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1 Harvard–Massachusetts Institute of Technology Division of Health Sciences and Technology, Harvard University, Cambridge, Massachusetts 02138; 2 Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
Submitted 16 September 2003; accepted in final form 16 December 2003
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: NONLINEAR KERNEL...ACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: NONLINEAR KERNEL...ACKNOWLEDGMENTSREFERENCES |
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; Guic-Robles et al. 1989). Because the vibrissa array is used to actively palpate textures in the natural environment, individual vibrissae undergo temporal patterns of mechanical deflections that reflect the properties of the surface. Although much is known about neuronal encoding in the vibrissa pathway through temporally isolated vibrissa deflections, the encoding of temporal patterns of deflection has been investigated in only a relatively small number of studies (Ahissar et al. 2001; Castro-Alamancos 2002a; Chung et al. 2002; Garabedian et al. 2003; Simons 1978). Our current understanding of the encoding dynamics of this sensory modality in the natural environment is thus quite limited.
The rat somatosensory cortex contains anatomically distinct clusters of cells, or "barrels," within which neurons respond with action potentials primarily to deflections of the corresponding primary whisker (PW) on the contralateral face (Welker 1976; Woolsey and van der Loos 1970). In a majority of studies of the encoding properties of this pathway, stereo-typed stimuli are created by isolated whisker deflections induced through piezo-electric actuation or air puffs. The neuronal responses to these stimuli are then subsequently used to characterize the fundamental functional properties of the pathway. Characteristic of responses to such punctate stimuli for both excitatory and inhibitory cortical cells is an initial quick excitatory response component followed by a prolonged, relatively pronounced inhibitory tail, which can extend past 100 ms (Carvell and Simons 1988; Higley and Contreras 2003; Moore and Nelson 1998; Zhu and Connors 1999). The importance of this interplay between excitatory and inhibitory dynamics has long been recognized as fundamentally important in the representation and transformation of sensory inputs by the thalamocortical pathway (Mountcastle et al. 1957). Because of the low firing rate of vibrissa-related cortical cells (Simons 1978), extracellular recordings in response to isolated deflections of the vibrissa do not generally reveal the suppressive component. However, a 2nd deflection of the vibrissa after an initial deflection serves as a probe for the relative level of suppression, and has been used previously in single (Fanselow and Nicolelis 1999; Kyriazi et al. 1994; Lee et al. 1994) and paired-whisker (Simons 1985; Simons and Carvell 1989) stimulation studies. The focus of the current study was to investigate how these interactions extend more broadly to aperiodic or periodic stimuli involving sequences of deflection patterns.
The aims of this study were 2-fold. To characterize the nature of the neuronal response to patterns of vibrissa deflections, we first stimulated the whiskers with a temporal probe involving tactile stimuli spaced at varying intervals in time. We hypothesized that temporal stimulus patterns would induce nonlinear interactions between excitatory and suppressive components of the neuronal response. This was confirmed through our experimental observations. Specifically, the stimulus-evoked suppression serves to "suppress the suppression" normally induced by subsequent stimuli, resulting in complex responses to stimulus patterns. Second, we hypothesized that these interactions could be predicted from the inferred postexcitatory suppression using nonlinear combinations of 1st- and 2nd-order dynamics. The steady-state frequency-response characteristics reported in a number of recent studies (Ahissar et al. 2000, 2001; Castro-Alamancos and Oldford 2002; Chung et al. 2002; Garabedian et al. 2003; Simons 1978) were, to a large degree, predicted through these nonlinear interactions. Finally, when extended to patterns in which deflections were randomly spaced in time, we found that predictions of higher-order interactions inferred from 2nd-order dynamics accurately captured the complex neuronal response observed experimentally.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: NONLINEAR KERNEL...ACKNOWLEDGMENTSREFERENCES |
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Eight female adult Sprague–Dawley rats weighing 250–300 g (Charles River Facility, Wilmington, MA) were used in the experiments. Animals were initially sedated with 2% vaporized isoflurane, anesthetized with an intraperitoneal injection of sodium pentobarbital (50 mg/kg), and transferred to a stereotaxic frame (Kopf Instruments, Tujunga, CA) for surgery and recording. Body temperature was maintained at 38° C with a heating pad. Atropine (0.09 mg/kg) was injected to keep the lungs clear of fluid and Ringers solution (1 ml) was periodically injected subcutaneously to keep the animal hydrated. Lidocaine was injected into the tissue on top of the head, and the skin, soft tissue, and left temporal muscle were resected. A craniotomy was performed on the left parietal bone, 1.0–4.0 mm caudal to the bregma and 3.0–7.0 mm lateral to the midline, to expose the vibrissa (barrel) region of the primary somatosensory cortex (Paxinos and Watson 1998). The edges of the craniotomy were sealed with bone wax to keep the opening clear and mineral oil was applied to cover and protect the exposed dura. ECG leads were attached to monitor the heart rate, and the physiological condition of the animals was assessed through the heart rate, respiratory rate, and pinch reflexes. After surgery, supplemental doses of pentobarbital (12.5 mg/kg) were administered as necessary to maintain a light level of anesthesia. At the termination of the experiment, the animals were killed with an overdose of sodium pentobarbital. All procedures were approved by the Animal Care and Use Committee at Harvard University and in accordance with National Institutes of Health guidelines.
