situated in such a way as to eliminate movements produced by pulsation of the nearby internal carotid artery. The nerves and ganglia were immersed in warm mineral oil to prevent dehydration.
Core temperature was continuously monitored via a rectal probe and stabilized at 38.5 ± 0.5°C using a heating blanket. This ensured that the mineral oil pool in the neck was maintained at ~37°C. End tidal CO2 was also monitored and stabilized at 4.5 ± 0.5% between periods of data recording. All animals received a physiological saline intravenous drip (~20 ml/h) during the surgical and recording periods.
Stimulus and recording techniques
The lungs were mechanically inflated with a standard ventilator
(Harvard Bioscience). Tracheal pressure (TP) was recorded at the
inspiratory sidearm of the trachea tube and used as a measure of the
mechanical deformation of the lungs. The electrical activity of
individual neurons was recorded at the level of their somata within the
nodose ganglion using a "floating electrode" configuration (Z ~ 1.0 M at 1 kHz) (Averill et al. 1984
).
Electrodes were inserted into the nodose ganglion manually without
removal of the connective tissue capsule, and a micromanipulator was
sometimes used to make fine adjustments to the electrode tip position.
PSRs were identified by their slowly adapting response to a step
increase in TP and faithful following of manipulations in TP.
Peripheral axonal conduction velocities were estimated by one of two
methods. The first method was by recording the response latency of
somatic potentials to electrical stimulation (0.2 ms, 1.5T,
every 1.5 s) of the vagus nerve and measuring the distance from
the stimulus electrodes to the recording site. Stimuli were delivered
via a pulse generator (Neurodata PG 4000) used to trigger isolated
stimulators (models A360 and A350D-A, WPI). The second method used was
(somatic) spike-triggered averaging of the vagal whole nerve recording
(see Backman et al. 1984) and subsequent measurement of
the distance between the nodose recording site and the vagal recording
electrodes. The times of the action potentials recorded in the nodose
ganglion were corrected relative to the TP stimulus by using the
conduction velocity of the PSRs' peripheral axon and the mean distance
from the nodose ganglion to the pulmonary tree/lungs. Due to the
possibility of varying conduction velocity along the peripheral axon,
as well as the rough estimate of the location of the PSR sensory
ending, there may be a slight error built into the corrected spike
times, but this is likely to be small compared with the sliding window
step size of 5 ms (see DATA DISCRETIZATION).
The TP stimulus waveform was generated via constant manual manipulation of both the rate (mean ~40 breaths/min) and volume (mean peak TP ~6 mmHg) of a mechanical ventilator. During single-unit recordings, the animals were paralyzed (vecuronium bromide, 70 µg iv boli, approximately every 45 min). Single units were recorded continuously for anywhere between 45 min up to 3 h, during which the TP rate and volume was constantly varied. The TP (200 samples/s), unit recording (8,000 or 10,000 samples/s), and VN recording (only for some cells; 4,000 samples/s) were stored on the hard drive of a PC (80486) via a data acquisition A/D board (National Instruments, AT-MIO-9) controlled by variable-speed, multi-channel data acquisition software (DataSponge version 3.02j, Bioscience Analysis Software). Following the recording sessions, the animals were killed by Saffan overdose.
Entropy and information computations were performed by custom-written software running on Microsoft Visual C++, and some postanalysis was performed using Microsoft Excel. All were executed on a conventional PC (Windows 95 OS). Preanalysis (spike time determination, waveform analysis, spike-triggered average of VN activity, and stimulus smoothing) and spike statistics (interspike interval analyses, 3-D data visualization, autocorrelation functions, data splicing, curvefitting, and variance analysis) were performed with the use of either custom-written or commercially available M-files written for Matlab toolboxes (version 5.2, The MathWorks) running on a Sun Enterprise 3000 workstation (Solaris OS, version 5.7) with 2.5 GB of RAM.
Spike train analysis
In this study, we calculate several measures related to the
information content within the spike trains of individual PSRs. Ultimately, we calculate the average information transmitted by the
spike train about the TP stimulus. We use a single metric to describe
this information quantity (i.e., Eq. 11). This derivation of
information is based on that proposed by Werner and Mountcastle (1965), itself rooted in the classical definition of mutual
information (Shannon 1948
).
DATA DISCRETIZATION. One of our main objectives was to quantify the amount of information that was transmitted by the PSR spike train regarding absolute TP values. We do not require a specific stimulus structure to do this, so we used a stimulus with a probability distribution similar to that which the animal would experience under a variety of naturally occurring respiratory behaviors, but one that was less temporally predictable. This allowed us to estimate how much information is transmitted by PSRs under natural physiological conditions.
