Peripheral and Central Determinants of a Nociceptive Reaction: An Approach to Psychophysics in the Rat

Background The quantitative end-point for many behavioral tests of nociception is the reaction time, i.e. the time lapse between the beginning of the application of a stimulus, e.g. heat, and the evoked response. Since it is technically impossible to heat the skin instantaneously by conventional means, the question of the significance of the reaction time to radiant heat remains open. We developed a theoretical framework, a related experimental paradigm and a model to analyze in psychophysical terms the “tail-flick” responses of rats to random variations of noxious radiant heat. Methodology/Principal Findings A CO2 laser was used to avoid the drawbacks associated with standard methods of thermal stimulation. Heating of the skin was recorded with an infrared camera and was stopped by the reaction of the animal. For the first time, we define and determine two key descriptors of the behavioral response, namely the behavioral threshold (Tβ) and the behavioral latency (Lβ). By employing more than one site of stimulation, the paradigm allows determination of the conduction velocity of the peripheral fibers that trigger the response (V) and an estimation of the latency (Ld) of the central decision-making process. Ld (∼130 ms) is unaffected by ambient or skin temperature changes that affect the behavioral threshold (∼42.2–44.9°C in the 20–30°C range), behavioral latency (<500 ms), and the conduction velocity of the peripheral fibers that trigger the response (∼0.35–0.76 m/s in the 20–30°C range). We propose a simple model that is verified experimentally and that computes the variations in the so-called “tail-flick latency” (TFL) caused by changes in either the power of the radiant heat source, the initial temperature of the skin, or the site of stimulation along the tail. Conclusions/Significance This approach enables the behavioral determinations of latent psychophysical (Tβ, Lβ, Ld) and neurophysiological (V) variables that have been previously inaccessible with conventional methods. Such an approach satisfies the repeated requests for improving nociceptive tests and offers a potentially heuristic progress for studying nociceptive behavior on more firm physiological and psychophysical grounds. The validity of using a reaction time of a behavioral response to an increasing heat stimulus as a “pain index” is challenged. This is illustrated by the predicted temperature-dependent variations of the behavioral TFL elicited by spontaneous variations of the temperature of the tail for thermoregulation.


Introduction
Pain is a subjective sensory and emotional experience, which can be estimated in non-communicating individuals, only by examining the reaction elicited by a presumed noxious stimulus (that is termed ''nociceptive''). The quantitative primary end-point for many behavioral tests of nociception in animals is the reaction time (t R ), i.e. the time lapse between the beginning of the application of a nociceptive stimulus and the evoked response. Its measurement is straightforward when short duration tightly time-locked stimuli are delivered, e.g. electrical stimuli. However, the situation is more complex when the stimulus intensity is gradually increased while the stimulus is being delivered. This is the case with thermal stimulation, which is the most frequently used in behavioral tests of nociception (e.g. the so-called ''tail-flick'' test) because it stimulates specific receptors, called nociceptors, in the skin [1]. Indeed, it is technically impossible to heat the skin instantaneously by conventional means: thermal stimulation is always progressive.
As any psychophysical process [2,3], a nociceptive behavioral response results from a series of sequential events, each with its own duration. To clarify this, we shall start by dissecting the series of events that lead eventually to a nociceptive withdrawal reaction (R) in response to a heat stimulus (Fig. 1). The reaction time t R is the sum of sequential physical (LQ), biophysical (Lt) and behavioral (Lb) latencies. LQ is the duration of the skin heating process starting at the initial skin temperature T 0 . Lt is the transduction period, i.e. the time required for heat to be transduced by nociceptors into neuronal spikes. Lb is the sum of: (1) the peripheral latency (Lp) required for the nociceptive information to reach the central nervous system (CNS) through the nerves; (2) the 'decisional' latency (Ld) required for central decision-making process (CDP), initiated by the arrival (and/or the accumulation) in the CNS, of a sufficient amount of nociceptive information to order the triggering of the withdrawal; and (3) the motor latency (Lm), time from motoneuron activation up to the shortening of the muscle. In other words, at the time when the 'behavioral threshold' (Tb) level of thermal stimulation is achieved, i.e. time t Tb = t R 2Lb, the flow of information transmitted by the nociceptors is sufficient to trigger the behavioral response at time t R . Whether a response occurs depends on the state of the CDP occupying a time Ld. Tb is defined by the occurrence of a behavioral response, itself depending on a go/nogo signal given by the CDP. Note that, the permissive state of the CDP is basically stochastic in nature and influenced by many factors, such as behavioral conditioning circumstances. We will use the term 'apparent threshold' (AT) for the skin temperature reached at the end of the sequence of latencies summing up to t R .
We aimed to develop an original paradigm to infer the latent psychophysical variables Lb and Tb, a priori unknown, from measurements of approachable quantities. To avoid the inconveniences of the conventional methods of thermal stimulation, a CO 2 laser was used [4]. We report here experiments performed on the tail of the rat because, although it is an organ for thermoregulation and balance, it is widely used for the study of pain. However, the paradigm can be applied to any other mammal on any part of its body. In a first series of experiments, the basal temperature of the tail was stabilized at 34uC -i.e. the highest value recorded when the tail vessels were dilated (see Supporting Video S1). In a second series, the tail temperatures were extended to a physiological range of temperatures. Finally, we propose, and verified experimentally, a simple model for computing the variations of t R , the so-called 'tail-flick latency' (TFL), elicited by changes in either the power of the radiant heat source, the initial temperature of the skin or the site of stimulation along the tail.