Vibrissa stimulation and electrophysiological recordings
The whisker deflection was controlled with a multilayered piezo-electric bending actuator (Polytech PI, Auburn, MA), calibrated using a photodiode circuit and an x–y–z positioner. The actuator was positioned at a location 10 mm from the face with the whisker inserted into a 4-cm section of a 20-µL glass pipette attached to the end of the actuator and maintained as close to resting position as possible. Neurons were stimulated with a square-wave deflection of the primary whisker, which was first filtered with a causal Gaussian function to minimize mechanical ringing of the actuator (Simons 1983). Vibrissae were always deflected in the rostral–caudal plane regardless of the directional tuning of the cell. The rostral (ON) deflection was therefore not always in the preferred direction, nor was the rostral (ON) response necessarily larger than the caudal (OFF) response. The deflections were 700 µm in amplitude with maximum velocities between 100 and 155 mm/s. Complete mathematical descriptions of the stimulus construction and the square-wave filtering have been previously reported (Stanley and Webber 2003).
The data collection and actuator were controlled using LabWindows acquisition/control software (National Instruments, Austin, TX). Neuronal signals were first amplified (A-M Systems, Sequim, WA), band-pass filtered between 300 Hz and 5 kHz, and then acquired at 20 kHz with 16-bit resolution. A sharp tungsten microelectrode (5–7 M
, FHC, Bowdoinham, ME) was slowly advanced through the dura and cortical tissue until a cell was encountered between 500 and 900 µm below the surface of the cortex, consistent with previous reports of the depth of the barrel field (Simons 1978). Single-unit activity was discriminated using standard template-matching techniques and physiologically plausible refractory periods (Lewicki 1998). The principal whisker (PW) was determined by manually deflecting vibrissae. The vibrissa that elicited the largest neuronal response was classified as the PW. We recorded from cells that typically responded only to the manual deflection of a single vibrissa (never more than 2 vibrissae), suggesting that the cell was located within a barrel. By selecting cells with narrowly focused receptive fields (typically single vibrissa), low spontaneous firing rates (typically <5 Hz), and action potential wave-forms of approximately 1.5-ms duration, we have limited the current study to that of excitatory regular spiking units (RSUs) within the barrel field (Brumberg et al. 1996; Simons 1978; Zhu and Connors 1999).
The PW was deflected from rest in the caudal–rostral direction with a 1-Hz smoothed square wave of 20% duty cycle (200-ms deflection duration, 800-ms rest duration) for 2 min to determine a baseline response. The stimulus was presented at an overall rate (1 Hz) at which previous reports show no significant adaptation (Ahissar et al. 2001; Chung et al. 2002). Subsequently, for 8 of the cells a square wave at a frequency of 1 Hz was presented for 2 min at duty cycles between 1 and 10% (Fig. 1A). This provided square pulses with pulse widths (t2 – t1) of 10 to 100 ms at 10-ms intervals and durations of "rest" between 900 and 990 ms. The remaining cells in the study (12) were stimulated with pulse widths of 20, 40, 60, 70, 80, 90, 100, 110, and 150 ms. At the end of this stimulation sequence, the whisker was again deflected with a 200-ms square pulse at 1 Hz to verify that the cellular response was stable over time. The whisker was then deflected in the opposite direction, rostral-to-caudal, beginning at the plateau position and returning to rest, using the same protocol as above (Fig. 1B). A 1-min rest period was used between all 2-min trials. For the 3-edge stimulation study, the PW was deflected by 2 square pulses with variable intervals between the 1st rostral deflection and caudal deflection, denoted by t2 – t1, as well as between the 1st caudal deflection and the 2nd rostral deflection, denoted by t3 – t2 (Fig. 1C). The interval between the 1st and 2nd deflection (t2 – t1) was varied between 20, 60, and 100 ms, and the interval between the 2nd and 3rd deflection (t3 – t2) was varied from 20 to 100 ms in 20-ms increments for each value of t2 – t1. Each of these stimuli was presented for 2 min at a frequency of 1 Hz with a 1-min rest period between trials. For the frequency-response study, the stimulus was a filtered square wave with a duty cycle of 50% and frequencies of 2, 4, 6, 8, and 16 Hz presented for 2 min. A random deflection stimulus was presented to a subset of the cells and consisted of rostral and caudal deflections at pulse widths drawn from a uniform distribution between 15 and 250 ms. This resulted in a stimulus containing frequencies effectively between 2 and 32 Hz (Fig. 1D) that was presented for 5 s followed by a rest period of 10 s and again repeated, for 6 min.