Figure 1 summarizes the general approach used in this study. The stimulus and spike train were discretized into continuous, overlapping time segments of equal length (i.e., a fixed-width sliding window), which was advanced in steps of 5 ms. At each successive window position, one specific attribute of both the TP stimulus (its mean value) and the spike train (total number of spikes in the window) was measured and used to increment the appropriate bin in a co-occurrence matrix, created as follows. First, the maximum and minimum values of these attributes were determined, and their distributions were divided into bins of equal width. The stimulus-response co-occurrence matrix could contain M response columns, which could vary from M = 2 up to M = n + 1, where n equals the maximum number of spikes observed in that sliding window width at any window location (i.e., 1 column for every response magnitude of 0, 1, 2, ... , n spikes/window). The stimulus distribution was divided equally into N rows, where N
|
SHANNON ENTROPY AND INFORMATION.
Once the co-occurrence matrix of size N × M was constructed, containing the number of
observations of all stimulus-response combinations, the normalized
stimulus entropy was calculated. This was done by first normalizing the
row sum values by the total stimulus occurrences (i.e., sum across each
entry in a row to obtain each row sum, RS, and then divide
each RS by the total number of stimulus observations). This
defines a probability of observing any given stimulus,
P(stimi), as simply
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
![]() |
(5) |
![]() |
(6) |
![]() |
(7) |
![]() |
(8) |
![]() |
(9) |
![]() |
(10) |
![]() |
(11) |
![]() |
(12) |
![]() |
(13) |
SPECIAL ANALYSES. PSRs display no change in firing rate below their threshold TP, even though the TP below these levels may be changing. For some cells, we examined what, if any, effects these silent periods exert on information transmission. To do this, we used two general approaches. In both cases, we chose a cutoff threshold just below that of the cell's TP threshold. In the "splice" method, we removed all TP data points and spikes (typically, <1% of the original spikes) that occurred during the times that TP was below threshold and spliced the remaining data together to make a new "continuous" stimulus and spike train. The other approach, which we term the "flatten" method, involves no removal of TP data points. Instead, we made all TP data points below the chosen TP threshold equal to that level, and used this new TP waveform with the original spike train, less the spikes that occurred during times when the stimulus was below threshold. In both cases, the stimulus traces were rescaled such that their dynamic ranges matched that of the original TP traces. The data were then analyzed as described above.
For multiple unit recordings, we created composite spike trains that contained the spike times of all the individual PSRs. Using the same stimulus waveform as in the analysis for individual PSRs, we calculated the same metrics as described above. These were then compared with the information and efficiency data calculated for the individual PSRs that went into the composite. We compared the sum of the information provided by the individual units to that provided by the composite spike train, by calculating raw information in bits/s (see above).Central delay determination
To determine a reasonable observation/integration time for the system, we evoked the Hering-Breuer reflex via electrical stimulation. One animal was bilaterally vagotomized, and the integrated voltage of the central end of the transected phrenic nerve was recorded (sampling rate, 4 kHz; integration time, 20-200 ms). The central end of the vagus nerve ipsilateral to the recorded phrenic nerve was stimulated (50 or 100 Hz; 50 µA, 0.2-ms width) during inspiration, and the stimulus trigger signals were digitized and recorded at a sampling rate of 4 kHz. We consider the delay between onset of the stimulus and onset of the diminution of phrenic output to be the maximal allowable time for spike train integration/interpretation by the brain stem and spinal cord structures. Therefore we hypothesize that a significant fraction of information about TP can be extracted from the spike train when both are considered using sliding window widths less than or equal to the central delay.
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Forty-two recordings were made from neurons classified as pulmonary stretch receptors. Of these, we selected 19 because their recordings were held long enough to perform the information theoretic analysis. The following characteristics of these neurons are given in the form of means ± SD (with the range in parentheses). Their maximum instantaneous firing frequencies were 205 ± 52 Hz (108-250 Hz). Their conduction velocities were 19.15 ± 6.78 m/s (8.01-39.09 m/s). Their TP thresholds were 1.87 ± 1.72 mmHg (0-5.5 mmHg). The central delay averaged 152.4 ± 5.4 ms (148-156 ms) for eight inductions of the Hering-Breuer reflex.