Behavioral threshold and behavioral latency
When the skin is exposed to a constant power source of infrared radiation, the temperature increases with the square root of time, according to the law of radiant heat transfer T = T 0 +a.t 0.5 or, expressed in terms of temperature variation [5]: DT = T2T 0 = a.t 0.5 ( Fig. 2A left). This quadratic relationship becomes linear in t by squaring the two terms of the equation: DT 2 = a 2 .t = a.t ( Fig. 2A right). Such temperature variations can be measured via a thermometric camera (see Fig. S1). Each behavioral trial can be summarized by four accessible quantities measured independently, namely the initial skin temperature T 0 , the apparent threshold AT, the reaction time t R and the slope a. From the behavioral standpoint, the process of heating can be described by three key moments ( Fig. 2A left):. the beginning of the stimulation, t = t 0 ;. the moment of the triggering of the reaction defined by t = t R 2Lb and Tb = T 0 +a.(t R 2Lb) 0.5 ;. the moment of the reaction defined by t = t R and AT = T 0 +a.t If T 0 remains stable during the experimental procedure, we can infer Tb the behavioral threshold temperature and Lb the behavioral latency -which are presumably constant -from a series of trials where the power of the radiant heat source varies to The behavioral response R results mainly from a serial processing along the dedicated pathways involving successive time epochs. As proposed by Luce [3], we will reserve the term latency (L) to an unobserved hypothetical time. Specifically, Lt is the time required for heat to be transduced by nociceptors into neuronal spikes, which in turn are transmitted toward and received by the CNS. Lp is the transit time for these spikes to reach the CNS. Ld is the ''decision'' time required by the CNS for interpreting and processing this information for an order to be sent to the motor system (CDP). Lm is the time required for a motor response to be triggered. However, as it is physically impossible to heat the skin instantaneously, the reaction time t R comprises an additional period of heating (LQ). LQ is completely dependent upon the heating rate and varies according to the experimental protocol. The other latencies are biological variables with Lt%Lp [11,55,56] and Lm%Ld [57]. doi:10.1371/journal.pone.0003125.g001 Figure 2. Theoretical analysis of a behavioral response to radiant heat. -A. When skin, at temperature T 0 , is exposed to a constant source of radiant heat, the temperature T increases with the square root of time (left graph). Expressed in terms of squared temperature variations, this relationship becomes linear (right graph). -B. If one varies the power source of radiation, a series of measures can be made, including the heating of the skin from the initial temperature T 0 up to the apparent threshold AT (left graph). In terms of squared temperature variations, the relationships are produce an appropriate range of a ( Fig. 2B & 3A). By adjusting the origin of the time scale of each individual heating curve to the actual time of the reaction, one can visualize the back-timing of events. In this temperature-time plane, each trajectory crosses every other (at a communal point) within a bounded region that allows determining Tb and Lb (Fig. 2B9 & 3A9).
Because of the stochastic nature of the psychophysical responses, the points of intersection of each curve with the others constituted a cluster in the temperature vs. time plot. Analyzing the cluster in terms of density of intersections revealed the highest density at coordinates Tb and 2Lb (Fig. 3B). One can also substitute a.t R of equation 2 in equation 1 thus obtaining after rearrangement: DAT 2 = DTb 2 +Lb.a [equation 3] (Fig. 2C). When DAT 2 was plotted as a function of a, a very strong linear relationship was observed; the slope and intercept of this relationship corresponded to Lb and DTb 2 , respectively (Fig. 3C). Thus, the paradigm and the mathematical processing of the data allow the determination of two key descriptors of the behavioral response to noxious heat, Tb and Lb, without any use of t R -each individual response being fully described by T 0 , AT and a. Tb and Lb were both determined by the site of stimulation (i.e. by D, the distance between the stimulation site and the dorsal horn entry zone) and the temperature of the skin (i.e. by T 0 ).
Role of the stimulation site and conduction velocity of the fibers that triggered the reaction A classical method for calculating nerve conduction velocities was used by changing the distance D between the stimulation site on the tail and the dorsal horn entry zone in the spinal cord. The cluster referred to above moved to the left as D increased (Fig. 4A) and the DAT 2 vs. a plots provided accurate calculations of Tb and Lb for each level of stimulation (Fig. 4B). On average, the behavioral threshold Tb was 44.9 (44.3-45.7)uC, but displayed a tendency to get lower for distal sites of stimulation in the 42.5-46.4uC range (Fig. 4C & 5A). It is possible that the thickness of the epidermis along the tail decreases, thus facilitating the heat transfer to the nociceptors (see section ''Measuring the temperature of the skin''). From a Darwinian perspective however, one can postulate that the extremities are more sensitive because of the survival value of protecting the more exposed parts of the body.
Lb was directly proportional to D (Fig. 4D). The reciprocal of the slope represents the conduction velocity of the fibers that triggered the reaction, in the part of their course within the tail. The linearity and parallelism of the relationships Lb = f(D) suggest homogeneity along the tail and across animals of the nerve fibers implicated in the reaction (Fig. 5B). The mean conduction velocity V t (695% CI) of these fibers in the section of their course traveling in the tail was 0.91 (0.81-1.01) m/s that fits with the 0.5-0.9 m/s of unmyelinated polymodal nociceptors recorded in the coccygeal nerve of the rat [6,7]. We did not see any evidence that Ad-fibers participate to the triggering of the tail withdrawal which confirms the conclusion of Danneman et al. [8]. Interestingly, Ad-fibers recorded from the coccygeal nerve did not respond to radiant heat stimulation, at least up to 50uC [7] and, from a more general standpoint, the threshold of activation of individual Ad-fiber nociceptors is higher than those for C-fiber nociceptors [9][10][11]. In any case, the possibility is now open for the non-invasive behavioral investigation of small primary afferent fibers, e.g. during the weeks or months periods of the full development of peripheral neuropathies such as those elicited by diabetes or chemotherapy [12].