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Characterization of postexcitatory suppression
Neuronal suppression after a rostral deflection of the stimulus was inferred from the activity of each cell in response to the subsequent caudal deflection of the stimulus (Kyriazi et al. 1994; stimulus shown in Fig. 1A). The AUC of the first 30 ms of the caudal PSTH was computed and normalized by the AUC of the PSTH obtained for the longest pulse width (200 ms). By computing this measure for varying pulse widths, the function 
(t2 – t1) was estimated, representing the normalized response to a caudal deflection at time t2 when preceded by a rostral deflection at time t1, referred to hereafter as the temporal tuning curve. This curve is the same as the response-suppression curves described in previous studies (Kyriazi et al. 1994; Simons 1985; Simons and Carvell 1989). An analogous temporal tuning curve, 
(t3 – t2), was computed for the case when a rostral deflection at time t3 is preceded by a caudal deflection at time t2 (Fig. 1B), to account for the difference in response characteristics of the neurons when deflected in the caudal direction versus the rostral direction. All error bars were determined through bootstrapping methods applied to the data set, and represent 1 SE above and below the mean.
A large subset of the neurons exhibited sigmoidally shaped temporal tuning curves, whereas a smaller subset of neurons exhibited a 2nd peak (or rebound excitation), as previously reported in both ventral posterior medial thalamus (VPM) and primary somatosensory cortex (SI) (Fanselow and Nicolelis 1999; Simons and Carvell 1989; Zhu and Connors 1999). These 2 types of tuning curves can be expressed mathematically as the product of a sigmoidal hyperbolic tangent function and a damped oscillation
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and t50, describing the rise time of the sigmoid and the time at which the function reaches 50% of the maximum value, respectively. The second part of the expression represents the step response of a 2nd-order linear system (Ogata 1978); 
n is the natural frequency (radians/s); 
is the damping ratio; and 
is the phase angle (radians). This function was fitted to the temporal tuning curves using a nonlinear least-squares search algorithm based on the Gauss–Newton method (Mathworks, Natick, MA). Functions were fitted to all of the cells to determine the 4 parameters and cells were categorized based on the shape of their temporal tuning curves and the parametric function fit. Specifically, neurons with tuning curves containing 2 or more points with lower error bounds above 1 and < 0.6 (significantly underdamped) were considered rebound cells. The rest of the neurons were classified as cells without significant rebound activity. Average tuning curves of cells with and without rebound excitation were also fitted with the parametric function (Fig. 3, A–D).
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To describe the nonlinear interactions induced by patterns of vibrissa deflections, we represented the dependence of the firing rate (PSTH) on the vibrissa deflections as a Volterra series, used to describe the dynamics of general nonlinear systems (Marmarelis and Marmarelis 1978). A more detailed discussion of the higher-order interactions and the relationship to nonlinear Volterra series representations are contained in the APPENDIX. This representation reduced to expressions containing combinations of the temporal tuning curves and the 1st-order Volterra series kernel, as presented below. Because barrel neurons respond primarily to the velocity of a whisker deflection rather than the amplitude (Pinto et al. 2000), we represented the ith deflection as a stimulus, si = 
(t – ti), consisting of a delta function (impulse) at time ti. For the 1st-order effect, rr(t|t3) reflected the response at time t, given a preceding rostral deflection at time t3. For this case, the response was expressed as the linear convolution of a 1st-order kernel with the stimulus
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t) given a previous rostral deflection at t3, normalized by the interval t. Furthermore, we defined the total response as the area under the PSTH in the 30 ms after deflection
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where the subscript 3|t2 denotes the response to a deflection at time t3 given a deflection at time t2. The neuronal response to a 2nd deflection was therefore predicted by scaling the 1st-order rostral kernel by the temporal tuning curve. Based on observations from the 3-deflection protocol, the 3rd-order interaction (Fig. 1C) was similarly formed as a function of the 1st-order kernel and the temporal tuning curves
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(t3 – t1) denotes the 2nd-order interaction between the response to the 1st rostral deflection at t1 and the response to the 2nd rostral deflection at t3, and eff(t3 – t2) represents the effective 2nd-order interaction between the response to the caudal deflection at t2 and the response to the rostral deflection at t3. This is an effective interaction because it is dictated by the previous deflection at t1. Boundary conditions for eff were determined from observations of the 3-deflection stimulus. With t2 – t1 large, the response to the 3rd deflection was mainly affected by the response to the 2nd deflection. With t2 – t1 small, there was no response to the 2nd deflection and less suppression of the response to the 3rd deflection. These constraints are satisfied by the following expression
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(t2 – t1) goes to 1, and eff becomes (t3 – t2), implying that the 1st deflection has no effect on the response to the 3rd. On the other hand, for small t2 – t1, (t2 – t1) is vanishingly small, and eff goes to 1, implying that the 1st deflection completely suppresses the response to the 2nd deflection. This is representative of the lifting of suppression we observed in the 3-deflection study. In addition, if t3 – t1 is sufficiently small, the 1st deflection also directly affected the response to the 3rd deflection. The prediction in Eq. 1 therefore contains a scaling term directly from the 1st stimulus, (t3 – t1), dependent on the time interval between the 1st and 3rd deflections.