Stimulus, response, and joint probability distributions from a typical
PSR recording, as well as the corresponding values of the respective
entropies are given in Fig. 2. In this
specific example, the width of the sliding time window was 150 ms. The dynamic range of the stimulus (mean TP within the window; 1.4-14.8 mmHg) and response (the number of spikes that occurred within the
window; 0-21) were discretized into 20 and 22 categories, respectively. For comparison, the joint probability distribution that
would yield
Hstim,resp(max) is
also plotted (P*stim,resp in
Fig. 2). As the figure shows, there is a marked difference between
P*stim,resp and
Pstim,resp. Compared to
P*stim,resp,
Pstim,resp shows a prominent
reduction in its dispersion of events (i.e., entropy) over the joint
probability surface, specifically exhibiting a diagonalization of
co-occurrences. Given this choice of parameters, the neuron's spike
train provides 0.201 (I) of the maximum theoretical information about the stimulus. For these same parameters,
D = 0.470 and E = 0.353.
|
Effects of analysis parameters
I, D, and E were influenced by the choice of SA and RA sizes. Here we describe how information varies as we demand finer discrimination of the stimulus by increasing the stimulus alphabet size, or as we require increased resolution of the spike response by increasing response alphabet size. Figure 3 shows the effects of systematic alteration of the SA and RA sizes on I, E, and D for a PSR using a sliding window width of 150 ms. I is relatively insensitive to changes in either SA or RA sizes (except when SA increases from 2 to 3), but tends to increase modestly with decreases in RA size. E increases as SA size and RA size increase and decrease, respectively, with a marked increase as RA size is reduced below 4. D increases modestly as SA size and RA size decrease and increase, respectively. In addition, unnormalized I (Eq. 9) increases with increases in SA and RA (data not shown).
|
Figure 4 shows the location, in SA-RA space, of the maximum I, E, and D values for all window widths and all PSRs. As Fig. 4 illustrates, the result shown in Fig. 3 is typical of all PSRs studied in that the most information is transmitted when the SA and RA sizes are relatively small, that is, when the system is not forced to provide information about the precise stimulus or response level. The maximum value of I tends to occur at small stimulus and response alphabet sizes (<6). Maximum values of E generally occur at a high stimulus resolution (20 categories) and a low response resolution (2 or 3 categories). The conspicuous increase in incidence of maximum E values at a stimulus alphabet size of 20 is due to the use of 20 categories as the most common upper limit on SA (see METHODS). Maximum D values tend to occur when the stimulus alphabet is small (2 or 3) and near maximum values of the response alphabet. The distribution along the response alphabet dimension is due to covariation of maximum D with the maximum spike resolution, which varies with window width.
|
The effect of the sliding window width on the information transmitted is of particular physiological importance. This analysis describes how the observation time of the stimulus and response influences the average information we gain by this observation. Figure 5 shows results for one PSR. Three common observations appear in this data. First, I saturates rapidly as window width is increased from 5 to ~50 ms (Fig. 5, top), and this is due to a saturating raw information function normalized by a saturating Hstim,resp(max) function (see Eq. 11). Thus there is limited advantage, in terms of increase in information transmitted, in observing/integrating over a long period (i.e., a wide window width), even though the dynamic range of the response increases substantially (as does Hresp; Fig. 5, middle) and the Hstim remains relatively constant (Fig. 5, bottom). Second, there is very little (typically <10%) information available when very narrow sliding windows (e.g., 5 ms) are used. Third, both D and E saturate rapidly as a function of window width, as do the D~1 and E~1 functions (Fig. 5, middle and bottom).
|
This general trend held true for all PSRs analyzed. The I and I~1 data for all PSRs studied are shown in Fig. 6. In this figure, the information measures have been normalized within each unit, expressing all values as a fraction of the neuron's maximum level. For I, the average ± SD percent of the cell's maximum value at 5, 50, and 150 ms was 11.3 ± 5.7%, 76.4 ± 13.3%, and 94.0 ± 5.2%, respectively. The same values for I~1 were 8.2 ± 4.5%, 76.5 ± 11.4%, and 94.4 ± 5.0%, respectively. For all 19 single units analyzed, the average global (i.e., over all window width-SA-RA combinations) maximum values ± SD were I = 0.309 ± 0.051 (0.195-0.380); I~1 = 0.234 ± 0.046 (0.124-0.298); D = 0.785 ± 0.066 (0.700-0.881); D~1 = 0.505 ± 0.130 (0.267-0.684); E = 0.774 ± 0.090 (0.494-0.918); and E~1 = 0.542 ± 0.134% (0.263-0.787). No statistically significant relationship between TP threshold and I or I~1 was found.