Ambient temperature as a key factor of variations
The experiments described above were performed at room temperature with the tail skin temperature intentionally set at 34uC. We replicated the experiments with the rat being introduced in a chamber where the ambient temperature was maintained stable during a given session, but changed over sessions in the 17-34uC range. The tail was not intentionally heated and was stimulated at 3-4 rostro-caudal levels. For each basal skin temperature, the corresponding DAT 2 = f(a) plots (Fig. 6A) allowed calculations of Lb and Tb for each level of stimulation on the tail. For a basal skin temperature, the straight lines had similar intercepts (DTb 2 ) but increasing slopes (Lb), as one would expect as the stimulation site moved distally. Comparing the bundles of straight lines over increasing skin temperatures, revealed a fall in both Lb and DTb 2 . The slope of the Lb = f(D) functions ( Fig. 6B) fell as the tail temperature increased, which was testimony to increases in the conduction velocities of the fibers that triggered the behavioral reaction (Fig. 6C). Finally, the calculations revealed that the behavioral threshold Tb increased slightly as the skin temperature T 0 increased (Fig. 6D).
The overall results are summarized in figure 7. The temperature of the tail T 0 was slightly above ambient temperature T a when T a #25uC, and several degrees above when T a $30uC (Fig. 7A). Between 25 and 30uC, the basal temperature of the tail was essentially variable, the vasomotor tone of the tail oscillating over time between vasoconstriction and vasodilatation phases (see Supporting Video S1). The individual and overall relationships V t = f(T 0 ) revealed a linear function best described by the equation V t = 0.041 T 0 20.471, with Q 10 between 20 and 30uC = 2.3 (1.9-2.7) (Fig. 7B). In the trunk, the corresponding conduction velocities of fibers at a core temperature T c can then be easily deduced: V c = V t +0.041 (T c 2T 0 ); on average, V c = 1.1 (1.0-1.2) m/s for a 38.1 (37.9-38.3)uC mean T c . The dependency of the conduction velocity of peripheral fibers on temperature is a classical electrophysiological notion [13,14]. During recordings of C-fibers in anesthetized cats, Q 10 = 1.8 and 2.1 were reported from the saphenous and aortic nerves, respectively [15,16], close to the value determined behaviorally in the present study. We shall give an idea of the tremendous variations introduced by this factor by noting the 2.7 times increase in the conduction velocity from 20 to linear and the constant term a can be calculated (right graph). This term reflects the density of power of the heating source. Overall, four quantities are potentially accessible to experimental measurements: T 0 , AT, t R and a. -B9. One can modify the representation by adjusting the time scale of each individual curve for heating to the actual moment of the reaction. Such a change of origin allows one to visualize the back timing of events and to identify on the abscissa the point 2Lb and on the ordinate Tb (left graph), or 2Lb and DTb 2 (right graph). Note that the latency artifact (AT-Tb) increases with the stimulus intensity. -C. Relationship between the apparent threshold and the slope a. The intercept and the slope of this linear function represent DTb 2 and Lb, respectively. -D. Theoretical relationship between the distance D, separating the site of stimulation on the tail from the dorsal root entry zone, and the behavioral latency Lb. The available experimental data from the tail at T 0 uC are shown as a large blue line. The reciprocal of the slope of this line corresponds to the conduction velocity V t of the fibers that triggered the reaction. However, the conduction velocity of these fibers increases when the coccygeal nerves travel through the core of the animal which is set at T c 38uC by thermoregulatory processes. The two components of the peripheral process are shown in blue and red, respectively, with Lp = Lp t +Lp c = D t /V t +D c /V c . The intercept y c = y t +D c (1/V t 21/V c ) of the red straight line with the ordinate represents the part of Lb that does not deal with the peripheral conduction, i.e. (Ld+Lm). doi:10.1371/journal.pone.0003125.g002    DAT 2 = f(a) plots. The tail was stimulated at three rostro-caudal levels (dark to light blue curves; distance D that separated the site of stimulation on the tail from the entry zone in the cord is indicated on the right side of individual curves) in three different ambient temperatures that maintained the mean T 0 at 18.5 (blue, left graph), 30.9 (green, middle graph) and 36.0uC (red, right graph). For a given skin temperature, the straight lines had close intercepts (DTb 2 ) but increasing gradients (Lb) as the stimulation site moved distally. Both Lb and DTb 2 fell as the ambient temperatures increased. The temporal evolution of the skin temperature of the corresponding individual heating curves can be seen in 34uC -temperatures achieved in the rat tail in an air-conditioned room during vasoconstriction and vasodilatation, respectively.
The behavioral temperature threshold Tb exhibited a large inter-individual variability with a significant tendency to increase with the basal tail temperature, best described by the equation Tb = 0.27 T 0 +36.8 (Fig. 7C). This is not trivial, as the behavioral threshold increased by 4uC between 20 and 34uC. This finding might be interpreted as resulting from a CNS build up process resulting from population coding. Indeed, if one considers the peripheral information emanating from the tail -which makes sense as the tail is obviously a specific organ that one can consider as a functional entity -one sees a huge imbalance between information from the tiny heated site (,9 mm 2 ) and the surrounding non-stimulated areas (tail ,5000 mm 2 ). Such imbalance is indisputably reflected in the firing of the corresponding populations of dorsal horn neurons, which means that the thermal picture of the tail received by the brain is more or less contrasted according to the basal temperature. Low background temperatures facilitate the detection of a nociceptive event -thus lowering the pain threshold -, while higher background temperatures blur the detection of a nociceptive event -thus increasing the pain threshold.
Such mechanisms could explain the tendency of the behavioral threshold to be lower for distal sites of stimulation (see above). During the experiments described in figures 4C & 5A, the temperature of a given site was carefully adjusted for each stimulation, but not during the inter-stimulus intervals. Knowing that the proximal areas are always warmer than the distal parts of the tail, it follows that during the total duration of an experiment, there was a rough gradient of temperature over the tail during the inter-stimulus periods. Such a gradient could have influenced the determination of threshold, although the experimenter eliminated The temperature of the tail T 0 was slightly above ambient temperature T a for T a #25uC (blue area) and several degrees above for T a $32uC (red area); the core temperature T c was significantly different between these two groups [37.7 (37.5-37.9) and 38.8 (38.4-39.3)uC, respectively; F 35,1 = 28.9, P,0.001]. Between 25 and 32uC, it was essentially variable but the behavioral tests were performed during vasodilatation phases (grey area). -B. Individual V t = f(T 0 ) plots, obtained from 9 animals (fine lines). The overall V t = f(T 0 ) plot (large line) includes the data obtained in the experiments with T 0 intentionally set at 34uC (black dots). A highly significant linear relationship was seen: V t = 20.471+0.041.T 0 (F 36,1 = 185.5; p,0.0001). A mean Q 10  it at that very moment. These hypotheses, which suggest that the recent history of the thermal status of the tail did influence a later measure, obviously need further investigations.
There is no reason to believe that the behavioral threshold corresponds to the mean threshold of primary nociceptors [17]. We are dealing here with a threshold corresponding to the amount of nociceptive information that is sufficient for triggering the reaction. This ''psychophysical'' threshold integrates both peripheral and central mechanisms of nociception and has therefore to be higher than the mean thresholds of individual nociceptors. In this respect, the mean thresholds for activation of nociceptors in the tail, the firing of sacro-coccygeal dorsal horn neurons and tail withdrawals elicited by immersion of the tail in a water bath, have been reported to be 40.0, 42.2 and 43.7uC, respectively in anesthetized rats [18,19]. This ranking exemplifies the increased level of convergence and summation needed to trigger neuronal events when one moves up the increasing complexity of the hierarchy of the nociceptive system. In spite of the wide range of thresholds (35-55uC) of single unmyelinated polymodal nociceptors in the coccygeal nerve [6,19], the global output of the peripheral system appears sufficiently reproducible to generate stable behavioral responses.
Using conventional radiant heat, thresholds in the 40-43uC range were reported [20,21], but the skin temperature was not measured in these studies. The observations that the (apparent) threshold and the magnitude of the tail-flick reflex increased when the power increased [22,23] fit the present proposal that a withdrawal is triggered following a delay Lb once the threshold Tb is reached. During this period, the temperature continues to grow and activates nociceptors increasingly. The integral of the variation of temperature (AT2Tb) during the period Lb will determines the ''total amount of nociceptive information'' generated by the nociceptors that elicits the strength of the response. Thus, the steeper the heating slope, the stronger will be the response with a shorter reaction time. In this respect, it is interesting to note that the tail-flick is classically described as a brief movement of the tail, with the reaction time being shorter [20][21][22][24][25][26][27] and the movement more vigorous [21,28] when the intensity of the source of radiant heat increases.
In addition, the reported thresholds were in fact what we call here the apparent thresholds, dependent upon the steepness of the heat ramp. Tsuruoka et al. [23] recorded the electromyogram of tail muscles as an indicator of the magnitude of the tail-flick reflex and measured the (apparent) threshold elicited by radiant heat on the blackened tail. They found that from a well-defined baseline this threshold was around 42uC for the lowest power beam; this increased when the power increased (see also [22]). In this respect, we can define the difference (AT2Tb) as the ''latency artifact'' (See Fig. 1A) because this is an obvious cause of variability in the determination of thermal thresholds. This was exemplified elegantly by the studies in humans by Yarnitsky and Ochoa [39,30]. They showed that the method of limits often used for the determination of thermal thresholds in which the stimulus is stopped by the subject, resulted in greater overestimations of threshold when temperature rose faster. By comparison, the method of levels, where the subject's response does not influence the stimulus duration, produced identical thresholds whatever the rate of temperature variations. The notions of (''true'') behavioral and apparent thresholds developed here in an animal study are fully compatible with these observations in man.