For a periodic or aperiodic stimulus sequence, as shown in Fig. 1D, the response to each deflection was affected by the response to prior deflections. Therefore the response to a deflection was expressed in a recursive fashion as a function of k1 and the average temporal tuning curves. Each deflection of the vibrissa, at time ti, was indexed and referred to as the ith deflection. For any rostral (odd) or caudal (even) deflection of the vibrissa, the response was determined by
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, 
, and 
. Error bars were obtained for all predictions by repeatedly drawing and values from a Gaussian distribution with mean and SD of the average curves. These values were used to predict the response and the predictions were averaged to obtain the mean and SE of the prediction. |
RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: NONLINEAR KERNEL...ACKNOWLEDGMENTSREFERENCES |
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Temporal interactions between paired deflections were used to infer underlying levels of neuronal suppression (Fanselow and Nicolelis 1999; Higley and Contreras 2003; Kyriazi et al. 1994; Simons 1985) and subsequently to predict nonlinear neuronal responses to complex tactile patterns. To this end, the primary whisker (PW) of anesthetized rats was stimulated with a 1-Hz mechanical deflection pattern consisting of filtered square pulses with duty cycles between 2 and 20% (Fig. 2, top stimulus pattern), while recording extracellular single-unit activity in barrel cortex. This resulted in square pulses with intervals between rostral and caudal deflections (pulse widths) of 20 to 200 ms, and pulse widths between the caudal and rostral deflections of 980 to 800 ms (long enough to prevent significant interactions between the caudal deflection and next rostral deflection). The effective presentation frequency (1 Hz) was also low enough to preclude significant adaptation (Ahissar et al. 2001; Castro-Alamancos and Oldford 2002; Chung et al. 2002). Twenty cells were presented with these paired stimuli and the response of a typical cell is shown (Fig. 2A).
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(t2 – t1), was formed. The temporal tuning curve is analogous to the normalized "OFF response" measures previously reported (Kyriazi et al. 1994), except that in this case the OFF deflection was always in the caudal direction. The function (Fig. 2B) is the normalized probability of neuronal firing in response to a caudal deflection at time t2 when preceded by a rostral deflection at time t1 (Fig. 1A). The data points in the curve at each pulse width (t2 – t1) were obtained by computing the area under the curve (AUC) of the PSTH for the first 30 ms after the caudal deflection for each pulse width, and normalizing by the AUC for a pulse width of 200 ms. These results are consistent with those found in previous studies (Fanselow and Nicolelis 1999; Kyriazi et al. 1994; Lee et al. 1994). The responses across all pulse widths were similar for trials with a spike elicited by the 1st deflection (
, solid line) and trials with no spike elicited by the 1st deflection (
, dashed line).
In addition to exhibiting poststimulus suppression, a subset of neurons recorded in this study (
35%) exhibited a rebound excitation after the suppressive portion of the neuronal response, consistent with previous findings in both thalamus and cortex (Fanselow and Nicolelis 1999; Higley and Contreras 2003; Zhu and Connors 1999). A single-cell example of this type of response is shown in Fig. 2C. The PSTH in the leftmost panel shows a rebound in activity in a time window centered about 90 ms after the initial deflection. The middle panel demonstrates that when the 2nd (caudal) deflection occurs within this time window after the 1st (rostral) deflection, the neuronal response is larger than that of the caudal response alone. The rightmost panel demonstrates the lack of response to a 2nd deflection with a short pulse width. The PSTHs for trials without a spike elicited by the 1st deflection are similar to those in Fig. 2C (data not shown). The temporal tuning curve for this cell, shown in Fig. 2D, illustrates a facilitated response for pulse widths in the region of rebound excitation (manifested in the sharp peak exceeding 1 at about 90 ms). Again, this cell exhibited similar responses to the 2nd deflection regardless of whether there was a spike evoked by the 1st deflection. Cumulative PSTHs for a 200-ms square pulse stimulus are shown for the group of neurons without rebound (Fig. 2E) and the group with a rebound in excitation at about 80 ms after a deflection (Fig. 2F). The bottom panels are shown with a smaller vertical axis to better illustrate the difference in response.
Because barrel neurons have been shown to exhibit different responses to rostral and caudal deflections, ascribed to directional tuning (sensitivity to direction of whisker deflection), we performed the same stimulus protocol with the opposite polarity (rostral-to-caudal) to characterize the suppression induced by the caudal deflection (Fig. 1B). The duty cycles of the 1-Hz stimulus ranged from 80 to 98%, resulting in pulse widths between the caudal and subsequent rostral deflections of 20 to 200 ms. The data again showed that as the pulse width was decreased, the neuronal response decreased and then disappeared altogether. A temporal tuning curve, (t3 – t2), was formed for the rostral deflection after a caudal deflection at various pulse widths and represents the normalized probability of firing in response to a rostral deflection at time t3 when preceded by a caudal deflection at time t2. The temporal tuning curve exhibits properties similar in nature (but not identical) to .