|
Multiple unit analysis
To determine the information available from simultaneously
recorded PSRs, we analyzed the information available in their spike trains alone and in combination by creating a composite spike train
that contained all the spike times from the individual units. Figure
7 shows these results for a two-unit
recording made from a single electrode in the nodose ganglion. The
maximum firing frequencies and TP thresholds for units A and B were 125 and 154 Hz, and 3.3 and 3.6 mmHg, respectively. The maximum mutual
information (Fig. 7, top left) transmitted by combined spike
train of units A and B (AB in Fig. 7) is greater than that transmitted
by either unit alone at all window widths compared with unit A, but not at all window widths for unit B. The average (over all sliding window
sizes) increase in I for AB compared with unit A is 0.074 and just 0.003 compared with unit B. However, when we force an increased resolution of the stimulus and response alphabets, as in the
I~1 metric (bottom left
of Fig. 7), there is an advantage in observing both spike trains when
compared with unit A at all window widths, but a disadvantage when
compared with unit B, save at window widths of 5, 10, 20, and 30 ms.
The average increase in I~1 for AB
compared with unit A is 0.056 and 0.021 compared with unit B.
|
Recalling that I is a normalized value, care must be taken not to assume that less raw information is transmitted by the combined spike train than the individual units. To evaluate this, the average raw information rate is calculated (Eq. 9), and we can then compare the sum of the information in the individual units (A+B in Fig. 7, bottom) to that transmitted by the combined spike train. The bottom plot of Fig. 7 shows that at window widths >50 ms, the information in unit A is completely redundant with that already present in unit B because the information provided by AB is essentially identical to that of unit B alone at these observation times. Thus one only gains an increase in information rates while observing both units if integration times are brief.
Maximum encoding efficiency, E, is greater when both spike trains are used than when that of either individual neuron is used for all window sizes except 5 and 10 ms (Fig. 7, top right), yielding an average increase of 0.121 over A and just 0.032 over B. By contrast, when stimulus and response resolutions are forced to higher level (E~1 ), coding efficiency using both sets of spikes is actually less than that for unit B alone, and greater than unit A alone, at all window sizes (Fig. 7, bottom right). On average, using the combined spike train response results in a 0.072 increase in E~1 when compared with unit A and a decrease of 0.151 when compared with E~1 of unit B alone.
Information in subthreshold data
Inclusion of time intervals when the TP stimulus was below threshold (i.e., the stimulus is changing while the response is constant) may yield lower average mutual information than those periods when both the stimulus and response were changing. To examine the possibility that our naturalistic stimulus is suboptimal in this regard, we examined the effects of either splicing or flattening the stimulus during these time intervals on information transfer (see METHODS for details). Figure 8 shows that the removal of these periods (splicing) results in less information than is transmitted by the original data, while flattening of the stimulus below threshold results in more information than the original data. The threshold chosen (TP = 3.0 mmHg) resulted in removal of 0.8% of the original spikes. Both the maximum (E) and forced resolution (E~1) encoding efficiencies were diminished by data splicing when compared with the original. Flattening the stimulus did not result in an appreciable increase in encoding efficiencies (Fig. 8, right). These results held true at all window sizes and for all neurons analyzed in this manner (n = 4).
|
Encoding and decoding
Using the conditional probability distributions, we may determine the encoding functions and the decoding functions for PSRs. Figure 9 (contour plots) shows the conditional probability distributions Pstim|resp and Presp|stim. Note that these are normalized, by definition, along each conditional value (e.g., the Pstim|resp distributions are normalized within each response value). As was the case for all PSRs, the conditional probability distributions in Fig. 9 are steep monotonic functions. Increased steepness of the conditional probabilities along the conditional dimension allows for greater accuracy when one uses the response to predict the stimulus (i.e., spike decoding using Pstim|resp) or when one uses the stimulus to predict the response (i.e., stimulus encoding using Presp|stim), shown in the bottom right of Fig. 9. These estimates were produced by calculating the weighted averages, along each condition, of the conditional probability distributions. The fact that the functions are monotonic permits assignment of a single, unambiguous best estimate for a given condition. The plots in this figure were produced from the same data set, with a window width of 120 ms. The encoding and decoding functions were consistently nonlinear, but they were not always as obviously related as the nonlinearities are in Fig. 9 (bottom right).