Decisional latency
The availability of a bundle of Lb = f(D) relationships with a large range of slopes presented an opportunity for the estimation of further, important, psychophysical variables. The first step was to estimate the length of the peripheral nerves where the conduction velocity increased (given that there would be a higher core temperature within the animal). The V t to V c (conduction velocity of the fibers that triggered the reaction in the part of their course within the core) change would occur approximately at the tailtrunk interface, but it can be determined statistically (D c = 90 mm) by considering the overall cluster of the intersection points of the straight lines Lb = f(D), obtained from different tail temperatures T 0 (Fig. 7D). The availability of both V c and D c provides keys for an estimation of the decisional latency Ld to be inferred from the intercept y c of the regression line Lb c = f(D) with the ordinate: y c = Ld+Lm = y t +D c (1/V t 21/V c ) (See Fig. 2D). Since Lm = 4 ms for the tail-flick response in rats [8], on average, Ld = 132 (117-146) ms. Importantly, it was confirmed that Ld was not influenced significantly by T 0 . During this short period, modulation processes, notably those from supraspinal origin, have the opportunity to modify the withdrawal response.
We provided evidence for the homogeneity of V along the tail and across animals, when T 0 is set at 34uC. We did correct the calculation of Ld by taking into account the increased temperature of the nerve in the core. A further cause of rise in conduction velocity, albeit probably minor, was neglected because of uncertainty: the possible confluence of peripheral fibers that gives rise to larger common branches [31,32]. Increased conduction velocity was reported 2-3 cm before reaching the L4-L5 dorsal root ganglion neurons [33,34]. Such uncontrolled factors of variations could add a few ms to the uncertainty of the estimation.

Modeling and simulation of the 'tail-flick latency'
The possibility is open of computing the variations of the reaction time t R (e.g. the so-called 'tail-flick latency', TFL), elicited by changing any of the parameters. In the classical tail-flick test, the principal source of variation introduced by experimenters is the power of heating of the electrical bulb used for achieving a predetermined range of TFL values -generally 2-4 seconds [35]. This corresponds here to variations in parameter a. As expected from the relation t R = DTb 2 /a+Lb, such computation produced hyperbolae when the basal temperature is stable (Fig. 8A, left graph), with the horizontal asymptote representing Lb. A simple anamorphous transformation [t R = f(a 21 )] linearized this relationship (Fig. 8A, right graph). The slope and the intercept with the ordinate of the straight line represent (Tb2T 0 ) 2 and Lb, respectively. The latter is the reaction time that one would expect following instantaneous heating (aR'). Note the low impact of the stimulation site (or peripheral conduction distance) and the very high impact of the basal skin temperature (Fig. 8B). These theoretical relationships were validated in another series of experiments where the stimulation laser power remained within a very limited range (16-20 mW) over a wide range of ambient temperatures (Fig. 8A9 & 8B9). In other words, the predictive model of t R was fully verified following variations of the radiant heat source or the basal temperature of the skin.
Incidentally, the predictive model of t R offers an explanation for the surprising negative correlations between the rostro-caudal position of the stimulation site and the TFL even though the pathway for the afferent signals increases [36,37]. This a priori paradoxical property was fully verified here by setting the parameters of the model to the values determined in the present study (see Fig. S3).