For each cell, the temporal tuning data were fitted with the following function, formed from the product of a sigmoid and a damped oscillation
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, t50, n, and determining the temporal progression of the suppression and possible rebound excitation. Cells were divided into groups with and without rebound excitation based on the shape of their temporal tuning curves and damping coefficient () values (see METHODS). For each group, cellular responses were averaged to obtain average temporal tuning properties (Fig. 3, A–D, ). Figure 3, A and B show the average temporal tuning properties for cells without rebound excitation, and Fig. 3, C and D show the average properties for cells exhibiting rebound excitation. The trials were again separated into those in which the 1st deflection elicited a spike and those in which it did not. Trials with no spike after the 1st deflection () exhibited temporal tuning properties that were not statistically different from the trials with a spike (). The average temporal tuning data were then fitted with the above parametric function, shown with the solid curve in each plot.
Figure 3E shows the rostral–caudal tuning properties of each neuron as the response (spikes/ms) after a caudal deflection versus the response after a rostral deflection. As shown, there is a slight bias toward the caudal direction in this sample of cells, but the tuning properties of neurons without rebound (
) do not appear to be different from the neurons with rebound (). However, other differences were observed between the 2 groups of cells. Spontaneous activity was slightly higher in cells exhibiting rebound excitation (Fig. 3F). Neurons with rebound excitation (Fig. 3, C and D) also exhibited a shorter time constant of suppression. Figure 3G summarizes the t50 parameter, the time at which the function reaches 50% of its maximum value, for the average temporal tuning curves with (black) and without (gray) rebound. For both curves, the time to 50% of the maximum response was significantly longer in neurons not exhibiting rebound excitation. Also there was a trend of longer t50 times for the curves (suppression induced by caudal deflection) relative to the curves (suppression induced by rostral deflection).
Higher-order interactions
The data presented above illustrate the effects of the postexcitatory suppression only on the excitatory activity induced by a subsequent deflection. To more fully determine the effect of the postexcitatory suppression evoked by the first deflection on the dynamics of the response to the 2nd deflection, we deflected the vibrissa with a stimulus consisting of 3 deflections. The 3rd deflection served as a probe of the postexcitatory suppression induced by the 2nd deflection. The interval between the 1st and 2nd deflections, t2 – t1, was held constant at one of 3 values (20, 60, or 100 ms) while varying the interval between the 2nd and 3rd deflections, t3 – t2, from 20 to 100 ms (Fig. 1C). Nonlinear responses were most apparent at t3 – t2 = 60 ms. Therefore all responses in Fig. 4 are shown with a fixed interval between t2 and t3 (responses to all tested intervals are summarized in Fig. 6). A typical single cell without rebound excitation is shown (Fig. 4A). With t2 – t1 long (100 ms), there was a large response to the 2nd deflection and no response to the 3rd deflection, as shown in the leftmost panel. When t2 – t1 was 60 ms, the response to the 2nd deflection disappeared, but an excitatory response to the 3rd deflection reappeared (middle panel). With t2 – t1 short (20 ms), there again was no neuronal response to the 2nd deflection and also no response to the 3rd deflection (rightmost panel). This behavior was consistent across the sample of cells. The cell shown in the bottom panels (Fig. 4B) responded to a single rostral deflection with rebound excitation at a delay of approximately 80–90 ms and displays activity typical of the group of neurons exhibiting rebound excitation. This neuron responded to the 1st and 2nd deflections for a long t2 – t1 (100 ms), but failed to respond to the 3rd deflection (left panel). For t2 – t1 = 60 ms (middle panel), there was no response to the 2nd deflection and a large response to the 3rd deflection. For a short t2 – t1 (20 ms), this neuron again did not respond to the 2nd deflection, but still responded to the 3rd deflection (rightmost panel). In this case the stimulus pattern placed the 3rd deflection in the middle of the rebound excitation induced by the 1st deflection, resulting in a slight facilitation of the response (Fig. 4B, right panel), in contrast to the pure suppression observed for the first cell (Fig. 4A, right panel). The 3rd peak in the PSTH (Fig. 4B, right panel) reflects rebound excitation from the response to the 3rd deflection. This phenomenon of response facilitation was seen for all cells in the group exhibiting rebound excitation.