|
Response mean and variance
To examine the magnitude of variance among responses to a given stimulus category, we analyzed the variance as a function of the mean. Figure 10 shows a scatterplot of response variance versus response mean for a PSRs at sliding window widths of 5, 10, 20, 30, 40, and 50 ms. The SA and RA sizes were set at the X~1 levels. Two important features are apparent in this data set. First, the variance lies close to the theoretical minimum for a response characterized by spike count, given by the scalloped function (- - -) in the top of Fig. 10. The shape of the minimum variance function is determined by the fact that individual responses are defined as integer spike counts.
|
The data indicate that for the most part, the response variances are
extremely small, particularly at small window sizes. The "spread"
to more response categories did not dramatically increase as response
strengths increased, indicating a consistency among responses even at
higher stimulus values. The second observation is that the response
variance increases much more slowly than the mean response as
observation time grows, as shown in the plot relating Fano factor
(variance/mean) with window width (Fig. 10, bottom). At
small window widths (e.g., 5 and 10 ms), the Fano factor is large only
because the mean response is small, i.e., the response variances lie
close to or on the initial portion of the theoretical minimum curve
(Fig. 10, top). As window widths increase, the response
variances may deviate from the theoretical minimum, but this is more
than offset by the increase in mean response, resulting in a decrease
in Fano factor. It is only at larger window widths (>100 ms) that the
rate of increase of response variance begins to exceed that of the mean
response. All window widths produce Fano factors that are statistically
different from that given by a Poisson process, characterized by a
variance equal to the mean (, Fig. 10, top). Without
exception, all instances where the response variance was greater than
or equal to the mean response occurred when the mean response (to a low
TP stimulus) is near zero. All PSRs showed the same relationships
between the means and variances of responses as well as between the
Fano factor and sliding window width.
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Our basic finding in this study is that, depending on our choice of parameters, individual PSR spike trains are capable of transmitting up to 38% (~31% on average) of the available information about TP, a surrogate measure of pulmonary distention or lung volume. In addition, we find that PSR encoding efficiency may reach 92%. When we force higher degrees of resolving power with regards to stimulus and response discretization, up to 30% (~23% on average) of all the available information is transmitted. Most importantly, nearly all the information that may be gleaned about the average TP value is done so by observing the spike train for periods less than the central delay between PSR activation and the phrenic motor command output. Before considering the implications for the ideal observer and actual biophysical system, it is appropriate to discuss the relevant analytical issues associated with our methods.
Methodological issues
In this study, we report the average information transmission and
coding efficiency of pulmonary stretch receptors using a continuous,
naturalistic stimulus. One specific concern we had was that, even
though the ventilator frequency and depth was continuously changed, the
stimulus was in part predictable, and that predictability could cause
overestimation in our normalized information data. We generated a
stimulus with a similar probability distribution and temporal structure
as the natural stimulus, including a wide range of combinations of
respiratory depth and frequency. Other studies used random-order
presentation of discrete stimuli to cutaneous mechanoreceptors
(Werner and Mountcastle 1965), or neurons in primary
visual cortex (Richmond and Optican 1990
) in efforts to
eliminate the effects of stimulus history. It is worth noting that
Werner and Mountcastle reported nearly identical encoding efficiencies
(and transmitted information) when stimuli were presented in random
order versus those when the stimuli were presented in seriatim. Even
so, these investigators allowed 3-5 s between stimulus presentations,
presumably to eliminate possible confounding effects of recent stimulus
history on the response of the neuron, which may occur in the case
where a continuous stimulus is used and only one attribute of that
stimulus is analyzed (e.g., TP only vs. TP-dTP/dt combinations). The
temporal effects of discretizing continuous data using a sliding window
may impose certain increased variability in the joint probability
distributions, especially considering that fact that PSRs are slowly
adapting mechanoreceptors.