Discussion
We have developed a psychophysical approach of nociceptive reactions, based on the joint analysis of the stimulus and the response and on the measurement of three observable variables, namely the initial temperature T 0 , the apparent threshold AT and the heating rate expressed as a. This paradigm allows one to reach the two key descriptors of the behavioral response to noxious heat in psychophysical terms without any use of the reaction time t R : the behavioral threshold Tb and the behavioral latency Lb, both of which are latent variables that are inaccessible with conventional methods. In addition, we calculated the conduction velocity of the peripheral fibers that trigger the reaction and proposed an estimate of the central decisional latency Ld, the most interesting part of Lb to be investigated [3]. The usefulness of such an approach was demonstrated by providing new fundamental findings: the skin temperature, itself dependent on ambient temperature, very markedly influenced the behavioral threshold, the behavioral latency and the conduction velocity, but not the latency of the central decision-making process. Finally, we proposed and verified experimentally a simple model for computing the variations of t R , the socalled 'tail-flick latency' (TFL), elicited by changes in either the power of the radiant heat source, the initial temperature of the skin or the site of stimulation along the tail.
In our experiments, the behavioral latency was always less than 500 ms. The remaining part of the reaction time t R mainly represents the time for the heating process LQ and, to a negligible extent, the transduction time Lt required to achieve the behavioral threshold of the reaction Tb. Lt can be inferred from isolated primary afferent neuron recordings: Cesare and McNaughton [38] reported a 35 ms time for half activation of the inward current elicited by pulses of noxious heat. The sum t R = (LQ+Lt+Lb corresponds to the ''latency'' usually observed in the conventional tail-flick test. In our experiments, the duration of heating was in the 0.25-2.5 seconds range (limited for technical reasons, see methods), whereas the current literature puts it in the 2-4 seconds range [35]. This undoubtedly means that most of this period is devoted to LQ, a simple physical process. This part increases when the reaction time increases by lowering either the power of the radiant heat source or the basal temperature of the skin. If t R is the only measured end-point elicited by a given power of the source, there is no way of knowing whether any variation was produced by changes of either T 0 or Tb or both. The test validity, i.e. the degree to which a test actually measures what it purports to measure, is undoubtedly one of the most difficult problems to resolve in pain research [39]. The question of the validity of using a reaction time of a behavioral response to an increasing heat stimulus as a ''pain index'' is challenged. To illustrate this statement, we will consider the spontaneous variations of the temperature of the tail of a rat placed in a conventional restrainer (See Supporting Video S1). The temporal evolution of the skin temperature recorded along the tail (Fig. 9A) revealed phases of vasoconstriction and vasodilatation throughout the 2 hours recording session, with large amplitude of variations (25.4-33.9uC) for the most distal parts of the tail. Figure 9B used the predictive model of t R to infer the corresponding predicted variations of the tail flick ''latency'', elicited from rostro-caudal levels on the tail identical to the temperature recording sites. Note the ,2 seconds range of variation for predicted TFL elicited from the tip of the tail. Measured in an identical experimental situation, behavioral TFL exhibited the predicted temperature-dependent variations (Fig. 9B).
In summary, we provide here both a theoretical framework and an experimental paradigm, based mainly on random variations of the stimulus, which for the first time enable the behavioral determinations of latent psychophysical (Tb, Lb, Ld) and neurophysiological (V t ) variables that have been inaccessible with conventional methods. We believe that such an approach satisfies the repeated requests for improving nociceptive tests [35,40] and offers a potentially heuristic progress for studying nociceptive behavior on more firm physiological and psychophysical grounds.

Animals
Experiments were performed on adult male Sprague-Dawley rats (Janvier, France) weighing 275-325 g in accordance with the National Institute of Health's 'Guide for the care and use of Laboratory animals', the European Communities Council Directive 86/609/EEC regulating animal research, and the ethics committee of the International Association for the Study of Pain [41,42]. Each animal was placed in a Plexiglas device composed of a parallelepiped tank (L = 20 cm, l = 6 cm, H = 25 cm) open at its upper and lower extremities in order to ventilate the animal and allow accessibility to the body regions to be stimulated. The tank was suspended a few millimeters above the surface on which the rat was placed, ensuring sufficient space for the passage of the tail. A cylinder, which could slide along the tank, made it possible to adjust the device to the size of the rat. The longitudinal insertion of removable Plexiglas stems prevented the animal from standing-up.
The tail was depilated by means of a cream (VichyH). Then the animal was habituated to this environment for one hour before the experiment. A colonic thermocouple was inserted 4 cm beyond the anal sphincter and fixed to the tank with adhesive tape. During the course of testing, stimulation was never applied during any behavior or postural adjustment of the animal.
At the end of the experiment, the animal was sacrificed by an overdose of pentobarbital and autopsied. The L3 vertebra was carefully identified and the distances (D) between the stimulation sites and this vertebra were measured. This level was considered as the main entry zone in the cord for afferent signals from the tail as: (1) the four major nerves innervating the tail, namely the dorsolateral and ventrolateral tail nerves, project to dorsal horn superficial laminae of S2-Co2 segments [43]; (2) these segments are located within vertebrae L2-L4 [44]; (3) the maximum Nwave dorsum potential elicited by electrical stimulation of the tail is found in the middle of a laminectomy of vertebrae L2-L4 [19,45].

Monitoring of the basal temperature of the tail
The tail, as a source of radiant heat, is an important organ for thermoregulation in the rat [46]. In the course of an experiment, the temperature of the tail varies spontaneously by several degrees (see Supporting Video S1). Since our approach was based on the measurement of the rise in temperature, it required the stability of the reference temperature of the zone being stimulated. We used an external device, consisting of a low power infrared lamp (50 W) coupled to a rheostat, to stabilize the temperature at 3461uC. This temperature is the highest spontaneously achieved value we recorded when the tail vessels were dilated. In some experiments, the rat was introduced into a chamber where, during a given session, the ambient temperature was maintained stable in the 17-34uC range without any other additional source of heating.