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Kernel estimation and response prediction
The hallmark of nonlinear dynamics is the violation of the principle of superposition, which states that the response to the sum of 2 inputs is equal to the sum of the responses to each input presented alone. Here, the observed neuronal responses to the combination of a rostral deflection and a caudal deflection were different from the sum of the responses obtained in isolation. Such behavior can be described generally through a Volterra series (Marmarelis and Marmarelis 1978; see METHODS and APPENDIX for a full description). In this series, the 1st-order effect described the firing rate at time t induced by a rostral deflection at time t3. The step input in deflection was equivalent to an impulse in velocity, for which the subsequent response was described by the convolution of the stimulus and the 1st-order kernel (see METHODS), resulting in
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This argument was extended to the 2nd-order interaction when the rostral deflection at time t3 was preceded by a caudal deflection at time t2 (Fig. 1B). Note that the response was different from what would be predicted by the superposition of the responses to the 2 deflections in isolation. This difference was represented by the 2nd-order kernel (see APPENDIX). In this case, we expressed the firing rate at time t as a function of the 1st-order kernel, kr1, and the temporal tuning curves. The AUC of the response was thus expressed as a scaled version of the AUC for the first 30 ms of kr1
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(t3 – t1) denotes the 2nd-order interaction between the 1st rostral deflection at t1 and the 2nd rostral deflection at t3, and eff(t3 – t2) takes the form given in Eq. 2, representing the effective 2nd-order interaction between the caudal deflection at t2 and the rostral deflection at t3. Predicting the response to a 3rd deflection
The nonlinear prediction method described above was based on the assumption that scaling the excitatory response of a neuron will also scale the suppressive portion of the response. This assumption was tested using the 3-deflection stimulus pattern. Figure 6 presents a summary of the actual () and predicted (solid line) responses for the averaged cells with and without rebound excitation for all of the t2 – t1 and t3 – t2 values used in the 3-deflection stimulus study. The full prediction was formed by estimating the 3rd-order response as a function of the average temporal tuning curves, as described in the previous section, and statistical significance was assessed using a two-tailed t-test.
The data points () represent the AUC for the first 30 ms after the 3rd deflection, normalized by the AUC determined from the paired stimulus with t3 – t2 = 200 ms. For the neurons without rebound, when t2 – t1 was long (100 ms), the prediction was not significantly different from the measured values (Fig. 6A). At medium values for t2 – t1 (60 ms, Fig. 6B), the response to the 1st deflection lifted the suppression induced by the 2nd deflection, resulting in larger responses for t3 – t2 in the range of 20–60 ms. The 3rd-order estimate (full prediction) predicted this trend and was not statistically different from the measured responses. Finally, for very short values of t2 – t1 (20 ms, Fig. 6C), the response to the 3rd deflection was directly suppressed by the response to the 1st deflection. This was well captured by the full prediction, with a large difference only at t3 – t2 = 40 ms. Similar results were seen for the neurons exhibiting rebound activity. For t2 – t1 = 60 ms (Fig. 6E), the response was large even for small t3 – t2 (20 ms), attributed to the rebound excitation induced by the 1st deflection. With short values for t2 – t1 (20 ms) the response at small t3 – t2 values was attenuated, but intermediate values (40 and 60 ms) place the 3rd deflection in the rebound portion of the response to the 1st deflection and a large response is seen (Fig. 6F). The predicted responses for all cases of neurons with rebound excitation were not statistically different from the measured responses (Fig. 6, D–F).
To illustrate the importance of the 3rd-order interactions, the same predictions were performed incorporating only 2nd-order interactions (dashed lines). In this case, only the effect of the directly preceding deflection (at t2) was taken into account in predicting the response to the rostral deflection at t3. We hypothesized that the 2nd-order interactions would not predict the lifting of suppression and therefore be less than the measured responses. Statistical significance was assessed using a one-tailed t-test. For neurons with and without rebound excitation, the 2nd-order prediction was not statistically different from the measured responses for t2 – t1 = 100 ms, except in the case of t3 – t2 = 100 ms for neurons without rebound. However, when both t3 – t2 and t2 – t1 were <80 ms, the 2nd-order prediction was significantly smaller (P < 0.05) than the measured responses in 7 out of 12 cases (Fig. 6, B, C, E, and F, *).
Prediction of neuronal response to general stimulus patterns
The previous sections have described a method for predicting neuronal responses to sequences of 2 and 3 deflections. This method was generalized to the case of an infinite train of stimuli. The response after an arbitrary deflection of a stimulus was influenced by the past history of deflections. For example, for any rostral deflection, the direct influence was experienced both through the immediately previous caudal deflection (primary effect from deflection i – 1), and at short pulse widths the rostral deflection before that (secondary effect from deflection i – 2). However, the response also had indirect influence from the entire past history of the stimulus. An analogous argument holds for the influence of past history on the response to an arbitrary caudal deflection in the sequence. The total response (AUC) of barrel neurons at the ith deflection of any stimulus was predicted by a set of recursive equations
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eff and eff represents the related deflection and 
, 
are the AUC for the first 30 ms of 
and 
(see METHODS for more detail). Note that both scaling terms reflect effects from all prior stimuli.