We determined that the stimulus history was not cause for significant concern in the interpretation of the present data because any temporal correlations in the stimulus were also present in the response, thereby affecting their entropies similarly since both were analyzed in the identical time window. Figure 11 shows the autocorrelation functions of the stimulus and response of a typical PSR during our experimental conditions (A) and during fixed ventilation parameters (B), a situation that is more closely matched to a natural breathing pattern. In both cases we see that the stimulus and response autocorrelations are well-matched at longer time scales (greater than or equal to the length of a typical breath; Fig. 11, top). The very short time scale response events are essentially uncorrelated under experimental conditions (0-30 ms; Fig. 11A, bottom), as are those in the fixed ventilation condition (0-10 ms; Fig. 11B, bottom). This lack of response autocorrelation at very short time scales is the sole mismatch with the stimulus autocorrelation, which is highly correlated at these offsets. The noncorrelated response behaviors 1) are in part due to the physical nature of the system, 2) diminish as the temporal resolution of the spike times decreases (i.e., increasing time/bin in the response autocorrelation; data not shown), and 3) are unavoidable when a continuous stimulus is utilized. Under natural conditions, the respiratory motor system normally generates a pulmonary distension stimulus that is both continuous and highly predictable, with TP dynamics that are slow compared with the average interspike interval of a typical PSR. This is one fundamental difference between the cardiorespiratory control system and other sensory systems (e.g., the primate visual or somatosensory system) in which the natural stimulus may be regarded as a series of temporally discrete events that are often initiated by motor commands (e.g., saccades or reaching movements). We have a very good notion of both the natural stimulus structure and behavioral motor output, both of which may be measured simultaneously in the cardiorespiratory control system. This may allow for a better contextual understanding of the activity of central neurons in this autonomic control system.
|
We are also concerned with the effects of discretizing the stimulus
because within a given category all TP stimuli are treated as identical
to others, despite any temporal history, differences in other aspects
of the stimulus (e.g., dTP/dt), or actual differences in the mean TP
values. Our variance-mean response analysis is striking in this regard
(Fig. 10). Specifically, we report that the variance is rather small
(near the theoretical limits), even given the sources of variability in
stimuli that were treated as identical. The finding of small response
variances is in accord with other reports of the variance of spike
responses of H1 neurons in the fly visual system to dynamic (de
Ruyter van Steveninck et al. 1997), and dynamic and static
(Warzecha and Egelhaaf 1999
) stimuli, as well as retinal
ganglion cells to random flicker stimuli (Berry et al.
1997
). These studies presented identical, temporally discrete
stimuli and examined the variability of the responses, bin by bin, from
one trial to another, a situation where one might expect less
variability in the responses than we observe in our data.
The second result is that PSR response variability does not increase at
the same rate as the mean when window widths are increased, as
de Ruyter van Steveninck et al. (1997) reported for
responses to static stimuli in H1 neurons. In both the present study
and in that study, one method of increasing the mean response was to
increase window width. But the result of de Ruyter van
Steveninck et al. (1997)
, as Warzecha and Egelhaaf
(1999)
have pointed out, is a methodological by-product of
increasing the mean response by increasing integration time, yet one
that we did not observe in our data. There are many factors that
contribute to the inconsistency between our results and the
Poisson-like relationship reported by de Ruyter van Steveninck
et al. (1997)
. We used a dynamic naturalistic stimulus, studied
mammalian primary afferent neurons, and only represented the response
by total spike count within a given time window. We do report an
initial decrease in Fano factor with window width, which is consistent
with what Warzecha and Egelhaaf (1999)
report for
constant and dynamic velocity stimuli. In our data, variances were
relatively small, such that the major determinant of Fano factor was
the mean response, which grows with window width. Other studies in
primate visual cortical areas (e.g., Gershon et al.
1998
) report that response variances to identical stimuli may
approach or even exceed the mean. Our results indicate that PSRs
respond quite reliably to the "same" stimulus, particularly considering the variability built into it.
Another issue we considered is the use of one information metric versus another. We chose to normalize our information values and report information as a fraction of the maximum possible. Our choice of reporting mutual information as a fraction of the maximum possible (Eq. 11) given the Pstim and Presp provides us with an internal normalization in comparing inter- and intra-neuronal information. Clearly, we could have reported information in standard units (e.g., bits), but this does not allow for ready comparisons as different SA-RA combinations and window widths permit different theoretical maximum information levels. The most important issue lies in the assumption of true mutual independence between the stimulus and the response. We note that the P*stim,resp and its associated entropy, Hstim,resp(max), assume mutual independence, and the degree to which this assumption is untrue provides us with a measure of information. It should also be noted the our Hstim is not a saturating function of window width, and thus does not serve to limit information transfer or decoding efficiency (Fig. 5).