The stimulus
We set aside the use of light bulbs because they give emissions in the visible and adjacent infrared parts of the electromagnetic spectrum, for which the skin is poorly absorbent and particularly reflective. In addition, by changing the voltage across the incandescent filament one changes also the emission spectrum of the electric light bulb. Since the transparency of skin to radiation depends on its wavelength [47], a modification of the emission spectrum influences all parameters of the skin heating process inclusive the volume of tissue affected by the heating. This point is insurmountable when one wishes to vary the intensity of heat stimulation using light bulbs. We used a laser stimulator (CO 2 LSD, SIFEC, Ferrière, Belgium) for the following reasons [4]: (1) it is an infrared monochromatic radiant source with a long wavelength (10.6 mm) for which the absorbance is almost total whatever the pigmentation of the skin and the incidence of the beam; (2) the transparency of skin is weak (,100 mm), so that the calorific energy absorbed at the level of the cutaneous surface propagates towards nerve endings sensitive to thermal variations, which are localized above the dermo-epidermic junction (60-120 mm depth); (3) the temporal and spatial profile of the calorific energy is well determined; (4) given the high power density of the laser, it is possible to apply abrupt heating. The surface area for stimulation was a circle determined by the Gaussian power profile of the laser beam (See Fig. S1). We chose a diameter of 3.4 mm, for which lateral diffusion of heat by conduction was negligible for at least 2.5 seconds (the longest reaction time recorded in the present study). Beyond this period, diffusion processes gradually and significantly thwart the temperature increase. In the last experiment (Fig. 9C), a 15 mm diameter was used in order to lower the power of the laser beam without such thwarting.

Monitoring the stimulus
During laser stimulation, the temperature increases proportionally with the square root of time according to the law of radiant heat transfer T = T 0 +a.t 0.5 (see Fig. 2A and 3A). The constant term a (a 2 = the slope a of the straight lines in the right graphs of Fig. 2A, 2B & 2B9) is proportional to the power density (Q) of the laser, according to the relation a = K.Q = K.q/S, where K is a composite constant grouping together the biophysical properties of the skin, S the stimulation surface area (mm 2 ) and q the laser power (mW).
In practice, for a given surface of stimulation and a constant angle of incidence of the laser beam to the skin, the term a is proportional to q. It follows that a is proportional to q 2 . To reach a satisfactory level of reproducibility, the laser beam must be perpendicular to the stimulated surface because the angle of Figure 9. Spontaneous variations of tail temperature influences tail flick latencies. -A. Thermographic film of the tail of a rat placed in a conventional restrainer, recorded with a 3206240 pixels resolution at 1 Hz during 135 minutes (see Supporting Video S1). Steel rings maintained its tail (picture to the right with base of tail up and tip of tail down). From left to right temporal evolution of ambient temperature (Ta, lower brown trace), central core temperature (Tc, upper black trace) and tail temperatures recorded from base to tip at six equally distant places (blue dots and corresponding traces). Note the phases of vasoconstriction and vasodilatation throughout the recording session. In the most distal part of the tail, temperature varied by 8.5uC. -B. By using the model equation presented in Figure 8, and the temporal variations of T 0 and T C shown in panel A, tail flick latencies (TFL) were computed (Ld = 134 ms; Lm = 4 ms; a = 0.1). Simulated TFLs obtained for the most distal part of the tail ranged from 1.9 to 3.9 seconds. -C. Four rats were placed in the same conditions as for the recordings of the thermographic film in panel A, except that their tail was free for a behavioral response to be elicited. Only responses triggered by heating ramps characterized by slopes of 0.09,a,0.11 were considered (red dots). The red line represents the t R = f(D) relationship for a single distal site of stimulation (D = 265 mm). Note that the behavioral TFL exhibited the temperature-dependent variations in the predicted range. doi:10.1371/journal.pone.0003125.g009 incidence influences power density. However, the skin is never flat and the tail is a conical cylinder. In order to minimize these sources of variability, the beam axis was targeted perpendicular to the axis of the tail. The beam was adjusted to 45u with respect to the vertical, in order to elicit a contralateral withdrawal movement. Doing this allowed stimulation of the right and left side of the tail at a given rostro-caudal level.
We determined a range of powers of stimulation (100-350 mW) provoking responses within less than 2.5 seconds without damaging the skin. In these conditions, the slope a was in the 0.07-3.2uC 2 /ms range and the maximum temperature reached at the actual moment of the reaction was always lower than 70uC.

Protocol for stimulation
Four (or 5) sites of stimulation, each 4 (or 3) cm from each other, were identified on each side of the tail. The left and the right side were stimulated alternatively with variable laser powers in the 100-350 mW range. These powers were delivered in a pseudorandom order, 15-20 times per stimulation site. The stimuli were directed successively to the 8 (or 10) sites. A minimum of 5 minutes passed between stimulation of a given site and the next stimulation at the same site. We checked that there were no significant differences between data computed separately from either side of the tail (Lb: p.0.8; Tb: p.0.8) or from the two temporal halves of the experiment (Lb: p.0.3; Tb: p.0.2).

Measuring the temperature of the skin
The measurement of temperature at the skin surface is justified by convenience of use, non-invasive character and possibility of extrapolating the underlying subcutaneous temperatures by modeling. This temperature represents only an approximation of the temperature reached at the level of the nociceptors, which are located at the dermo-epidermal junction, at an average depth of 100 mm [5,48,49]. The probes that might be used to make direct measurements (e.g. thermocouples ) have not yet been miniaturized to the extent that they would not disturb the thermal field. However, the temperature reached by the various layers within the skin can be estimated through modeling and simulation by means of the unidimensional heat conduction differential equation and biophysical parameters of skin reported in [5,48,49,51,52]. Figure 10A presents the results of such simulations. It shows the temporal evolution of skin temperature at the surface (d = 0) and at three depths in the skin (d = 50, 100 and 150 mm) using a mid-range laser power density of 0.025 W/mm 2 , stimulus duration of 2500 ms and initial skin temperature of 34uC. Figure 10B shows the temporal evolution of the difference between the temperature at a depth of 100 mm T d = 100 and surface skin temperature T d = 0 , i.e. DT d = T d = 100 2T d = 0 . It fits a rectangular hyperbola of the form DT d = DT max .t/(t DTmax/2 +t), where DT max represents the upper asymptote (i.e. maximum of DT) and t DTmax/2 represents the time t at DT max /2. The parameters are estimated by linearizing the rectangular hyperbola using the Hans-Woolf procedure [53]: The parameters are computed as follows: Which for the example given in Figure 10B yields: In other words, the difference in temperature between the skin surface and the dermo-epidermal junction at 100 mm depth is about 1uC and half of that difference is reached in about 100 ms. One can consider therefore that the temperature reached at the dermo-epidermal junction in our experiments was slightly lower, but indeed close to the measured surface temperature.
Finally, DT max (i.e. maximum of DT) increases linearly with laser power density Q. This allows to estimate easily, for a range of laser power densities used in the present experiments, the difference between skin surface temperature and the temperature at a skin depth of 100 mm by means of the following equation: DT max~5 :23 Q Since we derive the power of emitted laser radiation from the heating ramp by a = K.Q = a 0.5 where K is a lumped constant equating [2(12r 10.6 )/(p.k.r.c) 0.5 ] = 6.564 (see [54] and table 1), the maximum difference in temperature can also be expressed as DT max = 5.23 a 0.5 /K.
Although we filmed the full surface area of stimulation, we chose to investigate further the temporal evolution of the warmest pixel. From the physical standpoint, this choice was justified by the Gaussian profile of the beam, reflected by an equivalent spatial profile of the temperature increment (see Fig. S1). Knowing the highest value and the diameter of the temperature profile allows anyone to reconstruct the whole picture.