Barrel neurons exhibit frequency-response characteristics similar to low-pass filters; they respond well at low stimulation frequencies and less so at higher frequencies (Ahissar et al. 2001; Chung et al. 2002). Predicting the response to a periodic stimulus is a special case of the above method in which all the pulse widths are equal. The steady-state frequency response was determined by iterating through the above equations for subsequent deflections in the stimulus sequence until rri and rci converged, generally in <10 cycles. Given that the suppression induced from an isolated deflection was typically small by 
100 ms, the secondary effect from preceding deflections had the strongest influence for temporal frequencies exceeding 10 Hz. Figure 7 summarizes the measured () and predicted (
, solid line) responses to stimulation at frequencies between 2 and 16 Hz of the groups of neurons with (A) and without (B) rebound excitation (see METHODS for a full description of the periodic stimulus). The overall trend of the frequency response for both groups of neurons was well predicted (Fig. 7, A and B), capturing the low-pass nature of the frequency-response characteristics in both cases. This method was also able to capture the larger firing probability observed at the frequencies corresponding to the rebound excitation (4–8 Hz) for the 2nd group (Fig. 7B), resulting in a higher cutoff frequency for this class of cells.
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) are shown in Fig. 8C along with the predicted responses (, solid line). Measured and predicted responses to all of the caudal deflections in the stimulus are shown in Fig. 8D, presented in a separate panel for clarity. In both cases, the nonlinear prediction accurately captured the total neuronal response to a complex pattern of stimuli. Figure 8E shows the average PSTH (black) for a subsection of the aperiodic stimulus and the PSTH predicted from the temporal tuning curves and the 1st-order kernels (gray, reflected about the horizontal axis) for rostral and caudal deflections. The prediction captured the relative magnitude and shape of the responses, but was not designed to predict the slight shift in response latency observed at many of the deflections. In each case, the prediction was based on high-order nonlinear interactions, expressed as a function of the fundamental 2nd-order relationships inferred from the paired-stimulus study.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: NONLINEAR KERNEL...ACKNOWLEDGMENTSREFERENCES |
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Postexcitatory suppression and rebound excitation
The nature of the postexcitatory suppression in the thalamocortical pathway has been described in a number of studies of the somatosensory pathway in general (Hellweg et al. 1977; Laskin and Spencer 1979; Mountcastle et al. 1957) and more specifically in the rat vibrissa system (Fanselow and Nicolelis 1999; Kyriazi et al. 1994; Lee et al. 1994). By stimulating the primary whisker with paired deflections, the relative response to the 2nd deflection is scaled because of suppression and/or rebound excitation induced by the 1st deflection. Intracellular studies have shown that inhibition in the cortex is GABA-mediated, both in vitro (Agmon and Connors 1992) and in vivo (Carvell and Simons 1988; Moore and Nelson 1998; Zhu and Connors 1999). It has also been shown that administration of bicuculline methiodide (BMI), a GABAA inhibitor, leads to increased responses to the OFF deflection of a square pulse (Kyriazi et al. 1996). These findings suggest that GABA-mediated inhibition is partially responsible for the postexcitatory suppression we observed in this study. Previous studies have shown that the postexcitatory suppression described here is also present in the ventral posterior medial (VPM) thalamus (Fanselow and Nicolelis 1999; Kyriazi and Simons 1993; Lee et al. 1994). It is thought to originate in VPM through interactions with the nucleus reticularis thalami (nRt), and is perhaps further exaggerated in the cortex by thalamocortical circuitry (Barbaresi et al. 1986; Kyriazi and Simons 1993; Pinto et al. 1996).
Using the paired-deflection stimulus, we have shown that the postexcitatory suppression is not strongly dependent on the existence of a sensory-evoked action potential. This finding suggests that it is partly a network phenomenon, although intrinsic cellular properties cannot be ruled out. A subset (35%) of cells in this study exhibited rebound excitation after the suppression, as previously observed in VPM and cortex (Carvell and Simons 1988; Fanselow and Nicolelis 1999; Higley and Contreras 2003; Simons and Carvell 1989; Zhu and Connors 1999). Although intracellular studies have shown the presence of rebound excitation in most cells, we observed it in only a subset of cells. One possible explanation is that the rebound activity was not strong enough to produce a supra-threshold response in all cells. It has been postulated that this rebound excitation could be recurrent activation of excitatory cortical and subcortical circuitry (Armstrong-James et al. 1991; Zhu and Connors 1999). In our study, rebound excitation served to facilitate neuronal responsiveness to a stimulus presented during the rebound period. All cells in this study were stimulated in the rostral–caudal plane regardless of the directional tuning of the individual cell. The sample bias toward caudally tuned cells (Fig. 2E) resulted in more suppression after a caudal deflection than a rostral deflection, when averaged across all cells. This was observed as a trend of longer t50 times for the curves relative to the curves (Fig. 2G).