Finally, our choice of discretization parameters is arbitrary. The RA
is naturally limited by the maximum number of observed spikes in that
time window, since they are discrete measures. The SA size, on the
other hand, was chosen with no real guidance as to what the
cardiorespiratory control system "requires" in terms of pulmonary
stretch discrimination. Given this unknown, we placed an upper limit on
the SA that was the greater of either 1) the dynamic range
of the stimulus in 1-mmHg increments or 2) 20. Thus we
examined the information transmission over wide ranges of SA and RA
size combinations at a many integration times. Given these constraints,
we then reported the maximum mutual information at each window width
(I) as well as a measure when the maximum response
resolution and the stimulus resolution of 1-mmHg/category are fixed
(I~1). There is no "correct"
choice of parameters. Werner and Mountcastle's (1965)
results are given for SA sizes of up to 30, but apparently do not
include the joint occurrences when the stimulus was below the
mechanical threshold for a given fiber (Fig. 19 of that paper). It is
unclear as to how they changed the
Hstim in that study (via
manipulations in discretization or via a new series of stimuli), but in
the present study it is realized via changes in the SA and window size,
both of which effect Pstim, Pstim,resp, and
P*stim,resp.
Fundamental implications
IDEAL OBSERVER'S PERSPECTIVE: INFORMATION ABOUT "WHEN."
In this study, we report that the spike trains of single PSRs may
transmit on the order of 38% of the theoretical maximum information
about a given stimulus (and vice versa) with efficiencies approaching
92%. These values are in accord with those reported by Werner
and Mountcastle (1965), and Reike et al. (1995)
,
despite distinct differences in methods and cell types studied. In our results, this maximum information was usually transmitted when the
stimulus was discretized into between two and six categories (Fig. 4,
top). Based on these observations, we conclude that
individual PSRs provide the most information about when the TP crosses
between course levels. In this sense, the information is mainly of the "when," rather than the "what" form. Our results agree with
those of Reike et al. (1995)
, Warland et al.
(1997)
, Reinagel et al. (1999)
in that their
reconstructions fail to always capture the specifics of the stimulus
amplitude but do a good job of tracking its general dynamics, as in
Berry et al. (1997)
. It is unclear whether our results
are in agreement with the observations reported by Werner and
Mountcastle (1965)
, who noted an increase in (unnormalized) information and a decrease in encoding efficiency as stimulus uncertainty increases. However, they do report that on average the
afferents they studied would transmit 2.5 bits of information (i.e.,
discern between 6 or 7 categories) in a 250- to 1,000-ms window
reliably, which is greater discriminating power than we observe for
PSRs. Some investigators have argued that the role of PSR feedback lies
in terminating inspiration (or in prolonging expiration) (Clark
and von Euler 1972
; Cohen and Feldman 1977
) after a certain level of pulmonary distention, due to the nonlinear increase in PSR population activity as the stimulus increases. Execution of this physiological function would require at least a
binary representation of TP, and our data indicates that this could be
provided by just one PSR (Fig. 7, bottom) every 5 ms at
running integration times on the order of Hering-Breuer reflex delays.
OBSERVATION TIME AND INFORMATION.
The relationship between the sliding window width and information that
we report (Figs. 5-7) are related to, and in agreement with, the
effects of increasing observation time of the spike train following the
onset of the stimulus described by other investigators studying
different systems (e.g., Buracas et al. 1998;
Tovée et al. 1993
; Werner and
Mountcastle 1965
). Most importantly, the present results
reinforce one of the main findings in these studies: the vast majority
of the available information in the response of a single neuron may be
obtained by observing the system for considerably less than the
behavioral time delay, be it Hering-Breuer reflex, visual
saccades/recognition, or motion sensing. The present data indicate that
at window widths of 150 ms, the spike count of the average PSR is
providing ~94% of maximum information it can, and at 50-ms
integration times the PSRs provide ~76% of the information they can,
regardless of window width. Our results are also in general agreement
with optimal filter function widths, which extend for hundreds of
milliseconds, as reported by investigators using the stimulus
reconstruction method (see, e.g., Reinagel et al. 1999
;
Warland et al. 1997
). The investigators using this method have not described the effects of systematically restricting optimal filter widths on information rates, but our experience is that
the stimulus reconstruction diverges much more from the original
stimulus as filter widths are made progressively more narrow
(unpublished data).
Information transmission/signal processing in the biophysical system
The target neurons of PSRs have been established for some time. To
interpret our results, we would like to know exactly what effect the
PSR spikes have on the postsynaptic cell. Fortunately, this is also
known. PSR spike-triggered averaging of the membrane potentials of pump
or I cells has revealed that each PSR spike causes an average
excitatory postsynaptic potential (EPSP) of amplitude 30-265 µV
(Backman et al. 1984
) or 15-86 µV (Berger and
Dick 1987
) in I
neurons and 120-239 µV in pump cells
(Berger and Dick 1987
). In addition, the average
half-width durations of these events is 0.81-4.37 ms (Berger
and Dick 1987
) or 1.04-5.07 ms (Backman et al.