The thermometric camera
We used a JADE MWIR (3-5 mm optical bandpass) camera (CEDIP Infrared Systems, Croissy-Beaubourg, France) with a 500 ms integration time, which supplied images of 3206240 pixels at 172 Hz with a sensitivity of 0.02uC at 25uC. It was placed upright to the zone of stimulation and was controlled by the software Cirrus (CEDIP Infrared Systems, Croissy-Beaubourg, France). It was calibrated by means of a black body (CI SR80 CI Systems, Migdal Haemek, Israel). The software Altair (CEDIP Infrared Systems, Croissy-Beaubourg, France) allowed the monitoring of the spatial and temporal evolution of the temperature at the level of the stimulated surface area with 0.3 mm and 5.8 ms resolutions, respectively (See Fig. S2). The recording was triggered 0.2 second before the application of the stimulus.

Analysis of thermographic films
The analysis of the thermographic films was made by means of the software Altair (CEDIP Infrared Systems, Croissy-Beaubourg, France). This involved the following steps: (1) determination of the zone of interest in the recorded scene; (2) determination of the initial temperature T 0 in this zone; (3) calculation of the temporal evolution of the warmest pixel in this zone until the image preceding the movement -the ordinate of the ultimate point of this curve constituting the apparent threshold AT. This pixel corresponded to the top of the Gauss curve, which characterizes the spatial profile of the thermal rise (See Fig. S1B). The reaction time t R was measured by counting the number of pictures of the film where the temperature curve exceeded T 0 +1uC and multiplying this count by 1000/172. The analysis of an individual temperature curve included the following steps: (1) transforming the temperature in difference with regard to the initial temperature DT = f(t); (2) rising to the square DT 2 = f(t); (3) checking the linearity of this curve; and (4) determining the value of the slope a of this curve.

Analysis of data brought by a series of tests applied at a given site
At completion of the analysis of the thermographic films related to the tests applied at a given site, a set of temperature curves elicited by a series of laser powers was obtained. Each of these was summarized by the corresponding measured values of T 0 , AT and a the slope of the linear function DT 2 = a.t (see Fig. 2B). Since T 0 remained stable during the experimental procedure, we could infer the behavioral threshold Tb and the psychophysical latency Lbpresumably constant -by two approaches.  Fig. 2B9). This backward timing procedure is easily performed using the following computation for each individual trial. If x is the negative abscissa of the adjusted graphic (x = t2t R ), the equation . Such a change of origin allows one to visualize the back timing of events and to identify the point (2Lb on the abscissa and Tb on the ordinate. Because of the stochastic nature of the psychophysical responses, the points of intersection are distributed in the time vs. temperature plane. It was assumed that the peak value in the bivariate frequency histogram of intersections allowed an estimation of the coordinates 2Lb and Tb. In practice, the intersections of each curve with all the others were obtained by iterating the method of determinants for solving a system of two linear functions DT 2 (x) vs. x in two unknowns. Density distributions of the coordinates of intersections were computed with bin widths of 20 ms and 0.25uC, respectively. The value corresponding to the peak density of each distribution was considered as an estimate of the point having abscissa 2Lb and ordinate Tb. It was visualized on a bivariate frequency histogram as shown in Figure 3B and the right panels of Figure 4A.

Estimations of peripheral (Lp) and decisional (Ld) latencies
Knowing the distance D and the conduction velocity V of the fibers that trigger the reaction, one should be able to estimate the peripheral latency Lp = D/V. However, the conduction velocity of peripheral fibers is highly dependent on temperature, in a way which is equivalent across types, nerves and species [13,14]. We first measured V along the part of the coccygeal nerve that travels within the tail, maintained at T 0 = 34uC. But the temperature increases over the length (D c ) of the coccygeal nerve traveling within the core of the animal, which is maintained at 38uC by thermoregulation processes. Therefore, the decomposition of the peripheral latency into two components (Lp = Lp t +Lp c ) with different conduction velocities (V t ,V c ) appeared to be a reasonable assumption (see Fig.  S1E). The slope (1/V) of the straight line Lb = f(D) is reduced in the part of the nerve traveling in the core, with Lp = (D-D c )/V t +D c /V c . The intercept y of the straight line with the ordinate of the function Lb = f(D) is increased by as much : y c = y t +D c (1/V t -1/V c ). This intercept corresponds to the part of the psychophysical latency that is not devoted to the progress of nociceptive signals along afferent fibers. This is a composite time that includes two successive distinct periods, namely Ld and Lm. Since Lm has been estimated as 4 ms [8], one can finally estimate Ld as y c -4. In summary, both Lp and Ld   can be estimated if one knows the relative contribution of the afferent path within the core of the animal, which means estimation of D c and V c .
Data related to the afferent path within the core of the animal Estimation of both D c and V c is possible if data related to various initial temperatures of the skin T 0 are available. The experiments described above were replicated with the rat being introduced to a chamber where the ambient temperature T a was maintained stable during a given session, but changed over sessions in the 17-34uC range. The tail was not intentionally heated and was stimulated at 3-4 rostro-caudal levels. The temperature of the tail T 0 was slightly above ambient temperature T a for T a #25uC, and several degrees above for T a $32uC (See Fig. 7A). Between 25 and 30uC, the basal temperature was essentially variable, the vasomotor tone of the tail oscillating over time between vasoconstriction and vasodilatation phases (See Supporting Video S1). In these later cases, the behavioral tests were performed during vasodilatation phases. The results from this study revealed a very significant V t = f(T 0 ) linear function best described by the equation V t = 0.041 T 0 -0.471, which provided the possibility of calculating the conduction velocities of fibers at core temperature T c : V c = V t +0.041 (T c -T 0 ). The V t to V c change, which occurs at the tail-core interface where the temperature of the nerves increases from T 0 to the core temperature T c , can be estimated statistically in the D-Lb plot by considering the overall cluster of the intersection points of each straight line with the others (Fig. 7D). A crossed tabulation of the slope and intercept of each straight line was used to compute the median values of the coordinates of these intersection points, which are estimations of D c = length of the coccygeal nerve traveling within the trunk at core temperature and (Lp c +Ld+Lm), respectively.