Although it is possible that the nature of the response properties could be layer specific, a direct correlation between the presence or absence of rebound excitation and the recording site depth or response latency was not observed (data not shown). Previous studies have reported latency shifts on the order of 40 ms at increasing stimulation frequency in layers Va and II/III of the cortex, and smaller latency shifts on the order of a few milliseconds in layers IV and Vb (Ahissar et al. 2000, 2001). The latency shifts observed in this study were on the order of 5–10 ms, consistent with layer IV. Furthermore, all neurons in the study had small receptive fields, typically single whisker, consistent with recordings from within a barrel, and not within septa where receptive fields are typically multiwhisker. In addition, barbiturate anesthetics can in some cases induce exaggerated levels of cortical depression. However, the results we obtained closely mirror results in other studies where a myriad of anesthetic agents were used. Spontaneous activity was 2.0 ± 1.5 spikes/s, which is consistent with reports from other labs (Castro-Alamancos 2002b; Simons and Carvell 1989). The paired-stimulus results were very similar to those previously reported under fentanyl anesthesia (Kyriazi et al. 1994), as were the cortical frequency-response characteristics under urethane anesthesia (Ahissar et al. 2000; Chung et al. 2002).
Suppression of suppression
We found that the postexcitatory suppression not only affects the excitatory portion of a subsequent response, it also affects the suppressive portion. This has the effect of lifting the suppression from a future stimulus response (an effective disinhibition), as was verified experimentally using the 3-deflection stimulus patterns. Figure 4C demonstrates the lifting of suppression for the 2 groups of neurons. With a long interval between the first 2 deflections, the response to the 3rd deflection approximated the 2nd-order response, implying that the suppression was mainly attributable to the response to the 2nd deflection. As the interval between the first 2 deflections was shortened, the responses to the 3rd deflection increased, implying that there was a lifting of suppression. The normalized response for the neurons without rebound was <1 for all tested intervals between the first 2 deflections. This suggests that there is still some suppression present arising from the 2nd deflection. The 2nd-order predictions for the 3-deflection stimulus (Fig. 6, dashed lines) illustrate these findings. It should be noted that our method is purely functional and we do not attempt to divide the suppression into its various sources. Previous studies have shown that synaptic depression is present both at thalamocortical synapses (Chung et al. 2002) and at lemniscal synapses (Castro-Alamancos 2002a,b), causing suppression at cortical and thalamic levels. The synchrony of thalamic firing has been shown to affect the cortical response, with more thalamic synchrony resulting in larger cortical responses. More synchronous thalamic responses to the 3rd deflection could therefore contribute to the observed "suppression of suppression" phenomenon. Also, the barrel circuitry could serve to enhance the suppression properties seen in VPM (Kyriazi and Simons 1993; Kyriazi et al. 1994; Pinto et al. 1996, 2000; Simons and Carvell 1989). A recent intracellular study showed that deflecting remote whiskers 20 ms before a PW deflection suppressed the early PW excitatory postsynaptic potential (EPSP) as well as the following inhibitory PSP, although to a smaller extent (Higley and Contreras 2003). The observed suppression and "suppression of suppression" in this study is most likely a combination of these mechanisms of thalamic and cortical origin.
Nonlinear prediction method
The interactions observed in the 2- and 3-deflection stimulus protocols imply nonlinear relationships between the neuronal responses that we described through a Volterra series representation. The Volterra series is a general way to express nonlinear relationships between the input and output of a system through a sequence of kernels of increasing order (Marmarelis and Marmarelis 1978; Volterra 1959). In this case, the 1st-order kernel of the series is simply the PSTH in response to a caudal or rostral whisker deflection. Second-order kernels represent the postexcitatory suppression and/or facilitation, and 3rd-order kernels represent the lifting of suppression. The temporal tuning curves were generated by normalizing the response at a given pulse width to the response at the longest pulse width. Therefore the 2nd-order predictions are scaled versions of the 1st-order kernel (PSTH). Because the predictions were scaled versions of the PSTH for a single stimulus, subtle changes in dynamics such as the slight increase in response latency at pulse widths <80 ms (Fig. 5) were not captured. These latency shifts have been described in previous studies (Ahissar et al. 2001; Fanselow and Nicolelis 1999; Kyriazi et al. 1994) and could result from weaker subthreshold responses requiring more temporal summation (Kyriazi et al. 1994). We found that neuronal response depended on the past history of deflections. Response predictions based on only the preceding deflection significantly underestimated the experimental observations (Fig. 6), whereas the 3rd-order predictions were not statistically different from measured responses. The response to a sequence of stimuli was well predicted by the temporal tuning curves and the time course of these curves determined the suppression or facilitation of responses to subsequent stimuli.
Frequency-response characteristics
It has been shown that neurons in the barrel cortex of the anesthetized rat are not able to accurately follow stimulation at high frequencies, exhibiting low-pass filter characteristics (Ahissar et al. 2000, 2001; Castro-Alamancos and Oldford 2002; Chung et al. 2002; Garabedian et al. 2003). Based on the paired-stimulus results here, at high stimulation frequencies the response to each stimulus is affected by suppression induced by previous stimuli. This is supported by our prediction of the low-pass nature of steady-state frequency-response characteristics for cells without rebound. When present, we were also able to predict the increase in firing rate at frequencies corresponding to rebound excitation (4–8 Hz), resulting in a higher cutof