1984
) in I
cells and 0.65-1.25 ms (Berger and Dick
1987
) in pump cells. Based on these comparative synaptic strength data, we may hypothesize that the (subthreshold) membrane potentials of pump cells would transmit more information about the TP
stimulus than that of I
neurons. Given the above EPSP data and the
fact that both pump and I cells' membrane potentials so obviously
co-vary with TP, we surmise that these NTS neurons are the recipients
of large numbers of convergent PSR synaptic inputs.
Signal processing implies selective retention or extraction of certain types of information at the expense of others. This is true in the case of PSRs, which provide abundant transmission of TP information (present results) and much less information regarding dTP/dt (data not shown). A faithful representation of pulmonary distension in the spike trains of individual second-order neurons is not required to control respiration, since the activity of this neuronal population may be converged at a point in the system situated closer to the motor output (e.g., at the phrenic motor nucleus, CPG, or the respiratory musculature). However, the fact that these neurons are activated by inflation of the lungs suggests that they provide at least a 1-bit representation of this variable.
One of the most interesting observations in the present data is the emphasis that covariation and information are contextual. This is highlighted by the fact that removal of periods of the stimulus and response when covariation between stimulus and response is lacking ("spliced" data, Fig. 8) actually produces less information than when these periods are included ("original" or "flattened" data, Fig. 8). Although our finding that neuronal silence carries information may seem counterintuitive and has rarely been demonstrated, it illustrates the need to observe all portions of the spike response in context in order to ascertain their information value. In addition, our measures of E (Figs. 3, 5, and 6) indicate that PSRs may encode TP in their spike trains with great efficiency at window widths as narrow as 20 ms, implying that they generate little neural "noise" at the input level of this system.
The data in this report describe what and how much information is
present in one important input to the cardiorespiratory control
networks, and imply that more information is available to the animal in
spike trains of the entire population of hundreds of PSRs, even though
there is apparently great redundancy (see Fig. 7). Our data are limited
in that they cannot quantify channel capacity, defined as the maximum
raw information a PSR can carry using a stimulus of maximum entropy
(i.e., Gaussian-distributed), but rather how much they can carry under
naturalistic conditions. We are left with a picture of information
processing that is both massively parallel, at least partially
redundant, and both convergent and divergent. Based on the present
data, we expect that a very large proportion of the available
information about TP is present in the entire population of PSRs, even
using narrow observation windows. General principles of synaptic
integration support our use of a "combined spike train" (Fig. 7)
when analyzing the information present in multiple units. Our results
also show that a simple spike count code can convey information about
TP in individual PSR responses, but we recognize that a "pattern"
code may also carry information about TP. However, as window widths
narrow to levels approaching a typical EPSP duration, the difference
between rate and timing codes is blurred (cf., Reike et al.
1997), and it is at these integration times within which
individual neurons can operate.
In summary, our data 1) quantify the information present in individual PSR spike trains; 2) suggest, through redundancy measures, how much may be available in the PSR population; and 3) provide no real indication regarding how much of this information the nervous system actually uses. Only future analysis of the cardiorespiratory control system will provide answers to questions of information utilization and manipulation in this system. This system may serve as a good model system for integration, and the analysis we propose here can quantify how much information about a given peripheral input signal (e.g., TP, arterial blood pressure, blood gases) is present in activity of individual/populations of central neurons, providing us with a better picture of a true multiple-input/multiple-output neural system that provides homeodynamic control. In addition, our analysis may be used to quantify response multiplexing, where the same spike train is used to represent multiple features (e.g., TP, dTP/dt, arterial CO2 levels, arterial pressure) simultaneously.
![]() |
ACKNOWLEDGMENTS |
---|
The authors thank Drs. M. Weiner, B. Richmond, B. Ogunnaike, and A. Owens for helpful discussions and Dr. P. Reinagel for providing stimulus reconstruction software.
This work was supported by National Heart, Lung, and Blood Institute Grant 1-R01-HL-54194 and by the DuPont Company.
![]() |
FOOTNOTES |
---|
Present address and address for reprint requests: R. F. Rogers, 521 Jefferson Alumni Hall, Jefferson Medical College, Thomas Jefferson University, 1020 Locust St., Philadelphia, PA 19107-6799 (E-mail: Robert.Rogers{at}mail.tju.edu).
Received 1 February 2000; accepted in final form 14 September 2000.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|