Statistical analyses
Least squares linear regressions and one-way analyses of variance (ANOVA) were used for statistical purposes. Calculations were performed with the statistical software Staview TM 5.0 and Statgraphics TM Plus 5. Other calculations were made with the software Mathcad TM or Matlab TM . Results were considered significant at P,0.05. Data are expressed as means (6confident interval 95%). Figure S1 Analysis of thermal imaging of the skin during CO 2 laser stimulation. A. Examples of thermal imaging of the skin during CO 2 laser stimulation. The thermometric camera provided a picture every 5.8 ms (sampling rate 172 Hz). A sample of 5 pictures is shown, filmed 58, 122, 103, 308, and 477 ms following the triggering of the stimulus (a-e, from bottom to top; color temperature scale on the left; resolution 0.3 mm). The edges of the tail of the animal are indicated by the white dotted lines. Image e was recorded just before the withdrawal response -B. Spatial profiles of the pictures shown in A. They were calculated on the rostro-caudal line passing through the center of the stimulation spot. Note they all fit a Gaussian curve. One can compare the radius of the stimulation spot with the radius of the laser beam. The radius of the beam is defined as the distance separating its z axis from the zone where its power is reduced to 1/e 2 = 13.5% of its maximum. A corresponding radius of the stimulation spot resulted from these properties of the beam: the distance separating the z-axis of the heating spot from the zone where the difference of temperature is reduced to 13.5% of the maximum was 1.7 mm. The physical diameter of stimulation (3.4 mm) is shown as a yellow area. -C. Temporal evolution of the temperature of the warmest pixel during this laser stimulation. AT = apparent threshold; R = behavioral response; t R = reaction time. Found at: doi:10.1371/journal.pone.0003125.s001 (1.50 MB TIF) Figure S2 Individual heating curves corresponding to the example analyzed in Fig. 6. The tail was stimulated at three rostro-caudal levels (dark to light blue curves from top to bottom; distance D that separated the site of stimulation on the tail from the entry zone in the cord is indicated below curves) in three different ambient temperatures that maintained the mean T 0 at 18.5 (A, blue), 30.9 (B, green) and 36.0uC (C, red), respectively. Note the clear tendency of these curves to cross each other in a privileged zone (open circles) and the progressive shift of this zone both backward in time when the stimulation site moved from proximal to distal parts of the tail and upward when the ambient temperature increased (dashed lines). can be written: t R = f(D) = (Tb2T 0 ) 2 /a+(D290)/V t +90/V c +Ld+ Lm = (Tb2T 0 ) 2 /a+(D290)/V t +90/[V t +0.041 (T c 2T 0 )]+Ld+Lm. This opens the possibility of computing variations of t R introduced by moving the stimulation site along the tail. The graphs shows results of such a procedure, with the numerical values determined from the present study with the basal temperature of the tail stabilized at 34uC. The numerical values of a was chosen as 0.045uC 2 /s in order to achieve the 2-4 seconds range, shown as a yellow area, for the TFL commonly reported in the literature for the classical tail-flick test in control situations [35]. The other numerical values were: T c = 38uC, T 0 = 34uC, V t = 0.91 (0.81-1.01) m/s, Ld = 132 (117-146) ms, Lm = 4 ms. The results of computation are shown as means 695% C.I. (black, red and blue curves, respectively). -A. The behavioral threshold Tb is considered as invariant (value indicated in insert). In those cases, the TFL increases slightly when the stimulation site moves distally along the tail, as one would intuitively expect. -B. Since there was a significant proximo-distal shift of Tb (Fig. 5B), we introduced this factor of variation in the model (relation Tb = f(D) indicated in insert), transforming the linear function to a quadratic relationship. This modification skewed the D-t R relationship in such a way that t R was shortened when the site of stimulation moved distally. In other words, the model predicts a negative correlation between the rostro-caudal position of the stimulation site and the tail-flick ''latency'' (at least when a,0.13), even though the pathway for the afferent signals increases. In fact, this a priori paradoxical property has already been described [20,37], which a posteriori supports the present model. Video S1 A. An example of thermographic film of the tail of a rat placed in a conventional restrainer, recorded with a 3206240 pixels resolution at 1 Hz during 135 minutes. Steel rings maintained its tail. -B. Corresponding temporal evolution of the skin temperature recorded on 6 points distributed from the base to the tip of the tail (blue dots and corresponding traces). One observes phases of vasoconstriction and vasodilatation throughout the recording session. Note that the 10uC amplitude of the variations for the most distal points of the tail. As measured on an inert piece of wood (Ta, lower brown trace), the ambient temperature was stable around 25uC. The central core temperature (Tc) is shown as upper black trace. -C. The dissipation of heat is regulated by abrupt variations of blood flow in a system of arterio-venous anastomoses, which form a double ladder. Anatomy adapted from [58]. Found at: doi:10.1371/journal.pone.0003125.s004 (12.19 MB MOV)